factorization algebras in quantum field theory - northwestern

Factorization algebras in quantum field theory

Kevin Costello and Owen Gwilliam

K.C. and O.G. are partially supported by NSF grants DMS 0706945 and DMS 1007168,and K.C. is also partially supported by a Sloan fellowship. O.G. is supported by an NSF

postdoctoral fellowship DMS-1204826.

Contents

Chapter 1. Introduction and overview 11.1. Introduction 11.2. The motivating example of quantum mechanics 21.3. A preliminary definition of prefactorization algebras 61.4. Prefactorization algebras in quantum field theory 71.5. Classical field theory and factorization algebras 91.6. Quantum field theory and factorization algebras 111.7. The weak quantization theorem 111.8. The strong quantization conjecture 12

Part 1. Prefactorization algebras 15

Chapter 2. From Gaussian measures to factorization algebras 172.1. Gaussian integrals 172.2. Divergence in infinite dimensions 192.3. The prefactorization structure on observables 222.4. From quantum to classical 242.5. Correlation functions 262.6. Further results on free field theories 282.7. Interacting theories 29

Chapter 3. Prefactorization algebras and basic examples 333.1. Prefactorization algebras 333.2. Recovering associative algebras from prefactorization algebras on R 363.3. Comparisons with other axiom systems for field theories 373.4. A construction of the universal enveloping algebra 413.5. Some functional analysis 433.6. The factorization envelope of a sheaf of Lie algebras 51

Chapter 4. Factorization algebras and free field theories 574.1. The divergence complex of a measure 574.2. The factorization algebra of classical observables of a free field theory 604.3. Quantum mechanics and the Weyl algebra 704.4. Free field theories and canonical quantization 764.5. Quantizing classical observables 77

iii

iv CONTENTS

4.6. Correlation functions 814.7. Translation-invariant prefactorization algebras 82

Chapter 5. Holomorphic field theories and vertex algebras 955.1. Holomorphically translation-invariant factorization algebras 995.2. A general method for constructing vertex algebras 1065.3. The βγ system and vertex algebras 1185.4. Affine Kac-Moody algebras and factorization algebras 132

Part 2. Factorization algebras 147

Chapter 6. Factorization algebras: definitions and constructions 1496.1. Factorization algebras 1496.2. Locally constant factorization algebras 1546.3. Factorization algebras from cosheaves 1576.4. Factorization algebras from local Lie algebras 1606.5. Some examples of computations 162

Chapter 7. Formal aspects of factorization algebras 1697.1. Pushing forward factorization algebras 1697.2. The category of factorization algebras 1697.3. Descent 1727.4. Extension from a basis 1807.5. Pulling back along an open immersion 1877.6. Equivariant factorization algebras and descent along a torsor 187

Chapter 8. Structured factorization algebras and quantization 1898.1. Structured factorization algebras 1898.2. Commutative factorization algebras 1918.3. The P0 operad 1928.4. The Beilinson-Drinfeld operad 195

Part 3. Classical field theory 199

Chapter 9. Introduction to classical field theory 2019.1. The Euler-Lagrange equations 2019.2. Observables 2029.3. The symplectic structure 2029.4. The P0 structure 203

Chapter 10. Elliptic moduli problems 20510.1. Formal moduli problems and Lie algebras 20610.2. Examples of elliptic moduli problems related to scalar field theories 21010.3. Examples of elliptic moduli problems related to gauge theories 21210.4. Cochains of a local L∞ algebra 215

CONTENTS v

10.5. D-modules and local L∞ algebras 217

Chapter 11. The classical Batalin-Vilkovisky formalism 22511.1. The classical BV formalism in finite dimensions 22511.2. The classical BV formalism in infinite dimensions 22711.3. The derived critical locus of an action functional 23011.4. A succinct definition of a classical field theory 23511.5. Examples of field theories from action functionals 23711.6. Cotangent field theories 238

Chapter 12. The observables of a classical field theory 24312.1. The factorization algebra of classical observables 24312.2. The graded Poisson structure on classical observables 24412.3. The Poisson structure for free field theories 24512.4. The Poisson structure for a general classical field theory 248

Part 4. Quantum field theory 253

Chapter 13. Introduction to quantum field theory 25513.1. The quantum BV formalism in finite dimensions 25613.2. The “free scalar field” in finite dimensions 25813.3. An operadic description 26013.4. Equivariant BD quantization and volume forms 26113.5. How renormalization group flow interlocks with the BV formalism 26213.6. Overview of the rest of this Part 263

Chapter 14. Effective field theories and Batalin-Vilkovisky quantization 26514.1. Local action functionals 26614.2. The definition of a quantum field theory 26714.3. Families of theories over nilpotent dg manifolds 27614.4. The simplicial set of theories 28214.5. The theorem on quantization 285

Chapter 15. The observables of a quantum field theory 28915.1. Free fields 28915.2. The BD algebra of global observables 29315.3. Global observables 30015.4. Local observables 30315.5. Local observables form a prefactorization algebra 30515.6. Local observables form a factorization algebra 30815.7. The map from theories to factorization algebras is a map of presheaves 314

Chapter 16. Further aspects of quantum observables 32116.1. Translation-invariant factorization algebras from translation-invariant

quantum field theories 321

vi CONTENTS

16.2. Holomorphically translation-invariant theories and their factorizationalgebras 324

16.3. Renormalizability and factorization algebras 33016.4. Cotangent theories and volume forms 34316.5. Correlation functions 351

Part 5. Using the machine 353

Chapter 17. Noether’s theorem in classical field theory 35517.1. Symmetries of a classical field theory 35717.2. Examples of classical field theories with an action of a local L∞ algebra 36317.3. The factorization algebra of equivariant observable 36817.4. Inner actions 36817.5. Classical Noether’s theorem 37117.6. Conserved currents 37417.7. Examples of classical Noether’s theorem 376

Chapter 18. Noether’s theorem in quantum field theory 38118.1. Quantum Noether’s theorem 38118.2. Actions of a local L∞-algebra on a quantum field theory 38518.3. Obstruction theory for quantizing equivariant theories 38918.4. The factorization algebra associated to an equivariant quantum field theory 39218.5. Quantum Noether’s theorem 39318.6. The quantum Noether theorem and equivariant observables 40118.7. Noether’s theorem and the local index 40618.8. The partition function and the quantum Noether theorem 410

Part 6. Appendices 413

Appendix A. Background 415A.1. Simplicial techniques 415A.2. Operads and algebras 421A.3. Lie algebras and L∞ algebras 429A.4. Derived deformation theory 436A.5. Sheaves, cosheaves, and their homotopical generalizations 444A.6. Elliptic complexes 446

Appendix B. Homological algebra with differentiable vector spaces 453B.1. Diffeological vector spaces 453B.2. Differentiable vector spaces from sections of a vector bundle 461B.3. Differentiable vector spaces from holomorphic sections of a holomorphic

vector bundle 463B.4. Differentiable cochain complexes 464B.5. Pro-cochain complexes 468

CONTENTS vii

B.6. Differentiable pro-cochain complexes 469B.7. Differentiable cochain complexes over a differentiable dg ring 472B.8. Classes of functions on the space of sections of a vector bundle 472B.9. Derivations 478B.10. The Atiyah-Bott lemma 480

Appendix C. Homological algebra with topological vector spaces 483C.1. Bornological vector spaces 486C.2. Examples of bornological vector spaces 493C.3. Convenient cochain complexes 495C.4. Elliptic objects are closed under tensor product 496C.5. Convenient pro-cochain complexes 502C.6. Pro-cochain complexes over C[[h]] 504C.7. Observables as geometric pro-cochain complexes 506

Bibliography 511

CHAPTER 1

Introduction and overview

1.1. Introduction

This book will provide the analog, in quantum field theory, of the deformation quan-tization approach to quantum mechanics. In this introduction, we will start by recallinghow deformation quantization works in quantum mechanics.

The collection of observables in quantum mechanics form an associative algebra. Theobservables of a classical mechanical system form a Poisson algebra. In the deformationquantization approach to quantum mechanics, one starts with a Poisson algebra Acl , andattempts to construct an associative algebra Aq, which is an algebra flat over the ringC[[h]], together with an isomorphism of associative algebras Aq/h ∼= Acl . In addition, ifa, b ∈ Acl , and a, b are any lifts of a, b to Aq, then

limh→0

1h[a, b] = a, b ∈ Acl .

We will describe an analogous approach to studying perturbative quantum field the-ory. In order to do this, we need to explain the following.

• The structure present on the collection of observables of a classical field theory.This structure is the analog, in the world of field theory, of the commutative alge-bra which appears in classical mechanics. This structure we call a commutativefactorization algebra (section 8).• The structure present on the collection of observables of a quantum field theory.

This structure is that of a factorization algebra (section 6.1). We view our defini-tion of factorization algebra as a C∞ analog of a definition introduced by Beilin-son and Drinfeld. However, the definition we use is very closely related to otherdefinitions in the literature, in particular to the Segal axioms.• The extra structure on the commutative factorization algebra associated to a clas-

sical field theory which makes it “want” to quantize. This is the analog, in theworld of field theory, of the Poisson bracket on the commutative algebra of ob-servables.

1

2 1. INTRODUCTION AND OVERVIEW

• The quantization theorem we prove. This states that, provided certain obstruc-tion groups vanish, the classical factorization algebra associated to a classicalfield theory admits a quantization. Further, the set of quantizations is parametrized(order by order in h) by the space of deformations of the Lagrangian describingthe classical theory.

This quantization theorem is proved using the physicists’ techniques of perturbative renor-malization, as developed mathematically in [Cos11c]. We claim that this theorem is amathematical encoding of the perturbative methods developed by physicists.

This quantization theorem applies to many examples of physical interest, includ-ing pure Yang-Mills theory and σ-models. For pure Yang-Mills theory, it is shown in[Cos11c] that the relevant obstruction groups vanish, and that the deformation group isone-dimensional; so that there exists a one-parameter family of quantizations. A certaintwo-dimensional σ-model was constructed in this language in [Cos10, Cos11a]. Otherexamples are considered in [GG11] and [CL11].

Finally, we will explain how (under certain additional hypotheses) the factorizationalgebra associated to a perturbative quantum field theory encodes the correlation func-tions of the theory. This justifies the assertion that factorization algebras encode a largepart of quantum field theory.

1.1.1. Acknowledgements. We would like to thank Dan Berwick-Evans Damien Calaque,Vivek Dhand, Chris Douglas, John Francis, Dennis Gaitsgory, Greg Ginot, Ryan Grady,Theo Johnson-Freyd, David Kazhdan, Jacob Lurie, Takuo Matsuoka, Fred Paugam, NickRozenblyum, Graeme Segal, Josh Shadlen, Yuan Shen, Stephan Stolz, Dennis Sullivan,Hiro Tanaka and David Treumann for helpful conversations. K.C. is particularly gratefulto thank John Francis and Jacob Lurie for introducing him to factorization algebras (intheir topological incarnation [Lur09a]) in 2008; and to Graeme Segal, for many illuminat-ing conversations about QFT.

1.2. The motivating example of quantum mechanics

The model problems of classical and quantum mechanics involve a particle moving insome Euclidean space Rn under the influence of some fixed field. Our main goal in thissection is to describe these model problems in a way that makes the idea of a factorizationalgebra (section 6.1) emerge naturally, but we also hope to give mathematicians somefeeling for the physical meaning of terms like “field” and “observable.” We will not worryabout making precise definitions, since that’s what this book aims to do. As a narrativestrategy, we describe a kind of cartoon of a physical experiment, and we ask that physicistsaccept this cartoon as a friendly caricature elucidating the features of physics we mostwant to emphasize.

1.2. THE MOTIVATING EXAMPLE OF QUANTUM MECHANICS 3

1.2.1. A particle in a box. For the general framework we want to present, the detailsof the physical system under study are not so important. However, for concreteness, wewill focus attention on a very simple system: that of a single particle confined to someregion of space. We confine our particle inside some box and occasionally take measure-ments of this system. The set of possible trajectories of the particle around the box consti-tute all the imaginable behaviors of this particle; we might write this mathematically asMaps(I, box), where I ⊂ R denotes the time interval over which we conduct the experi-ment. In other words, the set of possible behaviors forms a space of fields on the timelineof the particle.

The behavior of our theory is governed by the action functional. The simplest case isthe action of the massless free field theory, whose value on a function f : I → box is

S( f ) =∫

I〈 f , ∆ f 〉 .

The aim of this section is to outline the structure one would expect the observables – thatis, the possible measurements one can make – should satisfy.

1.2.2. Classical mechanics. Let us start by considering the much simpler case, whereour particle is treated as a classical system. In that case, the trajectory of the particle isconstrained to be in a solution to the Euler-Lagrange equations of our theory. For example,if the action functional governing our theory is that of the massless free theory, then a mapf : I → box satisfies the Euler-Lagrange equation if it is a straight line.

We are interested in the observables for this classical field theory. Since the trajectoryof our particle is constrained to be a solution to the Euler-Lagrange equation, the onlymeasurements one can make are functions on the space of solutions to the Euler-Lagrangeequation.

If U ⊂ R is an open subset, we will let Fields(U) denote the space of fields on U, thatis, the space of maps f : U → box. We will let

EL(U) ⊂ Fields(U)

denote the subspace consisting of those maps f : U → box which are solutions to theEuler-Lagrange equation. As U varies, EL(U) forms a sheaf of spaces on R.

We will let Obscl(U) denote the space of functions on EL(U) (the precise class of func-tions we will consider will be discussed later). As U varies, the spaces Obscl(U) form acosheaf of commutative algebras on R. We will think of Obscl(U) as the space observ-ables for our classical system which only consider the behavior of the particle on timescontained in U.

Note that Obscl(U) is a cosheaf of commutative algebras on R.


1.2.3. Measurements in quantum mechanics. The notion of measurement is fraughtin quantum theory, but we will take a very concrete view. Taking a measurement meansthat we have physical measurement device (e.g., a camera) that we allow to interact withour system for a period of time. The measurement is then how our measurement devicehas changed due to the interaction. In other words, we couple the two physical systems,then decouple them and record how the measurement device has modified from its initialcondition. (Of course, there is a symmetry in this situation: both systems are affected bytheir interaction, so a measurement inherently disturbs the system under study.)

The observables for a physical system are all the imaginable measurements we couldtake of the system. Instead of considering all possible observables, we might also considerthose observables which occur within a specified time period. This period can be specifiedby an open interval U ⊂ R.

Thus, we arrive at the following principle.

Principle 1. For every open subset U ⊂ R, we have a set Obs(U) ofobservables one can make on U.

The superposition principle tells us that quantum mechanics (and quantum field the-ory) is fundamentally linear. This leads to

Principle 2. The set Obs(U) is a complex vector space.

We think of Obs(U) as being the space of ways of coupling a measurement device toour system on the region U. Thus, there is a natural map Obs(U) → Obs(V) if U ⊂ V isan open subset. This means that the space Obs(U) forms a pre-cosheaf.

1.2.4. Combining observables. Measurements (and so observables) differ qualita-tively in the classical and quantum settings. If we study a classical particle, the systemis not noticeably disturbed by measurements, and so we can do multiple measurementsat the same time. Hence, on each interval J we have a commutative multiplication mapObs(J)⊗Obs(J)→ Obs(J), as well as the maps that let us combine observables on disjointintervals.

For a quantum particle, however, a measurement disturbs the system significantly.Taking two measurements simultaneously is incoherent, as the measurement devices arecoupled to each other and thus also affect each other, so that we are no longer measuringjust the particle. Quantum observables thus do not form a cosheaf of commutative alge-bras on the interval. However, there are no such problems with combining measurementsoccurring at different times. Thus, we find the following.

1.2. THE MOTIVATING EXAMPLE OF QUANTUM MECHANICS 5

Principle 3. If U, U′ are disjoint open subsets of R, and U, U′ ⊂ V whereV is also open, then there is a map

? : Obs(U)⊗Obs(U′)→ Obs(V).

If O ∈ Obs(U) and O′ ∈ Obs(U′), then O ?O′ is defined by coupling oursystem to measuring device O for t ∈ U, and to device O′ for t ∈ U′.

Further, these maps are commutative, associative, and compatiblewith the maps Obs(U) → Obs(V) associated to inclusions U ⊂ V ofopen subsets. (The precise meaning of these terms is detailed in section3.1.)

1.2.5. Perturbative theory and the correspondence principle. In the bulk of this book,we will be considering perturbative quantum theory. For us, this means that we work overthe base ring C[[h]], where at h = 0 we find the classical theory. In perturbative theory,therefore, the space Obs(U) of observables on an open subset U is a C[[h]]-module, andthe product maps are C[[h]]-linear.

The correspondence principle states that the quantum theory, in the h→ 0 limit, mustreproduce the classical theory. Applied to observables, this leads to the following princi-ple.

Principle 4. The vector space Obsq(U) of quantum observables is a flatC[[h]]-module that, modulo h, is the space Obscl(U) of classical observ-ables.

These simple principles are at the heart of our approach to quantum field theory. Theysay, roughly, that the observables of a quantum field theory form a factorization algebra,which is a quantization of the factorization algebra associated to a classical field theory.The main theorem presented in this book is that one can use the techniques of perturba-tive renormalization to construct factorization algebras perturbatively quantizing a cer-tain class of classical field theories (including many classical field theories of physical andmathematical interest).

1.2.6. Associative algebras in quantum mechanics. The principles we have describedso far indicate that the observables of a quantum mechanical system should assign, to ev-ery open subset U ⊂ R, a vector space Obs(U), together with a product map

Obs(U)⊗Obs(U′)→ Obs(V)

if U, U′ are disjoint open subsets of an open subset V. This is the basic data of a factoriza-tion algebra (section 3.1).

It turns out that the factorization algebra produced by our quantization procedure ap-plied to quantum mechanics has a special property: it is locally constant (section 6.2). This


means that the map Obs((a, b)) → Obs(R) is an isomorphism for any interval (a, b). LetA be denote the vector space Obs(R); note that A is canonically isomorphic to Obs((a, b))for any interval (a, b).

The product map

Obs((a, b))⊗Obs((c, d))→ Obs((a, d))

(defined when a < b < c < d) becomes, when we perform this identification, a productmap

m : A⊗ A→ A.

The axioms of a factorization algebra imply that this multiplication turns A into an asso-ciative algebra.

This should be familiar to topologists: associative algebras are algebras over the op-erad of little intervals in R, and this is precisely what we have described. (As we will seelater (section ??), this associative algebra is the Weyl algebra one expects to find as thealgebra of observables of quantum mechanics.)

One important point to take away from this discussion is that associative algebras appearin quantum mechanics because associative algebras are connected with the geometry of R. Thereis no fundamental connection between associative algebras and any concept of “quanti-zation”: associative algebras only appear when one considers one-dimensional quantumfield theories. As we will see later, when one considers quantum field theories on n-dimensional space times, one finds a structure reminiscent of an En-algebra instead of anE1-algebra.

1.3. A preliminary definition of prefactorization algebras

Below (see section 3.1) we give a more formal definition, but here we provide the basicidea. Let M be a topological space (which, in practice, will be a smooth manifold).

1.3.0.1 Definition. A prefactorization algebra F on M, taking values in cochain complexes, is arule that assigns a cochain complex F (U) to each open set U ⊂ M along with

(1) a cochain map F (U)→ F (V) for each inclusion U ⊂ V;(2) a cochain map F (U1)⊗ · · · ⊗ F (Un) → F (V) for every finite collection of open sets

where each Ui ⊂ V and the Ui are disjoint;(3) the maps are compatible in a certain natural way. The simplest case of this compatibility

is that if U ⊂ V ⊂ W is a sequence of open sets, the map F (U) → F (W) agrees withthe composition through F (V)).

Remark: A prefactorization algebra resembles a precosheaf, except that we tensor the cochaincomplexes rather than taking their direct sum.

1.4. PREFACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY 7

The observables of a field theory (whether classical or quantum) form a prefactoriza-tion algebra on the spacetime manifold M. In fact, they satisfy a kind of local-to-globalprinciple in the sense that the observables on a large open set are determined by the ob-servables on small open sets. The notion of a factorization algebra (section 6.1) makes thislocal-to-global condition precise.

1.4. Prefactorization algebras in quantum field theory

The (pre)factorization algebras of interest in this book arise from perturbative quan-tum field theories. We have already discussed (section 1.2) how factorization algebrasappear in quantum mechanics. In this section we will see that this picture extends in avery natural way to quantum field theory.

The manifold M on which the prefactorization algebra is defined is the space-timemanifold of the quantum field theory. If U ⊂ M is an open subset, we will interpret F (U)as the space of observables (or measurements) that we can make, which only depend onthe behavior of the fields on U. Performing a measurement involves coupling a measuringdevice to the quantum system in the region U.

One can bear in mind the example of a particle accelerator. In that situation, one canimagine the space-time M as being of the form M = A× (0, t), where A is the interior ofthe accelerator and t is the duration of our experiment.

In this situation, performing a measurement on some open subset U ⊂ M is some-thing concrete. Let us take U = V × (ε, δ), where V ⊂ A is some small region in theaccelerator, and (ε, δ) is a short time interval. Performing a measurement on U amountsto coupling a measuring device to our accelerator in the region V, starting at time ε andending at time δ. For example, we could imagine that there is some piece of equipment inthe region V of the accelerator, which is switched on at time ε and switched off at time δ.

1.4.1. Interpretation of the prefactorization algebra axioms. Suppose that we havetwo different measuring devices, O1 and O2. We would like to set up our accelerator sothat we measure both O1 and O2.

There are two ways we can do this. Either we insert O1 and O2 into disjoint regionsV1, V2 of our accelerator. Then we can turn O1 and O2 on at any times we like, includingfor overlapping time intervals.

If the regions V1, V2 overlap, then we can not do this. After all, it doesn’t make senseto have two different measuring devices at the same point in space at the same time.


However, we could imagine inserting O1 into region V1 during the time interval (a, b);and then removing O1, and inserting O2 into the overlapping region V2 for the disjointtime interval (c, d).

These simple considerations immediately suggest that the possible measurements wecan make of our physical system form a prefactorization algebra. Let Obs(U) denote thespace of measurements we can make on an open subset U ⊂ M. Then, by combiningmeasurements in the way outlined above, we would expect to have maps

Obs(U)⊗Obs(U′)→ Obs(V)

whenever U, U′ are disjoint open subsets of an open subset V. The associativity and com-mutativity properties of a prefactorization algebra are evident.

1.4.2. The cochain complex of observables. In the approach to quantum field theoryconsidered in this book, the factorization algebra of observables will be a factorization al-gebra of cochain complexes. One can ask for the physical meaning of the cochain complexObs(U).

It turns out that the “physical” observables will be H0(Obs(U)). If O ∈ Obs0(U) is anobservable of cohomological degree 0, then the equation dO = 0 can often be interpretedas saying that O is compatible with the gauge symmetries of the theory. Thus, only thoseobservables O ∈ Obs0(U) which are closed are physically meaningful.

The equivalence relation identifying O ∈ Obs0(U) with O+dO′, where O′ ∈ Obs−1(U),also has a physical interpretation, which will take a little more work to describe. Often,two observables on U are physically indistinguishable (that is, they can not be distin-guished by any measurement one can perform). In the example of an accelerator outlinedabove, two measuring devices are equivalent if they always produce the same expectationvalues, no matter how we prepare our system, or no matter what boundary conditions weimpose.

As another example, in the quantum mechanics of a free particle, the observable mea-suring the momentum of a particle at time t is equivalent to that measuring the momen-tum of a particle at another time t′. This is because, even at the quantum level, momentumis preserved (as the momentum operator commutes with the Hamiltonian).

From the cohomological point of view, if O, O′ ∈ Obs0(U) are observables which arein the kernel of d (and thus “physically meaningful”), then they are equivalent in thesense described above if they differ by an exact observable.

It is a little more difficult to provide a physical interpretation for the non-zero coho-mology groups Hn(Obs(U)). The first cohomology group H1(Obs(U)) contains anom-alies (or obstructions) to lifting classical observables to the quantum level. For example,

1.5. CLASSICAL FIELD THEORY AND FACTORIZATION ALGEBRAS 9

in a gauge theory, one might have a classical observable which respects gauge symme-try. However, it may not lift to a quantum observable respecting gauge symmetry; thishappens if there is a non-zero anomaly in H1(Obs(U)).

The cohomology groups Hn(Obs(U)), when n < 0, are best interpreted as symme-tries, and higher symmetries, of observables. Indeed, we have seen that the physicallymeaningful observables are the closed degree 0 elements of Obs(U). One can construct asimplicial set, whose n-simplices are closed and degree 0 elements of Obs(U)⊗Ω∗(∆n).The vertices of this simplicial set are observables, the edges are equivalences betweenobservables, the faces are equivalences between equivalences, and so on.

The Dold-Kan correspondence tells us that the nth homotopy group of this simplicialset is H−n(Obs(U)). This allows us to interpret H−1(Obs(U)) as being the group of sym-metries of the trivial observable 0 ∈ H0(Obs(U)), and H−2(Obs(U)) as the symmetries ofthe identity symmetry of 0 ∈ H0(Obs(U)), and so on.

Although the cohomology groups Hn(Obs(U)) where n > 1 do not have such a clearphysical interpretation, they are mathematically very natural objects and it is importantnot to discount them. For example, let us consider a gauge theory on a manifold M,and let D be a disc in M. Then it is often the case that elements of H1(Obs(D)) can beintegrated over a circle in M to yield cohomological degree 0 observables (such as Wilsonoperators).

1.5. Classical field theory and factorization algebras

The main aim of this book is to present a deformation-quantization approach to quan-tum field theory. In this section we will outline how a classical field theory gives rise tothe classical algebraic structure we consider.

We use the Lagrangian formulation throughout. Thus, classical field theory means thestudy of the critical locus of an action functional. In fact, we use the language of derivedgeometry, in which it becomes clear that functions on a derived critical locus (section11.1) should form a P0 algebra (section 8.3), that is, a commutative algebra with a Poissonbracket of cohomological degree 1. (For an overview of these ideas, see the section 1.8.)

Applying these ideas to infinite-dimensional spaces, such as the space of smooth func-tions on a manifold, one runs into analytic problems. Although there is no difficulty inconstructing a commutative algebra Obscl of classical observabes, we find that the Poissonbracket on Obscl is not always well-defined. However, we show the following.

1.5.0.1 Theorem. For a classical field theory (section 11.4) on a manifold M, there is a sub-commutative factorization algebra Obs

cl of the commutative factorization algebra Obscl on which


the Poisson bracket is defined, so that Obscl

forms a P0 factorization algebra. Further, the inclusion

Obscl→ Obscl is a quasi-isomorphism of factorization algebras.

Remark: Our approach to field theory involves both cochain complexes of infinite-dimensionalvector spaces and families over manifolds (and dg manifolds). The infinite-dimensionalvector spaces that appear are of the type studied in functional analysis: for example,spaces of smooth functions and of distributions. One approach to working with suchvector spaces is to treat them as topological vector spaces. In this book, we will insteadtreat them as differentiable vector spaces. In particular, Obscl will be a factorization alge-bra valued in differentiable vector spaces. For a careful discussion of differential vectorspaces, see Appendix B. The basic idea is as follows: a differentiable vector space is a vec-tor space V with a smooth structure, meaning that we have a well-defined set of smoothmaps from any manifold X into V; and further, we have enough structure to be able todifferentiate any smooth map into V. These notions make it possible to efficiently studycochain complexes of vector spaces in families over manifolds.

1.5.1. A gloss of the main ideas. In the rest of this section, we will outline why onewould expect that classical observables should form a P0 algebra. More details are avail-able in section 9.

The idea of the construction is very simple: if U ⊂ M is an open subset, we willlet EL(U) be the derived space of solutions to the Euler-Lagrange equation on U. Sincewe are dealing with perturbative field theory, we are interested in those solutions to theequations of motion which are infinitely close to a given solution.

The differential graded algebra Obscl(U) is defined to be the space of functions onEL(U). (Since EL(U) is an infinite dimensional space, it takes some work to defineObscl(U). Details will be presented later (Chapter ??).

On a compact manifold M, the solutions to the Euler-Lagrange equations are the criti-cal points of the action functional. If we work on an open subset U ⊂ M, this is no longerstrictly true, because the integral of the action functional over U is not defined. However,fields on U have a natural foliation, where tangent vectors lying in the leaves of the foli-ation correspond to variations φ → φ + δφ, where δφ has compact support. In this case,the Euler-Lagrange equations are the critical points of a closed one-form dS defined alongthe leaves of this foliation.

Any derived scheme which arises as the derived critical locus (section 11.1) of a func-tion acquires an extra structure: it’s ring of functions is equipped with the structure of aP0 algebra. The same holds for a derived scheme arising as the derived critical locus ofa closed one-form define along some foliation. Thus, we would expect that Obscl(U) is

1.7. THE WEAK QUANTIZATION THEOREM 11

equipped with a natural structure of P0 algebra; and that, more generally, the commuta-tive factorization algebra Obscl should be equipped with the structure of P0 factorizationalgebra.

1.6. Quantum field theory and factorization algebras

Another aim of the book is to relate perturbative quantum field theory, as developedin [Cos11c], to factorization algebras. We give a natural definition of an observable of aquantum field theory, which leads to the following theorem.

1.6.0.1 Theorem. For a classical field theory (section 11.4) and a choice of BV quantization (sec-tion 14.2), the quantum observables Obsq form a factorization algebra over the ring R[[h]]. More-over, the factorization algebra of classical observables Obscl is homotopy equivalent to Obsq mod has a factorization algebra.

Thus, the quantum observables form a factorization algebra and, in a very weak sense,are related to the classical observables. The quantization theorems will sharpen the rela-tionship between classical and quantum observables.

The main result of [Cos11c] allows one to construct perturbative quantum field the-ories, term by term in h, using cohomological methods. This theorem therefore gives ageneral method to quantize the factorization algebra associated to classical field theory.

1.7. The weak quantization theorem

We have explained how a classical field theory gives rise to P0 factorization algebraObscl , and how a quantum field theory (in the sense of [Cos11c]) gives rise to a factor-ization algebra Obsq over R[[h]], which specializes at h = 0 to the factorization algebraObscl of classical observables. In this section we will state our weak quantization theorem,which says that the Poisson bracket on Obscl is compatible, in a certain sense, with thequantization given by Obsq.

This statement is the analog, in our setting, of a familiar statement in quantum-mechanicaldeformation quantization. Recall (section 1.1) that in that setting, we require that the asso-ciative product on the algebra Aq of quantum observables is related to the Poisson bracketon the Poisson algebra Acl of classical observables by the formula

a, b = limh→0

h−1[a, b]

where a, b are any lifts of the elements a, b ∈ Acl to Aq.


One can make a similar definition in the world of P0 algebras. If Acl is any com-mutative differential graded algebra, and Aq is a cochain complex flat over R[[h]] whichreduces to Acl modulo h, then we can define a cochain map

−,−Aq : Acl ⊗ Acl → Acl

which measures the failure of the commutative product on Acl to lift to a product on Aq,to first order in h. (A precise definition is given in section 8.3).

Now, suppose that Acl is a P0 algebra. Let Aq be a cochain complex flat over R[[h]]which reduces to Acl modulo h. We say that Aq is a weak quantization of Acl if the bracket−,−Aq on Acl , induced by Aq, is homotopic to the given Poisson bracket on Acl .

This is a very weak notion, because the bracket −,−Aq on Acl need not be a Poissonbracket; it is simply a bilinear map. When we discuss the notion of strong quantization(section 1.8), we will explain how to put a certain operadic structure on Aq which guaran-tees that this induced bracket is a Poisson bracket.

1.7.1. The weak quantization theorem. Now that we have the definition of weakquantization at hand, we can state our weak quantization theorem.

For every open subset U ⊂ M, Obscl(U) is a lax P0 algebra. Given a BV quantizationof our classical field theory, Obsq(U) is a cochain complex flat over R[[h]] which coincides,modulo h, with Obscl(U). Our definition of weak quantization makes sense with minormodifications for lax P0 algebras as well as for ordinary P0 algebras.

1.7.1.1 Theorem (The weak quantization theorem). For every U ⊂ M, the cochain complexObsq(U) of classical observables on U is a weak quantization of the lax P0 algebra Obscl(U).

1.8. The strong quantization conjecture

We have seen (section 1.7) how the observables of a quantum field theory are a quan-tization, in a weak sense, of the lax P0 algebra of observables of a quantum field theory.The definition of quantization appearing in this theorem is unsatisfactory, however, be-cause the bracket on the classical observables arising from the quantum observables is nota Poisson observable.

In this section we will explain a stronger notion of quantization. We would like toshow that the quantization of the classical observables of a field theory we construct liftsto a strong quantization. However, this is unfortunately is still a conjecture (except for thecase of free fields).

1.8. THE STRONG QUANTIZATION CONJECTURE 13

1.8.0.2 Definition. A BD algebra is a cochain complex A, flat over C[[h]], equipped with acommutative product and a Poisson bracket of cohomological degree 1, satisfying the identity

d(a · b) = a · (db)± (da) · b + ha, b.

The BD operad is investigated in detail in section 8.4. Note that, modulo h, a BDalgebra is a P0 algebra.

1.8.0.3 Definition. A quantization of a P0 algebra Acl is a BD algebra Aq, flat over C[[h]],together with an equivalence of P0 algebras between Aq/h and Acl .

More generally, one can (using standard operadic techniques) define a concept of ho-motopy BD algebra. This leads to a definition of a homotopy quantization of a P0 algebra.

Recall that the classical observables Obscl of a classical field theory have the structureof a P0 factorization algebra on our space-time manifold M.

1.8.0.4 Definition. Let F cl be a P0 factorization algebra on M. Then, a strong quantization ofF cl is a lift of F cl to a homotopy BD factorization algebra F q, such that F q(U) is a quantization(in the sense described above) of F cl .

We conjecture that our construction of the factorization algebra of quantum observ-ables associated to a quantum field theory has this structure. More precisely,

Conjecture. Suppose we have a classical field theory on M, and a BV quantization of the theory.Then, Obsq has the structure of a homotopy BD factorization algebra quantizing the P0 factoriza-tion algebra Obscl .

Part 1

Prefactorization algebras

CHAPTER 2

From Gaussian measures to factorization algebras

This chapter serves as a kind of second introduction. We will start by defining the ob-servables of a free field theory, motivated by thinking about finite-dimensional Guassianintegrals. We will see that for a free theory on a manifold M, there is a space of observ-ables associated to any open subset U ⊂ M. We will see that that the operations we canperform on these spaces of observables give us the structure of a prefactorization algebraon M. This example will serve as further motivation for the idea that observables of afield theory are described by a prefactorization algebra.

2.1. Gaussian integrals

In physics, a free field theory is one where the action functional is purely quadratic. Abasic example is the free scalar field theory on a manifold M, where the space of fields isthe space C∞(M) of smooth functions on M, and the action funtional is

S(φ) = 12

∫M

φ(4+ m2)φ.

Here4 refers to the Laplacian with the convention that it’s eigenvalues are non-negative.The positive real number m is the mass of the theory. The main quantities of interest inthe free field theory are the correlation functions, defined by the heuristic expression

〈φ(x1), . . . , φ(xn)〉 =∫

φe−S(φ)φ(x1) . . . φ(xn)

where x1, . . . , xn are points in M.

Our task is to explain how the language of prefactorization algebras provides a simpleand natural way to make sense of this expression.

Like in most approaches to quantum field theory, we will motivate our definition ofthe prefactorization algerbra of observables by studying finite dimensional Gaussian inte-grals. Thus, let A be an n× n symmetric positive-definite matrix, and consider Gaussianintegrals of the form ∫

x∈Rnexp

(− 1

2 ∑ xi Aijxj)

f (x)

where f is a polynomial function on Rn.

17

18 2. FROM GAUSSIAN MEASURES TO FACTORIZATION ALGEBRAS

Most textbooks on quantum field theory would, at this point, explain Wick’s lemma,which is a combinatorial expression for such integrals. Then, they would go on to definesimilar infinite-dimensional Gaussian integrals by the analogous combinatorial expres-sion.

We will take a different approach, however. Instead of focusing on the combinatorialexpression for the integral, we will focus on the divergence operator associated to theGaussian measure.

Let P(Rn) denote the space of polynomial functions on Rn. Let Vect(Rn) denote thespace of vector fields on Rn with polynomial coefficients. If dµ denotes the Lebesguemeasure on Rn, let ωA denote the measure

ωA = exp( 1

2 ∑ xi Aijxj)

dµ.

Then, the divergence operator DivωA associated to this measure is a linear map

DivωA : Vect(Rn)→ P(Rn),

defined abstractly by saying that if V ∈ Vect(Rn), then

LVωA = (DivωA V)ωA

whereLV refers to the Lie derivative. Thus, the divergence of V measures the infinitesimalchange in volume that arises when one applies the infinitesimal diffeomorphism V.

In coordinates, the divergence is given by the formula

(†) DivωA

(∑ fi

∂

∂xi

)= −∑

i,jfixj Aij + ∑

i

∂ fi

∂xi.

By the definition of divergence, it is immediate that∫ωA DivωA V = 0

for all polynomial vector fields V. By changing basis of Rn into one where A is diagonal,one sees that the image of the divergence map is a codimension 1 linear subspace of thespace P(Rn) of polynomials on Rn. (This statement is true as long as A is non-degenerate,positive-definiteness is not required).

Let us identify P(Rn)/ Im DivωA with R by taking the basis to be the image of thefunction 1. What we have shown so far can be summarized in the following lemma.

2.1.0.5 Lemma. The quotient map

P(Rn)→ P(Rn)/ Im DivωA = R

is the map that sends a function f to ∫Rn ωA f∫Rn ωA1)

.

2.2. DIVERGENCE IN INFINITE DIMENSIONS 19

One nice feature of this approach to finite-dimensional Gaussian integrals is that itworks over any ring det A is invertible in this ring (this follows from the explicit algebraicformula we wrote for the divergence of a polynomial vector field). This way of lookingat finite-dimensional Gaussian integrals was further analyzed in [?] by the second authorand Theo Johnson-Freyd, where it was shown that one can derive the Feynman rules forfinite-dimensional Gaussian integration from such considerations.

2.2. Divergence in infinite dimensions

So far, we have seen that finite-dimensional Gaussian integrals are entirely encoded inthedivergence map from the Gaussian measure. In our approach to infinite-dimensionalGaussian integrals, the fundamental object we will define will be the divergence operator.We will recover the usual formulae for infinite-dimensional Gaussian integrals (in termsof the propagator) from our divergence operator. Further, we will see that analyzing thecokernel of the divergence operator will lead naturally to the notion of prefactorizationalgebra.

For concreteness, we will work with the free scalar field theory on a Riemannian man-ifold M which may or may not be compact. We will define a divergence operator for theputative Gaussian measure on C∞(M) associated to the quadratic form 1

2

∫M φ(4+ m2)φ.

Before we define the divergence operator, we need to define spaces of polynomialfunctions and of polynomial vector fields. Let U ⊂ M be an open subset. We will definepolynomial functions and vector fields on the space C∞(U).

The space of all continuous linear functionals on C∞(U) is the space Dc(U) of com-pactly supported distributions on U. In order to define the divergence operator, we needto consider functionals with more regularity. The space of linear functionals we will con-sider will be C∞

c (U), where every element f ∈ C∞c (U) defines a linear functional on

C∞(U) by the formula

φ 7→∫

Uf φd Vol

where d Vol refers to the Riemannian volume form on U.

As a first approximation, we will define the space of polynomial functions on C∞c (U)

to be the spaceP(C∞(U)) = Sym∗ C∞

c (U),

i.e. the symmetric algebra on C∞c (U). An element of P(C∞

c (U)) which is homogeneous ofdegree n can be written as a finite sum of monomials f1 . . . fn where the fi ∈ C∞

c (U). Sucha monomial defines a function on the space C∞(U) of fields by sending

φ 7→∫

x1,...,xn

f1(x1)φ(x1) . . . fn(xn)φ(xn).


Note that because C∞c (U) is a topological vector space, it might make more sense to use

an appropriate completion of Symn C∞c (U) instead of just the algebraic symmetric power.

We will introduce the appropriate completion shortly, but for the moment we will just usethe algebraic symmetric power. We use the notation P because this version of the algebraof polynomial functions is a little less natural than the completed version which we willcall P.

We will define the space of polynomial vector fields in a similar way. Recall that ifV is a finite-dimensional vector space, then the space of polynomial vector fields on V isP(V)⊗V, where P(V) is the space of polynomial functions on V.

In the same way, we let

Vect(C∞(U)) = P(C∞(U))⊗ C∞(U).

This is not, however, the vector fields we are really interested in. The space C∞(U) has afoliation, coming from the linear subspace C∞

c (U) ⊂ C∞(U). We are interested in vectorfields along this foliation. Thus, we let

Vectc(C∞(U)) = P(C∞(U))⊗ C∞c (U).

Again, it is more natural to use a completion of this space which takes account of thetopology on C∞

c (U). We will discuss such completions shortly.

Any element of Vectc(C∞(U)) can be written as an finite sum of monomials of theform

f1 . . . fn∂

∂φ

for fi, φ ∈ C∞c (U). By ∂

∂φ we mean the constant-coefficient vector field given by infinitesi-mal translation in the direction φ in C∞(U).

Vector fields act on functions, in the usual way: the formula is

f1 . . . fn∂

∂φ(g1 . . . gm) = f1 . . . fn ∑ g1 . . . gi . . . gm

∫U

gi(x)φ(x)d Vol

where d Vol is the Riemannian volume form on U.

2.2.0.6 Definition. The divergence operator associated to the quadratic form∫

φ(4 + m2)φ isthe linear map

Vectc(C∞(U))→ P(C∞c (U))

defined by

(‡) Div(

f1 . . . fn∂

∂φ

)= − f1 . . . fn(4+ m2)φ +

n

∑i=1

f1 . . . fi . . . fn

∫x∈U

φ(x) fi(x).

2.2. DIVERGENCE IN INFINITE DIMENSIONS 21

Note that this formula is entirely parallel to the formula for divergence of a Gaussianmeasure in finite dimensions, given in formula (†).

2.2.1. As we mentioned above, it is probably more natural to use a completion ofthe spaces P(C∞(U)) and Vectc(C∞(U)) of polynomial functions and polynomial vectorfields.

Any element F ∈ C∞c (Un) defines a polynomial function on C∞(U) by

φ 7→∫

UnF(x1, . . . , xn)φ(x1) . . . φ(xn).

This functional doesn’t change if we permute the arguments of F by an element of thesymmetric group Sn, so that this only depends on the image of F in the coinvariants ofC∞

c (Un) by the symmetric group action. This, of course, is the same as the symmetricgroup invariants.

Therefore we letP(C∞(U)) = ⊕n≥0C∞

c (Un)Sn ,where the subscript indicates coinvariants. A dense subspace of C∞

c (Un)Sn is given by thesymmetric power Symn C∞

c (U). Thus, P(C∞(U)) is a dense subspace of P(C∞(U)).

In a similar way, we can define Vectc(C∞(U)) by

Vectc(C∞(U)) = ⊕nC∞c (Un+1)Sn

where the symmetric group acts on the first n factors. A dense subspace of C∞c (Un+1)Sn is

given by Symn C∞c (U)⊗C∞

c (U). Thus, Vectc(C∞(U)) is a dense subspace of Vectc(C∞(U)).

2.2.1.1 Lemma. The divergence map

Div : Vectc(C∞(U))→ P(C∞(U))

extends continuously to a map

Vectc(C∞(U))→ P(C∞(U)).

PROOF. Suppose that

F(x1, . . . , xn+1) ∈ C∞c (Un+1)Sn ⊂ Vectc(C∞(U)).

The divergence map in equation (‡) extends to a map which sends F to

−4xn+1 F(x1, . . . , xn+1) + ∑i

∫xi∈U

F(x1, . . . , xi, . . . , xn, xi).

Now we can define the quantum observables of a free field theory.


2.2.1.2 Definition. For an open subset U ⊂ M, we let

H0(Obsq(U)) = P(C∞(U))/ Im Div .

In other words, we let H0(Obsq(U)) be the cokernel of the operator Div. Later we wilsee that there is a cochain complex of quantum observables whose zeroth cohomology iswhat we just defined (this is why we write H0).

Let us explain why we should interpret this space as a space of quantum observables.We expect that an observable in a field theory is a function on the space of fields. Anobservable on a field theory on an open subset U ⊂ M is a function which only dependson the behaviour of the fields on U. Morally speaking, the expectation value of the ob-servable is the integral of this function against the “functional measure” on the space offields.

Our approach is that we will not try to define the functional measure, we will insteaddefine the divergence operator. If we have some functional of C∞(U) which is the di-vergence of an appropriate vector field, then the expectation value of the correspondingobservable is zero. Thus, we would expect that the observable given by a divergence isnot a physically measurable quantity, and so should be set to zero.

The appropriate vector fields on C∞(U) – the ones for which our divergence opera-tor makes sense – are vector fields along the foliation of C∞(U) by compactly supportedfields. Thus, the quotient of functions on C∞(U) by the subspace of divergences of suchvector fields gives a definition of observables.

2.3. The prefactorization structure on observables

Suppose that we have a Gaussian measure on Rn. Then, every function on Rn withpolynomial growth is integrable, and this space of functions forms an algebra. In ourapproach to quantum field theory, we defined the space of observables to be the quotientof a space of polynomial functions by the subspace of functions which are divergences.This subspace is not an ideal. Thus, we would not expect the space of observables to be acommutative algebra.

However, we will find that some shadow of this commutative algebra structure ex-ists, which allows us to combine observables on disjoint subsets. This residual structurewill give the spaces H0(Obsq(U)) of observables associated to open subset U ⊂ M thestructure of a prefactorization algebra.

Let us make these statements precise. Note that P(C∞(V)) is a commutative algebra,as it is a space of polynomial functions on C∞(V). Further, if U ⊂ V there is a mapof commutative algebras P(C∞(U)) → P(C∞(V)), obtained by composing a polynomial

2.3. THE PREFACTORIZATION STRUCTURE ON OBSERVABLES 23

map C∞(U) → R with the restriction map C∞(V) → C∞(U). This map is injective. Wewill sometimes refer to an element of the subspace P(C∞(U)) ⊂ P(C∞(V)) as an elementof P(C∞(V)) which is supported on U.

2.3.0.3 Lemma. The product map

P(C∞(V))⊗ P(C∞(V))→ P(C∞(V))

does not descend to a map

H0(Obs(V))⊗ H0(Obs(V))→ H0(Obs(V)).

If U1, U2 are disjoint open subset of M, both contained in V, then we have a map

P(U1)⊗ P(U2)→ P(V)

obtained by combining the inclusion maps P(Ui) → P(V) with the product map on P(V). Thismap does descend to give a map

H0(Obs(U1))⊗ H0(Obs(U2))→ H0(Obs(V)).

In other words, although the product of general observables does not make sense, theproduct of observables with disjoint support does.

PROOF. Let U1, U2 be disjoint open subsets of M, both contained in an openV. Let usview the spaces Vectc(C∞(Ui)) and P(C∞(Ui)) as subspaces of Vectc(C∞(V)) and P(C∞(V))respectively.

On an ordinary finite-dimensional manifold, the divergence with respect to any vol-ume form has the following property: for a vector field X and a function f , we have

Div( f X)− f Div X = X f ,

where X f is the action of X on f .

The same equation holds for the divergence operator we have defined in infinite di-mensions. If X ∈ Vectc(C∞(V)) and G ∈ P(C∞

c (V)) then

Div(GX)− G DivX = X(G).

This tells us that the image of Div is not an ideal, as X(G) is not necessarily in the imageof Div.

However, suppose that G is in P(C∞(U1)) and that X is in Vectc(C∞(U2)). Then,X(G) = 0. This is clear from the fact that Vectc(C∞(U2)) is the space of polynomial vectorfields along the foliation of C∞(U2) by compactly supported functions. If X ∈ Vectc(U2)and φ ∈ C∞(U2), let Xφ ∈ C∞

c (U2) be the tangent vector associated to X at the point φ.The infinitesimal symmetry X sends φ to φ + εXφ. Since Xφ has compact support in U2,φ and φ + εXφ agree outside a compact set in U2. (This is just rephrasing the fact that X


is a vector field on C∞(U2) along the foliation coming from compactly-supported smoothfunctions).

If φ ∈ C∞(V), then G(φ) only depends on the behaviour of φ on C∞(U1), whereasφ + εXφ is equal to φ outside a compact set in U2, and in particular is equal to φ on U1which is disjoint from U2. Therefore

G(φ) + ε (XG) (φ) = G(φ + εXφ) = G(φ).

That is, XG = 0.

This shows that, if X and G have disjoint support,

Div(GX) = G Div X.

This is enough to show that the product map

P(U1)⊗ P(U2)→ P(V)

descends to a map

H0(Obsq(U1))⊗ H0(Obsq(U2))→ H0(Obsq(V)).

Indeed, if Gi ∈ P(U2) and Xi ∈ Vectc(Ui) then

G2 Div(X1) = Div(GX2) ∈ Im Div

and similarly, Div(X2)G1 ∈ Im Div.

In a similar way, if U1, . . . , Un are disjoint opens all contained in V, then the map

P(U1)⊗ · · · ⊗ P(Un)→ P(V)

descends to a map

H0(Obsq(U1))⊗ · · · ⊗ H0(Obsq(Un))→ H0(Obsq(V)).

Thus, we see that the spaces H0(Obsq(U)) for open sets U ⊂ M are naturally equippedwith the structure maps necessary to define a prefactorization algebra. (See section 1.3 fora sketch of the definition of a factorization algebra, and section 3.1 for more details on thedefinition). It is straightforward to check that these structure maps satisfy the necessarycompatibility conditions to define a prefactorization algebra.

2.4. From quantum to classical

Our general philosophy is that the quantum observables of a field theory are a factor-ization algebra which deform the classical observables. Classical observables are definedto be functions on the space of solutions to the equations of motion.

2.4. FROM QUANTUM TO CLASSICAL 25

Let us see why this holds for a class of measures in finite dimensions. Let S be apolynomial function on Rn. Let d Vol denote the Lebesgue measure on Rn, and considerthe measure

ω = e−S/hd Volwhere h is a small parameter. The divergence with respect to ω satisfies

Divω

(∑ fi

∂

∂xi

)= ∑

∂ fi

∂xi− 1

h ∑ fi∂S∂xi

.

Let P(Rn) denote the space of polynomial functions on Rn and let Vect(Rn) denote thespace of polynomial vector fields. We view the divergence operator Divω as a map Vect(Rn)→P(Rn). Note that the image of Divω and of h Divω is the same as long as h→ 0. As h→ 0,the operator h Divω becomes the operator

∑ fi∂

∂xi→ −∑ fi

∂S∂xi

.

Therefore, the h → 0 limit of the image of Divω is the ideal in P(Rn) which cuts out thecritical locus of S.

Let us now check the analogous property for the observables of a free scalar field the-ory on a manifold M. We will consider the divergence for the putative Gaussian measure

exp(− 1

h

∫M

φ(4+ m2)φ

)d Vol

on C∞(M). For any open subset U ⊂ M, this divergence operator gives us a map

Divh : Vectc(C∞(U))→ P(C∞(U))

which sends

f1 . . . fn∂

∂φ→∑ f1 . . . fi

∫M

f φ− 1h f1 . . . fn(4+ m2)φ.

As before, in the h → 0 limit, the second term dominates; so that the h → 0 limit ofthe image of Divh is the closed subspace of P(C∞(U)) spanned by functionals of theform f1 . . . fn(4 + m2)φ, where fi and φ are in C∞

c (U)). This is the topological ideal inP(C∞(U)) generated by linear functionals of the form (4+ m2) f where f ∈ C∞

c (U). IfS(φ) =

∫φ(4 + m2)φ is the action functional of our theory, then this is precisely the

topological ideal generated by∂S∂φ

for φ ∈ C∞c (U). In other words, it is the ideal cut out by the Euler-Lagrangian equations.

Let us call this ideal IEL.

A more precise statement of what we have just sketched is the following. Define aprefactorization algebra H0(Obscl(U)) (the superscript cl stands for classical) which as-signs to U the quotient of P(C∞(U)) by the ideal IEL. Thus, H0(Obscl(U)) should be


thought of as the space of polynomial functions on the space of solutions to the Euler-Lagrangian equations. Note that each constituent space H0(Obscl(U)) in this prefactor-ization algebra has the structure of a commutative algebra, and the structure maps are allmaps of commutative algebras. This means that H0(Obscl) forms a commutative prefactor-ization algebra. Heuristically, this means that the product map defining the factorizationstructure is defined for all pairs of opens U1, U2 ⊂ V, and not just when U1 and U2 aredisjoint.

2.4.0.4 Lemma. There is a prefactorization algebra H0(Obsqh) over C[h] which when specialized

to h = 1 is H0(Obsq) and to h = 0 is H0(Obscl).

This prefactorization algebra assigns to an open set the quotient of the map

h Divh : Vectc(C∞(M))[h]→ P(C∞(M)j)[h]

where Divh is the map discussed earlier.

We will see later that the R[h]-module H0(Obsqh(U)) is free (although this is a special

property of free theories and is not always true for an interacting theory.).

2.5. Correlation functions

We have seen that the observables of a free scalar field theory on a manifold M giverise to a factorization algebra. In this section, we will explain how the structure of a factor-ization algebra is enough to define correlation functions of observables. We will calculatecertain correlation functions explicitly, and find the same answers physicists normallyfind.

Let us suppose that M is a compact Riemannian manifold, and as before, let us con-sider the observables of the free scalar field theory on M with mass m > 0. Then, we havethe following.

2.5.0.5 Lemma. If the mass m is positive, then H0(Obsq(M)) = R.

We should compare this with the statement that if we have a Gaussian measure on Rn,the image of the divergence map is of codimension 1 in the space of polynomial functionson Rn. The assumption that the mass is positive is necessary to ensure that the quadraticform

∫M φ(4+ m2)φ is non-degenerate.

This lemma will follow from our more detailed analysis of free theories in chap-ter 4. The main point is that there is a family over R[h] connecting H0(Obsq(M)) andH0(Obscl(M)). Since the only solution to the equations of motion in the case that m > 0 isthe function φ = 0, the algebra H0(Obscl(M)) is R. To conclude that H0(Obsq(M)) is also

2.5. CORRELATION FUNCTIONS 27

R, we need to show that H0(Obsqh(M)) is flat over R, which will follow from a spectral

sequence analysis we will perform later.

There is always a canonical observable 1 ∈ H0(Obsq(U)) for any open subset U ⊂ M.This is defined to be the image of the function 1 ∈ P(C∞(U)). We identify H0(Obsq(M))with R by taking the observable 1 to be a basis.

2.5.0.6 Definition. Let U1, . . . , Un ⊂ M be disjoint open subsets. The correlator is the prefac-torization structure map

〈−, . . . ,−〉 : H0(Obsq(U1))⊗ · · · ⊗ H0(Obsq(Un))→ H0(Obsq(M)) = R

We should compare this definition with what happens in finite dimensions. If we havea Gaussian measure on Rn, then the space of polynomial functions modulo divergencesis one-dimensional. If we take the image of a function 1 to be a basis of this space, thenwe get a map

P(Rn)→ R

from polynomial functions to R. This map is the integral against the Gaussian measure,normalized so that the integral of the function 1 is 1.

In our infinite dimensional situation we are doing something very similar. Any rea-sonable definition of the correlation function of functions O1, . . . , On with Oi ∈ P(C∞(Ui))should only depend on the product function O1 . . . On ∈ P(C∞(M)). Thus, the correlationfunction map should be a linear map P(C∞(M))→ R. Further, it should send divergencesto zero. We have seen that up to scale there is only one such map.

Next we will check explicitly that this correlation function map really matches up withwhat physicists expect. Let fi ∈ C∞

c (Ui) be compactly-supported smooth functions on theopen sets U1, U2 ⊂ M. Let us view each fi as a linear function on C∞(Ui), and so as anelement of P(C∞(Ui)).

Let G ∈ D(M×M) be the unique distribution on M×M with the property that

(4x + m2)G(x, y) = δDiag.

In other words, if we apply the operator 4+ m2 to the first factor of G, we find thedelta function on the diagonal. Thus, G is the kernel for the operator (4+ m2)−1. In thephysics literature, G is called the propagator, and in the math literature it is called theGreen’s function for the operator4+ m2. Note that G is smooth away from the diagonal.

Then,

2.5.0.7 Lemma.〈 f1, f2〉 =

∫x,y∈M

f1(x)G(x, y) f2(y).


Note that this is exactly what a physicist would say.

PROOF. Later we will give a slicker and more general proof of this kind of statement.Here we will give a simple proof to illustrate how the Green’s function arises from ourhomological approach to defining functional integrals.

Letφ = (4+ m2)−1 f2 ∈ C∞(M).

Thus,

φ(x) =∫

y∈MG(x, y) f2(y).

Consider the vector field

f1∂

∂φ∈ Vectc(C∞(M)).

Note that

Div(

f1∂

∂φ

)=∫

x∈Mf1(x)φ(x)− f1

((4+ m2)φ

)=

(∫M

f1(x)G(x, y) f2(y))· 1− f1 f2.

The element f1 f2 ∈ P(C∞(M)) represents the factorization product of the observablesfi ∈ H0(Obsq(Ui)) in H0(Obsq(M)). The displayed equation tells us that

f1 f2 =

(∫M

f1(x)G(x, y) f2(y))· 1 ∈ H0(Obsq(M)).

Since the observable 1 is chosen to be the basis element identifying H0(Obsq(M)) with R,the result follows.

2.6. Further results on free field theories

In this chapter, we showed that, if we define the observables of a free field theory asa cokernel of a certain divergence operator, then these spaces of observables form a pref-actorization algebra. We also showed that this prefactorization algebra contains enoughinformation to allow us to define the correlation functions of observables, and that forlinear observables we find the same formula that physicists would write.

In chapter 3 we will show that a certain class of factorization algebras on the real lineare equivalent to associative algebras together with a derivation. The derivation comesfrom infinitesimal translation on the real line.

In chapter 4 we will analyze the factorization algebra of free field theories in moredetail. We will show that if we consider the free field theory on R, the factorization algebra

2.7. INTERACTING THEORIES 29

H0(Obsq) corresponds (under the relationship between factorization algebras on R andassociative algebras) to the Weyl algebra. The Weyl algebra is generated by observablesp, q corresponding to position and momentum satisfying [p, q] = 1. If we consider insteadthe family over R[h] of factorization algebras H0(Obsq

h) discussed above, then we find thecommutation relation [p, q] = h. This, of course, is what is traditionally called the algebraof observables of quantum mechanics. In this case, we will further see that the derivationof this algebra (corresponding to inner time translation) is an inner derivation, given bybracketing with the Hamiltonian

H = p2 −m2q2

which is the standard Hamiltonian for quantum mechanics.

More generally, we can consider a free scalar free theory on N ×R where N is a com-pact Riemannian manifold. This gives rise to a factorization algebra on R which assignsto an open subset U ⊂ R the space H0(Obsq(N ×U)). We will see that there is a densesub-factorization algebra of this factorization algebra on R which corresponds to an as-sociative algebra. This associative algebra is a tensor product of Weyl algebras, with oneWeyl algebra arising from each eigenspace of the operator 4 + m2 on C∞(N). In otherwords, we find quantum mechanics on R with values in the infinite-dimensional spaceC∞(N), where the λ eigenspace of the operator4+ m2 is given mass

√λ. This is entirely

consistent with physics expectations.

2.7. Interacting theories

In any approach to quantum field theory, free field theories are easy to construct. Thechallenge is always to construct interacting theories. The core results of this book showhow to construct the factorization algebra corresponding to interacting field theories, de-forming the factorization algebra for free field theories discussed above.

Let us explain a little bit about the challenges we need to overcome in order to dealwith interacting theories, and how we overcome these challenges.

Consider an interacting scalar field theory on a Riemannian manifold M. For instance,we could consider an action functional of the form

S(φ) = − 12

∫M

φ(4+ m2)φ +∫

Mφ4.

In general, the action functional must be local: it must arise as the integral over M of somepolynomial in φ and its derivatives.

We will let I(φ) denote the interacting term in our field theory, which consists of thecubic and higher terms in S. In the above example, I(φ) =

∫φ4. We will always assume


that the quadratic term in S is of the form − 12

∫M φ(4+ m2)φ. (Of course, our techniques

apply to a very general class of interacting theories, including gauge theories).

If U ⊂ M is an open subset, we can consider, as before, the spaces Vectc(C∞(M))and P(C∞(M)) of polynomial functions and vector fields on M. By analogy with thefinite-dimensional situation, one can try to define the divergence for the putative measureexp S(φ)/hdµ (where dµ refers to the “Lebesgue measure” on C∞(M))) by the formula

Divh

(f1 . . . fn

∂

∂φ

)= 1

h f1 . . . fn∂S∂φ

+ ∑ f1 . . . fi . . . fn

∫M

fiφ.

This is the same formula we used earlier when S was quadratic.

Now we see that a problem arises. We defined P(C∞(U)) as the space of polynomialfunctions whose Taylor terms are given by integration against a smooth function on Un.That is,

P(C∞(U)) = ⊕nC∞c (Un)Sn .

If φ ∈ C∞c (U)), then ∂S

∂φ is not necessarily in this space of functions. For instance, if I(φ) =∫φ4 is the interaction term in the example of the φ4 theory, then

∂I∂φ0

(φ) =∫

Mφ0φ3.

This is a cubic function on the space C∞(U) but it is not given by integration against anelement in C∞

c (U3). Instead it is given by integrating against a distribution on U3, namely,the delta-distribution on the diagonal.

We can try to solve this by using a larger class of polynomial functions. Thus, wecould let

P(C∞(U)) = ⊕nDc(Un)Sn

whereDc(Un) is the space of compactly supported distributions on U. Similarly, we couldlet

Vectc(U) = ⊕nDc(Un+1)Sn .

The spaces P(C∞(U)) and Vectc(C∞(U)) are dense subspaces of these spaces.

If φ0 ∈ Dc(U) is a compactly-supported distribution, then for any local functional S,∂S∂φ is a well-defined element of P(C∞(U)). Thus, it looks like this has solved our problem.

However, using this larger space of functions gives us an additional problem: thesecond term in the divergence operator now fails to be well-defined. For example, iff , φ ∈ Dc(U), then we have

Divh f φ = 1h f

∂S∂φ

+∫

Mf φ.

2.7. INTERACTING THEORIES 31

Now,∫

M f φ doesn’t make sense, because it involves pairing the distribution f φ on thediagonal in M2 with the delta-function on the diagonal.

If we take the h→ 0 limit of h Divh we find, however, a well-defined operator

Vectc(C∞(U))→ P(C∞(U))

X → XS

which sends a vector field X to its action on the local functional S.

The cokernel of this operator is the quotient of space P(C∞(U)) by the ideal IEL gen-erated by the Euler-Lagrange equations. We thus let

H0(ObsclS (U)) = P(C∞(U))/IEL.

As U varies, this forms a factorization algebra on M, which we call the factorization alge-bra of classical observables associated to the action functional S.

2.7.0.8 Lemma. If S = − 12

∫φ(4+ m2)φ, then this definition of classical observables coincides

with the one we discussed earlier:

H0(ObsclS (U)) = H0(Obscl(U))

where H0(Obscl(U)) is defined, as earlier, to be the quotient of the space P(C∞(M)) by the idealof Euler-Lagrange equations.

This result is a version of elliptic regularity.

Now we can see the challenge we have. If S is the action functional for the free fieldtheory, then we have a factorization algebra of classical observables. This factorizationalgebra deforms in two ways: first, we can deform it into the factorization algebra ofquantum observables for a free theory. Second, we can deform it into the factorizationalgebra of classical observables for an interacting field theory.

The challenge is to perform both of these deformations simultaneously.

The technique we use to construct the observables of an interacting field theory usesthe renormalization technique of [Cos11c]. In [Cos11c], the first author gives a definitionof a quantum field theory and a cohomological method for constructing field theories. Afield theory as defined in [Cos11c] gives us (essentially from the definition) a family ofdivergence operators

Div[L] : Vectc(M)→ P(C∞(M)),one for every L > 0. These divergence operators, for varying L, can be conjugated to eachother by linear continuous isomorphisms of Vectc(M)) and P(C∞(M)). These divergenceoperators do not map the space Vectc(U)) to the space P(C∞(U)), for an open subsetU ⊂ M. However (roughly speaking) for L small the operator Div[L] only increases the


support of an element of Vectc(U)) by a small amount. This turns out to be enough todefine the factorization algebra of quantum observables.

This construction of quantum observables for an interacting field theory is given inchapter 15, which is the most technically difficult chapter of the book. In the preceedingchapters, we will develop some aspects of the theory of factorization algebras in general;analyze in more detail the factorization algebra associated to a free theory; construct andanalyze factorization algebras associated to vertex algebras such as the Kac-Moody vertexalgebra; develop classical field theory using a homological approach arising from the BVformalism; and flesh out the description of the factorization algebra of classical observ-ables we have sketched here.

CHAPTER 3

Prefactorization algebras and basic examples

In this chapter we will give a formal definition of the notion of prefactorization al-gebra. We also explain how to construct a factorization algebra from any sheaf of Liealgeras on a manifold M. This construction is called the factorization envelope, and is re-lated to the universal enveloping algebra of a Lie algebra as well as to Beilinson-Drinfeld’s[BD04] chiral envelope. Although the factorization envelope construction is very simple,it plays an important role in field theory. For example, the factorization algebra for anyfree theories is a factorization envelope, as is the factorization algebra corresponding tothe Kac-Moody vertex algebra. More generally, factorization envelopes play an importantrole in our formulation of Noether’s theorem for quantum field theories.

3.1. Prefactorization algebras

Let M be a topological space and let (C,⊗) be a symmetric monoidal category. We areparticularly interested in the case where M is a smooth manifold and C is Vect or dgVectwith the usual tensor product as the monoidal product. In this section we will give aformal definition of the notion of a prefactorization algebra.

3.1.1. The essential idea of a prefactorization algebra. A prefactorization algebra Fon M, taking values in cochain complexes, is a rule that assigns a cochain complex F (U)to each open set U ⊂ M along with

• a cochain map F (U)→ F (V) for each inclusion U ⊂ V;• a cochain map F (U1)⊗ · · · ⊗ F (Un) → F (V) for every finite collection of open

sets where each Ui ⊂ V and the Ui are pairwise disjoint;• the maps are compatible in the obvious way (e.g. if U ⊂ V ⊂ W is a sequence of

open sets, the map F (U)→ F (W) agrees with the composition through F (V));

Thus F resembles a precosheaf, except that we tensor the cochain complexes rather thantake their direct sum. These axioms imply that F (∅) is a commutative algebra; we saythat F is a unital prefactorization algebra if F (∅) is a unital commutative algebra. Inpractise, F (∅) is one of C, R, C[[h]], R[[h]].

33

34 3. PREFACTORIZATION ALGEBRAS AND BASIC EXAMPLES

The crucial example to bear in mind is an associative algebra. Every associative alge-bra A defines a prefactorization algebra FA on R, as follows. To each open interval (a, b),we set FA((a, b)) := A. To any open set U = äj Ij, where each Ij is an open interval, weset F (U) :=

⊗j A.1 The structure maps simply arise from the multiplication map for A.

Figure (1) displays the structure of FA. (Notice the resemblance to the notion of an E1 orA∞ algebra.)

FIGURE 1. Structure of FA

In the remainder of this section, we describe two other ways of phrasing this idea, butthe reader who is content with this definition and eager to see examples should feel freeto jump to section ??, referring back as needed.

3.1.2. Prefactorization algebras in the style of algebras over an operad.

3.1.2.1 Definition. Let FactM denote the following multicategory associated to M.

• The objects consist of all connected open subsets of M.• For every (possibly empty) finite collection of open sets Uαα∈A and open set V, there is

a set of maps FactM(Uαα∈A, V). If the Uα are pairwise disjoint and all are containedin V, then the set of maps is a single point. Otherwise, the set of maps is empty.• The composition of maps is defined in the obvious way.

Remark: By “multicategory” we mean what is also called a colored operad or a pseudo-tensor category. In [Lei04], there is an accessible discussion of multicategories; in Lein-ster’s terminology, we work with “fat symmetric multicategories.”

3.1.2.2 Definition. Let C be a multicategory. A prefactorization algebra on M taking values inC is a functor (of multicategories) from FactM to C.

Since symmetric monoidal categories are special kinds of multicategories, this defini-tion makes sense for symmetric monoidal categories.

Remark: In other words, a prefactorization algebra is an algebra over the colored operadFactM.

Remark: If C is a symmetric monoidal category under coproduct, then a precosheaf on Mwith values in C defines a prefactorization algebra valued in C. Hence, our definition

1One can take infinite tensor products of unital algebras (see, for instance, exercise 23, chapter 2 [AM69]).The idea is simple. Given an infinite set I, consider the poset of finite subsets of I, ordered by inclusion. Foreach finite subset J ⊂ I, we can take the tensor product AJ :=

⊗j∈J A. For J → J′, we define a map AJ → AJ′

by tensoring with the identity 1 ∈ A for every j ∈ J′\J. Then AI is the colimit over this poset.

3.1. PREFACTORIZATION ALGEBRAS 35

broadens the idea of “inclusion of open sets leads to inclusion of sections” by allowingmore general monoidal structures to “combine” the sections on disjoint open sets.

Note that if F is any prefactorization algebra, then F (∅) is a commutative algebraobject of C.

3.1.2.3 Definition. We say a prefactorization algebra F is unital if the commutative algebraF (∅) is unital.

3.1.3. Prefactorization algebras in the style of precosheaves. Any multicategory Chas an associated symmetric monoidal category SC , which is defined to be the univer-sal symmetric monoidal category equipped with a functor of multicategories C → SC .Concretely, an object of SC is a formal tensor product a1 ⊗ · · · ⊗ an of objects of C . Mor-phisms in SC are characterized by the property that for any object b in C, the set ofmaps SC (a1 ⊗ · · · ⊗ an, b) in the symmetric monoidal category is exactly the set of mapsC (a1, . . . , an, b) in the multicategory category C .

We can give an alternative definition of prefactorization algebra by working with thesymmetric monoidal category S FactM rather than the multicategory FactM.

3.1.3.1 Definition. Let S FactM denote the following symmetric monoidal category.

• The objects of S FactM consist of topological spaces U equipped with a map U → Mwhich, on each connected component of U, is an open embedding embedding.• A map from U → M to V → M is a commutative diagram

U i−→ V↓ M

where the map i is an embedding.• The symmetric monoidal structure on S FactM is given by disjoint union.

3.1.3.2 Lemma. S FactM is the universal symmetric monoidal category containing the multicat-egory FactM.

The alternative definition of prefactorization algebra is as follows.

3.1.3.3 Definition. A prefactorization algebra with values in a symmetric monoidal categoryC is a symmetric monoidal functor

S FactM → C .

Remark: Although “algebra” appears in its name, a prefactorization algebra only allowsone to “multiply” elements that live on disjoint open sets. The category of prefactorizationalgebras (taking values in some fixed target category) has a symmetric monoidal product,


so we can study commutative algebra objects in that category. As an example, we willconsider the observables for a classical field theory (chapter ??).

3.2. Recovering associative algebras from prefactorization algebras on R

We explained above how an associative algebra provides a prefactorization algebraon the real line. There are, however, prefactorization algebras on R that do not comefrom associative algebras. Here we will characterize those that do arise from associativealgebras.

3.2.0.4 Definition. Let F be a prefactorization algebra on R taking values in the category ofvector spaces (without any grading). We say F is locally constant if the map F (U)→ F (V) isan isomorphism whenever the inclusion of opens U ⊂ V is a homotopy equivalence.

3.2.0.5 Lemma. Let F be a locally constant, unital prefactorization algebra on R taking values invector spaces. Let A = F (R). Then A has a natural structure of an associative algebra.

Remark: Recall that F being unital means that the commutative algebra F (∅) is equippedwith a unit. We will find that A is an associative algebra over F (∅).

PROOF. For any interval (a, b) ⊂ R, the map

F ((a, b))→ F (R) = A

is an isomorphism. Thus, we have a canonical isomorphism

A = F ((a, b))

for all intervals (a, b).

Notice that if (a, b) ⊂ (c, d) then the diagram

A∼= //

Id

F ((a, b))

i(a,b)(c,d)

A∼= // F ((c, d))

commutes.

The product map m : A⊗ A → A is defined as follows. Let a < b < c < d. Then, theprefactorization structure on F gives a map

F ((a, b))⊗F ((c, d))→ F ((a, d)),

and so, after identifying F ((a, b)), F ((c, d)) and F ((a, d)) with A, we get a map

A⊗ A→ A.

This is the multiplication in our algebra.

3.3. COMPARISONS WITH OTHER AXIOM SYSTEMS FOR FIELD THEORIES 37

It remains to check the following.

(1) This multiplication doesn’t depend on the intervals (a, b) q (c, d) ⊂ (a, d) wechose, as long as (a, b) < (c, d).

(2) This multiplication is associative and unital.

This is an easy (and instructive) exercise.

3.3. Comparisons with other axiom systems for field theories

Now that we have explained carefully what we mean by a prefactorization algebra,let us say a little about the history of this concept, and how it compares to other axiomia-tizations of quantum field theory.

3.3.1. Factorization algebras in the sense of Beilinson-Drinfeld. For us, one sourceof inspiration is Beilinson-Drinfeld’s book [BD04]. These authors gave a geometric refor-mulation of the theory of vertex algebras in terms of an algebro-geometric version of theconcept of factorization algebra. For Beilinson and Drinfeld, a factorization algebra on analgebraic curve X is a sheaf on a certain auxiliary space, the Ran space Ran(X) of X. TheRan space Ran(X) is the set of all finite non-empty subset of X, equipped with a certainnatural structure of ind-scheme.

A factorization algebra is in particular a quasi-coherent sheaf F on Ran(X). Thus, Fis something which assigns to every finite set I ⊂ X a vector space FI (the fibre of F at I)which varies in an algebraic way with I. In addition, for F to be a factorization algebra,we must have natural isomorphisms

FIqJ∼= FI ⊗FJ .

for disjoint finite sets I, J ⊂ X. These isomorphisms must vary nicely with the sets I andJ and must satisfy natural associativity and commutativity properties similar to the oneswe have considered.

Let us explain the heuristic dictionary between the definition used by Beilinson andDrinfeld and the one used here. We will sketch show how factorization algebras (in thesense used in this book) on a manifold M can be turned into Beilinson-Drinfeld stylefactorization algebras, and conversely.

Suppose we have a factorization algebra on M in the sense used in this book. At leastheuristically, one gets an infinite-rank vector bundle on Ran(M) as follows. Given finitesubset I ⊂ M, one can define FI to be the costalk of F at I:

FI = holimI⊂U F (U).


The symbol holim indicates the homotopy limit, which is necessary because the spacesF (U) are cochain complexes. Since we are only sketching a relationship between ourpicture and Beilinson-Drinfeld’s, we will not go into details about homotopy limits.

Since we interpet F (U) as the observables of a quantum field theory on U, we shouldinterpret FI as the observables supported on the finite set I ⊂ U.

The factorization structure maps give us maps

FI ⊗FJ → FIqJ

if I, J ⊂ M are disjoint finite subsets. The assumption that F is a factorization algebra andnot just a prefactorization algebra will imply that these maps are isomorphisms.

From this presentation it’s not entirely obvious that the spaces FI vary nicely as Ivaries, especially as some points in I collide. This is necessary in Beilinson-Drinfeld’salgebro-geometric picture to show that we have a quasi-coherent sheaf on the Ran space.In fact, in our main examples of factorization algebras (arising from field theories), wewere unable to show that the spaces FI vary nicely as the points in I collide, which is whywe chose the particular axiom system for factorization algebras that we did.

A simpler relationship between factorization algebras as developed in this book andthe Ran space is as follows. We will show how to construct a precosheaf on the Ran spaceof M from a factorization algebra on M.

If U ⊂ M is an open subset, let Ran(U) ⊂ Ran(M) be the open subset of Ran(M)consisting of finite subsets of U. More generally, if U1, . . . , Un ⊂ M are disjoint opensubsets, we have a map

Ran(U1)× · · · × Ran(Un) ⊂ Ran(M)

which sends a collection of finite subsets Ii ⊂ Ui to their disjoint union. This map isinjective and the image is an open subset. These sets form a basis for a certain naturaltopology on Ran(M) (which is the topology we will consider).

If we have a factorization algebra F on M, we define a precosheaf Ran(F ) on Ran Min this topology by declaring that

Ran(F )(Ran(U1)× · · · × Ran(Un)) = F (U1)⊗ · · · ⊗ F (Un).

This precosheaf has a certain multiplicative property. We say two open subsets W1, W2 ⊂Ran(M) are strongly disjoint if for every pair of finite sets I1, I2 ⊂ M with Ii ∈ Wi, I1 andI2 are disjoint. If W1, W2 are strongly disjoint, we let W1 ∗W2 denote the open subset ofthe Ran space consisting of those finite sets J ⊂ M which are a disjoint union J = I1 q I2where I1 ∈W1 and I2 ∈W2. The factorization structure on F induces a map

Ran(F )(W1)⊗ Ran(F )(W2)→ Ran(F )(W1 ∗W2).

The factorization axiom will imply that this map is an isomorphism.

3.3. COMPARISONS WITH OTHER AXIOM SYSTEMS FOR FIELD THEORIES 39

The codescent axiom we impose on factorization algebras is closely related to thecodescent axiom for cosheaves on the Ran space.

3.3.2. Segal’s axioms for quantum field theory. Segal has developed and studiedsome very natural axioms for quantum field theory []. These axioms were first studiedin the world of topological field theory by Atiyah, Segal, Witten ... and in conformal fieldtheory by Segal [].

According to Segal’s philosophy, a d-dimensional quantum field theory (in Euclideansignature) is a symmetric functor from the category CobRiem

d of d-dimensional Riemann-ian cobordisms. An object of the category CobRiem

d is a compact d− 1-manifold togetherwith a germ of a d-dimensional Riemannian structure. A morphism is a d-dimensionalRiemannian cobordism. The symmetric monoidal structure arises from disjoint union. Asdefined, this category does not have identity morphisms, but they can added in formally.

3.3.2.1 Definition. A Segal field theory is a symmetric monoidal functor from CobRiemd to the

category of (topological) vector spaces.

We won’t get into details about what kind of topological vector spaces one shouldconsider, because our aim is just to sketch a formal relationship between Segal’s pictureand our picture.

Segal’s axioms admit obvious variants where the cobordisms are decorated with othergeometric structures. The case relevant to our story is cobordisms embedded in an ambi-ent manifold M. Thus, if X is a d-manifold, let Cobd(X) denote the category whose objectsare compact codimension 1 submanifolds of M, and whose morphisms are cobordismsembedded in X, that is, compact codimension 0 submanifolds of X with boundary. Thesymmetric monoidal structure on CobRiem

d is defined by disjoint union. However, disjointunion only makes sense for objects and morphisms of Cobd(X) which are disjoint. So wecan endow Cobd(X) with a partially-defined symmetric monoidal structure by disjointunion of submanifolds which are disjoint.

In Segal’s formalism, one could define a field theory on X to be a symmetric monoidalfunctor from the partially-defined symmetric monoidal category Cobd(X) to topologicalvector spaces.

It turns out that a variant of this construction will give us something closely relatedto prefactorization algebras. Recall that FactX is the multicategory whose objects are opensubsets in X, and where a multimorphism from U1, . . . , Un → V exists when the Ui aredisjoint and contained in V.

Say an open subset U ⊂ X is nice if it is the interior of a compact submanifold M ⊂ Xwith boundary. Let us define a sub-multicategory Factnice

X ⊂ FactX be whose objects are


nice open sets U ⊂ X. Suppose U1, . . . , Un, V are nice open subsets of X, and the Ui aredisjoint and contained in V. Then, the corresponding multimorphism in FactX defines amultimorphism in Factnice

X if either

(1) n = 1 and U1 = V.(2) Or, the closures Ui of the sets Ui in X are disjoint and contained in V.

Note that 2 happens if the disjoint union of the Ui define a submanifold with boundary ofV, and the boundary of each Ui is disjoint from the boundary of V.

3.3.2.2 Lemma. There is a natural functor of multicategories

FactniceX → Cobd(X).

PROOF. The functor sends an object U ∈ FactniceX to the boundary ∂U of its closure. By

assumption, U is a submanifold with boundary, so this makes sense. The functor sends anon-trivial inclusion U1 q . . . Un ⊂ V to the cobordism

V \ (U1 q · · · qUn)

which, by assumption, is a cobordism from the disjoint union of the submanifolds ∂Ui to∂V.

The construction in this lemma identifies FactniceX with the multicategory Cobd

X /∅whose objects are submanifolds of X equipped with a cobordism from the emptyset,or equivalently, of submanifolds which are presented as the boundary of codimension0 manifold.

A factorization algebra by definition gives a functor of multicategories from FactX →Vect. A Segal-style field theory on X gives a functor of multicategories from Cobd(X) →Vect. Both, therefore, restrict to give functors of multicategories Factnice

X → Vect.

If we consider factorization algebras and not just prefactorization algebras, there islittle difference between working with the subcategory Factnice

X and with FactX. The pointis that the value of a factorization algebra on any open set is determined by its value on asufficiently fine cover, and we can always make a sufficiently fine cover out of nice opensubsets.

What this discussion shows is that the factorization algebras on X are essentially thesame as solutions to Segal’s axioms for a field theory on X where we only consider codi-mension 1 submanifolds N ⊂ X which are presented as the boundary of an open region.

Further, the physical interpretation we give for factorization algebras is essentially thesame as the physical interpretation of Segal’s axioms. If N ⊂ X is a codimension 1 sub-manifold, Segal tells us that the vector space associated to N in his axiom system is the

3.4. A CONSTRUCTION OF THE UNIVERSAL ENVELOPING ALGEBRA 41

Hilbert space of the theory on N. Standard quantization yoga tells us that we should con-struct the Hilbert space as follows. The space of germs on N of solutions to the equationsof motion is a symplectic manifold, which we can call EOM(N) (see Segal [] and Peierls).If N is the boundary of M ⊂ X, then the space EOM(M) of solutions to the equationsof motion on M is a Lagrangian submanifold if EOM(N). The Hilbert space associatedto N should be defined by geometric quantization. In general, if we have a symplecticmanifold Y, then philosophy of geometric quantization tells us that the classical limit of astate in the Hilbert space is a Lagrangian submanifold L ⊂ Y.

FINISH THIS???

3.3.3. Locally constant factorization algebras. A factorization algebra F on a mani-fold M, valued in cochain complexes, is called locally constant if, for any two discs D1 ⊂ D2in M, the mapF (D1)→ F (D2) is a quasi-isomorphism. A theorem of Lurie [] shows that,given a locally constant factorization algebra F on Rn, the complex F (D) has the struc-ture of an En algebra.

3.3.4. Algebraic quantum field theory. FINISH THIS SUBSECTIO

3.4. A construction of the universal enveloping algebra

As an example of this relationship, we will present a construction of a factorizationalgebra on R from a Lie algebra whose corresponding associative algebra is the universalenveloping algebra of g. This construction is a basic special case of the “factorization en-velope” construction, which appears in our formulation of Noether’s theorem, and whichproduces many interesting examples of factorization algebras (such as the Kac-Moodyvertex algebra). Another motivation for considering this example is that the argument weuse to analyze this factorization algebra are a precursor to arguments we will see whenwe analyze the factorization algebra for quantum mechanics.

Let gR denotes the cosheaf on R that assigns (Ω∗c (U)⊗ g, ddR) to each open U, with ddRthe exterior derivative. This is a cosheaf of cochain complexes, but it is only a precosheafdg Lie algebras. This is because the cosheaf axiom involves the use of coproducts, and thecoproduct in the category of dg Lie algebras is not given by direct sum.

Let C∗h denote the Chevalley-Eilenberg complex for Lie algebra homology, written asa cochain complex. In other words, C∗h is the graded vector space Sym(h[1]) with adifferential determined by the bracket of h.

Our main result shows how to construct the universal enveloping algebra Ug usingC∗gR.


3.4.0.1 Proposition. Let H denote the cohomology prefactorization algebra of C∗gR. That is, wetake the cohomology of every open and every structure map, so

H(U) = H∗(C∗gR(U))

for any open U. Then H is locally constant, and the corresponding associative algebra is theuniversal enveloping algebra Ug of g.

PROOF. Local constancy of H is immediate from the fact that, if I ⊂ J is an inclusionof intervals, the map of dg Lie algebras

Ω∗c (I)⊗ g→ Ω∗c (J)⊗ g

is a quasi-isomorphism. We let Ag be the associative algebra constructed from H bylemma ??.

The underlying vector space of Ag is the spaceH(I) for any interval I. To be concrete,we will use the interval R, so that we identify

Ag = H(R) = H∗(C∗(Ω∗c (R)⊗ g)).

Note that the dg Lie algebra Ω∗c (R) ⊗ g is quasi-isomorphic to H∗c (R) ⊗ g = g[−1].This is Abelian because the cup product on H∗c (I) is zero.It follows that C∗(Ω∗c (R) ⊗ g)is quasi-isomorphic to chains of the Abelian Lie algebra g[−1], which is simply Sym∗ g.Thus, as a vector space, Ag is isomorphic to the symmetric algebra Sym∗ g.

There is a mapΦ : g→ Ag

which sends an element X ∈ g to Xε where ε ∈ H1c (I) is a basis for the compactly sup-

ported cohomology of the interval I whose integral is 1. It suffices to show that this is amap of Lie algebras where Ag is given the Lie bracket coming from the associative struc-ture.

Let us check this explicitly. Let δ > 0 be small number, and let f0 ∈ C∞c (−δ, δ) be a

compactly supported smooth function with∫

f0dx = 1. Let ft(x) = f0(x− t). Note that ftis supported on the interval (t− δ, t+ δ). If X ∈ g, a cochain representative for Φ(X) ∈ Ag

is provided byX f0dx ∈ Ω1

c((−δ, δ))⊗ g.

Because the ft for varying t are all cohomologous in Ω1c(R), the elements X ft are all

cochain representatives of Φ(X).

Given elements α, β ∈ Ag, the product α · β is defined as follows.

(1) We choose intervals I, J with I < J.

3.5. SOME FUNCTIONAL ANALYSIS 43

(2) We regard α as an element of H(I) and β as an element of H(J) using the in-verses to the isomorphisms H(I) → H(R) and H(J) → H(R) coming from theinclusions of I and J into R.

(3) The product α · β is defined by taking the image of α⊗ β under the factorizationstructure map

H(I)⊗H(J)→ H(R) = Ag.

Let us see how this works in our example. The cohomology class [X ftdx] ∈ H(t− δ, t+ δ)becomes Φ(X) under the natural map from ch(t − δ, t + δ) to ch(R). If we take δ to besufficiently small, the intervals (−δ, δ) and (1− δ, 1 + δ) are disjoint. It follows that theproduct Φ(X)Φ(Y) is represented by the cocycle

(X f0dx)(Y f1dx) ∈ Sym2(Ω1c(R)⊗ g) ⊂ C∗(Ω∗c (R)⊗ g).

Similarly, the commutator [Φ(X), Φ(Y)] is represented by the expression

(X f0dx)(Y f1dx)− (X f0dx)(Y f−1dx).

It suffices to show that this cocycle in C∗(Ω∗c (R)⊗ g) is cohomologous to Φ([X, Y]).

Note that the 1-form f1dx − f−1dx has integral 0. It follows that there exists a com-pactly supported function h ∈ C∞

c (R) with

ddRh = f−1dx− f1dx.

We can assume that h takes value 1 in the interval (−δ, δ).

We will calculate the differential of the element

(X f0dx)(Yh) ∈ C∗(Ω∗c (R)⊗ g).

We have

d ((X f0dx)(Yh)) = (X f0dx)(YddRh) + [X, Y] f0hdx

= (X f0dx)(Y( f−1 − f1)dx) + [X, Y] f0hdx.

Since h takes value 1 on the interval (−δ, δ), f0h = f0. This equation tells us that a repre-sentative for [Φ(X), Φ(Y)] is cohomologous to Φ([X, Y]).

3.5. Some functional analysis

Almost all of the examples of factorization algebras we will consider in this book willassign to an open subset U ⊂ M a cochain complex built from vector spaces of analyticalprovenance: for example, smooth sections of a vector bundle, distributions on a manifold,etc. Such vector spaces are best viewed as being equipped with an extra structure (such asa topology) reflecting their analytical origin. In this section we will briefly sketch a flexiblemulticategory of vector spaces equipped with an extra “analytic” structure. Many moredetails are contained in the appendices ??.


Probably the commonest way to encode the analytic structure on a vector space suchas the space of smooth functions on a manifold is to endow it with a topology. Homolog-ical algebra with topological vector spaces is not easy, however (for instance, topologicalvector spaces do not form an abelian category). To get around this issue, we will workwith differentiable vector spaces. Let us first define the slightly weaker notion of diffeologi-cal vector space.

3.5.0.2 Definition. Let Man be the site of smooth manifolds, i.e. the category of smooth manifoldsand smooth maps between them equipped with a structure of site where a cover is an open cover inthe usual sense. Let C∞ denote the sheaf of rings on Man which assigns to any manifold M thecommutative algebra C∞(M).

A diffeological vector space is a sheaf of C∞-modules on the site Man.

For example, if V is any topological vector space, then there is a natural notion ofsmooth map from any manifold M to V (see e.g. [KM97]). The space C∞(M, V) of suchsmooth maps is a module over the algebra C∞(M). Since smoothness is a local conditionon M, sending M to C∞(M, V) gives a sheaf of C∞-modules on the site Man.

As an example of this construction, let us consider the case when V is the space ofsmooth functions on a manifold N (equipped with its natural Frechet topology). Onecan show that C∞(M, C∞(N)) is naturally isomorphic to the space C∞(M× N) of smoothfunctions on M× N.

As we will see shortly, we lose very little information when we view a topologicalvector space as a diffeological vector space.

Sheaves of C∞-modules on Man which arise from topological vector spaces are en-dowed with an extra structure: we can always differentiate smooth maps from a manifoldM to a topological vector space V. Differentiation can be viewed as an action of vectorfields on M on the space C∞(M, V), or dually, as coming from a connection

∇ : C∞(M, V)→ Ω1(M, V)

where Ω1(M, V) is defined to be the tensor product

Ω1(M, V) = Ω1(M)⊗C∞(M) C∞(M, V).

This tensor product is just the algebraic one, which is a reasonable thing to do becauseΩ1(M) is a direct summand of a free C∞(M)-module of finite rank.

This connection is flat, in the usual sense that the curvature

F(∇) = ∇ ∇ : C∞(M, V)→ Ω2(M, V).

This leads us to the following definition.


3.5.0.3 Definition. Let Ω1 denote the diffeological vector space which assigns Ω1(M) to a mani-fold M.

If F is a diffeological vector space, the diffeological vector space of k-forms valued in F is thespace Ωk(F) which assigns to a manifold M, the tensor product

Ωk(M, F) = Ωk(M)⊗C∞(M) F(M).

A connection on a diffeological vector space F is a map (of sheaves on the site Man)

∇ : F → Ω1(F),

which satisfies the Leibniz rule on every manifold M. A connection is flat if it is flat on everymanifold M.

A differentiable vector space is a diffeological vector space equipped with a flat connection.

We say a sheafF on the site of smooth manifolds is concrete if the natural mapF (M)→Hom(M,F (∗)) (where F (∗) is the value of F on a point) is injective. Almost all of thedifferentiable vector spaces we will consider are concrete. For this reason, we will nor-mally think of a differentiable vector space as being an ordinary vector space (the valueon a point) together with an extra structure. We will refer to the value of the sheaf on amanifold M as the space of smooth maps to the value on a point. If V is a differentiablevector space, we often write C∞(M, V) for the space of smooth maps from a manifold M.

As we have seen, every topological vector space gives rise to a differentiable vectorspace. For our purposes, differentiable vector spaces are much easier to use than topolog-ical vector spaces, because they are (essentially) sheaves on a site. Homological algebrafor such objects is very well-developed.

3.5.0.4 Definition. A differentiable cochain complex is a cochain complex in the category of differ-entiable vector spaces. A map V →W of differentiable cochain complexes is a quasi-isomorphism ifthe map C∞(M, V)→ C∞(M, W) is a quasi-isomorphism for all manifolds M. This is equivalentto asking that the map be a quasi-isomorphism at the level of stalks.

3.5.1. We have defined the notion of prefactorization algebra with values in any mul-ticategory. In order to discuss factorization algebras valued in differentiable vector spaces,we need to define a multicategory structure on differentiable vector spaces. Let us firstdiscuss the multicategory structure on the diffeological vector spaces.

3.5.1.1 Definition. Let V1, . . . , Vn, W be differentiable vector spaces. A smooth multi-linear map

Φ : V1 × . . . Vn →W


is a C∞-multilinear map Φ of sheaves, which satisfies the following Leibniz rule with respect to theconnections on the Vi and W. For every manifold M, and for every vi ∈ Vi(M), we require that

∇Φ(v1, . . . , vn) = ∑ i = 1nΦ(v1, . . . ,∇vi, . . . , vn) ∈ Ω1(M, W).

We let HomDVS(V1 × · · · ×Vn, W) denote this space of smooth multilinear maps.

In more down-to-earth terms, such a Φ gives a C∞(M)-multilinear map V1(M) ×. . . Vn(M) → W(M) for every manifold M, in a way compatible with the connectionsand with the maps Vi(M)→ Vi(N) associated to a map f : N → M of manifolds.

The category of differentiable cochain complexes acquires a multicategory structurefrom that on differentiable vector spaces, where the multi-maps are smooth multilinearmaps which are compatible (in the usual way) with the differentials.

3.5.1.2 Definition. A differentiable prefactorization algebra is a prefactorization algebra valuedin the multicategory of differentiable cochain complexes.

A differentiable prefactorization algebra F on a manifold X will asign a differeniablecochain complex F (U) to every open subset U ⊂ X, and smooth multi-linear maps (com-patible with the differentials)

F (U1)× · · · × F (Un)→ F (V)

whenever U1, . . . , Un are disjoint opens contained in V.

3.5.2. As we have seen, every topological vector space gives rise to a differentiablevector space. There is a rather beautiful theory developed in [KM97] concerning theprecise relationship between topological vector spaces and differentiable vector spaces.These results are disucssed in much more detail in the appendix ??: we will briefly sum-marize them now.

Let LCTVS denote the category of locally-convex Hausdorff topological vector spaces,and continuous linear maps. Let BVS denote the category with the same objects, butwhose morphisms are bounded linear maps. Every continuous linear map is bounded, butnot conversely. These categories have natural enrichments to multicategories, where themulti-maps are continuous (respectively, bounded) multi-linear maps. The category BVSis equivalent to a full subcategory of LCTVS whose objects are called bornological vectorspaces.

Theorem. The functor LCTVS → DVS extends to a functor BVS → DVS, which embeds BVSas a full sub multicategory of DVS.

In other words: if V, W are topological vector spaces, and if Φ(V), Φ(W) denote thecorresponding differentiable vector spaces, then maps from Φ(V) to Φ(W) are the same


as bounded linear maps from V →W. More generally, if V1, . . . , Vn and W are topologicalvector spaces, bounded multi-linear maps V1 × · · · × Vn → W are the same as smoothmulti-linear maps Φ(V1)× · · · ×Φ(Vn)→ Φ(W).

This theorem tells us that we lose very little information if we think of a topologicalvector space as being a differentiable vector space.

So far, we have not, however, discussed how completeness of topological vector spacesappears in theory. We need a notion of completeness for a topological vector space whichonly depends on smooth maps to that vector space. The relevant concept was developedin [?]. We will view the category BVS as being a full subcategory of DVS.

3.5.2.1 Definition. We say a topological vector space V ∈ BVS is c∞-complete, or convenientif every smooth map R→ V has an antiderivative.

The category of convenient vector spaces, and bounded linear maps, will be called ConVS.

This completeness condition is a little weaker than the one normally studied for topo-logical vector spaces. That is, every complete topological vector space is c∞-complete.

Proposition. The full subcategory ConVS ⊂ DVS is closed under the formation of all limits andcountable coproducts.

We give ConVS the multicategory structure inherited from BVS (given by boundedmultilinear maps). Since BVS is a full sub multicategory of DVS, so is ConVS.

Theorem. The multicategory structure on ConVS is represented by a symmetric monoidal struc-ture.

This symmetric monoidal structure is called the completed bornological tensor prod-uct. If E, F ∈ ConVS the bornological tensor product is written as E⊗βF. The state-ment that this represents the multicategory structure means that smooth (or equivalentlybounded) bilinear maps E1 × E2 → F are the same as bounded linear maps E1⊗βE2 → F,for objects E1, E2, F of ConVS.

If it will cause no confusion, we will often use the symbol ⊗ instead of ⊗β for thistensor product.

3.5.3. Let us now give some examples of differentiable vector spaces. These exam-ples will include the basic building blocks for most of the factorization algebras we willconsider.


Let E be a vector bundle on a manifold X, and let U be an open subset of X. We letE (U) denote the space of smooth sections of E on U, and we let Ec(U) denote the spaceof compactly supported sections of E on U.

Let us give these vector spaces the structure of differentiable vector spaces, as follows.If M is a manifold, we say a smooth map from M to E (U) is a smooth section of thebundle π∗XE on M × X. We call this space C∞(M, E (U)). Sending M to C∞(M, E (U))defines a sheaf of C∞-modules on the site of smooth manifolds with a flat connection, andso a differentiable vector space.

Similarly, we say a smooth map from M to Ec(U) is a smooth section of the bundle π∗XEon M×X, whose support maps properly to M. Let us denote this space by C∞(M, Ec(U));this defines, again, a sheaf of C∞-modules on the site of smooth manifolds with a flatconnection.

Theorem. With this differentiable structure, the spaces E (U) and Ec(U) are in the full subcat-egory ConVS of convenient vector spaces. Further, this differentiable structure isthe same as theone that arises from the natural topologies on E (U) and Ec(U).

The proof (like the proofs of all results in this section) are contained in the appendix,and based heavily on the book [KM97].

3.5.4. The category of differentiable vector spaces has internal Hom’s, and internalmulti-morphism spaces. If V, W are differentiable vector spaces, then a smooth map froma manifold M to the space HomDVS(V, W) is by definition an element of HomDVS(V, C∞(M, W)).The space C∞(M, W) is given the structure of differentiable vector space by saying that asmooth map from N to C∞(M, W) is an element of C∞(N ×M, W).

Theorem. The category ConVS has internal Hom’s and a Hom-tensor adjunction. That is, ifE, F, G are objects of ConVS, then the object HomDVS(E, F) of DVS is actually in the full sub-category ConVS. Further, we have a natural isomorphism

Hom(E⊗βF, G) = Hom(E, Hom(F, G)).

3.5.5. Let E be a vector bundle on a manifold X, and let U be an open subset of X.Throughout this book, we will often use the notation E (U) to denote the distributionalsections on U, defined by

E (U) = E (U)⊗C∞(U) D(U),

where D(U) is the space of distributions on U. Similarly, let E c(U) denote the compactlysupported distributional sections of E on U. There are natural inclusions

Ec(U) → E c(U) → E (U),

Ec(U) → E (U) → E (U),


by viewing smooth functions as distributions.

If, as above, E is a graded vector bundle on M, let E! = E∨ ⊗ DensM. We give E !

a differential that is the formal adjoint to Q on E. Let E !(U), E !c (U) denote the cochain

complexes of smooth and compactly supported sections of E!, and let E!(U) and E

!c(U)

denote the cochain complexes of distributional and compactly-supported distributionalsections of E!.

Note that E c(U) is the continuous dual to E !(U), and that Ec(U) is the continuousdual to E

!(U).

We need to give the spaces E c(U) and E (U) differentiable structures. We will needthe following proposition.

Proposition. There is are natural identifications

HomDVS(E!

c (U), R) = E (U)

HomDVS(E!(U), R) = E c(U).

This proposition, together with the fact that differentiable (and convenient) vectorspaces have internal Hom’s, give us a differentiable (and convenient) structure on thespaces E (U) and E c(U). With these differentiable structures, a smooth map from a man-ifold M to E (U) is a map E !

c (U) → C∞(M) of differentiable vector spaces, and similarlyfor E (U).

This proposition should be compared with the fact that E (U) and E c(U) are the con-tinuous linear duals of E !

c (U) and E !(U). The content of the proposition is that everysmooth linear functional on E !

c (U) or E !(U) is actually continuous. In the appendix weshow that this differentiable structure on the spaces E (U) and E c(U) coincides with theone coming from the standard topologies on these spaces.

3.5.6. The prefactorization algebras we will use for most of the book are built as al-gebras of functions or symmetric algebras of the convenient vector spaces E (U) or Ec(U).Recall that E (U) and E c(U) are both convenient vector spaces, and that, in the full sub-category ConVS ⊂ DVS of convenient vector spaces, the multicategory is represented bya symmetric monoidal structure called the completed bornological tensur product anddenoted by ⊗β.

Proposition. Let X, Y be manifolds, and let E, F be vector bundles on X and Y respectively.

C∞(X, E)⊗βC∞(Y, F) = C∞(X×Y, E F)

C∞c (X, E)⊗βC∞

c (Y, F) = C∞c (X×Y, E F).


Remark: An alternative approach to the one we’ve taken is to use the category of nucleartopological vector spaces, with the completed projective tensor product, instead of thecategory of convenient (or differentiable) vector spaces. Using nuclear spaces raises anumber of technical isssues, but one immediate one is the following: although it is truethat C∞(X)⊗πC∞(Y) = C∞(X × Y) (where ⊗π refers to the completed projective tensorproduct, it is not true (or at least not obviously true) that the same statement holds ifwe use compactly supported smooth functions. The problem stems from the fact thatthe projective tensor product does not commute with colimits, whereas the bornologicaltensor product we use does.

We can define symmetric powers of convenient vector spaces using the symmetricmonoidal structure we have described. If, as before, E is a vector bundle on X and U is anopen subset of X, this proposition allows us to identify

Symn(Ec(U)) = C∞c (Un, En).

The symmetric algebra Sym∗ Ec(U) is defined as usual to be the direct sum of the sym-metric powers. It is an algebra in the symmetric monoidal category of convenient vectorspaces.

A related construction is the algebra of functions on a differentiable vector space. IfV is a differentiable vector space, we can define, as we have seen, the space of linearfunctionals on V to be the space of maps HomDVS(V, R). Because the category DVS hasinternal Hom’s, this is again a differentiable vector space. In a similar way, we can definethe space of polynomial functions on V homogeneous of degree n to be the space

Pn(V) = HomDVS(V × · · · ×V, R)Sn .

In other words, we take smooth multilinar maps from n copies of V to R, and then takethe Sn coinvariants. This acquires the structure of differentiable vector space, by sayingthat a smoooth map from a manifold M to Pn(V) is

C∞(M, Pn(B)) = HomDVS(V × · · · ×V, C∞(M))Sn .

One can then define the algebra of functions on V by

O(V) = ∏n

Pn(V).

(We take the product rather than the sum, so that O(V) should be thought of as a space offormal power series on V). The space O(V) is a commutative algebra in a natural way.

This construction is a very general one, of course: one can define the algebra of func-tions on any object in any multicategory in the same way.

An important example is the following.

3.5.6.1 Lemma. Let E be a vector bundle on a manifold X. Then, Pn(C∞(X, E)) is the Sn covari-ants of the space of compactly supported distributional sections of E! on Xn.

3.6. THE FACTORIZATION ENVELOPE OF A SHEAF OF LIE ALGEBRAS 51

PROOF. We know that the multicategory structure on the full subcategory ConVS ⊂DVS is represented by a symmetric monoidal category, and that in this symmetric monoidlacategory,

C∞(X, E)⊗βn = C∞(Xn, En).It follows form this that the space Pn(C∞(X, E)) is the Sn covariants of the space of smoothlinear maps

C∞(Xn, En)→ R.We have seen that this space of smooth linear maps is (with its differentiable structure)the same as the space Dc(Xn, (E!)n) of compactly supported distributional sections ofthe bundle (E!)n.

Note that O(E (U) is naturally the same as HomDVS(Sym∗ E (U), R), i.e. it’s the dualof the symmetric algebra of E (U).

Remark: It is not true (at least, not obviously true) that Pn(E (U)) is the n’th symmetricpower of the space P1(E (U)) = E

!c(U).

3.6. The factorization envelope of a sheaf of Lie algebras

In this section, we will introduce an important class of examples of prefactorization al-gebras. We will show how to construct, for every sheaf of Lie algebras L on a manifold M,a prefactorization algebra which we call the factorization envelope. If L is a homotopy sheafthen the factorization envelope is a factorization algebra and not just a prefactorizationalgebra; we will often restrict attention to homotopy sheaves for this reason.

This construction is our version of Beilinson-Drinfeld’s chiral envelope [BD04]. Theconstruction can also be viewed as a natural generalization of the universal envelopingalgebra of a Lie algebra. A special case of this construction yields the universal envelopingalgebra of a Lie algebra (see 3.4).

The factorization envelope plays an important role in our story.

(1) The factorization algebra associated to a free field theory is an example of a(twisted) factorization envelope.

(2) In section 5.4, we will show (following Beilinson and Drinfeld) that the Kac-Moody vertex algebra arises as a (twisted) factorization envelope.

(3) The most important appearance of factorization envelopes appears in our treat-ment of Noether’s theorem at the quantum level. We will show in section 18.5that, if a sheaf of Lie algebras L acts on a quantum field theory on a manifoldM, then there is a homomorphism from a twisted factorization envelope of L tothe quantum observables of the field theory. This construction is very useful. Forexample, we show in section ?? TK that this construction gives rise to our version


of the Segal-Sugawara construction: it allows us to construct a homomorphismfrom the factorization algebra for the Virasoro vertex algebra to the factorizationalgebra associated to a class of chiral conformal field theories.

3.6.1. Thus, let M be a manifold. Let L be a sheaf of dg Lie algebras on M. Let Lcdenote the cosheaf of compactly supported sections of L.

Remark: Note that, although Lc is a cosheaf of cochain complexes, and a precosheaf ofdg Lie algebras, it is not a cosheaf of dg Lie algebras. This is because colimits of dg Liealgebras are not the same as colimits of cochain complexes.

We can view Lc as a prefactorization algebra valued in the category of dg Lie algebraswith symmetric monoidal structure given by direct sum. Indeed, if Ui are disjoint opensin M contained in V, there is a natural map of dg Lie algebras

⊕Lc(Ui) = Lc(⊕Ui)→ Lc(V)

giving the factorization product.

Taking Chevalley chains is a symmetric monoidal functor from dg Lie algebras, equippedwith the direct sum monoidal structure, to cochain complexes.

3.6.1.1 Definition. If L is a sheaf of dg Lie algebras on M, define the factorization envelopeUL to be the prefactorization algebra obtained by applying the Chevalley-Eilenberg chain functorto Lc, viewed as a factorization algebra valued in dg Lie algebras.

Concretely, UL assigns to an open subset V ⊂ M the complex

U(L)(V) = C∗(Lc(V))

where C∗ is the Chevalley-Eilenberg chain complex. The product maps are defined byapplying the functor C∗ to the dg Lie algebra map ⊕Lc(U)i → Lc(V) associated to aninclusion of disjoint opens Ui into V.

We will see later ?? that – under the hypothesis that L is a homotopy sheaf – then thisprefactorization algebra is a factorization algebra.

3.6.2. In practice, we will need an elaboration of this construction which involves asmall amount of analysis.

3.6.2.1 Definition. Let M be a manifold. A local dg Lie algebra on M consists of the followingdata.

(1) A graded vector bundle L on M, whose sheaf of smooth sections will be denoted L.(2) A differential operator d : L → L, of cohomological degree 1 and square 0.


(3) An alternating bi-differential operator

[−,−] : L⊗2 → Lwhich endows L with the structure of a sheaf of dg Lie algebras.

Remark: This definition will play an important role in our approach to classical field theoryas detailed in Chapter ??.

In section 3.5, we explained how spaces of sections of vector bundles on a manifoldare, in a natural way, differentiable vector spaces. We also explained that they live inthe full subcategory of convenient vector spaces, and that the multicategory structure ondifferentiable vector spaces is represented by a symmetric monoidal structure on the fullsubcategory of convenient vector spaces.

Therefore, U ⊂ M, the space L(U) is a convenient graded vector space. We wouldlike to form, as above, the Chevalley-Eilenberg chain complex C∗(Lc(U)). The under-lying vector space of C∗(Lc(U) is the (graded) symmetric algebra on Lc(U)[1]. As weexplained in section 3.5, we need to take account of the differentiable structure on Lc(U)when we take the tensor powers of Lc(U). We define (Lc(U))⊗n to be the tensor powerdefined using the completed bornological tensor product on the convenient vector spaceLc(U). Concretely, if Ln denotes the vector bundle on Mn obtained as the external tensorproduct, then

(Lc(U))⊗n = Γc(Un, Ln)

is the space of compactly supported smooth sections of Ln on Un. Symmetric (or exte-rior) powers of Lc(U) are defined by taking coinvariants of Lc(U)⊗n with respect to theaction of the symmetric group Sn. The completed symmetric algebra on Lc(U)[−1] that isthe underlying graded vector space of C∗(Lc(U)) is defined using these completed sym-metric powers. The Chevalley-Eilenberg differential is continuous, and therefore definesa differential on the completed symmetric algebra of Lc(U)[−1], giving us the cochaincomplex C∗(Lc(U)).

3.6.2.2 Definition. The factorization envelope of the local dg Lie algebra L is the factorizationalgebra which assigns to an open subset U ⊂ M the chain complex C∗(Lc(U)). This is a factor-ization algebra in the multicategory of differentiable vector spaces.

Example: Let g be a Lie algebra. There is a local Lie algebra on R given by the sheaf Ω∗R⊗ g.As we showed in section 3.4.0.1, the factorization envelope of Ω∗R ⊗ g is a locally constantfactorization algebra and so corresponds to an associative algebra. This associative alge-bra is the universal enveloping algebra of g.

In the same way, for any Lie algebra g we can construct a factorization algebra onRn as the factorization envelope of Ω∗Rn ⊗ g. The resulting factorization algebra is stilllocally constant: it has the property that the inclusion map from one disc to another isa quasi-isomorphism. A theorem of Lurie [?] tells us that locally-constant factorization


algebras on Rn are the same as En algebras. The En algebra we have constructed is theEn-enveloping algebra of g.

3.6.3. Many interesting factorization algebras (such as the Kac-Moody factorizationalgebra, and the factorization algebra associated to a free field theory) can be constructedfrom a variant of the factorization envelope construction, which we call the twisted factor-ization envelope.

3.6.3.1 Definition. Let L be a local dg Lie algebra on a manifold M. A local k-shifted centralextension of L is a dg Lie algebra structure on the precosheaf

Lc = C[k]⊕Lc

(where C[k] is the constant precosheaf which assigns to every open set the vector space C in degree−k), such that

(1) C[k] is central, and the sequence

0→ C[k]→→ Lc → L → 0

is an exact sequence of dg Lie algebras.(2) The differential and bracket maps from Lc(U) → C[k] and Lc(U)⊗2 → C[k] defining

the central extension are local, meaning that they can be represented as compositions

Lc(U)→ ωc(U)[−k]∫−→ C[k]

Lc(U)⊗2 → ωC(U)[−k]∫−→ C[k]

where the map in the first line is a differential operator, and the second is a bidifferentialoperator.

In Chapter 10, subsection ??, we will analyze the complex “local Lie algebra cochains”and see that the L∞-version of local central extensions are classified by the cohomology ofthis cochain complex.

3.6.3.2 Definition. In this situation, define the twisted factorization envelope to be the prefac-torization algebra U(L) which sends an open set U to C∗(Lc(U)) ( as above, we use the completedtensor product as above).

The chain complex C∗(Lc(U)) is a module over chains on the Abelian Lie algebra C[k] forevery open subset U. Thus, we will view the twisted factorization envelope as a prefactorizationalgebra in modules for C[c] where c has degree −k− 1.

This is a factorization algebra over the base ring C[c]. Of particular interest is the casewhen k = −1, so that the central parameter c is of degree 0.


Let us now introduce some important examples of this construction.

Example: Let g be a simple Lie algebra, and let 〈−,−〉g denote an invariant pairing on g.Let us define the Kac-Moody factorization algebra as follows.

Let Σ be a Riemann surface, and consider the local Lie algebra Ω0,∗ ⊗ g on Σ, whichsends an open subset U to the dg Lie algebra Ω0,∗(U)⊗ g. There is a −1-shifted centralextension of Ω0,∗

c ⊗ g defined by the cocycle

ω(α, β) =∫

U〈α, ∂β〉g

where α, β ∈ Ω0,∗c (U)⊗ g and ∂ : Ω0,∗ → Ω1,∗ is the holomorphic de Rham operator. Note

that this is a −1-shifted cocycle because ω(α, β) is zero unless deg(α) + deg(β) = 1.

The Kac-Moody factorization algebra on Σ is the twisted universal enveloping algebraUω(Ω0,∗ ⊗ g). We will analyze this example in more detail in chapter 5.

Example: In this example we will define a higher-dimensional analog of the Kac-Moodyvertex algebra.

Let X be a complex manifold of dimension n. Let φ ∈ Ωn−1,n−1(X) be a closed form.

Then, given any Lie algebra equipped with an invariant pairing, we can construct a−1-shifted central extension of Ω0,∗

c ⊗ g, defined as above by the cocycle

α⊗ β 7→∫

X〈α, ∂β〉g ∧ φ,

It is easy to verify that this is a cocycle. (The case of the Kac-Moody extension is whenn = 1 and φ is a constant). The cohomology class of this cocycle is unchanged if wechange φ to φ + ∂ψ where ψ ∈ Ωn−1,n−2(X) satisfies ∂ψ = 0.

Let F denote the twisted factorization envelope of this local dg Lie algebra. Thefactorization algebra F is closely related to the Kac-Moody algebra. For instance, ifX = Σ × Pn−1 where Σ is a Riemann surface, and the form φ is the volume form onPn−1, then the push-forward of this factorization algebra to Σ is quasi-isomorphic to theKac-Moody factorization algebra described above. (The push-forward is defined by

(p∗F )(U) = F (p−1(U))

for U ⊂ Σ).

An important special case of this construction is when dimC(X) = 2, and the form φ isthe curvature of a connection on the canonical bundle of X (so that φ represents c1(X)). Aswe will show when we discuss Noether’s theorem at the quantum level, if we have a fieldtheory with an action of a local dg Lie algebra L then a twisted factorization envelopesof L will map to observables of the theory. One can show that the local dg Lie algebra


Ω0,∗X ⊗ g acts on a twisted N = 1 gauge theory with matter, and (following Johansen [?])

that the twisted factorization envelope – with central extension determined by c1(X) –maps to observables of this theory.

CHAPTER 4

Factorization algebras and free field theories

4.1. The divergence complex of a measure

We will start by motivating this construction of this cochain complex by consideringGaussian integrals in finite dimensions. Let M be a background and let ω0 be a measure onM. Let f be a function on M. (For example, M could be a vector space, ω0 the Lebesguemeasure and f a quadratic form). The divergence operator for the measure e f /hω0 is amap

Divh : Vect(M)→ C∞(M)

X 7→ h−1(X f ) + Divω0 X.

One way to define the divergence operator is to use the volume form e f /hω0 to iden-tify Vect(M) with Ωn−1(M), and C∞(M) with Ωn(M) (where n = dim M). Under thisidentification, the divergence operator is simply the de Rham operator from Ωn−1(M) toΩn(M).

The de Rham operator, of course, is part of the de Rham complex. In a similar way,we can define the divergence complex, as follows. Let

PVi(M) = C∞(M,∧iTM)

denote the space of polyvector fields on M. The divergence complex is the complex

· · · → PVi(M)Divh−−→ PV1(M)

Divh−−→ PV0(M)

where the differentialDivh : PVi(M)→ PVi−1(M)

is defined so that the diagram

PVi(M)∨e f /hω0 //

Divh

Ωn−i(M)

ddR

PVi−1(M)∨e f /hω0// Ωn−i+1(M)

commutes.

57

58 4. FACTORIZATION ALGEBRAS AND FREE FIELD THEORIES

Thus, after contracting with the volume e f /hω0, the divergence complex becomes thede Rham complex. It is easy to check that, as maps from PVi(M) to PVi−1(M), we have

Divh = ∨h−1d f + Divω0

where ∨d f is the operator of contracting with d f . In the h → 0 limit, the dominant termis ∨h−1d f .

More precisely, there is a flat family of cochain complexes over the algebra C[h] whichat h 6= 0 is isomorphic to the divergence complex, and at h = 0 is the complex

→ PV2(M)∨d f−−→ PV1(M)

∨d f−−→ PV0(M).

Note that this complex is a differential graded algebra (which is not the case for the diver-gence complex).

The image of the map ∨d f : PV1(M) → PV0(M) is the ideal cutting out the criticallocus. Indeed, this whole complex is the Koszul complex for the equations cutting out thecritical locus. This leads to the following definition.

4.1.0.3 Definition. The derived critical locus of f is the dg manifold whose functions are PV∗(M)with differential contracting with d f .

Remark: Since the purpose of this section is motivational, we will not go into any details onthe theory of dg manifolds. Dg manifolds are one way to think about derived geometry:more details on derived geometry (from a different point of view) will be discussed inchapter ??.

Let Γ(d f ) ⊂ T∗M denote the graph of d f . The ordinary critical locus of f is theintersection of Γ(d f ) with the zero-section M ⊂ T∗M. The derived critical locus is definedto be the derived intersection. In derived geometry, functions on derived intersections aredefined by derived tensor products:

C∞(Crith( f )) = C∞(Γ(d f ))⊗C∞(T∗M) C∞(M).

By using a Koszul resolution of C∞(M) as a module for C∞(T∗M), one sees finds a quasi-isomorphism of dg algebras between this derived intersection and the complex PV∗(M)with differential ∨d f .

Thus, we find that the divergence complex has a h→ 0 limit which is functions on thederived critical locus of f .

An important special case of this is when the function f is zero. In that case, thederived critical locus of f has functions the algebra PV∗(M) with zero differential. Thiscan be viewed as the functions on the graded manifold T∗[−1]M. The derived criticallocus for a general function f can be viewed as a deformation of T∗[−1]M obtained byintroducing a differential given by contracting with d f .

4.1. THE DIVERGENCE COMPLEX OF A MEASURE 59

4.1.1. We will define the factorization algebra of observables of a free scalar fieldtheory as a divergence complex, just like we defined H0 of observables to be given byfunctions modulo divergences in chapter 2. It turns out that there is a slick way to writethis factorization algebra as a twisted factorization envelope of a certain sheaf of Heisen-berg Lie algebras. We will explain this point in a finite-dimensional toy model, and thenuse the factorization envelope picture to define the factorization algebra of observables ofthe field theory.

Let V be a vector space, and let q : V → R be a quadratic function on V. Let ω0be the Lebesgue measure on V. We want to understand the divergence complex for themeasure eq/hω0. The construction is quite general: we do not need to assume that q isnon-degenerate.

The derived critical locus of the function q is a linear dg manifold. Linear dg mani-folds are the same thing as cochain complexes: any cochain complex B∗ gives rise to a dgmanifold whose functions are the symmetric algebra on the dual of B∗.

The derived critical locus of q is described by the cochain complex

W = V → V∗[−1]

where the differential sends v ∈ V to the linear functional ∂∂q v.

Note that W is equipped with a graded anti-symmetric pairing of cohomological de-gree −1, defined by pairing V and V∗. In other words, W has a symplectic pairing ofcohomological degree −1. We let

HW = C · h[−1]⊕W

where C · h indicates a one-dimensional vector space with basis h. We give the cochaincomplexHW a Lie bracket by saying that

[w, w′] = h⟨w, w′

⟩where 〈−,−〉 denotes the pairing on W. Thus, HW is a shifted-symplectic version of theHeisenberg Lie algebra of an ordinary symplectic vector space.

Consider the Chevalley-Eilenberg chain complex C∗HW . This is defined to be thesymmetric algebra ofHW [1] = W[1]⊕C · h equipped with a certain differential. Since thepairing on W identifies W[1] = W∗, we can identify

C∗(HW) = Sym∗ (W∗) [h]

with a differential. Since, as a graded vector space, W = V ⊕ V∗[−1], we have a naturalidentification

C∗(HW) = PV∗(V)[h]where PV∗(V) refers to polyvector fields on V witih polynomial coefficients, where asbefore we place PVi(V) in degree −i.


4.1.1.1 Lemma. The differential on C∗(HW) is, under this identification, the operator

h Diveq/hω0: PVi(V)[h]→ PVi−1(V)[h],

where ω0 is the Lebesgue measure on V, and q is the quadratic function on V used to define thedifferential on the complex W.

PROOF. The proof is an explicit calculation, which we leave to the interested reader.The calculation is facilitated by choosing a basis of V in which we can explicitly writeboth the divergence operator and the differential on the Chevalley-Eilenberg complexC∗(HW).

In what follows, we will define the dg factorization algebra of observables of a freefield theory as a Chevalley-Eilenberg chain complex of a certain Heisenberg Lie algebra,constructed as in this lemma.

4.2. The factorization algebra of classical observables of a free field theory

In this section, we will construct the prefactorization algebra associated to any freefield theory. We will concentrate, however, on the free scalar field theory on a Riemannianmanifold. We will show that, for one-dimensional manifolds, this prefactorization algebrarecovers the familiar Weyl algebra, the algebra observables for quantum mechanics. Ingeneral, we will show how to construct correlation functions of observables of a free fieldtheory and check that these agree with how physicists define correlation functions.

4.2.1. Defining the prefactorization algebra. We will start by defining the prefactor-ization algebra of classical observables of a free scalar field theory.

Let M be a Riemannian manifold, and so M is equipped with a natural density, arisingfrom the metric. We will use this natural density to integrate functions, and also to providean isomorphism between functions and densities that we use implicitly from hereon. Thefield theory we will discuss has as fields φ ∈ C∞(M) and has as action functional

S(φ) = 12

∫M

φ4φ,

where4 is the Laplacian on M. (Normally we will reserve the symbol4 for the Batalin-Vilkovisky Laplacian, but that’s not necessary in this section.)

If U ⊂ M is an open subset, then the space of solutions to the equation of motion onU is the space of harmonic functions on U.

In this book, we will always consider the derived space of solutions of the equation ofmotion. For more details about the derived philosophy, the reader should consult Chapter

4.2. THE FACTORIZATION ALGEBRA OF CLASSICAL OBSERVABLES OF A FREE FIELD THEORY 61

10. In this simple situation, the derived space of solutions to the free field equations, onan open subset U ⊂ M, is the two-term complex

E (U) =

(C∞(U)0 4−→ C∞(U)1

),

where the superscripts indicate the cohomological degree.

The classical observables of a field theory on an open subset U ⊂ M are functionson the derived space of solutions to the equations of motion on U. As we explained insection 3.5, the space E (U) has the structure of differentiable cochain complex (essentially,it’s a sheaf on the site of smooth manifolds). We define the space of polynomial functionshomogeneous of degree n on E (U) to be the space

Pn(E (U)) = HomDVS(E (U)× · · · × E (U), R)Sn .

In other words, we take smooth multi-linear maps from n copies of E (U), and thentake the Sn-coinvariants. The algebra of all polynomial functions on E (U) is the spaceP(E (U)) = ⊕nPn(E (U)).

As we discussed in section 3.5, we can identify

Pn(E (U)) = Dc(Un, (E!)n)Sn

as the Sn-coinvariants of the space of compactly supported distributional sections of thebundle (E!)n on Un. In general, if E (U) is sections of a graded bundle E, then E! isE∨ ⊗Dens. In the case at hand, the bundle E! is two copies of the trivial bundle, one indegree −1 and one in degree 0.

For example, the space P1(E (U)) = E (U)∨ of smooth linear functionals on E (U) isthe space

E (U)∨ =

(Dc(U)−1 4−→ Dc(U)0

),

where Dc(U) indicates the space of compactly supported distributions on U.

Using the space of all polynomial functionals will lead to difficulties defining thequantum observables. When we work with an interacting theory, these difficulties canonly be surmounted using the techniques of renormalization. For a free field theory,though, there is a much simpler solution.

We haveE !

c (U) =(

C∞c (U)−1 → C∞

c (U)0)

,

using our identification between densities and functions. Note that E (U)∨ is the complexE

!c(U) of compactly supported distributional sections of the bundle E!. There is therefore

a natural cochain map E !c (U)→ E

!c(U).


The observables we will work with is the space of “smeared observables”, defined by

Obscl(U) = Sym∗(E !c (U)) = Sym∗(C∞

c (U)−1 4−→ C∞c (U)0),

the symmetric algebra on E !c (U). As we explained in section 3.5, this symmetric algebra is

defined using the natural symmetric monoidal structure on the full subcategory ConVS ⊂DVS of convenient vector spaces. Concretely, we can identify

Symn(E !c (U)) = C∞

c (Un, (E!)⊗n)Sn .

In other words, Symn E !c (U) is the subspace of Pn(E (U)) defined by taking all distribu-

tions to be smooth functions with compact support.

4.2.1.1 Lemma. The map Obscl(U)→ P(E (U)) is a homotopy equivalence of cochain complexesof differentiable vector spaces.

PROOF. It suffices to show that the map Symn E !c (U) → Pn(E (U)) is a smooth homo-

topy equivalence. For this, it suffices to show that the map

C∞c (U, (E!)⊗n)→ Dc(U, (E!)⊗n)

is a smooth homotopy equivalence.

This is a special case of a general result proved in the appendix B.10, which says thatthe spaces of smooth and distributional sections of any elliptic complex are homotopyequivalent.

Note that by smooth homotopy equivalence we mean that there is a smooth inversemap E (U)∨ → E !

c (U) and smooth cochain homotopies between the two composed mapsand the identity maps. Smooth means that all maps are in the category DVS of differen-tiable vector spaces; it suffices to construct a continuous homotopy equivalence.

This lemma says that, since we are working homotopically, we can replace a distribu-tional observable (given by integration against some distribution on Un) by a smooth ob-servable (given by integration against a smooth function on Un). We will think of smoothlinear observables as “smeared observables”.

4.2.2. Let us describe the cochain complex Obscl(U) more explicitly, in order to clar-ify the relationship with what we discussed in chapter 2. The complex Obscl(U) lookslike

· · · → ∧2C∞c (U)⊗ Sym∗ C∞

c (U)→ C∞c (U)⊗ Sym∗ C∞

c (U)→ Sym∗ C∞c (U).

All tensor products appearing in this expression are completed tensor products in thecategory of convenient vector spaces.


We should interpret Sym∗ C∞c (U) as being an algebra of polynomial functions on

C∞(U), using (as we explained above) the Riemannian volume form on U to identifyC∞

c (U) with a subspace of the dual of C∞(U). There is a similar interpretation of the otherterms in this complex using the geometry of the space C∞(U) of fields. Let TcC∞(U) referto the subbundle of the tangent bundle of C∞(U) given by the subspace C∞

c (U) ⊂ C∞(U).An element of a fibre of TcC∞(U) is a first-order variation of a field which is zero outsideof a compact set. This subspace TcC∞(U) defines an integrable foliation on C∞(U), andthis foliation can be defined if we replace C∞(U) by any sheaf of spaces.

Then, we can interpret C∞c (U) ⊗ Sym∗ C∞

c (U) as a space of polynomial sections ofTcC∞(U). Similarly, ∧kC∞

c (U)⊗ Sym∗ C∞c (U) should be interpreted as a space of polyno-

mial sections of the bundle ∧kTcC∞(U).

That is, if PVc(C∞(U)) refers to polynomial polyvector fields on C∞(U) along thefoliation given by TcC∞(U), we have

Obscl(U) = PVc(C∞(U)).

So far, this is just an identification of graded vector spaces. We need to explain howto identify the differential. Roughly speaking, the differential on Obscl(U) correspondsto the differential on PVc(C∞(U)) obtain this complex is given by contracting with theone-form dS, where

S = 12

∫φ D φ

is the action functional.

Let us explain the precise sense in which this is true. Note that the functional S isnot well-defined for all fields φ (because the integral may not converge). However, theexpression

∂S∂φ0

(φ)

make sense for any φ ∈ C∞(U) and φ0 ∈ C∞c (U). This means that we can make sense

of the one-form dS, not as a section of the cotangent bundle of C∞(U), but as a sectionof the space T∗c C∞(U), the dual of the subbundle TcC∞(U) ⊂ TC∞(U) describing vectorfields along the leaves. This leafwise one-form is closed. Such one-forms are the kindsof things we can contract with elements of PVc(C∞(U)). The differential on PVc(C∞(U))

which matches the differential on Obscl is given by contracting with dS.

4.2.3. General free field theories. Let’s give an abstract definition of a free theory ingeneral (although we will mostly focus on free scalar field theories in this chapter). Moremotivation for this general definition is presented in Chapter ??, where we introduce theclassical BV formalism from the point of view of derived geometry.


4.2.3.1 Definition. Let M be a manifold. A free field theory on M is the following data:

(1) A graded vector bundle E on M, whose sheaf of sections will be denoted E , and whosecompactly supported sections will be denoted Ec.

(2) A differential operator d : E → E , of cohomological degree 1 and square zero, making Einto an elliptic complex.

(3) Let E! = E∨ ⊗DensM, and let E ! be the sections of E!. Let d be the differential on E !

which is the formal adjoint to the differential on E . Note that there is a natural pairingbetween Ec(U) and E !(U), and this pairing is compatible with differentials.

Then, we require an isomorphism of bundles E → E![−1] compatible with differ-entials, in the sense that the induced map E (U) → E !(U)[−1] is an isomorphism ofcochain complexes. We further require that the induced pairing of cohomological degree−1 on each Ec(U) is graded anti-symmetric.

Note that the equations of motion for a free theory are always linear, so that the spaceof solutions is a vector space. Similarly, the derived space of solutions of the equationsof motion of a free field theory is a cochain complex, which is a linear derived stack.The cochain complex E (U) should be thought of as the derived space of solutions to theequations of motion on an open subset U. As we will explain in Chapter ??, the pairing onEc(U) arises from the fact that the equations of motion of a field theory are not arbitrarydifferential equations, but describe the critical locus of an action functional.

For example, for the free scalar field theory on a manifold M with mass m, we have,as above,

E = C∞M4+m2

−−−→ C∞M[−1].

Our convention is that 4 is a non-negative operator, so that on Rn, 4 = −∑ ∂∂xi

2. The

pairing on Ec is defined by ⟨φ0, φ1

⟩=∫

Mφ0φ1

for φi ∈ C∞(M)[−i].

As another example, let us describe Abelian Yang-Mills theory (with gauge group R)in this language. Let M be a manifold of dimension 4. If A ∈ Ω1(M) is a connection onthe trivial R-bundle on a manifold M, then the Yang-Mills action functional applied to Ais

SYM(A) = − 12

∫M(dA) ∗ (dA) = 1

2

∫M

A(d ∗ d)A.

The equations of motion are that d ∗ dA = 0. There is also gauge symmetry, given byX ∈ Ω0(M), which acts on A by A→ A + dX. The complex E describing this theory is

E = Ω0(M)[1] d−→ Ω1(M)d∗d−−→ Ω3(M)[−1] d−→ Ω4(M)[−2].


We will explain how to derived this statement in Chapter ??. For now, note that H0(E ) isthe space of those A ∈ Ω1(M) which satisfy the Yang-Mills equation d ∗ dA = 0, modulogauge symmetry.

For any free field theory with complex of fields E , we can define the classical observ-ables of the theory as

Obscl(U) = Sym∗(E !c (U)) = Sym∗(Ec(U)[1]).

It is clear that classical observables form a prefactorization algebra. Indeed, Obscl(U)is a commutative differential graded algebra. If U ⊂ V, there is a natural algebra homo-morphism

iUV : Obscl(U)→ Obscl(V),

which on generators is just the natural map C∞c (U)→ C∞

c (V), extension by zero.

If U1, . . . , Un ⊂ V are disjoint open subsets, the prefactorization structure map is thecontinuous multilinear map

Obscl(U1)× · · · ×Obscl(Un) → Obscl(V)

(α1, . . . , αn) 7→ ∏ni=1 iUi

V αi.

4.2.4. The one-dimensional case, in detail. Let us compute the space of classical ob-servables for a free scalar theory in dimension 1 (i.e. for free quantum mechanics).

4.2.4.1 Lemma. If U = (a, b) ⊂ R is an interval in R, then the commutative algebra of classicalobservables for the free field with mass m ≥ 0 has cohomology

H∗(Obscl((a, b))) = R[p, q],

the polynomial algebra in two variables.

PROOF. The picture is the following. The equations of motion for free quantum me-chanics on the interval (a, b) are that the field φ satisfies (D+m2)φ = 0. This space istwo dimensional, spanned by the functions e±mx (if m > 0) and by the functions 1, x ifm = 0. Classical observables are functions on the space of solutions to the equations ofmotion. We would this expect that classical observables are a polynomial algebra in twogenerators.

We need to be a little more careful, however, because we used the derived version ofthe space of solutions to the equations of motion. We will show that the complex

E !c ((a, b)) =

(C∞

c ((a, b))−1 4+m2

−−−→ C∞c ((a, b))0

)


is smoothly homotopy equivalent to the complex R2 situated in degree 0. Since the alge-bra Obscl((a, b)) of observables is defined to be the symmetric algebra on this complex,this will imply the result. Without loss of generality, we can take a = −1 and b = 1.

First, let us introduce some notation. If m = 0, let φq = 1 and φp = x. If m > 0, let

φq =12

(emx + e−mx)

φp = 12m

(emx − e−mx) .

For any value of m, the functions φp and φq are annhilated by the operator −∂2x + m2,

and they form a basis for the kernel of this operator. Further, φp(0) = 1 and φq(0) = 0,whereas φ′p(0) = 0 and φ′q(0) = 1. Finally, φq = φ′p.

Define a mapπ : E !

c ((−1, 1))→ Rp, qto the vector space spanned by p, q, by sending

g 7→ π(g) = q∫

g(x)φqdx + p∫

g(x)φpdx.

This is a cochain map, because if g = (D+m2) f , where f has compact support, thenπ(g) = 0. This map is easily seen to be surjective.

We need to construct a contracting homotopy on the kernel of π. That is, if Ker π0 ⊂C∞

c ((−1, 1)) refers to the kernel of π in cohomological degree 0, we need to construct aninverse to the differential

C∞c ((−1, 1)) = Ker π−1 D+m2

−−−→ Ker π0.

This is defined as follows. Let G(x) ∈ C0(R) be the Green’s function for the operatorD+m2. Explicitily, we have

G(x) =

m2 e−m|x| if m > 0− 1

2 |x| if m = 0.

Then (D+m2)G is the delta function at 0. The inverse map sends a function

f ∈ Ker π0 ⊂ C∞c ((−1, 1))

toG ? f =

∫y

G(x− y) f (y)dy.

The fact that∫

f φq = 0 and∫

f φp = 0 implies that G ? f has compact support. The fatthat G is the Green’s function implies that this operator is the inverse to D+m2. It is clearthat the operator of convolution with G is smooth (and even continuous), so the resultfollows.


4.2.5. The Poisson bracket. We now return to the general case.

Suppose we have any free field theory on a manifold M, with complex of fields E .Classical observables are the symmetric algebra Sym Ec(U)[1]. Recall that the complexEc(U) is equipped with an antisymmetric pairing of cohomological degree −1. Thus,Ec(U)[1] is equipped with a symmetric pairing of degree 1.

4.2.5.1 Lemma. There is a unique smooth Poisson bracket on Obscl(U) of cohomological degree1, with the property that for α, β ∈ Ec(U)[1], we have

α, β = 〈α, β〉 .

Recall that “smooth” means that the Poisson bracket is a smooth bilinear map

−,− : Obscl(U)×Obscl(U)→ Obscl(U)

as defined in section 3.5.

PROOF. The argument we will give is very general, and applies in any reasonablesymmetric monoidal category. Recall that, as stated in section 3.5, the category of conve-nient vector spaces is a symmetric monoidal category with internal Hom’s and a Hom-tensor adjunction.

If A is any commutative algebra object in the category ConVS∗ of convenient cochaincomplexes, and M is an A-modules, then we can define Der(A, M) to be the space ofalgebra homomorphisms A→ A⊕M which are trivial modulo the ideal M. (Here A⊕Mis given the square-zero algebra structure, where the product of any two elements in M iszero and the product of an element in A with one in M is the module structure.)

Since the category of convenient cochain complexes has internal Hom’s, the cochaincomplex Der(A, M) is again an A-module in ConVS∗.

The commutative algebra

Obscl(U) = Sym∗ E !c (U)

is the universal commutative algebra in the category ConVS∗ of convenient cochain com-plexes equipped with a smooth linear cochain map E !

c (U) → Obscl(U). It follows fromthis that, for any module M over Sym∗ E !

c (U),

Der(Sym∗ E !c (U), M) = Hom(E !

c (U), M)

A Poisson bracket on Sym∗ E !c (U) is in particular a biderivation. A biderivation is some-

thing that assigns to an element of Sym∗ E !c (U) a derivation of the algebra Sym∗ E !

c (U).Thus, the space of biderivations is the space

Der(

Sym∗ E !c (U), Der

(Sym∗ E !

c (U), Sym∗ E !c (U)

)).


What we have said so far identifies this space of biderivations with

Hom(E !c (U), Hom(E !

c (U), Sym∗ E !c (U))) = Hom(E !

c (U)⊗ E !c (U), Sym∗ E !

c (U)).

In this line we have used the Hom-tensor adjunction in the category ConVS.

The Poisson bracket we are constructing corresponds to the biderivation which is thepairing on E !

c (U) viewed as a map

E !c (U)⊗ E !

c (U)→ R = Sym0 E !c (U).

This biderivation is antisymmetric and satisfies the Jacobi identity. Since Poisson brack-ets are a subspace of biderivations, we have proved both the existence and uniquenessclauses.

Note that for U1, U2 disjoint open subsets of V and for observables αi ∈ Obscl(Ui), wehave

iU1V α1, iU2

V α2 = 0.

That is, observables coming from disjoint open subsets commute with respect to the Pois-son bracket. This means that Obscl(U) defines a P0 prefactorization algebra. (We will seelater in section 6.3 that this prefactorization algebra is actually a factorization algebra.)

In the case of classical observables of the free scalar field theory, we can think ofObscl(U) as a certain space of polyvector fields on C∞(U) along the foliation of C∞(U)defined by the subspace C∞

c (U) ⊂ C∞(U). The Poisson bracket we have just defined isthe Schouten bracket on polyvector fields.

4.2.6. Quantum observables. In Chapter 2, we construct a prefactorization algebrawhich we called H0(Obsq) of quantum observables of a free scalar field theory on a mani-fold. This space is defined as a space of functions on the space of fields, modulo the imageof a certain divergence operator. The aim of this section is to lift this vector space to acochain complex. As we explained in section 4.1, this complex will be the analog of thedivergence complex of a measure in finite dimensions.

In section 4.1, we explained that for a quadratic function q on a vector space V, thedivergence complex for the measure eq/hω0 on V (where ω0 is the Lebesgue measure) canbe realized as the Chevalley-Eilenberg chain complex of a certain Heisenberg Lie algebra.

The prefactorization algebra Obsq(U) is an example of a twisted prefactorization en-velope of a sheaf of Lie algebras. Let

Ec(U) = Ec(U)⊕R · h


where Rh is situated in degree 1. We give Ec(U) a Lie bracket by saying that, for α, β ∈Ec(U),

[α, β] = h 〈α, β〉 .

Thus, Ec(U) is a graded version of a Heisenberg algebra, centrally extending the abeliandg Lie algebra Ec(U).

LetObsq(U) = C∗(Ec(U)),

where C∗ denotes the Chevalley-Eilenberg complex for the Lie algebra homology of Ec(U),defined using the completed tensor product on the category of convenient vector spaces(as discussed in section 3.5). Thus,

Obsq(U) =(

Sym∗(Ec(U)[1]

), d)

=(

Obscl(U)[h], d)

where the differential arises from the Lie bracket and differential on Ec(U). The symbol[1] indicates a shift of degree down by one. (Note that we always work with cochaincomplexes, so our grading convention of C∗ is the negative of one popular convention.)

As we discussed in section 4.2, we can view classical observables on an open set U forthe free scalar theory as a certain complex of polyvector fields on the space C∞(U):

Obscl(U) = (PVc(C∞(U)),∨dS) .

By PVc(C∞(U)) we mean polylvector fields defined using the foliation TcC∞(U) ⊂ TC∞(U)of compactly-supported variants of a field. Concretely,

Obscl(U) = . . . ∨dS−−→ ∧2C∞c (U)⊗Sym∗ C∞

c (U)∨dS−−→ C∞

c (U)⊗Sym∗ C∞c (U)

∨dS−−→ ∨dS−−→ Sym∗ C∞c (U)

where all tensor products appearing in this expression are completed bornological tensorproducts.

In a similar way, quantum observables look like

Obsq(U) = . . . ∨dS+h Div−−−−−−→ ∧2C∞c (U)⊗Sym∗ C∞

c (U)[h] ∨dS+h Div−−−−−−→ C∞c (U)⊗Sym∗ C∞

c (U)∨dS−−→ ∨dS+h Div−−−−−−→ Sym∗ C∞

c (U).

The operator Div is the extension to all polyvector fields of the operator we defined inChapter 2 as a map from polynomial vector fields on C∞(U) to polynomial functions.Thus, H0(Obsq(U)) is the same vector space (and same prefactorization algebra) that wedefined in Chapter 2.

Since this is an example of the general construction we discussed in section 3.6 we seethat Obsq(U) has the structure of a prefactorization algebra.


4.2.7. As we explained in section ??, our philosophy is that we should take a P0 pref-actorization algebra and deform it into a BD prefactorization algebra. In the situation weare considering in this section, we will construct a prefactorization algebra of quantumobservables Obsq with the property that, as a vector space,

Obsq(U) = Obscl(U)[h],

but with a differential d such that

(1) modulo h, d coincides with the differential on Obscl(U), and(2) the equation

d(a · b) = (da) · b + (−1)|a|a · b + ha, bholds.

Here, · indicates the commutative product on Obscl(U). These properties imply that Obsq

defines a BD prefactorization algebra quantizing the P0 prefactorization algebra Obscl .

4.2.8. Let us identify Obsq(U) = Obscl(U)[h] as above. Then, Obsq(U) acquires aproduct and Poisson bracket from that on Obscl(U). Further, the differential on Obsq(U)satisfies the BD-algebra axiom

d(ab) = (da)b± a(db) + ha, b.

Thus, Obsq defines a prefactorization BD algebra quantizing Obscl .

It follows from the fact that Obscl is a factorization algebra (which we prove in section6.3) that Obsq is a factorization algebra over R[h].Remark: Those readers who are operadically inclined might notice that the Lie algebrachain complex of a Lie algebra g is the E0 version of the universal enveloping algebra ofa Lie algebra. Thus, our construction is an E0 version of the familiar construction of theWeyl algebra as a universal enveloping algebra of a Heisenberg algebra.

4.3. Quantum mechanics and the Weyl algebra

We will now show that our construction of the free scalar field on R recovers the Weylalgebra, which is the associative algebra of observables in quantum mechanics.

First, we must check that this prefactorization algebra is locally constant, and so givesus an associative algebra.

4.3.0.1 Lemma. The prefactorization algebra on R constructed from the free scalar field theorywith mass m is locally constant.

4.3. QUANTUM MECHANICS AND THE WEYL ALGEBRA 71

PROOF. Let Obsq denote this prefactorization algebra. Recall that Obsq(U) is theLie algebra chains on the Heisenberg Lie algebra H(U) built as a central extension of

C∞c (U)

4+m2

−−−→ C∞c (U)[−1]. Let us filter Obsq(U) by saying that

F≤i Obsq(U) = Sym≤i(H(U)[1]).

The associated graded for this filtration is Obscl(U)[h]. Thus, to show that H∗Obsq islocally constant, it suffices (by considering the spectral sequence associated to this filtra-tion) to show that H∗Obscl is locally constant. We have already seen (in lemma 4.2.4.1)that H∗(Obscl(a, b)) = R[p, q] for any interval (a, b), and that the inclusion maps (a, b)→(a′, b′) induces isomorphisms. Thus, the cohomology of Obscl is locally constant, as de-sired.

It follows that the cohomology of this prefactorization algebra is an associative alge-bra, which we call Am. We will show that Am is the Weyl algebra.

In fact, we will show a little more. The prefactorization algebra for the free scalar field

theory on R is built as Chevalley chains of the Heisenberg algebra based on C∞c (U)

4+m2

−−−→C∞

c (U)[−1]. The operator ∂∂x act on C∞

c (U) and commute with the operator4+m2. It alsopreserve the cocycle defining the central extension, and therefore acts naturally on theChevalley chains of the Heisenberg algebra. One can check that this operator is a deriva-tion for the prefactorization product. That is, the operator ∂

∂x commutes with inclusionsof one open subset into another, and if U, V are disjoint and α ∈ Obsq(U), β ∈ Obsq(V),we have

∂

∂x(α · β) = ∂

∂x(α) · β + α · ∂

∂x(β) ∈ Obsq(U qV).

(Derivations are discussed in more detail in section 4.7).

It follows immediately that ∂∂x defines a derivation of the associative algebra Am com-

ing from the cohomology of Obsq((0, 1)).

4.3.0.2 Definition. The Hamiltonian H is the derivation of the associative Am arising from thederivation − ∂

∂x of the prefactorization algebra Obsq of observables of the free scalar field theorywith mass m.

4.3.0.3 Proposition. The associative algebra Am coming from the free scalar field theory withmass m is the Weyl algebra, generated by p, q, h with the relation [p, q] = h and all other commu-tators being zero. The Hamiltonian H is the derivation

H(a) = 12h [p

2 −m2q2, a].


PROOF. We will start by writing down elements of Obsq corresponding to positionand momentum. Recall that

Obsq((a, b)) = Sym∗(

C∞c ((a, b))−1 ⊕ C∞

c ((a, b))0)[h]

with a certain differential.

We let φq, φp ∈ C∞(R) be the functions introduced in the proof of lemma 4.2.4.1.Explicitly, if m = 0, then φq = 1 and φp = x, whereas if m 6= 0 we have

φq =12

(emx + e−mx)

φp = 12m

(emx − e−mx) .

Thus, φq, φp are both in the kernel of the operator −∂2 + m2, with the properties that∂φp = φq and that φq is symmetric under x 7→ −x, whereas φp is antisymmetric.

Choose a function f0 ∈ C∞c ((− 1

2 , 12 )) which is symmetric under x 7→ −x and has the

property that ∫ ∞

−∞f0(x)φq(x)dx = 1.

The symmetry of f0 implies that the integral of f0 against φp is zero.

Let ft ∈ C∞c ((t− 1

2 , t + 12 )) be ft(x) = f0(x− t). We define observables Pt, Qt by

Qt = ft

Pt = − f ′tThe observables Qt, Pt are in the space C∞

c (I)[1]⊕ C∞c (I) of linear observables, where I =

((t − 12 , t + 1

2 ). They are also of cohomological degree zero. If we think of observablesas functionals of a field in C∞(I)⊕ C∞(I)[−1], then these are linear observables given byintegrating ft or − f ′t against the field φ.

Thus, Qt and Pt represent average measurements of positions and momenta of thefield φ in a neighborhood of t.

Because the cohomology classes [P0], [Q0] generate the commutative algebra H∗(Obscl(R)),it is automatic that they still generate the associative algebra H0 Obsq(R). We thus needto show that they satisfy the Heisenberg commutation relation

[[P0], [Q0]] = h

for the associative product on H0 Obsq(R), which is an associative algebra by virtue of thefact that H∗Obsq is locally constant. (The other commutators automatically vanish).

The Hamiltonian acting on [P0] gives the t-derivative of [Pt] at t = 0, and similarly forQ0. Note that

ddt Pt = − d

dt f ′0(x− t) = f ′′0 (x− t).


Since, in cohomology, the image of −∂2x + m2 is zero, we see that

ddt [Pt] = m2[ f0(x− t)] = m2[Qt].

In particular, when m = 0, [Pt] is independent of t: this is conservation of momentum.Similarly, d

dt Qt = Pt.

Thus, the Hamiltonian satisfies

H([P0]) = m2[Q0]

H([Q0]) = [P0].

If we assume that the commutation relation [[P0], [Q0]] = h holds, this implies that

H(a) = 12h [[P0]

2 −m2[Q0]2, a]

as desired.

Therefore, to prove the proposition, it suffices to prove the commutation relation be-tween [P0] and [Q0].

The first lemma we need is the following.

4.3.0.4 Lemma. Let a(t) be any function which satisfies a′′(t) = m2a(t). Let Q−1t denote the

observable in cohomological degree −1 given by ft ∈ C∞c ((t− 1

2 , t + 12 )[1]. Then,

∂

∂t(a(t)Pt − a′(t)Qt

)= −d(a(t)Q−1

t )

where d is the differential on observables.

PROOF. Let us prove this equation explicitly.

∂

∂t(a(t)Pt − a′(t)Qt

)= −a′(t)

∂

∂xft(x)− a(t)

∂

∂t∂

∂xft(x)− a′′(t) ft(x)− a′(t)

∂

∂tft(x)

= a(t)∂

∂x

2

ft(x)− a′′(t) ft(x)

= a(t)∂

∂x

2

ft(x)−m2a(t) ft(x)

= −d(a(t)Q−1t ).

We will define modified observables Pt, Qt which are independent of t at the coho-mological level. We let

Pt = φq(t)Pt − φ′q(t)Qt

Qt = φp(t)Pt − φ′p(t)Qt.


Since φp and φq are in the kernel of the operator −∂2 + m2, the observables Pt, Qt areindependent of t at the level of cohomology. In the case m = 0, then φq(t) = 1 so that Pt =Pt. The statement that Pt is independent of t corresponds, in this case, to conservation ofmomentum.

In general, P0 = P0 and Q0 = Q0. We also have

∂

∂tPt = −d(φq(t)Q−1

t )

and similarly for Qt.

It follows from the lemma that if we define a linear degree −1 observable hs,t by

hs,t =∫ t

u=sφq(u)Q−1

u (x)du,

thendhs,t = Ps −Pt.

Note that if |t| > 1, the observables Pt, Qt and P0, Q0 have disjoint support. Thismeans that we can use the prefactorization structure map

Obsq((− 12 , 1

2 ))⊗Obsq((t− 12 , t + 1

2 ))→ Obsq(R)

to define a product observable

Q0 ·Pt ∈ Obsq(R).

We will let ? denote the associative multiplication on H0 Obsq(R). We defined thismultiplication as follows. If α, β ∈ H0(Obsq(R)), we represent α by an element of H0(Obsq(I))and β by an element of H0(Obsq(J)), where I, J are disjoint intervals with I < J. The map

H0(Obsq(I))→ H0(Obsq(R))

(and similarly for J) is an isomorphism, which allows us to choose such representationsof α and β.

A representative of [P0] which is supported in the interval (t − 12 , t + 1

2 ) is given byPt. This follows from the fact that the cohomology class of Pt is independent of t, andthat P0 = P0.

Thus, the products between [P0] and [Q0] are defined by

[Q0] ? [P0] = [Q0 ·Pt] if t > 1

[P0] ? [Q0] = [Q0 ·Pt] if t < −1.

Thus, it remains to show that, if t > 1,

[Q0 ·Pt]− [Q0 ·P−t] = h.


We will construct an observable whose differential is the difference between the left andright hand sides. Consider the observable of cohomological degree 1 defined by

S = f0(x)h−t,t(y) ∈ C∞c (R)⊗ C∞

c (R)[1],

where the functions f and h−t,t were defined above. We view h−t,t as being of cohomo-logical degree −1, and f as being of cohomological degree 0.

Recall that the differential on Obsq(R) has two terms: one coming from the Laplacian4+m2 mapping C∞

c (R)−1 to C∞c (R)0, and one arising from the bracket of the Heisenberg

Lie algebra. The second term maps

Sym2(

C∞c (R)−1 ⊕ C∞

c (R)0)→ Rh.

Applying this differential to the observable S, we find that

(dS) = f0(x)(−∂2 + m2)h−t,t(y) + h∫

Rh−t,t(x) f0(x)dx

=Q0 · (P−t −Pt) + h∫

f0(x)h−t,t(x).

Therefore

[Q0Pt]− [Q0P−t] = h∫

f0(x)h−t,t(x).

It remains to compute the integral. This integral can be rewritten as∫ t

u=−t

∫ ∞

x=−∞f0(x) f0(x− u)φq(u)du.

Note that the answer is automatically independent of t for t sufficiently large, becausef (x) is supported near the origin so that f0(x) f0(x− u) = 0 for u sufficiently large. Thus,we can sent t→ ∞.

Since f0 is also symmetric under x → −x, we can replace f0(x− u) by f0(u− x). Wecan perform the u-integral by changing coordinates u → u− x, leaving the integrand asf0(x) f0(u)φq(u + x). Note that

φq(u + x) = 12

(em(x+u) + e−m(x+u)

).

Now, by assumption on f0, ∫f0(x)emx =

∫f0(x)e−mx = 1.

It follows that ∫ ∞

u=−∞φq(u + x) f0(u)du = φq(x).

We can then perform the remaining x integral∫

φq(x) f0(x)dx, which gives 1, as desired.


Thus, we have proven[[Q0], [P0]] = h,

as desired.

4.4. Free field theories and canonical quantization

Consider the free scalar field theory on a manifold of the form N×R, with the productmetric. We assume for simplicity that N is compact. Let Obsq denote the prefactorizationalgebra of observables of the free scalar field theory with mass m on N × R. Let π :N × R → R be the projection map. There is a push-forward prefactorization algebraπ∗Obsq, defined by

(π∗Obsq)(U) = Obsq(π−1(U)).In this section, we will explain how to relate this prefactorization algebra on R to aninfinite tensor product of the prefactorization algebras associated to quantum mechanicson R.

Let ei be an ortho-normal basis of eigenvectors of the operator4+m2 on C∞(N), witheigenvalue λi. The space ⊕R · ei is a dense subspace of C∞(N).

For m ∈ R, let Am denote the cohomology of the prefactorization algebra associatedto the free one-dimensional scalar field theory with mass m. Thus, Am is the Weyl al-gebra R[p, q, h] with commutator [p, q] = h. The dependence on m is only through theHamiltonian.

We can form the tensor product

⊗R[h]A√λi= A√λ1

⊗R[h] A√λ2⊗R[h] . . .

(The infinite tensor product is defined to be the colimit of the finite tensor products, wherethe maps in the colimit come from the unit in each algebra).

4.4.0.5 Proposition. There is a dense sub-factorization algebra of π∗Obsq which is locally con-stant, and has cohomology ⊗R[h]A√λi

.

Remark: The prefactorization algebra π∗Obsq has a derivation, the Hamiltonian, comingfrom infinitesimal translation in R. The prefactorization algebra A√λi

also has a Hamil-tonian, given by bracketing with 1

2h [p2 − λiq2,−]. The map from ⊗A√λi

to H∗(π∗Obsq)intertwines these derivations.

PROOF. The prefactorization algebra π∗Obsq on R assigns to an open subset U ⊂ R

the Chevalley chains of a Heisenberg Lie algebra given by a central extension of

C∞c (U × N)

4+m2

−−−→ C∞(U × N)[−1].

4.5. QUANTIZING CLASSICAL OBSERVABLES 77

A dense subcomplex of this is

(†) ⊕i

(C∞

c (U) · ei4R+λi−−−−→ ⊕C∞

c (U) · ei[−1])

.

Let Fi be the prefactorization algebra on R associated to quantum mechanics withmass

√λi. This is the prefactorization algebra associated to Heisenberg central extension

of

C∞c (U) · ei

4R+λi−−−−→ ⊕iC∞c (U) · ei[−1]

Note that Fi is a prefactorization algebra over R[h]. We can define the tensor productprefactorization algebra

⊗R[h]Fi = F1 ⊗R[h] F2 ⊗R[h] . . .

to be the colimit of the finite tensor products of the Fi under the inclusion maps comingfrom the unit 1 ∈ Fi(U) for any open subset.

We can, equivalently, view this tensor product as being associated to the Heisenbergcentral extension of the complex (†) above.

Because the complex (†) is a dense subspace of the complex whose Heisenberg exten-sion defines π∗Obsq, we see that there’s a map of prefactorization algebras with denseimage

⊗R[h]Fi → π∗Obsq

Passing to cohomology, we have a map

⊗R[h]H∗(Fi)→ H∗(π∗Obsq).

The prefactorization algebra H∗(Fi) is A√λi.

4.5. Quantizing classical observables

We have given an abstract definition of the prefactorization algebra of quantum ob-servables of a free field theory, as the prefactorization envelope of a certain Heisenbergdg Lie algebra. The prefactorization algebra of quantum observables Obsq, viewed as agraded prefactorization algebra with no differential, coincides with Obscl [h]. The onlydifference between Obsq and Obscl [h] is in the differential.

In this section we will give an alternative, but equivalent, description of Obsq. We willconstruct an isomorphism of precosheaves Obsq ∼= Obscl [h] which is compatible with dif-ferentials. This isomorphism is not, however, compatible with the prefactorization prod-uct. Thus, this isomorphism induces a deformed prefactorization product on Obscl [h]corresponding to the prefactorization product on Obsq.


In other words, instead of viewing Obsq as being obtained from Obscl by keeping theprefactorization product fixed but deforming the differential, we will show that it can beobtained from Obscl by keeping the differential fixed but deforming the product.

One advantage of this alternative description is that it is easier to construct correlationfunctions and vacua in this language.

4.5.1. The isomorphism we will construct between the cochain complexes of quan-tum and classical observables relies on a Green’s function for the Laplacian.

4.5.1.1 Definition. A Green’s function is a distribution G on M×M which is symmetric, andwhich satisfies

(4⊗ 1)G = δ4where δ4 is the δ-distribution on the diagonal.

A Green’s function for the Laplacian with mass satisfies the equation

(4⊗ 1)G + m2G = δ4.

(The convention is that4 has positive eigenvalues, so that on Rn,4 = −∑ ∂∂xi

2.)

If M is compact, then there is no Green’s function for the Laplacian. Instead, there is aunique function G satisfying

(4⊗ 1)G = δ4 − π

where π is the kernel for the operator of projection on to harmonic functions.

However, if we introduce a non-zero mass term, then the operator4+ m2 on C∞(M)is an isomorphism, so that there is a unique Green’s function.

If M is non-compact, then there can be a Green’s function for the Laplacian withoutmass term. For example, if M is Rn, then a choice of Green’s function is

G(x, y) =

1

4πd/2 Γ(d/2− 1) |x− y|2−n if n 6= 2− 1

2π log |x− y| if n = 2.

Let us now turn to the construction of the isomorphism of graded vector spaces be-tween Obscl(U) and Obsq(U) in the presence of a Green’s function.

The underlying graded vector space of Obsq(U) is Sym∗(C∞c (U)−1 ⊕ C∞

c (U))[h]. Ingeneral, for any vector space V, any element P ∈ (V∨)⊗2 defines a differential operator∂P of order two on Sym∗ V, uniquely characterized by the condition that it is zero onSym≤1 V and that on Sym2 V it is given by pairing with P. The same holds when wedefine the symmetric algebra using the completed tensor product.

4.5. QUANTIZING CLASSICAL OBSERVABLES 79

In the same way, for every distribution P on U ×U, we can define a continuous order2 differential operator on Sym∗(C∞

c (U)−1 ⊕ C∞c (U)0) which is uniquely characterized by

the properties that on Sym≤1 it is zero, that on Sym2(C∞c (U)−1 ⊕ C∞

c (U)0) it is zero onelements of cohomological degree less than zero, and that for φ, ψ ∈ C∞

c (U)0 we have

∂P(φψ) =∫

P(x, y)φ(x)ψ(y).

Let us choose a Green’s function G for the Laplacian on M. Then, G restricts to aGreen’s function for the Laplacian on any open subset U of M.

Therefore, we can define an order two differential operator ∂G on Sym∗(C∞c (U)−1 ⊕

C∞c (U)0). Since we have an identification (as graded vector spaces)

Obsq(U) = Obscl(U)[h] = Sym∗(C∞c (U)−1 ⊕ C∞

c (U)0)[h]

we can extend this by R[h]-linearity to an operator on the graded vector space Obsq(U).

Now, the differential on Obsq(U) = Obscl(U)[h] can be written as d = d1 + d2, whered1 is the differential on classical observables (which cohomologically imposes the equa-tions of motion), and d2 is the quantum correction which corresponds to divergence. Notethat d1 is a first-order differential operator and that d2 is a second-order operator. The op-

erator d1 is the derivation arising from the differential on the complex C∞c (U)

4−→ C∞c (U).

The operator d2 is the term arising from the Lie bracket on the Heisenberg dg Lie algebra;it is a continuous h-linear order two differential operator uniquely characterized by theproperty that d2(φ−1φ0) = h

∫φ−1φ0 for φi in the copy of C∞

c (U) in degree i, sitting insideof Obsq(U).

The Green’s function G satisfies

((4+ m2)⊗ 1)G = δ4

where δ4 is the Green’s function on the diagonal.

It follows that

[h∂G, d1] = d2.

Indeed, both sides of this equation are order two differential operators, so to check theequation, it suffices to calculate how the act on an element of Sym2(C∞

c (Rn)[1]⊕C∞c (Rn)).


If φi for i = −1, 0 are elements of the copy of C∞c (Rn) in degree i, we have

h∂Gd1(φ0φ−1) = h∂G(φ

0(4+ m2)φ−1)

=∫

G(x, y)φ0(x)((4+ m2)φ−1)(y)

=∫((4y + m2)G(x, y))φ0(x)φ−1(y)

=∫

φ0(x)φ−1(x) = d2(φ0φ−1).

On the fourth line, we have used the fact that 4y + m2 applied to G(x, y) is the delta-distribution on the diagonal.

It is also immediate that ∂G commutes with d2. Thus, if we let

W(α) = eh∂G(α)

(d1 + d2)W(α) = W(d1α).In other words, W is a cochain map from the complex Obsq(U) with differential d1 to thesame graded vector space with differential d1 + d2. Since the differential on Obscl(U) =Sym∗(C∞

c (M)[1]⊕ C∞c (U)) is d1, we see that, as desired, W gives an R[h]-linear cochain

isomorphismW : Obscl(U)[h]→ Obsq(U).

Note that W is not a map of prefactorization algebras. Thus, W induces a prefac-torization product on classical observables which quantizes the original prefactorizationproduct. Let us denote this quantum product by ?h, whereas the original product on clas-sical observables will be denoted by a dot. Then, if U, V are disjoint open subsets of M,and α ∈ Obscl(U), β ∈ Obscl(V), we have

α ?h β = e−h∂G(

eh∂G α)·(

eh∂G β)

This can be compared to the Moyal formula for the product on the Weyl algebra.

We have described this construction for the case of a free scalar field theory. Thisconstruction can be readily generalized to the case of an arbitrary free theory. Supposewe have such a theory on a manifold M, with space of fields E (M) and differential d.Instead of a Green’s function, we require a symmetric and continuous linear operatorG : (E (M)[1])⊗2 → R such that

Gd(e1 ⊗ e2) = 〈e1, e2〉where 〈−,−〉 is the pairing on E (M) which is part of the data of a free field theory. Inthe case that M is compact and H∗(E (M)) = 0, the propagator of the theory satisfies thisproperty. If M = Rn, then we can generally construct such a G from the Green’s functionfor the Laplacian. In this context, the operator eh∂G is, in the terminology of [Cos11c], therenormalization group flow operator from scale zero to scale ∞.



Let us suppose we have a free field theory on a compact manifold M, with the prop-erty that H∗(E (M)) = 0. As an example, consider the massive scalar field theory on Mwhere

E (M) = C∞(M)4+m2

−−−→ C∞(M)[−1].Our conventions are such that the eigenvalues of4 are non-negative. Adding a non-zeromass term m2 gives an operator with no zero eigenvalues, so that this complex has nocohomology.

As above, let E (M) be the Heisenberg dg Lie algebra whose underlying cochain com-plex is E (M) ⊕ R · h[−1], where the central element h is in degree 1. The inclusionR · h[−1] → E (M) is a quasi-isomorphism of dg Lie algebras. It follows that there isan isomorphism of R[h]-modules

H∗(Obsq(M)) ∼= R[h].

Let us normalize this isomorphism by asking that the element 1 in

Sym0(E [1]) = R ⊂ C∗(E [1])

gets send to 1 ∈ R[h].

4.6.0.2 Definition. In this situation, define the correlation functions of the free theory as follows.If U1, . . . , Un ⊂ M are disjoint opens, and Oi ∈ Obsq(Ui) are closed elements, then we define

〈O1, . . . , On〉 = [O1 . . . , On] ∈ H∗(Obsq(M)) = R[h].

Here O1 . . . On ∈ Obsq(M) is defined by the product structure of the prefactorization algebra.

The map W constructed in the previous section allows us to calculate correlation func-tions. Since we have a non-zero mass term and M is compact, there is a unique Green’sfunction for the operator4+ m2.

4.6.0.3 Lemma (Wick’s lemma). Let

αi ∈ Obscl(Ui) = Sym∗(C∞c (U)i[1]⊕ C∞

c (Ui))

be classical observables, and let

W(αi) = eh∂G αi ∈ Obsq(Ui)

be the corresponding quantum observables under the isomorphism of cochain complexes W fromclassical to quantum observables. Then,

〈W(α1), . . . , W(αn)〉 = W−1 (W(α1) · · ·W(αn)) (0) ∈ R[h]

On the right hand side W(α1) ·W(α2) indicates the product in the algebra Sym∗(C∞(M)[1]⊕C∞(M))[h]. The symbol (0) indicates evaluating a function on C∞(M)[1]⊕ C∞(M)[h] at zero,that is, taking the term in Sym0.


PROOF. The map W gives an isomorphism of cochain complexes between Obscl(U)[h]and Obsq(U) for every open subset U of M. As above, let ?h denote the prefactorizationproduct on Obscl [h] corresponding, under the isomorphism W, to the prefactorizationproduct on Obsq. Then,

α1 ?h · · · ?h αn = e−h∂G((eh∂G α1) · · · (eh∂G αn)

).

Since W gives an isomorphism of prefactorization algebras between (Obscl [h], ?h) andObsq, the correlation functions of the observables W(αi) is the cohomology class of α1 ?h

· · · ?h αn in Obscl(M)[h]. The map Obscl(M)[h]→ R[h] sending an observable α to α(0) isan R[h]-linear cochain map inducing an isomorphism on cohomology, and it sends 1 to 1.There is a unique (up to cochain homotopy) such map. Therefore,

〈W(α1), . . . , W(αn)〉 = (α1 ?h · · · ?h αn) (0).

This lemma gives us an explicit formula in terms of the Green’s functions for the cor-relation functions of any observable.

As an example, let us suppose that we have two linear observables α1, α2 of cohomo-logical degree 0, defined on open sets U, V. Thus, α1 ∈ C∞

c (U) and α2 ∈ C∞c (V). Then,

W(α1) = α1, butW−1(α1α2) = α1α2 − h∂G(α1α2).

Further,

∂G(α1α2) =∫

M×MG(x, y)α1(x)α2(y).

It follows that〈W(α1), W(α2)〉 = −h

∫M×M

G(x, y)α1(x)α2(y)

which is up to sign what a physicist would write down to the expectation value of twolinear observables.

Remark: We have set things up so that we are computing the functional integral againstthe measure which is eS/hdµ where S is the action functional and dµ is the (non-existent)“Lebesgue measure” on the space of fields. Physicists often use the convention where weuse e−S/h or eiS/h. One goes between these different conventions by a change of coordi-nates h→ −h or h→ ih.

4.7. Translation-invariant prefactorization algebras

In this section we will analyze in detail the notion of translation-invariant prefactoriza-tion algebras on Rn. Most theories from physics possess this property. For one-dimensionalfield theories, one often uses the phrase “time-independent Hamiltonian” to indicate this

4.7. TRANSLATION-INVARIANT PREFACTORIZATION ALGEBRAS 83

property, and in this section we will explain, in examples, how to relate the Hamiltonianformalism of quantum mechanics to our approach.

4.7.1. We now turn to the definition of a translation-invariant prefactorization alge-bra. If U ⊂ Rn and x ∈ Rn, let

Tx(U) := y : y− x ∈ U

denote the translate of U by x.

4.7.1.1 Definition. A prefactorization algebra F on Rn is discretely translation-invariant ifwe have isomorphisms

Tx : F (U) ∼= F (Tx(U))

for all x ∈ Rn and all open subsets U ⊂ Rn. These isomorphisms must satisfy a few conditions.First, we require that Tx Ty = Tx+y for every x, y ∈ Rn. Second, for all disjoint open subsetsU1, . . . , Uk in V, the diagram

F (U1)⊗ · · · ⊗ F (Uk)Tx //

F (TxU1)⊗ · · · ⊗ F (TxUk)

F (V)

Tx // F (TxV)

commutes. (Here the vertical arrows are the structure maps of the prefactorization algebra.)

Example: Consider the prefactorization algebra of quantum observables of the free scalarfield theory on Rn, as defined in section 4.2.6. This has complex of fields

E =

C∞ 4−→ C∞[−1]

,

where the superscript indicates cohomological degree.

By definition, Obsq(U) is the Chevalley-Eilenberg chains of a −1-shifted central ex-tension Ec(U) of Ec(U), with cocycle defined by

∫φ0φ1 where φi ∈ C∞

c (U)i.

This Heisenberg algebra is defined only using the Riemannian structure on Rn, andis therefore automatically invariant under all isometries of Rn. In particular, the resultingprefactorization algebra is discretely translation-invariant.

We are interested in a refined version of this notion, where the structure maps of theprefactorization algebra depend smoothly on the position of the open sets. It is a bit subtleto talk about “smoothly varying an open set,” and in order to do this, we introduce somenotation.


Firstly, we need to introduce the notion of a derivation of a prefactorization algebra ona manifold M. We will construct a differential graded Lie algebra of derivations of anyprefactorization algebra.

4.7.1.2 Definition. A degree k derivation of a prefactorization algebra F is a collection of mapsDU : F (U) → F (U) of cohomological degree k for each open subset U ⊂ M, with the propertythat, if U1, . . . , Un ⊂ V are disjoint, and αi ∈ F (Ui), then

DVmU1,...,UnV (α1, . . . , αn) = ±∑ mU1,...,Un

V (α1, . . . , DUi αi, . . . , αn),

where ± indicates the usual Koszul rule of signs.

Example: Let us consider, again, the prefactorization algebra of the free scalar field onRn. Observables on U are Chevalley-Eilenberg chains of the Heisenberg algebra Ec(U).Suppose that X is a Killing vector field on Rn (i.e. X is an infinitesimal isometry). Forexample, we could take X to be a translation vector field ∂

∂xj . The Heisenberg dg Liealgebra has a derivation which sends φi → Xφi, for i = 0, 1, and is zero on the centralelement h. (Recall that φi is notation for an element of the copy of C∞

c (U) situated indegree i). The fact that X is a Killing vector field on Rn implies that it commutes with thedifferential on the Heisenberg algebra, and the equality∫

(Xφ0)φ1 +∫

φ0(Xφ1)

implies that X is a derivation of dg Lie algebras.

By naturality, X extends to an endomorphism of Obsq(U) = C∗(Ec(U)). This endo-morphism defines a derivation of the prefactorization algebra Obsq of observables of thefree scalar field theory.

Let Derk(F ) denote the derivations of degree k; it is easy to verify that Der∗(F ) formsa differential graded Lie algebra. The differential is defined by (dD)U = [dU , DU ], wheredU is the differential on F (U). The Lie bracket is defined by

[D, D′]U = [DU , D′U ].

The concept of derivation allows us to talk about the action of a dg Lie algebra ona prefactorization algebra F . Such an action is simply a homomorphism of differentialgraded Lie algebras

g→ Der∗(F ).

Next, let us introduce some notation which will help us describe the smoothness con-ditions for a discretely translation-invariant prefactorization algebra.

Let U1, . . . , Uk ⊂ V be disjoint open subsets. Let W ⊂ (Rn)k be the set of thosex1, . . . , xk such that the sets Tx1(U1), . . . , Txk(Uk) are all disjoint and contained in V. It


parametrizes the way we can move the open sets without causing overlaps. Let us assumethat W has non-empty interior, which happens when the closure of the Ui are disjoint andcontained in V.

Let F be any discretely translation-invariant prefactorization algebra. Then, for each(x1, . . . , xk) ∈W, we have a multilinear map obtained as a composition

mx1,...,xk : F (U1)× · · · × F (Uk)→ F (Tx1U1)× · · · × F (Txk Uk)→ F (V),

where the second map arises from the inclusion

Tx1U1 q · · · q Txk Uk → V.

4.7.1.3 Definition. A discretely translation-invariant prefactorization algebra F is smoothlytranslation-invariant if the following conditions hold.

(1) The map mx1,...,xk above depends smoothly on (x1, . . . , xk) ∈W.(2) The prefactorization algebra F is equipped with an action of the Abelian Lie algebra Rn

of translations. If v ∈ Rn, we will denote the corresponding action maps by

ddv

: F (U)→ F (U).

We view this Lie algebra action as an infinitesimal version of the global translation in-variance.

(3) The infinitesimal action is compatible with the global translation invariance in the follow-ing sense. If v ∈ Rn, let vi ∈ (Rn)k denote the vector with v placed in the ith positionand 0 in the other k− 1 slots. If αi ∈ F (Ui), then we require that

ddvi

mx1,...,xk(α1, . . . , αk) = mx1,...,xk

(α1, . . . ,

ddv

αi, . . . , αk

).

When we refer to a translation-invariant prefactorization algebra without further qual-ification, we will always mean a smoothly translation-invariant prefactorization algebra.

Example: We have already seen that the prefactorization algebra of the free scalar fieldtheory on Rn is discretely translation invariant, and is equipped with an action of theAbelian Lie algebra Rn by derivations. It is easy to verify that this prefactorization algebrais smoothly translation-invariant.

Example: Suppose that F is a locally-constant, smoothly translation invariant prefactor-ization algebra on R, valued in vector spaces. Then, A = F ((0, 1)) has the structure of anassociative algebra.

For any two intervals (0, 1) and (t, t+ 1), there is an isomorphismF ((0, 1)) ∼= F ((t, t+1)) coming from the isomorphismF ((a, b))→ F (R) associated to inclusion of an intervalinto R.


The fact thatF is translation invariant means that there is another isomorphismF ((0, 1))→F ((t, t + 1)) for any t ∈ R. Composing these two isomorphism yields an action of thegroup R on A = F ((0, 1)). One can check (excercise!) that this is an action on R byassociative algebras.

The fact that F is smoothly translation-invariant means that the action of R on A issmooth, and differentiates to an infinitesimal action of the Lie algebra R on A by deriva-tions. The basis element ∂

∂x of R becomes a derivation H of A, called the Hamiltonian.

In the case that F is the cohomology of the prefactorization algebra of observables ofthe free scalar field theory on R with mass m, we have seen in section 4.3 that the algebraA is the Weyl algebra, generated by p, q, h with commutation relation [p, q] = h. TheHamiltonian is given by

H(a) = 12h [p

2 −m2q2, a].

Remark: As always, we work with prefactorization algebras taking values in the categoryof differentiable cochain complexes. Generalities about differentiable cochain complexesare developed in appendix B. There we explain what it means for a smooth multilinearmap between differentiable cochain complexes to depend smoothly on some parameters.

4.7.2. Next, we will explain how to think of the structure of a translation-invariantprefactorization algebra on Rn in more operadic terms. This description has a lot in com-mon with the En algebras familiar from topology.

Let r1, . . . , rk, s ∈ R>0. Let

Discsn(r1, . . . , rk | s) ⊂ (Rn)k

be the (possibly empty) open subset consisting of x1, . . . , xk ∈ Rn with the property thatthe closures of the balls Bri(xi) are all disjoint and contained in Bs(0) (where Br(x) denotesthe open ball of radius r around x).

4.7.2.1 Definition. Let Discsn be the R>0-colored operad in the category of smooth manifoldswhose space of k-ary morphisms is the space Discsn(r1, . . . , rk | s) between ri, s ∈ R>0 describedabove.

Note that a colored operad is the same thing as a multicategory (recall remark 3.1.2).An R>0-colored operad is thus a multicategory whose set of objects is R>0.

The essential data of the colored operad structure on the spaces Discsn(r1, . . . , rk | s)is the following. We have maps

i : Discsn(r1, . . . , rk | ti)×Discsn(t1, . . . , tm | s)

→ Discsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).


This map is defined by inserting the outgoing ball (of radius ti) of a configuration x ∈Discsn(r1, . . . , rk | ti) into the ith incoming ball of a point y ∈ Discsn(t1, . . . , tk | s).

These maps satisfy the natural associativity and commutativity properties of a multi-category.

4.7.3. Next, let F be a translation-invariant prefactorization algebra on Rn. Let

Fr = F (Br(0))

denote the cochain complexF that assigns to a ball of radius r. This notation is reasonablebecause translation invariance gives us an isomorphism between F (Br(0)) and F (Br(x))for any x ∈ Rn.

The structure maps for a translation-invariant prefactorization algebra yield, for eachp ∈ Discsn(r1, . . . , rk | s), multiplication operations

m[p] : Fr1 × · · · × Frk → Fs.

The map m[p] is a smooth multilinear map of differentiable spaces; and furthermore, thismap depends smoothly on p.

These operations make the complexes Fr into an algebra over the R>0-colored operadDiscsn(r1, . . . , rk | s), valued in the multicategory of differentiable cochain complexes.In addition, the complexes Fr are endowed with an action of the Abelian Lie algebraRn. This action is by derivations of the Discsn-algebra F compatible with the action oftranslation on Discsn, as described above.

4.7.4. Now, let us unravel explicitly what it means to be such a Discsn algebra.

The first property is that, for each p ∈ Discsn(r1, . . . , rk | s), the map m[p] is a multilin-ear map, of cohomological degree 0, compatible with differentials.

Second, let N be a manifold and let fi : N → F diri be smooth maps into the space F di

ri

of elements of degree di. The smoothness properties of the map m[p] mean that the map

N ×Discsn(r1, . . . , rk | s)→ Fs

(x, p) 7→ m[p]( f1(x), . . . , fk(x))

is smooth.

Next, note that a permutation σ ∈ Sk gives an isomorphism

σ : Discsn(r1, . . . , rk | s)→ Discsn(rσ(1), . . . , rσ(k) | s).

We require that, for each p ∈ Discsn(r1, . . . , rk | s) and each αi ∈ Fri ,

m[σ(p)](ασ(1), . . . , ασ(k)) = m[p](α1, . . . , αk).


Finally, we require that the maps m[p] are compatible with composition, in the follow-ing sense. For p ∈ Discsn(r1, . . . , rk | ti), q ∈ Discsn(t1, . . . , tl | s), αi ∈ Fri , and β j ∈ Ftj , werequire that

m[q] (β1, . . . , βi−1, m[p](α1, . . . , αk), βi+1, . . . , βl)

= m[q i p](β1, . . . , βi−1, α1, . . . , αk, βi+1, . . . , βl).

In addition, the action of Rn on each F r is compatible with these multiplication maps, inthe way described above.

4.7.5. Let us give one more equivalent way of rewriting these axioms, which willbe useful when we discuss the holomorphic context. These alternative axioms will saythat the spaces C∞(Discsn(r1, . . . , rk | s)) form an R>0-colored co-operad when we use theappropriate completed tensor product (we will use the completed tensor product on thecategory of convenient vector spaces). Since we know how to tensor a differentiable vectorspace with the space of smooth functions on a manifold, it makes sense to talk about analgebra over this colored co-operad in the category of differentiable cochain complexes.

The smoothness axiom for the product map

m[p] : Fr1 ⊗ · · · ⊗ Frk → Fs,

where p ∈ Discsn(r1, . . . , rk | s), can be rephrased as follows. For any differentiable vec-tor space V and smooth manifold M, we use the notation V ⊗ C∞(M) interchangeablywith the notation C∞(M, V); both indicate the differentiable vector space of smooth mapsM → V. The smoothness axiom states that the map above extends to a smooth map ofdifferentiable spaces

µ(r1, . . . , rk | s) : Fr1 × · · · × Frk → Fs ⊗ C∞(Discsn(r1, . . . , rk | s)).

In general, if V1, . . . , Vk, W are differentiable vector spaces and if X is a smooth mani-fold, let

C∞(X, Hom(V1, . . . , Vk |W))

denote the space of smooth multilinear maps

V1 × · · · ×Vk → C∞(X, W).

Note that there is a natural gluing map

i : C∞(X, Hom(V1, . . . , Vk |Wi))× C∞(Y, Hom(W1, . . . , Wl | T))

→ C∞(XxY, Hom(W1, . . . , Wi−1, V1, . . . , Vk, Wi+1, . . . , Wl | T)).

With this notation in hand, there are elements

µ(r1, . . . , rk | s) ∈ C∞ (Discs(r1, . . . rk | s), Hom(Fr1 , . . . ,Frk | Fs))

with the following properties.


(1) µ(r1, . . . , rk | s) is closed under the natural differential, arising from the differen-tials on the cochain complexes Fri .

(2) If σ ∈ Sk, then

σ∗µ(r1, . . . , rk | s) = µ(rσ(1), . . . , rσ(k) | s)

where

σ∗ : C∞(Discs(r1, . . . rk | s), Hom(Fr1 , . . . ,Frk | Fs))

→ C∞(Discs(rσ(1), . . . rσ(k) | s), Hom(Frσ(1) , . . . ,Frσ(k) | Fs))

is the natural isomorphism.(3) As before, let

i : Discsn(r1, . . . , rk | ti)×Discsn(t1, . . . , tm | s)

→ Discsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).

denote the gluing map. Then, we require that

∗i µ(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tl) = µ(r1, . . . , rk | ti) i µ(t1, . . . , tl | s).

These elements equip the Fr with the structure of an algebra over the colored co-operad,as stated earlier.

4.7.6. Let us write down explicit formula for these product maps in the case of thefree massless scalar field theory. Let G be the Green’s function for the Laplacian on Rn.Thus, G =

(4πd/2)−1 Γ(d/2 − 1) |x− y|2−n for n 6= 2, and G = −(2π)−1 log |x− y| if

n = 2.

We have seen in section 4.5 that the choice of a Green’s function G leads to an iso-morphism of cochain complexes W : Obscl(U)[h] → Obsq(U)[h], for every open subsetU. This allows us to transfer the product in the prefactorization algebra Obsq(U) to adeformed product ?h in the prefactorization algebra Obscl(U)[h], defined by

α ?h β = W−1(W(α) ·W(β))

where on the right hand side · indicates the product in the prefactorization algebra Obsq.

This leads to a completely explicit description of the product maps

µsr1,...,rk

: Fr1 ⊗ · · · ⊗ Frk → C∞(P(r1, . . . , rk | s),Fs)

discussed above, in the case that F arises from the prefactorization algebra of quantumobseravbles of a free scalar field theory, or equivalently from the prefactorization algebra(Obscl [h], ?h).


For example, on R2, let α1, α2 be compactly supported smooth functions on discsD(0, ri) of radii ri around 0. Let us view each α as a a cohomological degree 0 elementof

Fri = Obscl(D(0, ri))[h] = Sym∗(C∞c (D(0, ri))[1]⊕ C∞

c (D(0, ri)))[h].

Let Txαi denote the translate of αi to an element of C∞c (D(x, ri)).

Then, for x1, x2 ∈ R2 which are such that D(xi, ri) are disjoint and contained in D(0, s),we have

µsr1,r2

(α1, α2) = Tx1 α1 ?h Tx2 α2

= (Tx1 α1) · (Tx2 α2)− h∫

u1,u2∈R2α1(u1 + x1)α2(u2 + x2) log |u1 − u2| .

4.7.7. Vacua.

4.7.7.1 Definition. Let F be a smoothly translation-invariant prefactorization algebra on Rn,over a ring R (which in practice is R, C or R[[h]], C[[h]]).

A state for F is a smooth linear map

〈−〉 : H0(F (Rn))→ R

(or to C if we work with complex coefficients).

The smooth means the following. Because F (Rn) is a differentiable vector space, it is a sheafon the site of smooth manifolds, so that H∗(F (Rn)) is a presheaf on the site of smooth manifolds:that is, if M is a smooth manifold, and α ∈ H∗(C∞(M,F (Rn)), then we require that 〈α〉 is asmooth map from M to R.

A state 〈−〉 is translation-invariant if it commutes with the action of both the group Rn andof the infinitesimal action of the Lie algebra Rn, where Rn acts trivially on R

A state 〈−〉 allows us to define correlation functions of observables. If Oi ∈ F (D(0, ri))are cohomologically closed observables, then we can construct TxiOi ∈ F (D(xi, ri). If xiare such that the discs D(xi, ri) are disjoint – so that (x1, . . . , xk) ∈ P(r1, . . . , rk | ∞)– thenwe can define the correlation function

〈O1(x1), . . . , On(xn)〉 ∈ R.

by applying 〈−〉 to the cohomology class of the product observable Tx1O1 . . . TxnOn ∈F (Rn). The fact that we have a smoothly translation invariant prefactorization algebra,and that the vacuum is assumed to be a smooth map, implies that 〈O1(x1), . . . , On(xn)〉is a smooth function of the xi. Further, this function is invariant under simultaneoustranslation of all the points xi.


4.7.7.2 Definition. A translation-invariant state 〈−〉 is a vacuum if it satisfies the cluster de-composition principle, which states that, in the situation above, for all cohomology classes O1, O2where Oi ∈ H∗(F (D(0, ri)) we have

〈O1(0), O2(x)〉 − 〈O1〉〈O2〉 → 0 as x → ∞.

A vacuum is massive if 〈O1(0), O2(x)〉 − 〈O1〉〈O2〉 tends to zero exponentially fast.

Example: Consider the free scalar field theory on Rn with mass m. We have seen that thechoice of a Green’s function G for the operator 4+ m2 leads to, for every open subsetsU ⊂ Rn, an isomorphism of cochain complexes Obscl(U)[h] ∼= Obsq(U); this becomes anisomorphism of prefactorization algebras if we endow Obscl [h] with a deformed prefac-torization product ?h defined using the Green’s function.

If m > 0, there is a unique Green’s function G of the form G = f (x − y) where fis a distribution on Rn which is smooth away from the origin, and which tends to zeroexponentially fast at infinity. For example, if n = 1, the function f is

f (x) =1

2me−m|x|.

If m = 0, there is a canonically-defined Green’s function which, if n 6= 2, is G(x, y) =1

4πd/2 Γ(d/2− 1) |x− y|2−n and is −(2π)−1 log |x− y| if n = 2.

Because Obscl(Rn) is, as a cochain complex, the symmetric algebra on the complex

C∞c (Rn)[1]

4+m2

−−−→ C∞c (Rn), there is a map from Obscl(Rn) to R which is the identity on

Sym0 and sends Sym>0 to 0. This map extends to an R[h]-linear cochain map

〈−〉 : Obscl(Rn)[h]→ R[h].

Clearly, this map is translation invariant and smooth. Thus, because we have a cochainisomorphism between Obscl(Rn)[h] and Obsq(Rn), we have produced a translation-invariantstate.

4.7.7.3 Lemma. If m > 0, this state is a massive vacuum. If m = 0, this state is a vacuum ifn > 2, otherwise it does not satisfy the cluster decomposition principle.

PROOF. Let F1 ∈ C∞c (D(0, r1))

⊗k1 and F2 ∈ C∞c (D(0, r2))⊗k2 . We will view F1, F2 as

observables in Obscl(D(0, ri)) by using the natural map from C∞c (U)⊗k to the coinvariants

Symk C∞c (U). Let TcF2 be the translation of F2 by c, where c is sufficiently large so that

the discs D(0, r1) and D(c, r2) are disjoint. Explicitly, TcF2 is represented by the functionF2(y1 − c, . . . , yk2 − c). We are interested in computing the expectation value 〈F1, TcF2〉.


We gave, above, an explicit formula for the quantum prefactorization product ?h onObscl [h]. In this case, it leads to the formula

F1 ?h TcF2 =min(k1,k2)

∑r=0

hr ∑1≤i1<···<ir≤k11≤j1<···<jr≤k2

∫xi1 ,...,xir∈Rn

yi1 ,...,yir∈Rn

F1(x1, . . . , xk1)F2(y1 − c, . . . , yk2 − c)G(xi1 , yj1) . . . G(xir , yjr)dxi1 . . . dxir dyi1 . . . dyir .

Note that after performing the integral, we are left with a function of the k1 + k2 − 2rcopies of Rn we have not integrated over. This function is then viewed as an observablein Symk1+k2−2r C∞

c (Rn).

If k1, k2 > 0, then 〈F1〉 = 0 and 〈F2〉 = 0. Further, 〈F1, TcF2〉 selects the constant term inthe expression for F1 ?h TcF2. There is only a non-zero constant term if k1 = k2 = k. In thatcase, the constant term is (up to a combinatorial factor)

〈F1, TcF2〉 = hk∫

xi ,yi∈RnF1(x1, . . . , xk)F2(y1 − c, . . . , yk − c)G(x1, y1) . . . G(xk, yk) ∈ R[h].

To check whether the cluster decomposition principle holds, we need to check whether ornot

〈F1, TcF2〉 = 〈F1, TcF2〉 − 〈F1〉〈F2〉tends to zero as c→ ∞. If m > 0, we know that G(x, y) tends to zero exponentially fast asx− y→ ∞. This implies immediately that we have a massive vacuum in this case.

If m = 0, the Green’s function G(x, y) tends to zero like the inverse of a polynomial aslong as n > 2. In this case, we have a vacuum. If n = 1 or n = 2, then G(x, y) does nottend to zero, so we don’t have a vacuum.

Example: We can give a more abstract construction for the vacuum of associated to a mas-

sive scalar field theory on Rn. Let Ec = C∞c4+m2

−−−→ C∞c be, as before, the complex of

fields. Let Ec be the Heisenberg central extension Ec = Ec ⊕R · h where h has degree 1.We defined the complex of observables on U as the Chevalley-Eilenberg chain complex ofEc(U).

Let ES (Rn) be the complex S(Rn)

4+m2

−−−→ S(Rn), where S(Rn) is the space of Schwartzfunctions on Rn. (Recall that a smooth function is Schwartz if it and all its derivativestends to zero at ∞ faster than the reciprocal of any polynomial).

The Heisenberg dg Lie algebra can be defined using ES (Rn) instead of Ec(Rn). We let

ES (Rn) be ES (R

n)⊕R · h[−1], with bracket defined by [φ0, φ1] = h∫

φ0φ1. Here φi are


Schwartz functions of degrees 0 and 1. This makes sense, because the product of any twoSchwartz function is Schwartz and Schwartz functions are integrable.

Schwartz functions have a natural topology, so we will view them as being a conve-nient vector space. Since the topology is nuclear Frechet, a result discussed in appendix Ctells us that the tensor product in the category of convenient vector spaces coincides withthat in the category of nuclear Frechet spaces. A result of Grothendieck [Gro52] tells usthat

S(Rn)⊗S(Rm) = S(Rn+m),

and similarly for Schwartz sections of vector bundles.

This allows us to define the Chevalley chain complex

ObsqS (R

n)def= C∗(ES (Rn))

of the Heisenberg algebra based on Schwartz functions; as usual, we use the completedtensor product on the category of convenient vector spaces when defining the symmetricalgebra.

There’s a map

Obsq(Rn)→ ObsqS (R

n)

given by viewing a compactly supported function as a Schwartz function.

Now,

4.7.7.4 Lemma. The cohomology of ObsqS (R

n) is R[h].

PROOF. The complex

ES (Rn) = S(Rn)

4+m2

−−−→ S(Rn)

has no cohomology. We can see this using Fourier duality: the Fourier transform is anisomorphism on the space of Schwartz functions, and the Fourier dual of the operator4+ m2 is the operator p2 + m2, where p2 = ∑ p2

i and pi are coordinates on the Fourier

dual Rn. (Note that our convention is that the Laplacian is −∑ ∂∂xi

2). The operator of

multiplication by p2 + m2 is invertible on the space of Schwartz functions, just becauseif f is a Schwartz function then so is (p2 + m2)−1 f . (At this point we need to use ourassumption that m 6= 0). The inverse is a smooth linear map, so that this complex issmoothly homotopy equivalent to the zero complex.

This implies immediately that the Chevalley chain complex of ES (Rn) has cohomol-

ogy the same as that of the Abelian Lie algebra R · h, i.e. R[h] as desired.


Thus, we have a translation-invariant state

H∗(Obsq(Rn))→ H∗(ObsqS (R

n)) = R[h].

This state is characterized uniquely by the fact that it is defined on Schwartz observables.Since the state constructed more explicitly above also has this property, we see that thesetwo states coincide, so that this state is a massive vacuum.

CHAPTER 5

Holomorphic field theories and vertex algebras

This chapter serves two purposes. On the one hand, we develop several examples thatexhibit how to understand the observables of a two-dimensional theory from the point ofview of factorization algebras and how this approach recovers standard examples of ver-tex algebras. On the other hand, we provide a precise definition of a factorization alge-bra on Cn whose structure maps vary holomorphically, much as we defined translation-invariant factorization algebras in section 4.7. We then give a proof that when n = 1 andthe factorization algebra possesses a U(1)-action, we can extract a vertex operator alge-bra. For n > 1, the structure we find is a higher-dimensional analog of a vertex algebra.Such higher-dimensional vertex algebras appear, for example, as the factorization algebraof observables of partial twists of supersymmetric gauge theories.

5.0.8. Reminder on vertex algebras. In mathematics, the notion of a vertex algebrasis a standard formalization of the observables of a chiral conformal field theory (a theoryon the complex plane C). Before embarking on our own approach, we recall the definitionof a vertex algebra and various properties as given in [FBZ04].

5.0.8.1 Definition. Let V be a vector space. An element a(z) = ∑n∈Z anz−n in End V[[z, z−1]]is a field if, for each v ∈ V, there is some N such that ajv = 0 for all j > N.

Remark: The usage of the term “field” in the theory of vertex operators often provokesconfusion. In this book, the term field is used to refer to a configuration in a classical fieldtheory: for example, in a scalar field theory on a manifold M, a field is an element ofC∞(M). The term “field” as used in the theory of vertex algebras is not related to thisusage.

5.0.8.2 Definition (Definition 1.3.1, [FBZ04]). A vertex algebra is the following data:

• a vector space V over C (the state space);• a nonzero vector |0〉 ∈ V (the vacuum vector);• a linear map T : V → V (the shift operator);• a linear map Y(−, z) : V → End V[[z, z−1]] sending every vector v to a field (the vertex

operation);

subject to the following axioms:

95

96 5. HOLOMORPHIC FIELD THEORIES AND VERTEX ALGEBRAS

• (vacuum axiom) Y(|0〉, z) = 1V and Y(v, z)|0〉 ∈ v + zV[[z]] for all v ∈ V;• (translation axiom) [T, Y(v, z)] = ∂zY(v, z) for every v ∈ V and T|0〉 = 0;• (locality axiom) for any pair of vectors v, v′ ∈ V, there exists a nonnegative integer N

such that (z− w)N [Y(v, z), Y(v′, w)] = 0 as an element of End V[[z±1, w±1]].

The vertex operation is best understood in terms of the following intuition. The vectorspace V represents the set of pointwise measurements one can make of the fields, andone should imagine labeling each point z ∈ C by a copy of V, which we’ll denote Vz.Moreover, in a disk D containing the point z, the measurements at z are a dense subspaceof the measurements one can make in D. We’ll denote the observables on D by VD. Thevertex operation is a way of combining pointwise measurements. Let D be a disk centeredon the origin. For z 6= 0, we can multiply observables to get a map

Yz : V0 ⊗Vz → VD,

and it should vary holomorphically in z ∈ D \ 0. In other words, we should get some-thing with properties like the formal definition Y(−, z) above. (This picture clearly resem-bles the “pair of pants” product from two-dimensional topological field theories.)

An appealing aspect of our approach to observables is that this intuition becomesexplicit and rigorous. Our procedure describes the observables on every disk and givesthe structure maps in a coordinate-free way. By choosing a coordinate z on C, we recoverthe usual formulas for vertex algebras.

This is seen in the main theorem in this chapter, which we will now state. The theoremconnects a certain class of factorization algebras on C with vertex algebras. The factoriza-tion algebras of interest are holomorphically translation invariant factorization algebra.We will give a precise definition of what this means shortly; but here is a rough definition.Let Fr denote the cochain complex F (D(0, r)) that F assigns to a disc of radius r. Recall,as we explained in section 4.7 of chapter 4, if F is smoothly translation invariant then wehave a operator product map

Fr1 ⊗ · · · ⊗ Frn → C∞(Discs(r1, . . . , rk | s),Fs)

where Discs(r1, . . . , rk | s) refers to the open subset of Ck consisting of points z1, . . . , zksuch that the discs of radius ri around zi are all disjoint and contained in the disc of radius saround the origin. If F is holomorphically translation invariant, then this lifts to a cochainmap

Fr1 ⊗ · · · ⊗ Frn → Ω0,∗(Discs(r1, . . . , rk | s),Fs)

where on the right hand side we have used the Dolbeault complex of the complex mani-fold Discs(r1, . . . , rk | s). We also require some compatibility of these lifts with composi-tions, which we will detail later.

In other words, being holomorphically translation invariant means that the operatorproduct map is holomorphic (up to homotopy) in the location of the discs.

5. HOLOMORPHIC FIELD THEORIES AND VERTEX ALGEBRAS 97

5.0.8.3 Theorem. Let us suppose that F is a holomorphically translation invariant factorizationalgebra on C. Let us suppose that F is also invariant under the action of S1 on C by rotation. LetF k

r denote the weight k eigenspace of the S1 action on the complex Fr. Let us assume that the mapsF k

r → F ks (for r < s) associated to the inclusion D(0, r) ⊂ D(0, s) are quasi-isomorphisms.

Then, the spaceV = ⊕k H∗(F k

r )

has the structure of a vertex algebra. The vertex algebra structure map

V ⊗V → V[[z, z−1]]

is the Laurent expansion of operator product map

H∗(F k1r1)⊗ H∗(F k2

r2)→ Hol(Discs(r1, r2 | s), H∗(Fs)).

On the right hand side, Hol denotes the space of holomorphic maps.

In other words, the intuition that the vertex algebra structure map is the operatorproduct expansion is made precise in our formalism.

This result should be compared with the classic result of Huang [], who relates chiralconformal field theories at genus 0 in the sense used by Segal with vertex algebras. Aswe have seen in section 3.3 of chapter 3.3, the axioms for factorization algebras are veryclosely related to Segal’s axioms. Our axioms for a holomorphically translation invariantfield theory is similarly related to Segal’s axioms for a two-dimensional chiral field theory.Although our result is closely related to Huang’s, it is a little different because of thetechnical differences between a factorization algebra and a Segal-style chiral conformalfield theory.

One nice feature of our definition of holomorphically translation-invariant factoriza-tion algebra is that it makes sense in any complex dimension. The structure present on thecohomology of a higher-dimensional holomorphically translation-invariant factorizationalgebra is a higher dimensional version of the axioms of a vertex algebra. This structurewas discussed briefly in [?].

Another nice feature of our approach is that our general construction of field theoriesallows one to construct vertex algebras by perturbation theory, starting with a Lagrangian.This should lead to the construction of many interesting vertex algebras.

5.0.9. Before we turn to a more detailed discussion of the main theorems in this chap-ter, we will recollect some further properties of vertex algebras.

Although vertex algebras are not (typically) associative algebras, they possess an im-portant “associativity” property:


5.0.9.1 Proposition. Let V be a vertex algebra. For any v1, v2, v3 ∈ V, we have the followingequality in V((w))((z− w)):

Y(v1, z)Y(v2, w)v3 = Y(Y(v1, z− w)v2, w)v3.

Finally, there is a powerful “reconstruction” theorem that provides simple criteria touniquely construct a vertex algebra given “generators and relations.” We will exploit thistheorem to verify that we have indeed recovered the standard vertex algebras in our ex-amples.

5.0.9.2 Theorem (Reconstruction, Theorem 4.4.1, [FBZ04]). Let V be a complex vector spaceequipped with a nonzero vector |0〉, an endomorphism T, a countable ordered set aαα∈S of vec-tors, and fields

aα(z) = ∑n∈Z

aα(n)z

−n−1 ∈ End(V)[[z, z−1]]

such that

(1) for all α, aα(z)|0〉 = aα + O(z);(2) T|0〉 = 0 and [T, aα(z)] = ∂zaα(z) for all α;(3) all fields aα(z) are mutually local;(4) V is spanned by the vectors

aα1(j1)· · · aαm

(jm)|0〉

with the ji < 0.

Then, using the formula

Y(aα1(j1)· · · aαm

(jm)|0〉, z) :=

1(−j1 − 1)! · · · (−jm − 1)!

: ∂−j1−1z aα1(z) · · · ∂−jm−1

z aαm(z) :

to define a vertex operation, we obtain a well-defined and unique vertex algebra (V, |0〉, T, Y)satisfying conditions (1)-(4) and Y(aα, z) = aα(z).

Here : a(z)b(w) : denotes the normally ordered product of fields, defined as

: a(z)b(w) := a(z)+b(w) + b(w)a(z)−

where

a(z)+ := ∑n≥0

anzn and a(z)− := ∑n<0

anzn.

Normal ordering eliminates various “divergences” that appear in naively taking productsof fields.

5.1. HOLOMORPHICALLY TRANSLATION-INVARIANT FACTORIZATION ALGEBRAS 99

5.0.10. Organization of this chapter. We will start this chapter by stating and provingour main theorem. The first thing we define, in section 5.1, is the notion of a holomorphi-cally translation invariant factorization algebra on Cn for every n ≥ 1. Holomorphictranslation invariance guarantees that the operator products are all holomorphic. We willthen show to construct from such an object in dimension 1 a vertex algebra.

The rest of this chapter is devoted to analyzing examples. In section 5.3, we discussthe factorization algebra associated to a very simple two-dimensional chiral conformalfield theory: the free βγ system. We will show that the vertex algebra associated to thefactorization algebra of observables of this theory is an object called the βγ vertex algebrain the literature. Then, in section 5.4, we will construct a factorization algebra encodingthe affine Kac-Moody algebra. This factorization algebra again encodes a vertex algebra,which is the standard Kac-Moody vertex algebra.

5.1. Holomorphically translation-invariant factorization algebras

In this section we will analyze in detail the notion of translation-invariant prefactoriza-tion algebras on Cn. On Cn we can ask for a translation-invariant prefactorization algebrato have a holomorphic structure; this implies that all structure maps of the prefactorizationalgebra are (in a sense we will explain shortly) holomorphic. There are many natural fieldtheories where the corresponding prefactorization algebra is holomorphic: for instance,chiral conformal field theories in complex dimension 1, and minimal twists [Cos11c] ofsupersymmetric field theories in complex dimension 2.

5.1.1. We now explain what it means for a (smoothly) translation-invariant prefac-torization algebra F on Cn to be holomorphically translation-invariant. For this definitionto make sense, we require that F is defined over C: that is, the vector spaces F (U) arecomplex vector spaces and all structure maps are complex linear.

Recall that such a factorization algebra has, as part of its structure, an action of thereal Lie algebra R2n = Cn by derivations. This action is as a real Lie algebra; since F isdefined over C, the action extends to an action of the complexified translation Lie algebraR2n ⊗R C. We will denote the action maps by

∂

∂zi,

∂

∂zj: F (U)→ F (U).

5.1.1.1 Definition. A translation-invariant prefactorization algebra F on Cn is holomorphi-cally translation-invariant if it is equipped with derivations ηi : F → F of cohomologicaldegree −1, for i = 1 . . . n, with the property that

dηi =∂

∂zi∈ Der(F ).


Here, d refers to the differential on the dg Lie algebra Der(F ).

We should understand this definition as saying that the vector fields ∂∂zi

act homotopi-cally trivially onF . In physics terminology, the operators ∂

∂xi(where xi are real coordinates

on Cn = R2n) are related to the energy-momentum tensor. We are asking that the com-ponents of the energy momentum tensor in the zi directions are exact for the differentialon observables (in physics, this might be called “BRST exact”). This is rather similar toa phenomenon that sometimes appears in the study of topological field theory [Wit88],where a topological theory depends on a metric, but the variation of the metric is exactfor the BRST differential.

5.1.2. Now we will interpret holomorphically translation-invariant prefactorizationalgebras in the language of R>0-colored operads. When we work in complex geometry, itis better to use polydiscs instead of balls, as is standard in complex analysis.

Thus, if z ∈ Cn, let

PDr(z) = w ∈ Cn | |wi − zi| < r

be the polydisc of radius r around z. Let

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k

be the space of z1, . . . , zk ∈ Cn with the property that the closures of the polydiscs PDri(zi)are disjoint and contained in the polydisc PDs(0).

It is clear that the spaces PDiscsn(r1, . . . , rk | s) form a R>0-colored operad in thecategory of complex manifolds.

Now, let F be a holomorphically translation-invariant prefactorization algebra on Cn.Let Fr denote the differentiable cochain complex F (PDr(0)) associated to the polydisc ofradius r.

Then, as above, for each p ∈ PDiscsn(r1, . . . , rk | s) we have a map

m[p] : Fr1 × · · · × Frk → Fs.

This map is smooth, multilinear, and compatible with the differential. Further, this mapvaries smoothly with p.

The fact that F is a holomorphically translation-invariant prefactorization algebrameans that these maps are equipped with extra structure. We have derivations ηj of Fwhich make the derivations ∂

∂zjhomotopically trivial.


For i = 1, . . . , k and j = 1, . . . , n, let zij, zij refer to coordinates on (Cn)k, and so on theopen subset

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k.Thus, we have operations

∂

∂zijm[p] : Fr1 × · · · × Frk → Fs.

obtained by differentiating the operation m[p], which depends smoothly on p, in the di-rection zij.

Let m[p] i ηj denote the operation

m[p] i ηj : Fr1 × · · · × Frk → Fs

α1 × · · · × αk 7→ ±m[p](α1, . . . , ηjαi, . . . , αk)

(where ± indicates the usual Koszul rule of signs).

The axioms of a (smoothly) translation invariant factorization algebra tell us that

∂

∂zijm[p] = m[p] i

∂

∂zj

where ∂∂zj

is the derivation of the factorization algebra F .

This, together with the fact that [d, ηi] =∂

∂zj, tells us that

∂p∂zij

m[p] =[d, m[p] i ηj

]holds. This tells us that the product map m[p] is holomorphic in p, up to a homotopygiven by ηi.

5.1.3. In the smooth case, we saw that we could describe the structure as that of an al-gebra over a R>0-colored co-operad built from smooth functions on the spaces Discsn(r1, . . . , rk |s). In this section we will see that there is an analogous story in the complex world, wherewe use the Dolbeault complex of the spaces PDiscsn(r1, . . . , rk | s).

Let us first introduce some notation. For any complex manifold X, and any collectionV1, . . . , Vk, W of differentiable cochain complexes over C, let

Ω0,∗(X, Hom(V1, . . . , Vk |Wi))

denote the cochain complex of smooth multilinear maps

V1 × · · · ×Vk → Ω0,∗(X, W).

Recall thatΩ0,∗(X, W) = C∞(X, W)⊗C∞(X) Ω0,∗(X);


as we see in the appendix, a differentiable vector space W has enough structure to de-fine the ∂ operator on Ω0,∗(X, W). The differential on Ω0,∗(X, Hom(V1, . . . , Vk | Wi)) is acombination of the Dolbeault differential on X with the differentials on the differentiablecochain complexes Vi, W.

Since the spaces PDiscsn(r1, . . . , rk | s) form an colored operad in the category of com-plex manifolds, it is automatic that the Dolbeault complexes of these spaces form a coloredcooperad in the category of convenient cochain complexes. This is because the contravari-ant functor sending a complex manifold to its Dolbeault complex, viewed as a convenientvector space, is symmetric monoidal:

Ω0,∗(X×Y) = Ω0,∗(X)⊗Ω0,∗(Y)

where ⊗ denotes the symmetric monoidal structure on the category of convenient vectorspaces.

Explicitly, the colored co-operad structure is given as follows. The operad structureon the complex manifolds PDiscsn(r1, . . . , rk | ti) is given by maps

i : PDiscsn(r1, . . . , rk | ti)× PDiscsn(t1, . . . , tm | s)

→ PDiscsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).

We let

∗i : Ω0,∗(PDiscsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s))

→ Ω0,∗(PDiscsn(r1, . . . , rk | ti)× PDiscsn(t1, . . . , tm | s)

be the corresponding pullback map on Dolbeault complexes.

The factorization algebras we are interested in take values in the category of differen-tiable vector spaces. We want to say that if F is a holomorphically translation invariantfactorization algebra on Cn, then it defines a coalgebra over the colored co-operad givenby the Dolbeault complex of the spaces PDiscsn. A priori, this doesn’t make sense, becausethe category of differentiable vector spaces is not a symmetric monoidal category; only itsfull subcategory of convenient vector spaces has a tensor structure.

However, in order to make this definition, all we need to be able to do is to tensor dif-ferentiable vector spaces with the Dolbeault complexes of complex manifolds. We knowhow to do this: of X is a complex manifold and V a differentiable vector space, we canview Ω0,∗(X, V) as a tensor product of Ω0,∗(X) with V. This tensor product is functorialfor maps Ω0,∗(X) → Ω0,∗(Y) which arise from holomorphic maps Y → X, and for ar-bitrary smooth maps between differentiable vector spaces. Since the co-operad structuremaps in our colored dg cooperad Ω0,∗(PDiscsn) arise from maps of complex manifolds,it makes sense to ask for a coalgebra over this co-operad in the category of differentiablevector spaces.


5.1.3.1 Proposition. Let F be a holomorphically translation-invariant factorization algebra onCn. Then, F defines a coalgebra over the dg co-operad Ω0,∗(PDiscsn). More precisely, the productmaps

m[p] : Fr1 × · · · × Frk → Fs

for p ∈ PDiscsn(r1, . . . , rk | s) lift to smooth multilinear maps

µ∂(r1, . . . , rk | s) ∈ Fr1 × . . .Frk → Ω0,∗(PDiscsn(r1, . . . , rk | s),Fs))

which are compatible with differentials, and which satisfy the usual properties needed to define acoalgebra over a cooperad.

Explicitly, these properties are follows.

(1) Let dF denote the differential on the cochain complexes Fr. Let fi denote ele-ments of Fri . Then,

∑±µ∂(r1, . . . , rk | s)( f1, . . . , dF fi, . . . , fk) = (dF + ∂)µ∂(r1, . . . , rk | s)( f1, . . . , fk).

(2) Let σ ∈ Sk. Then, as in the smooth case,

µ∂(rσ(1), . . . , rσ(k) | s)( fσ(1), . . . , fσ(k)) = (σ−1)∗µ∂(r1, . . . , rk | s)( f1, . . . , fk)

where σ−1 is the isomorphism

σ−1 : PDiscs(r1, . . . , rk | s)→ PDiscs(rσ(1), . . . , rσ(k) | s).

(3) The usual associativity rule relating composition in the cooperad Ω0,∗(PDiscsn)

and of the multilinear maps µ∂ holds. For example,

∗2 µ∂(t1, r1, r2 | s)( f , g1, g2) = µ∂(t1, t2 | s)(

f , µ∂(r1, r2 | t2)(g1, g2))

∈ Ω0,∗ (PDiscsn(t1, t2 | s)× PDiscsn(r1, r2 | t2)) .

The interested reader should consult, for example, [Get94] and [Cos07] for similar ax-iom systems in the context of topological field theory, and [Seg04] for a related systemof axioms for chiral conformal field theories. This construction is also closely related tothe construction of “descendents” that appears in the study of topological field theory inthe physics literature (see for example [Wit88, Wit91, ?]). Suppose one has a field theorywhich depends on a metric, but for where the variation with respect to the metric is exactfor what the physicists call the BRST operator (which corresponds to the differential onobservables in our language). The metric-dependent functional which makes the varia-tion of the original action functional BRST exact can be viewed as a one-form on somemoduli space of metrics. Higher homotopies yield forms on the moduli space of metrics,or, in two dimensions, on the moduli of conformal classes of metrics; i.e. the moduli ofRiemann surfaces.


In our approach, because we are working with holomorphic rather than topologicaltheories, we find elements of the Dolbealt complex of appropriate moduli spaces of com-plex manifolds, which in our simple case are the spaces PDiscsn.

PROOF OF THE PROPOSITION. Giving a smooth multilinear map

Fr1 × · · · × Frk → Ω0,∗(PDiscsn(r1, . . . , rk | s),Fs)

which is compatible with differentials is equivalent to giving an element of

Ω0,∗ (PDiscsn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | Fs))

which is closed with respect to the differential ∂ + dF , where dF refers to the naturaldifferential on the differentiable cochain complex Hom(Fr1 , . . . ,Frk | Fs) of smooth mul-tilinear maps

Fr1 × · · · × Frk → Fs.

We will produce the desired element

µ∂(r1, . . . , rk | s) ∈ Ω0,∗ (PDiscsn(r1, . . . , rk | s), Him(Fr1 , . . . ,Frk | Fs)

starting from the operations

µ0(r1, . . . , rk | s) ∈ C∞(PDiscs(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s))

that we already have because F is a smoothly translation-invariant factorization algebra.

First, we need to introduce some notation. Recall that

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k

is an open (possibly empty) subset. Thus,

Ω0,∗ (PDiscsn(r1, . . . , rk | s)) = Ω0,0 (PDiscsn(r1, . . . , rk | s))⊗C[dzij]

where the dzij are commuting variables of cohomological degree 1, with i = 1, . . . , k andj = 1, . . . , n. We let ∂

∂(dzij)denote the graded derivation which removes dzij.

As before, let ηj : Fr → Fr denote the derivation which cobounds the derivation ∂∂zj

.We can compose any element

(†) α ∈ Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s))

with ηj acting on Fri , to get

α i ηj ∈ Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s)).


We use the shorthand notations:

lij(α) = α i ηj

rij(α) = α i∂

∂zj

where ∂∂zj

refers to the derivation acting on Frj .

Note that[dF , lij] = rij,

if dF is the differential on the graded vector space (†) above arising from the differentialson the spaces Fri , Fs.

Then, the cochains µ∂(r1, . . . , rk | s) are defined by

µ∂(r1, . . . , rk | s) = exp(−∑ dzijlij)µ0(r1, . . . , rk | s).

Here dzijlij denotes the composition of wedging with dzij with the operator lij. Note thatthese two operators graded commute.

In what follows, we will write simply µ∂ instead of µ∂(r1, . . . , rk | s), and similarly forµ0.

Next, we need to verify that (∂+dF )µ∂ = 0, where ∂+dF is the differential on gradedvector space (†) above which arises by combining the ∂ operator with the differentialarising from the complexes Fri , Fs.

We need the following identities:

[dF , ∑ dzijlij] = ∑ dzijrij

rijµ0 =

∂

∂zijµ0

dFµ0 = 0.

The second and third identities are part of the axioms for a smoothly translation-invariantfactorization algebra.

These identities allows us to calculate that

(∂ + dF )µ∂ = (∂ + dF ) exp(−∑ dzijlij)µ0

= −(exp(−∑ dzijlij)

) (∑ dzijrij

)µ0 +

(exp(−∑ dzijlij)

)(∂ + dF )µ0

=(exp(−∑ dzijlij)

) (−∑ dzij

∂

∂zij+ ∂

)µ0

= 0.


Thus shows that µ∂ is closed.

It is straightforward to verify that the elements µ∂ are compatible with compositionand with the symmetric group actions.

5.2. A general method for constructing vertex algebras

In this section we will prove that the cohomology of a holomorphically translation in-variant prefactorization algebra on C with a compatible circle action gives rise to a vertexalgebra. Together with the main result of this book, which allows one to construct fac-torization algebras by obstruction theory starting from the Lagrangian of a classical fieldtheory, this gives a general method to construct vertex algebras.

5.2.1. Equivariant factorization algebras. In section 4.7 we gave a definition of smoothlytranslation invariant prefactorization algebra on Rn. In the course of constructing the ver-tex algebra associated to a factorization algebra, we will need to discuss factorizationalgebras invariant under the action of a more general Lie group.

5.2.1.1 Definition. Let M be a manifold with an action of a group G. Let F be a factorizationalgebra on M. We say F is G-equivariant, if for all g ∈ G and all open subset U ⊂ M we aregiven isomorphisms

σg : F (U) ∼= F (g(U))

such that

(1) σg σh = σgh : F (U)→ F (gh(U)).(2) Tg respects the factorization product. If U1, . . . , Uk are disjoint opens contained in V,

then the diagram

F (U1)⊗ · · · ⊗ F (Un)

// F (g(U1))⊗ · · · ⊗ F (g(Un))

F (V) // F (g(V))

commutes.

We will now define what it meanas for a factorization algebra F to be smoothly equi-variant under the action of a Lie group G on a manifold M. We need to introduce somenotation before making this definition. Let U1, . . . , Uk be disjoint subsets of V. Let W ⊂ Gk

be the open subset consisting of those g1, . . . , gk where g1(U1), . . . , gk(Uk) continue to bedisjoint and contained in W. If (g1, . . . , gk) ∈W, we have a map

mg1,...,gk : F (U1)⊗ · · · ⊗ F (Uk)→ F (V)

5.2. A GENERAL METHOD FOR CONSTRUCTING VERTEX ALGEBRAS 107

defined by the composition

⊗F(Ui)⊗σgi−−→ ⊗F (gi(Ui))→ F (V)

where the second map is the prefactorization multiplication.

5.2.1.2 Definition. F is a smoothly G-equivariant prefactorization algebra if the followingconditions hold.

(1) mg1,...,gk depends smoothly on (g1, . . . , gk) ∈W ⊂ Gk.(2) There’s an action of the Lie algebra g of G on F by derivations.(3) This action is compatible with the action of G, in the following sense. For all X ∈ g, all k

and all i with 1 ≤ i ≤ k, we require that

∂

∂Ximg1,...,gk(α1, . . . , αk) = mg1,...,gk(α1, . . . , X(αi), . . . , αk).

On the left hand side ∂∂Xi

indicates the action of the left-invariant vector field associatedto X on the ith factor of Gk.

The case of interest is the action of the group S1 n C of isometries of C on C.

5.2.1.3 Definition. A holomorphically translation-invariant prefactorization algebra F onC with a compatible S1 action is a smoothly S1 ×R2-invariant factorization algebra F , definedover the base field of complex numbers, together with an extension of the action of the complex Liealgebra

LieC(S1 n R2) = C ∂θ , ∂z, ∂z(where ∂θ is a basis of LieC(S1)) to an action of the dg Lie algebra

C ∂θ , ∂z, ∂z ⊕Cη

where η is of cohomological degree −1, and the differential is

dη = ∂z.

In this dg Lie algebra, all commutators involving η vanish except for

[∂θ , η] = −η.

Note that, in particular, F is a holomorphically translation invariant factorization al-gebra on C.

5.2.2. Let us now turn to the theorem relating S1-equivariant holomorphically-translationinvariant factorization algebras with vertex algebras. Before stating the theorem, we needa few technical remarks.


Every differentiable vector space is a sheaf on the site of smooth manifolds. If E isa differentiable vector space, we denote the sections of this sheaf on a manifold M byC∞(M, E): we think of this as the space of smooth maps from M to E. If E is a complexof differentiable vector spaces, then sending an open subset U of M to C∞(U, E) definesa sheaf of cochain complexes on M. The cohomology of a sheaf of cochain complexes isa presheaf of graded vector spaces, which we can sheafify to get a sheaf of graded vectorspaces. We let C∞(U, H∗(E)) denote the sections on U of these cohomology sheaves.

We are allowed to differentiate smooth maps to a differentiable vector spaces. Thus, ifM is a complex manifold and E is a cochain complex of differentiable vector spaces overC, we have, for every open U ⊂ M, a cochain map

∂ : C∞(U, E)→ Ω0,1(U, E).

These maps form a map of sheaves of cochain complexes on M. Taking cohomology, wefind a map of sheaves of graded vector spaces

∂ : C∞(U, H∗(E))→ Ω0,1(U, H∗(E)).

The kernel of this map defines a sheaf on M, whose sections we denote by Hol(U, H∗(E)).

The theorem on vertex algebras will require an extra hypothesis regarding the S1 ac-tion on the factorization algebra. Note that for any compact Lie group G, the space D(G)of distributions on G is an algebra under convolution. Since spaces of distributions arenaturally differentiable vector spaces, and the product map D(G) × D(G) → D(G) issmooth, this forms an algebra in the category of differentiable vector spaces. There is amap

G → D(G)

sending an element g to the δ-distribution at g. This is a smooth map, and is also a homo-morphism of monoids.

Given a complex E of differentiable vector spaces with a G action, we can ask that itextends to a smooth action of the algebra D(G). That is, we can ask for a smooth bilinearmap, compatible with differentials,

D(G)× E→ E

which extends the smooth map G × E → E, and which defines an action of the algebraD(G).

Let us now specialize to the case when G = S1, which is the case which is relevant forthe theorem on vertex algebras. The irreducible representation ρk of S1 where λ ∈ S1 ⊂C× acts by λk lifts to a representation ρk of the algebra D(S1), defined by

ρk(φ) =⟨

φ, λk⟩

where 〈−,−〉 indicates the pairing between distributions and functions.


Let Ek ⊂ E denote the subspace on which D(S1) acts by ρk. We call this the weight keigenspace for the S1-action on E.

In the algebraD(S1), the element λk (viewed as a distribution on S1) is an idempotent.If we denote the action of D(S1) on E by ∗, then the map

λ−k∗ : E→ E

defines a projection from E onto Ek. We will denote this projection by πk.

(Of course, all this holds for a general compact Lie group where instead of the λk weuse the characters of irreducible representations).

Now we can state the main theorem of this section.

5.2.2.1 Theorem. Let F be a unital S1-equivariant holomorphically translation invariant prefac-torization algebra on C valued in differentiable vector spaces. Assume that, for each disc D(0, r)around the origin, the action of S1 on F (D(0, r) extends to a smooth action of the algebra D(S1)of distributions on S1.

Let Fk(D(0, r)) be the kth eigenspace of the S1 action on F (D(0, r)). Let us make the follow-ing additional assumptions.

(1) Assume that, for r < r′, the map

Fk(D(0, r))→ Fk(D(0, r′))

is a quasi-isomorphism.(2) The vector space H∗(Fk(D(0, r)) is zero for k 0.(3) For each k and r, we require that H∗(Fk(D(0, r)) is isomorphic, as a sheaf on the site of

smooth manifolds, to a countable sequential colimit of finite-dimensional graded vectorspaces.

Let Vk = H∗(Fk(D(0, r)), and let V = ⊕Vk. This space is independent of r by assumption.

Then, V has the structure of a vertex algebra.

Remark: (1) If V is not concentrated in cohomological degree 0, then it will havethe structure of a vertex algebra valued in the symmetric monoidal category ofgraded vector spaces, that is, the Koszul rule of signs will apear in the axioms.

(2) We will often deal with factorization algebras F equipped with a complete de-creasing filtration FiF , so that F = limF/FiF . In this situation, to constructthe vertex algebra we need that the properties listed in the theorem hold on eachgraded piece Grl F . This implies, by a spectral sequence, that they hold on eachF/FiF , allowing us to construct an inverse system of vertex algebras associatedto the prefactorization algebra F/FiF . The inverse limit of this system of vertexalgebras is the factorization algebra associated to F .


♦

The conditions of the theorem are always satisfied in practise by factorization algebrasarising from quantizing a holomorphically translation invariant classical field theory.

Because our factorization algebraF is translation invariant, the cochain complexF (D(z, r))associated to a disc of radius r is independent of z. We can therefore use the notation F (r)for F (D(z, r)). We will use this notation when r = ∞ to denote F (C).

Let Fk(r) denote the kth weight space of the S1 action on F . The S1 action on eachF (r) extends to an action of D(S1). If λ denotes an element of S1 ⊂ C×, and ∗ denotesthis action, then λ−k∗ gives a projection map

F (r)→ Fk(r).

At the level of cohomology, it gives a map

πk : H∗(F (r))→ H∗(Fk(r)) = Vk

of differentiable vector spaces, splitting the natural inclusion.

Further, the map H∗(F (r))→ H∗(F (r′)) associated to the inclusion D(0, r) → D(0, r′)is the identity on V.

By assumption, the natural differentiable vector space structure on Vk = H∗(Fk(r))has the property that a smooth map from a manifold M to Vk is a map which locallyon M is given by a smooth (in the ordinary sense) map to a finite dimensional subspaceof Vk. Thus, Vk is the colimit in the category of differentiable vector spaces of all of itsfinite-dimensional subspaces.

LetV = ∏ Vk

where the product is taken in the category of differentiable vector spaces.

There is a natural mapH∗(F (r))→ V

given by the product of all the projection maps.

The strategy of the proof is as follows. We will analyze the structure on V given to usby the axioms of a translation-invariant factorization algebra. The factorization productwill become the operator product expansion or state-field map. The locality axiom of avertex algebra will follow from the associativity axioms of the factorization algebra.

Let us start by analyzing the structure on V given by the factorization (or operator)product. Let r1, . . . , rk, s ∈ R>0. Recall that we have a complex manifold

Discs(r1, . . . , rk | s) = z1, . . . , zk ∈ C | D(z1, r1)q · · · q D(zk, rk) ⊂ D(0, s) ⊂ Ck.


Note that C acts on Discs(r1, . . . , rk | ∞) by translation.

As we have seen, the factorization (or operator) product is a multilinear map, compat-ible with differentials,

µ∂(r1, . . . , rk | ∞) : F (r1)× · · · × F (rk)→ Ω0,∗(Discs(r1, . . . , rk | ∞),F (∞)).

Note that for any complex manifold X there is a map

H∗(Ω0,∗(X,F (∞))

)→ Hol(X, H∗(F (∞))).

If α ∈ Ω0,∗(X,F (∞)) is a closed element, this map is defined by first extracting the com-ponent α0 of α which is in Ω0,0(X,F (∞)). The fact that α is closed means that α0 is closedfor the differential DF on F (∞) and that ∂α0 is exact. Thus, the cohomology class [α0] is asmooth map from X to F (∞), and ∂[α0] = 0 so that [α0] is holomorphic.

This means that the operator product map, at the level of cohomology, gives a smoothmultilinear map

mz1,...,zk : H∗(F (r1))× · · · × H∗(F (rk))→ Hol(Discs(r1, . . . , rk | ∞), H∗(F (∞))).

Here zi indicate the positions of the centers of the discs in Discs(r1, . . . , rk | ∞).

If r′i < ri, thenDiscs(r1, . . . , rk | ∞) ⊂ Discs(r′1, . . . , r′k | ∞).

There is also a natural map F (r′) → F (r), given by including D(0, r′) → D(0, r). Thefollowing diagram commutes:

H∗(F (r′1))⊗ · · · ⊗ H∗(F (r′k)) //

Hol(Discs(r′1, . . . , r′k | ∞), H∗(F (∞)))

H∗(F (r1))⊗ · · · ⊗ H∗(F (rk)) // Hol(Discs(r1, . . . , rk | ∞), H∗(F (∞))).

Note that V maps to limr→0 H∗(F (r)). Therefore, the operator product, when restrictedto V, gives a map

mH∗(F (∞))z1,...,zk : V⊗k → lim

r→0Hol(Discs(r, . . . , r | ∞), H∗(F (∞))) = Hol(Confk(C), H∗(F (∞)))

where Confk(C) is the configuration space of k ordered distinct points in C.

If all the zi lie in a disc D(0, r), this lifts to a map

mH∗(F (r))z1,...,zk : V⊗k → Hol(Confk(D(0, r)), H∗(F (r))).

Composing the factorization product map mH∗(F (∞))z1,...,zk with the map

∏ πk : H∗(F (∞))→ V = ∏ Vk


gives a map

mz1,...,zk : V⊗k → Hol(Confk(C), V) = ∏l

Hol(Confk(C), Vl).

This map does not involve the spaces F (r) anymore, only the space V and its naturalcompletion V. If there is potential confusion, we will refer to this version of the operatorproduct map by mV

z1,...,zkinstead of just mz1,...,zk .

The operator product will, of course, be constructed from the maps mz1,...,zk . We canconsider the map

mz,0 : V ⊗V →∏l

Hol(C×, Vl)

where we have restricted the map mVz,w to the locus where w = 0. Since each space Vl is

a discrete vector space, that is, a colimit of finite dimensional vector spaces, we can formthe ordinary Laurent expansion of an element in Hol(C×, Vl) to get a map

LzmVz,0 : V ⊗V → V[[z, z−1]].

5.2.2.2 Lemma. The image of LzmVz,0 is in the subspace V((z)).

PROOF. The map mVz,0 is S1-equivariant, where S1 acts on V and C× in the evident

way. Therefore so is LzmVz,0. Since every element in V ⊗ V is in a finite sum of the S1-

eigenspaces, the image of LzmVz,0 is in the subspace of V[[z, z−1]] spanned by finite sums

of eigenvectors. An element of V[[z, z−1]] is in the kth eigenspace of the S1 action if it is ofthe form

∑ zk−lvl

where vl ∈ Vl . Since Vl = 0 for l 0, every such element is in V((z)).

Let us now define the structures on V which will correspond to the vertex algebrastructure.

(1) The vacuum element 1 ∈ V: By assumption, F is a unital prefactorization alge-bra. Therefore, the commutative algebra F (∅) has a unit element 1. The prefac-torization structure mapF (∅)→ F (D(0, r)) for any r gives an element 1 ∈ F (r).This element is automatically S1-invariant, and therefore in V0.

(2) The translation map T : V → V: The structure of holomorphically translationfactorization algebra onF includes a derivation ∂

∂z corresponding to infinitesimaltranslation in the (complex) direction z. The fact thatF has a compatible S1 actionmeans that, for all r, the map ∂

∂z maps

Fk(r)→ Fk−1(r).


Therefore, passing to cohomology, it becomes a map ∂∂z : Vk → Vk−1. We let T be

the map V → V which on Vk is ∂∂z .

(3) The state-field map Y : V → End(V)[[z, z−1]] : We let

Y(v, z)(v′) = Lzmz,0(v, v′) ∈ V((z)).

Note that Y(v, z) is a field in the sense used in the axioms of a vertex algebra,because Y(v, z)(v′) has only finitely many negative powers of z.

It remains to check the axioms of a vertex algebra. We need to verify the

(1) Vacuum axiom: Y(1, z)(v) = v.(2) Translation axiom:

Y(Tv1, z)(v2) =∂

∂zY(v1, z)(v2).

and T1 = 0.(3) Locality axiom:

(z1 − z2)N [Y(v1, z1), Y(v2, z2)] = 0

for N sufficiently large.

The vacuum axiom follows from the fact that the unit 1 ∈ F (∅), viewed as an element ofF (D(0, r)), is a unit for the factorization product.

The translation axiom follows immediately from the corresponding axiom of factor-ization algebras, which is built into our definition of holomorphic translation invariance:namely,

∂

∂zmz,0(v1, v2) = mz,0(∂zv1, v2).

The fact that T1 = 0 follows from the fact that the derivation ∂z ofF gives a derivationof the commutative algebra F (∅), which must therefore send the unit to zero.

It remains to prove the locality axiom. This will follow from the associativity propertyof factorization algebras. The rest of this section will be devoted to the proof.

5.2.3. Proof of the locality axiom. Let us write down some useful properties of theoperator product maps mz1,...,zk . Firstly, the map mz1,...,zk is S1-covariant, where we use thediagonal S1 action on V⊗k, and the S1 action on the right hand side comes from the naturalS1 action on Confk(C) by rotation coupled to the S1 action on V coming from the structureof S1-invariant factorization algebra. It is also Sk-covariant, where Sk acts on V⊗k and on


Confk(C) in the evident way. It is also invariant under translation, in the sense that forarbitrary vi ∈ V,

mz1+λ,...,zk+λ(v1, . . . , vk) = ρλ (mz1,...,zk(v1, . . . , vk)) ∈ V

where ρλ denotes the action of C on V which integrates the translation action of the Liealgebra C.

Let us writeLzmz,0(v1, v2) = ∑

kzkmk(v1, v2)

where mk(v1, v2) ∈ V. As we have seen, mk(v1, v2) is zero for k 0.

The key proposition which will allow us to prove the locality axiom is the following.

5.2.3.1 Proposition. Let Uij ⊂ Confk(C) be the open subset where∣∣zj − zi∣∣ < ∣∣zj − zl

∣∣for l 6= i, j.

Then, for (z1, . . . , zk) ∈ Uij, we have the following identity:

mz1,...,zk(v1, . . . , vk) = ∑(zi − zj)nmz1,...,zj,...,zk

(v1, . . . , vi−1, mn(vi, vj), . . . , vj, . . . , vk)

∈ Hol(Uij, V).

The sum on the right hand side converges.

In this expression, zj and vj indicate that we skip these entries.

When n = 3, this identity allows us to expand mz1,z2,z3 in two different ways. We findthat

mz1,z2,z3(v1, v2, v3) =

∑(z2 − z3)kmz1,z3(m

k(v1, v2), v3) if |z2 − z3| < |z1 − z3|∑(z1 − z3)kmz2,z3(v2, mk(v1, v3)) if |z2 − z3| > |z1 − z3| .

This should be compared to the associativity axiom in the theory of vertex algebras.

PROOF. By symmetry, we can reduce to the case when i = 1 and j = 2.

If all of the zi lie in a particular disc D(0, r) ⊂ C, then the map mz1,...,zk lifts to a map

mH∗(F (r))z1,...,zk : Hol(Confk(D(0, r)), H∗(F (r))).

Further, if we take some ε > 0 and consider the open subset

U ⊂ Confk(C)


where the zj for j 6= 1 are disjoint from the disc of radius ε around z1, then the map mz1,...,zk

extends to a mapH∗(F (ε))⊗V · · · ⊗V → Hol(U, V).

The axioms of a prefactorization algebra tell us that the following associativity condi-tion holds:

(†) mz1,z2,...,zk(v1, . . . , vk) = mz2,z3,...,zk(mz1,z2(v1, v2), v3, . . . , vk)

in the following sense. First, we must choose ε > 0 and restrict the zi to lie in theopen set where the discs the discs D(zi, ε) are all disjoint. Further, we ask that D(z2, δ)(where δ > ε) contains D(z1, ε) and is disjoint from each D(zi, ε) for i > 2. (Here ε andδ can be chosen arbitrarily small). Then, we are viewing mz1,z2(v1, v2) as an element ofH∗(F (D(z2, δ))), depending holomorphically on z1, z2 which lie in the open set whereD(z1, ε) is disjoint from D(z2, ε) and contained in D(z2, δ). By translating z2 to 0, we iden-tify H∗(F (D(z2, δ))) with H∗(F (δ)). This associativity property is an immediate conse-quence of the axioms of a prefactorization algebra.

By taking ε → 0, we see that this identity holds when the zi are restricted to lie in theset Uδ

12 where |z2 − z1| < δ and |z2 − zi| > δ for i > 2. Note that the union of the sets Uδ12

as δ varies is U12.

We can, without loss of generality, assume that each vi is homogeneous of weight |vi|under the S1 action on V.

Let πk : H∗(F (δ)) → Vk be the projection onto the eigenspace Vk for the S1 action.Recall that D(S1) acts on F (δ) and so on H∗(F (δ)). If this action is denoted by ∗, and ifλ ∈ S1 ⊂ C× denotes an element of the circle viewed as a complex number, then

πk( f ) = λ−k ∗ f .

Note that S1-equivariance of the operator product map means that we can write the Lau-rent expansion of mz,0(v1, v2) as

mz,0(v1, v2) = ∑ zkmk(v1, v2) = ∑ π|v1|+|v2|−kmz,0(v1, v2).

That is, zkmk(v1, v2) is the projection of mz,0(v1, v2) onto the |v1 + v2| − k eigenspace of V.

We want to show that

∑(z1 − z2)nmz2,z3,...,zk(m

n(v1, v2), . . . , vk) = mz1,...,zk(v1, . . . , vk)

where the zi lie in Uδij. This is equivalent to showing that

∑n

mz2,z3,...,zk(πnmz1,z2(v1, v2), . . . , vk) = mz1,...,zk(v1, . . . , vk).

Indeed, the Laurent expansion of mz1,z2(v1, v2) and the expansion in terms of eigenspacesof the S1 action on V differ only by a reordering of the sum.


Fix v1, . . . , vn ∈ V. Define a map

Φ : D(S1)→ Hol(Uδij, V)

Φ(α) = mz2,z3,...,zk(α ∗mz1,z2(v1, v2), v3, . . . , vn).

Here α∗ refers to the action of D(S1) on H∗(F (δ)).

Note that Φ is a smooth map. Note also that the associativity identity (†) of factoriza-tion algebras implies that

Φ(δ1) = mz1,...,zk(v1, . . . , vk).The point is that δ1∗ is the identity on H∗(F (δ)).

To prove the proposition, it now suffices to prove that

Φ

(∑

n∈Z

λn

)= Φ(δ1).

In the space D(S1) with its natural topology the sum ∑n∈Z λn converges to δ1 (this issimply the Fourier expansion of the delta-function). So, to prove the proposition, it suf-fices to prove that Φ is continuous, where the spaces D(S1) and Hol(Uδ

ij, V) are endowedwith their natural topologies. (In the topology on Hol(Uδ

ij, V), a sequence converges if itsprojection to each Vk converges uniformly, with all derivatives, on compact sets of Uδ

ij)).

We know that Φ is smooth. The spaces D(S1) and Hol(Uδij, V) both lie in the essential

image of the functor from locally-convex topological vector spaces to differentiable vectorspaces. A result of [KM97] tells us that smooth linear maps between topological vectorspaces are bounded. Therefore, the map Φ is a bounded linear map of topological vectorspaces.

In lemma C.2.0.8 in Appendix C we show (using results of [KM97]) that the space ofcompactly supported distributions on any manifold has the bornological property, mean-ing that a bounded linear map from it to any topological vector space is the same as acontinuous linear map. It follows that Φ is continuous, thus completing the proof.

As a corollary, we find the following.

5.2.3.2 Corollary. For v1, . . . , vk ∈ V, mz1,...,zk(v1, . . . , vk) has finite order poles on every diago-nal in Confk(C). That is, for some N sufficiently large,

∏i,j(zi − zj)

Nmz1,...,zk(v1, . . . , vk)

extends to an element of Hol(Ck, F).

PROOF. This is an immediate corollary of the previous proposition.


Now we are ready to prove the locality axiom.

5.2.3.3 Proposition. The locality axiom holds:

(z1 − z2)N [Y(v1, z1), Y(v2, z2)] = 0

for N 0.

PROOF. For any holomorphic function F(z1, . . . , zk) of variables z1, . . . , zk ∈ C×, welet Lzi F denote the Laurent expansion of F in the variable zi. This is an expansion thatconverges when |zi| <

∣∣zj∣∣ for all j. It can be defined by fixing the values of zj with j 6= i,

then viewing F as a function of zi on the punctured disc where 0 < |zi| <∣∣zj∣∣, and taking

the usual Laurent expansion. We can define iterated Laurent expansions. For example, ifF is a function of z1, z2 ∈ C×, we can define

Lz2Lz1 ∈ C[[z±11 , z±1

2 ]]

by first taking the Laurent expansion with respect to z1, yielding a series in z1 whose coef-ficients are holomorphic functions of z2 ∈ C×; and then applying the Laurent expansionwith respect to z2 to each of the coefficient functions of the expansion with respect to z1.

Recall that we define

Y(v1, z)(v2) = Lzmz,0(v1, v2) ∈ V((z)).

We define mk(v1, v2) so that

Lzmz,0(v1, v2) = ∑ zkmk(v1, v2).

Note that, by definition,

Y(v1, z1)Y(v2, z2)(v3) = Lz1 mz1,0 (v1,Lz2 mz2,0(v2, v3))

= ∑Lz1 ∑ zn2 mz1,0 (v1, mn(v2, v3)) ∈ V[[z±1

1 , v±12 ]].

Proposition 5.2.3.3 tells us that

zn2 mz1,0

(v1, ∑ mn(v2, v3)

)= mz1,z2,0(v1, v2, v3)

as long as |z2| < |z1|. Thus,

Y(v1, z1)Y(v2, z2)(v3) = Lz1Lz2 mz1,z2,0(v1, v2, v3).

Similarly,

Y(v2, z2)Y(v1, z1)(v3) = Lz2Lz1 mz2,z2,0(v2, v1, v3)

= Lz2Lz1 mz1,z2,0(v1, v2, v3).

Therefore,

[Y(v1, z1), Y(v2, z2)](v3) = (Lz1Lz2 −Lz2Lz1)mz1,z2,0(v1, v2, v3).


Since Lz1 and Lz2 are maps of C[z1, z2] modules, we have

(z1 − z2)N (Lz1Lz2 −Lz2Lz1)mz1,z2,0(v1, v2, v3)

= (Lz1Lz2 −Lz2Lz1) (z1 − z2)Nmz1,z2,0(v1, v2, v3)

Finally, we know that for N sufficiently large, (z1 − z2)NzN1 zN

2 mz1,z2,0 has no poles, and soextends to a function on C2. It follows from the fact that partial derivatives commute that

(Lz1Lz2 −Lz2Lz1) (z1 − z2)NzN

1 zN2 mz1,z2,0(v1, v2, v3) = 0.

Since Laurent expansion is a map of C[z1, z2]-modules, and since z1, z2 act invertibly onC[[z±1

1 , z±12 ]], the result follows.

Remark: In the vertex algebra literature, the idea that Y(v1, z1)Y(v2, z2)(v3) and Y(v2, z2)Y(v1, z1)(v3)arise as expansions of a holomorphic function of z1, z2 ∈ C× in the regions when |z1| <|z2| and |z2| < |z1| is often cited as a heuristic justification of the locality axiom for a vertexalgebra. Our approach makes this idea rigorous.

5.3. The βγ system and vertex algebras

This section focuses on one of the simplest holomorphic field theories, the free βγsystem. Our goal is to study it just as we studied the free particle in section ??. Followingthe methods developed there, we will construct the factorization algebra for this theory,show that it is holomorphically translation-invariant, and finally show that the associatedvertex algebra is what is known in the vertex algebra literature as the βγ system. Alongthe way, we will compute the simplest operator product expansions for the theory usingpurely homological methods.

5.3.1. The βγ system. Let M = C and let E =(

Ω0,∗M ⊕Ω1,∗

M , ∂)

be the Dolbeault com-plex resolving holomorphic functions and holomorphic 1-forms as a sheaf on M. Follow-ing the convention of physicists, we denote by γ an element of Ω0,∗ and by β an elementof Ω1,∗. The pairing 〈−,−〉 is

〈−,−〉 : Ec ⊗ Ec → C,

(γ0 + β0)⊗ (γ1 + β1) 7→∫

Cγ0 ∧ β1 + β0 ∧ γ1.

Thus we have the data of a free BV theory. The action functional for the theory is

S(γ, β) = 〈γ + β, ∂(γ + β)〉 = 2∫

Mβ ∧ ∂γ

The Euler-Lagrange equation is simply ∂γ = 0 = ∂β. One should think of E as the“derived space of holomorphic functions and 1-forms on M.” Note that this theory is

5.3. THE βγ SYSTEM AND VERTEX ALGEBRAS 119

well-defined on any Riemann surface, and one can study how it varies over the modulispace of curves.

Remark: One can add d copies of E (equivalently, tensor E with Cd) and let Sd be the d-fold sum of the action S on each copy. The Euler-Lagrange equations for Sd picks out“holomorphic maps γ from M to Cd and holomorphic sections β of Ω1

M(γ∗TCd).”

5.3.2. The quantum observables of the βγ system. To construct the quantum observ-ables, following section ??, we start by defining a certain graded Heisenberg Lie algebraand then take its Chevalley-Eilenberg complex for Lie algebra homology.

For each open U ⊂ C, we set

H(U) = Ω0,∗c (U)⊕Ω1,∗

c (U)⊕ (Ch)1,

where Ch is situated in cohomological degree 1. The Lie bracket is simply

[µ, ν] = h∫

Uµ ∧ ν,

so H is a central extension of the abelian dg Lie algebra given by all the Dolbeault forms(with ∂ as differential).

The factorization algebra Obsq of quantum observables assigns to each open U ⊂ C,the cochain complex C∗(H(U)), which we will write as

Obsq(U) :=(

Sym(

Ω1,∗c (U)[1]⊕Ω0,∗

c (U)[1])[h], ∂ + h∆

).

The differential has a component ∂ arising from the underlying cochain complex ofH anda component arising from the Lie bracket, which we’ll denote ∆. It is the BV Laplacian forthis theory.

Below we unpack what information Obsq encodes by examining some simple opensets and the cohomology H∗Obsq on those open sets. As usual, the meaning of a complexis easiest to garner through its cohomology. First, though, we discuss how this example isholomorphically translation invariant (as defined in section 5.1).

Everything in this construction is manifestly translation-invariant, so it remains toverify that the action of ∂

∂z is homotopically trivial. Consider the operator

η =d

d(dz),

which acts on the space of fields E . The operator η maps Ω1,1 to Ω1,0 and Ω0,1 to Ω0,0.Then we see that

[∂ + h∆, η] =ddz

,

so that the action of d/dz is homotopically trivial, as desired.


We would like to apply the result of theorem 5.2.2.1, which shows that a holomorphi-cally translation invariant factorization algebra with certain additional conditions givesrise to a vertex algebra. We need to check the conditions for the example of interest. Theconditions are the following.

(1) The factorization algebra must have an action of S1 covering the action on C byrotation.

(2) For every disc D(0, r) ⊂ C (including r = ∞), the S1 action on Obsq(D(0, r))must extend to an action of the algebra D(S1) of distributions on S1.

(3) If Obsqk(D(0, r)) denotes the kth eigenspace of the S1 action, then we require that

the mapH∗(Obsq

k(D(0, r)))→ H∗(Obsqk(D(0, s)))

is an isomorphism for r < s.(4) Finally, we require that the space H∗(Obsq

k(D(0, r))) is a discrete vector space,that is, a colimit of finite dimensional vector spaces.

The first condition is obvious in our example: the S1 action arises from the natural ac-tion of S1 on Ω0,∗

c (C) and Ω1,∗c (C). The second condition is also easy to check: if f ∈

C∞c (D(0, r)) then the expression

z 7→∫

λ∈S1φ(λ) f (λz)

makes sense for any distribution φ ∈ D(S1), and defines a continuous and hence smoothmap

D(S1)× C∞c (D(0, r))→ C∞

c (D(0, r)).To check the remaining two conditions, we need to analyze H∗(Obsq(D(0, r))) more ex-plicitly.

5.3.3. Analytic preliminaries. In Chapter ??, we showed that for any free field theoryfor which there exists a Green’s function for the differential defining the elliptic complexof fields, then there is an isomorphism of differentiable cochain complexes

Obscl(U)[h] ∼= Obsq(U)

for any open subset U in space time. In our example, we want to understand H∗(Obsq(D(0, r)))as a differentiable cochain complex, and in particular, understand its decomposition intoeigenspaces for the action of S1. This remark shows that it suffices to understand the co-homology of the corresponding complex of classical observables. We will do this in thissubsection.

We remind the reader of some facts from the theory of several complex variables (ref-erences for this material are [GR65], [For91], and [Ser53]). We then use these facts todescribe the cohomology of the observables.


5.3.3.1 Proposition. Every open set U ⊂ C is Stein [For91]. As the product of Stein manifoldsis Stein, every product Un ⊂ Cn is Stein.

Remark: Behnke and Stein [BS49] proved that every noncompact Riemann surface is Stein,so the arguments we develop here extend farther than we exploit them.

We need a particular instance of Cartan’s theorem B about coherent analytic sheaves[GR65].

5.3.3.2 Theorem (Cartan’s Theorem B). Let X be a Stein manifold and let E be a holomorphicvector bundle on X. Then,

Hk(Ω0,∗(X, E), ∂) =

0, k 6= 0Hol(X, E), k = 0,

where Hol(X, E) denotes the holomorphic sections of E on X.

We use a corollary first noted by Serre [Ser53]; it is a special case of the Serre dualitytheorem. (Nowadays, people normally talk about the Serre duality theorem for compactcomplex manifolds, but in Serre’s original paper he proved it for noncompact manifoldstoo, under some additional hypothesis that will be satisfied on Stein manifolds).

Note that we use the Frechet topology on Hol(X, E), obtained as a closed subspace ofC∞(X, E). We let E! be the holomorphic vector bundle E∨ ⊗ KX where KX is the canonicalbundle of X.

5.3.3.3 Corollary. For X a Stein manifold of complex dimension n, the compactly-supportedDolbeault cohomology is

Hk(Ω0,∗c (X, E), ∂) =

0, k 6= n(Hol(X, E!)∨, k = n,

where (Hol(X, E!)∨ denotes the continuous linear dual to Hol(X, E!).

PROOF. The Atiyah-Bott lemma (see lemma B.10) shows that the inclusion

(Ω0,∗c (X, E), ∂) → (Ω0,∗

c (X, E), ∂)

is a chain homotopy equivalence. (Recall that the bar denotes “distributional sections.”)

As Ω0,kc (X, E) is the continuous linear dual of Ω0,n−k(X, E!), it suffices to prove the desired

result for the continuous linear dual complex.

Consider the acyclic complex

0→ Hol(X, E)i→ C∞(X, E) ∂→ Ω0,1(X, E)→ · · · → Ω0,n(X)→ 0.

Our aim is to show that the linear dual of this complex is also acyclic. Note that this is acompex of Frechet spaces. The result follows from the following lemma.


5.3.3.4 Lemma. If V∗ is an acyclic cochain complex of Frechet spaces, then the dual complex(V∗)∨ is also acyclic.

PROOF. Let di : Vi → Vi+1 denote the differential. We need to show that the sequence

(Vi+1)∨ → (Vi)∨ → (Vi−1)∨

is exact in the middle. That is, we need to show that if α : Vi → C is a continuous linearmap, and if α di−1 = 0, then there exists some β : Vi+1 → C such that α = β di.

Note that α is zero on Im di−1 = Ker di, so that α descends to a linear map

Vi/ Ker di = Im di → C.

Since the complex is acyclic, Im di = Ker di+1 as vector spaces. However, it is not auto-matically true that they are the same as topological vector spaces, where we view Im di as aquotient of Vi and Ker di+1 as a subspae of Vi+1. This is where we use the Frechet hypoth-esis: the open mapping theorem holds for Frechet spaces, and tells us that any surjectivemap between Frechet spaces is open. Since Ker di+1 is a closed subspae of Vi+1, it is aFrechet space. The map Vi → Ker di+1 is surjective, and therefore open. It follows thatIm di = Ker di+1 as topological vector spaces.

From this, we see that our α : Vi → C descends to a continuous linear functional onKer di+1. Since this is a closed subspace of Vi+1, the Hahn-Banach theorem tells us that itextends to a continuous linear functional on Vi+1.

These lemmas allow us to understand the cohomology of classical observables justas a vector space. Since we treat classical observables as a differentiable vector space,i.e. a sheaf of D-modules on the site of smooth manifolds, we are really interested in itscohomology as a sheaf on the site of smooth manifolds. It turns out (perhaps surprisingly)that the isomorphism

Hn(Ω0,∗c (X, E)) = Hol(X, E!)∨

(where X is a Stein manifold of dimension n) is not an isomorphism of sheaves o n thesmooth site, where we define a smooth map from a manifold M to Hol(X, E!)∨ to be acontinuous linear map Hol(X, E)→ C∞(M). However, we have a different interpretationof compactly-supported Dolbeault cohomology that will give a description of it as a sheafon the site of smooth manifolds.

We will present this description for polydiscs (although it works more generally). If0 < r1, . . . , rn ≤ ∞, let Dr ⊂ C be the disc of radius r, and let

Dr1,...,rn = Dr1 × · · · × Drk ⊂ Cn

be the corresponding polydisc.


We can view Dr as an open subset in P1. Let O(−1) denote the holomorphic linebundle on P1 consisting of functions vanishing at ∞.

In general, if X is a complex manifold and C ⊂ X is a closed subset, we can define

Hol(C) = colimC⊂U

Hol(U)

to be the germs of holomorphic functions on C. The space Hol(C) has a natural structureof differentiable vector space, where we view Hol(U) as a differentiable vector space andtake the colimit in the category of differentiable vector spaces. The same definition holdsfor the space Hol(C, E) of germs on C of holomorphic sections of a holomorphic vectorbundle on E.

We have the following theorem, describing compactly supported Dolbeault cohomol-ogy as a differentiable vector space.

5.3.3.5 Theorem. For a polydisc Dr1,...,rn ⊂ Cn, we have a natural isomorphism of differentiablevector spaces

Hn(Ω0,∗c (Dr1,...,rn))

∼= Hol((P1 \ Dr1)× . . . (P1 × Drn), O(−1)n).

Further, all other cohomology groups of Ω0,∗c (Dr1,...,rn) are zero as differentiable vector spaces.

Note that O(−1)n denotes the line bundle on (P1)n consisting of functions whichvanish at infinity in each variable. This isomorphism is invariant under holomorphicsymmetries of the polydisc Dr1,...,rk , and in particular, under the actions of S1 by rotationin each coordinate, and under the action of the symmetric group (if the ri are all the same).

Before we prove this theorem, we need a technical result.

5.3.3.6 Proposition. For any complex manifold X and holomorphic vector bundle E on X, andany manifold M, the complex C∞(M, Ω0,∗(X, E)) is a fine resolution of the sheaf on M × Xconsisting of smooth sections of the bundle π∗XE which are holomorphic in X.

Further, if we assume that Hi(Ω0,∗(X, E)) = 0 for i > 0, then

Hi(C∞(M, Ω0,∗(X, E))) =

C∞(M, Hol(X, E)) if i = 00 if i > 0.

PROOF. Note that the first statement follows from the second statement. As, locallyon X, the Dolbeault-Grothendieck lemma tells us that the sheaf Ω0,∗(X, E) has no highercohomology, and the sheaves C∞(M, Ω0,i(X, E)) are certainly fine.

To prove the second statement, consider the exact sequence

0→ Hol(X, E)→ Ω0,0(X, E)→ Ω0,1(X, E) · · · → Ω0,n(X, E)→ 0.


This is an exact sequence of Frechet spaces. Theorem A1.6 of [?] tells us that the com-pleted projetive tensor product of nuclear Frechet spaces is an exact functor, that is, takesexact sequences to exact sequences. A result of Grothendieck [Gro52] tells us that forany complete locally convex topological vector space F, C∞(M, F) is naturally isomorphicto C∞(M)⊗π F, where ⊗π denotes the completed projective tensor product. The resultfollows.

PROOF OF THE THEOREM. Let M be a smooth manifold. We need to produce an iso-morphism, natural in M,

C∞(M, Hn(Ω0,∗c (Dr1,...,rn)))

∼= C∞(M, Hol((P1 \ Dr1)× . . . (P1 × Drn), O(−1)n)).

We also need to show that C∞(M, Hi(Ω0,∗c (Dr1,...,rn))) are zero for i < n. Note that we are

taking the cohomology sheaves, i.e. the sheafification of the presheaf on M which sendsU to

Hi(C∞(U, Ω0,∗c (Dr1,...,rn))).

The first thing to observe is that, for any complex manifold X and holomorphic vectorbundle E on X, and any open subset U ⊂ X, there is an exact sequence of sheaves ofcochain complex on M

0→ C∞(M, Ω0,∗c (U, E))→ C∞(M, Ω0,∗(X, E))→ C∞(M, Ω0,∗(X \U, E))→ 0.

Indeed, if the Ki are an exhausting family of compact subsets of U, we define Ω0,∗Ki(U, E)

to be the kernel of the map

Ω0,∗(U, E)→ Ω0,∗(U \ Ki, E).

This coincides with Ω0,∗Ki(X, E). The sequence

0→ C∞(M, Ω0,∗Ki(X, E))→ C∞(M, Ω0,∗(X, E))→ C∞(M, Ω0,∗(X \ Ki, E))

is therefore exact. It is not necessarily exact on the right.

Now, let us take the colimit of this sequence as i → ∞, where the colimit is taken inthe category of sheaves on M. We find an exact sequence

0→ C∞(M, Ω0,∗c (U, E))→ C∞(M, Ω0,∗(X, E))→ C∞(M, Ω0,∗(X \U, E)).

Now, we claim that this sequence is exact on the right, as a sequence of sheaves on M. Thepoint is that, locally on M, every smooth function on M × (X \U) extends to a smoothfunction on some M× (X \ Ki) and by applying a bump function which is 1 on a neigh-bourhood of M × (X \ U) and zero on the interior of Ki we can extend it to a smoothfunction on M× X.

Let’s apply this to the case when X = (P1)n, U = Dr1,...,rn and E = O(−1)n. Notethat in this case E is trivialized on U, so that we find, on taking cohomology, an exact


sequence of sheaves on M

· · · → C∞(M, Hi(Ω0,∗c (Dr1,...,rn))→ C∞(M, Hi(Ω0,∗((P1)n, (Ø(−1))n))→ C∞(M, Hi(Ω0,∗((P1)n \Dr1,...,rn , O(−1)n))→ . . .

Note that the Dolbeault cohomology of (P1)n with coefficients in O(−1)n vanishes. Thisimplies, using the previous proposition, that the middle term in this exact sequence van-ishes, so that we get an isomorphism

C∞(M, Hi(Ω0,∗c (Dr1,...,rn))) = C∞(M, Hi−1(Ω0,∗((P1)n \ Dr1,...,rn , O(−1)n))).

We need to compute the right hand side of this. We can view the complex appearing onthe right hand side as the colimit, as ε→ 0, of

C∞(M, Ω0,∗((P1)n \ Dr1−ε,...,rn−ε, O(−1)n)).

Here D indicates the closed disc.

We have a sheaf on M ×((P1)n \ Dr1−ε,...,rn−ε

)), which sends an open set U × V to

C∞(U, Ω0,∗(V, O(−1)n)). This is a cochain complex of fine sheaves. We are interestedin the cohomology of global sections. We can compute this sheaf cohomology using thelocal-to-global spectral sequence associated to a Cech cover of (P1)n \Dr1−ε,...,rn−ε. We usethe Cech cover is given by setting

Ui = P1 × . . . (P1 \ Dri−ε) · · · ×P1.

We take the corresponding cover of M×((P1)n \ Dr1−ε,...,rn−ε

)) given by the opens M×

Ui.

Note that, for k < n, we have

H∗(Ω0,∗(Ui1,...,ik , O(−1)n)) = 0.

The previous proposition implies that we also have

H∗(C∞(M, Ω0,∗(Ui1,...,ik , O(−1)n))) = 0.

The local-to-global spectral sequence then tells us that we have a natural isomorphism

H∗(C∞(M, Ω0,∗(P1 \ Dr1−ε,...,rn−ε, O(−1)n))) = C∞(M, Hol(U1,...,n, O(−1)n))[−n].

That is, all cohomology groups on the left hand side of this equation are zero, except forthe top cohomology, which is equal to the vector space on the right.

Note thatU1,...,n = (P1 \ Dr1−ε)× · · · × (P1 \ Drn−ε).

Taking the colimit as ε → 0, combined with our previous calculations, gives the desiredresult. Note that this colimit must be taken in the category of sheaves on M. We are alsousing the fact that sequential colimits commute with formation of cohomology.


5.3.4. A description of observables. This analytic discussion will allow us to under-stand both classical and quantum observables of the theory we are considering. Let usfirst define some basic classical observables.

5.3.4.1 Definition. On any disk D(x, r) centered at the point x, let cn(x) denote the linear clas-sical observable

cn(x) : γ ∈ Ω0,0(D(x, r)) 7→ 1n!(∂n

z γ)(x).

Likewise, for n > 0, let bn(x) denote the linear functional

bn(x) : β dz ∈ Ω1,0(D(x, r)) 7→ 1(n− 1)!

(∂n−1z β)(x).

These observables descent to elements of H0(Obscl(D(0, r)).

Let us introduce some notation to deal with products of these. If K = (k1, . . . , kn) orL = (l1, . . . , lm) is a multi-index, we let

bK(x) = bk1(x) . . . bkn(x) ∈ H0(Obscl(D(x, r))

cL(x) = cl1(x) . . . clm(x).

Of course this makes sense only if ki > 0 for all i. Note that under the natural S1 action onH0(Obscl(D(x, r)), bK(x) and cL(x) are of weights − |K| and − |L|, where |K| = ∑ ki andsimilarly for L.

5.3.4.2 Lemma. (1) The cohomology groups Hi(Obscl(D(x, r)) vanish (as differentiablevector spaces) unless i = 0.

(2) The monomials bK(x)cL(x) form a basis for the weight space H0|K|+|L|(Obscl(D(x, r)).

Further, they form a basis in the sense of differentiable vector spaces, meaning that anysmooth map

f : M→ H0n(Obscl(D(x, r))

can be expressed uniquely as a sum

f = ∑|K|+|L|=n

fKLbK(x)cL(x)

where the fKL ∈ C∞(M) and locally on M all but finitely many of the fKL are zero.(3) More generally, any smooth map

f : M→ H0(Obscl(D(x, r))

can be expressed uniquely as a sum (convergent in a natural topology on C∞(M, H0(Obscl(D(x, r))))

f = ∑ fKLbk(x)cL(x)

where the coefficient functions fKL have the following properties:(a) Locally on M, fk1,...,kn,l1,...,lm = 0 for n + m 0.


(b) For fixed n and m, the sum

∑ fk1,...,kn,l1,...,lm z−k11 . . . z−kn

n w−l1−11 . . . w−lm−1

1

is absolutely convergent when |zi| ≥ r and∣∣wj∣∣ ≥ r, in the natural topology on

C∞(M).

This lemma then gives an explicit description of the spaces C∞(M, H( Obscl(D(x, r)))).

PROOF. We will set x = 0. By definition, Obscl(D(0, r)) is a direct sum, as differen-tiable cochain complexes,

Obscl(U) = ⊕n(Ω0,∗

c (D(0, r)n, En)[n])

Sn

where E is the holomorphic vector bundle O ⊕ K on C.

It follows from theorem 5.3.3.5 that, as differentiable vector spaces,

Hi(Ω0,∗(D(0, r)n, En) = 0

unless i = n, and that

Hn(Ω0,∗(D(0, r)n, En) = Hol((P1 \ D(0, r)), (O(−1)⊕O(−1)dz)n).

Theorem 5.3.3.5 was stated only for the trivial bundle; but the bundle E is trivial onD(x, r). It is not, however, S1-equivariantly trivial. The notation O(−1)dz indicates thatwe change the S1-action on O(−1).

It follows that we have an isomorphism, of S1-equivariant differentiable vector spaces,

H0(Obscl(D(0, r)) ∼= ⊕nH0((P1 \ D(0, r))n, (O(−1)⊕O(−1)dz)n)Sn .

Under this isomorphism, the observable bK(0)cL(0) goes to the function

1(2πi)n+m z−k1

1 dz1 . . . z−knn dznw−l1−1

1 . . . w−lm−1m

Everything in the statement is now immediate; the topology we use on H0(Obscl(D(0, r))is the one arising as the colimit of the topologies on holomorphic functions on (P1 \D(0, r− ε))n as ε→ 0.

5.3.4.3 Lemma. We have

H∗(Obsq(U)) = H∗(Obscl(U))[h]

as S1-equivariant differentiable vector spaces. This isomorphism is compatible with maps inducedfrom inclusions U → V of open subsets of C (but not compatible with the factorization productmap).


PROOF. This follows, as explained in Chapter 4 from the existence of a Green’s func-tion for the ∂ operator, namely

G(z1, z2) =dz1 − dz2

z1 − z2∈ E (C)⊗ E (C)

where E denotes the complex of fields of the β−γ system. We will make this more explicitlater.

We will use the notation bK(x)cL(x) for the quantum observables on D(x, r) whicharise from the classical observables discussed above, using the isomorphism given by thislemma.

5.3.4.4 Corollary. The properties listed in lemma 5.3.4.2 also hold for quantum observables. Asa result, all the conditions of theorem 5.2.2.1 are satisfied, so that the structure of factorizationalgebra leads to a C[h]-linear vertex algebra structure on the space

V = ⊕k H∗k (Obsq(D(0, r))

where H∗l (Obsq(D(0, r)) indicates the kth eigenspace of the S1 action.

PROOF. The only conditions we have not so far checked are that the inclusion maps

H∗k (Obsq(D(0, r)))→ H∗k (Obsq(D(0, s)))

for r < s are quasi-isomorphisms, and that the differentiable vector spacesVk = H∗k (Obsq(D(0, r)))are countable colimits of finite-dimensional vector spaces, in the category of differentiablevector spaces. Both of these conditions follow immediately from the analog of lemma5.3.4.2 that applies to quantum observables.

5.3.5. An isomorphism of vertex algebras. Our goal in this subsection is to demon-strate that the vertex algebra constructed by theorme 5.2.2.1 is from the quantum observ-ables of the β− γ-system is isomorphic to a vertex algebra considered in the physics liter-ature called the β− γ vertex algebra.

We will first describe the β− γ vertex algebra. We follow [FBZ04], notably chapters11 and 12, to make the dictionary clear.

Let W denote the space of polynomials C[an, a∗m] where generated by variables an, a∗mn < 0 and m ≤ 0.

5.3.5.1 Definition. The βγ vertex algebra has state space W, vacuum vector 1, translationoperator T the map sending

ai → −iai−1

a∗i → −(i− 1)a∗i−1,


and the vertex operator satisfies

Y(a−1, z) = ∑n<0

anz−1−n + ∑n≥0

∂

∂a∗−nz−1−n

andY(a∗0 , z) = ∑

n≤0a∗nz−n − ∑

n>0

∂

∂a−nz−n

By the reconstruction theorem 5.0.9.2, these determine the vertex algebra.

The main theorem of this section is the following.

5.3.5.2 Theorem. Let Vh=2πi denote the vertex algebra constructed from quantum observables ofthe β− γ system, specialized to h = 2πi. Then, there is an S1-equivariant isomorphism of vertexalgebras

Vh=2πi∼= W.

The circle S1 acts on W by giving ai, a∗j weights i, j respetively.

PROOF. Note that Vh=2πi is the polynomial algebra on the generators bn, cm wheren ≥ 1 and m ≥ 0. Also bn, cm have weights −n,−m respectively, under the S1-action. Wedefine an isomorphism Vh=2πi to W by sending bn to a−n and cm to a∗−m, and extending itto be an isomorphism of commutative algebras. By the reconstruction theorem, it sufficesto calculate Y(b1, z) and Y(c0, z).

By the way we defined the vertex algebra associated to the factorization algebra ofquantum observables in theorem 5.2.2.1, we have

Y(b1, z)(α) = Lzmz,0(b1, α) ∈ Vh=2πi((z))

where Lz is denotes Laurent expansion, and

mz,0 : Vh=2πi ⊗Vh=2πi → V h=2πi

is the map associated to the factorization product coming from the inclusion of the disjointdiscs D(z, r) and D(0, r) into D(0, ∞) (where r can be taken to be arbitrarily small).

We are using the Green’s function for the ∂ operator to identify classical and quantumobservables. Let us recall how the Green’s function leads to an explicit formula for thefactorization product.

Let E denote the sheaf on C of fields of our theory, so that

E (U) = Ω0,∗(U, O ⊕ K).

This is sections of a graded bundle E on C. Let E ! = E [1] denote sections of E∨⊗ω whereω is the bundle of 2-forms on C. We let E denote the sheaf of distributional sections, andEc denote compactly-supported sections.


The propagator (or Green’s function) is

P =dz1 ⊗ 1− 1⊗ dz2

2πi(z1 − z2)

∈ E (C)⊗πE (C)

= D(C2, E E)

where D denotes the space of distributional sections.

We can also view at as being a symmetric and smooth linear map

(†) P : E !c (C)⊗ E !

c (C) = C∞C (C2, (E!)2)→ C.

Here ⊗ denotes the completed bornological tensor product on the category of convenientvector spaces.

Recall that we identify

Obscl(U) = Sym∗ E !c (U) = Sym∗ Ec(U)[1]

where the symmetric algebra is defined using the completed tensor product on the cate-gory of convenient vector spaces.

We have an order two differential operator

∂P : Obscl(U)→ Obscl(U)

for every U ⊂ C, characterized by the fact that it is a smooth (or, equivalently, continuous)order two differential operator, which is zero on Sym≤1 and on Sym2 is given by applyingthe map in (†).

For example, if x, y ∈ U, then

∂P(bi(x)cj(y)) =1

(i− 1)!j!∂i−1

∂i−1x∂j

∂jy1

2πi(x− y)

We can identify Obsq(U) as a graded vector space with Obscl(U)[h], with differentiald = d1 + hd2 a sum of two terms, where d1 is the differential on Obscl(U) and d2 isthe differential arising from the Lie bracket in the shifted Heisenberg Lie algebra whoseChevalley chain complex defines Obsq(U).

It is easy to check that[h∂P, d1] = d2.

This follows immediately from the fact that (1, 1)-current ∂P on C2 is the delta-current onthe diagonal.


As we explained in section ??, we get an isomorphism of cochain complexes

W : Obscl(U)[h] 7→ Obsq(U)

α 7→ eh∂P α.

Further, the factorization product map

?h : Obscl(U1)[h]×Obscl(U2)[h]→ Obscl(V)[h]

(if U1, U2 are disjoint and in V) which arises from that on Obsq the identification W isgiven by the formula

α ?h β = e−h∂P(

eh∂P α)·(

eh∂P β)

.

Here · refers to the commutative product on classical observables.

Let us apply this formula to α = b1(z) and β in the algebra generated by bi(0) andcj(0). First note that since b1(z) is linear, h∂Pb1(z) = 0. Note also that [∂P, b1(z)] commuteswith ∂P. Thus, we find that

b1(z) ?h β = e−h∂P(

b1(z)eh∂P β)

= b1(z)β− [h∂P, b1(z)]β.

Note that [∂P, b1(z)] is an order one operator, and so a derivation. So it suffices to calculatewhat it does on generators. We find that

[∂P, b1(z)]cj(0) =1

2πi1j!

∂j

∂jw1

z− wevaluated at w = 0

=1

2πiz−j−1.

[∂P, b1(z)]bj(0) = 0.

In other words,

[∂P, b1(z)] =1

2πi ∑ z−j−1 ∂

∂cj(0).

Note also that we can expand the cohomology class b1(z) ∈ H0(Obscl(D(0, r)) (for |z| < r)as a sum

b1(z) =∞

∑n=0

bn+1(0)zn.

Indeed, for a classical field γ ∈ Ω1hol(D(0, r)) solving the equations of motion,

b1(z)(γ) = γ(z)

= ∑ zn 1n!

γ(n)(0)

= ∑ znbn1(0)(γ).


Putting all this together, we find that, for β in the algebra generated by cj(0), bi(0), wehave

b1(z) ?h β =

(∞

∑n=0

bn+1(0)zn +h

2πi

∞

∑m=0

∂

∂cm(0)z−m−1

)β ∈ H0(Obscl(D(0, r))[h].

Thus, if we set h = 2πi, we see that the operator product on the space Vh=2πi matches theone on W if we sent bn(0) to a−n and cn(0) to a∗−n. A similar calculation of the operatorproduct of c0(0) completes the proof.

5.4. Affine Kac-Moody algebras and factorization algebras

In this section, we will construct a holomorphically translation invariant factorizationalgebra whose associated vertex algebra is the affine Kac-Moody vertex algebra. Thisconstruction is an example of the twisted factorization envelope construction, which alsoproduces the factorization algebras for free field theories (see section 3.6). This construc-tion is our version of Beilinson-Drinfeld’s [BD04] chiral envelope construction. We willthings up in somewhat greater generality then needed for this theorem.

The input data is the following:

• a Riemann surface Σ ;• a Lie algebra g (for simplicity, we stick to ordinary Lie algebras like sl2);• a g-invariant symmetric pairing κ : g⊗2 → C.

From this data, we obtain a cosheaf on Σ,

gΣ : U 7→ (Ω0,∗c (U)⊗ g, ∂),

where U denotes an open in Σ. Note that gΣ is a cosheaf of dg vector spaces and merely aprecosheaf of dg Lie algebras. When κ is nontrivial (though not necessarily nondegener-ate), we obtain an interesting −1-shifted central extension on each open:

gΣκ : U 7→ (Ω0,∗

c (U)⊗ g, ∂)⊕C · c,

where C denotes the locally constant cosheaf on Σ and c is a central element of cohomo-logical degree 1. The bracket is defined by

[α⊗ X, β⊗Y]κ := α ∧ β⊗ [X, Y]− 12πi

(∫U

∂α ∧ β

)κ(X, Y)c,

with α, β ∈ Ω0,∗(U) and X, Y ∈ g. (These constants are chosen to match with the use of κfor the affine Kac-Moody algebra below.)

Remark: As discussed in section 10.1, every dg Lie algebra g has a geometric interpretationas a formal moduli space Bg. The dg Lie algebra gΣ(U) in fact possesses a natural geometricinterpretation: it describes “deformations with compact support in U of the trivial G-bundle

5.4. AFFINE KAC-MOODY ALGEBRAS AND FACTORIZATION ALGEBRAS 133

on Σ.” Equivalently, it describes the moduli space of holomorphic G-bundles on U whichare trivialized outside of a compact set. For U a disc, it is closely related to the affineGrassmannian of G. The affine Grassmannian is defined to be the space of algebraic bun-dles on a formal disc trivialized away from a point, whereas our formal moduli spacedescribes G-bundles on an actual disc trivialized outside a compact set.

The choice of κ has the interpretation of a line bundle on the formal moduli problemBgΣ(U) for each U. In general, −1-shifted central extensions of a dg Lie algebra g are thesame as L∞-maps g→ C, that is, as rank-one representations. Rank-one representations ofa group are line bundles on the classifying space of the group. In the same way, rank-onerepresentations of a Lie algebra are line bundles on the formal moduli problem Bg.

As explained in section ??, we can form the twisted factorization envelope of Ω0,∗c ⊗ g.

Concretely, this factorization algebra assigns to an open subset U ⊂ Σ, the complex

F κ : U 7→ C∗(gΣκ (U)) = (Sym(Ω0,∗

c (U)⊗ g[1])[c], ∂ + dCE),

where c now has cohomological degree 0 in the Lie algebra homology complex. This is afactorization algebra in modules for the ring C[c] generated by the central parameter. Weshould therefore think of it as a family of factorization algebras depending on the centralparameter c.

Remark: Given a dg Lie algebra (g, d), we interpret C∗g as the “distributions with supporton the closed point of the formal space Bg.” Hence, our factorization algebras F κ(U) de-scribes the κ-twisted distributions supported at the point in BunG(U) given by the trivialbundle on U.

This description is easier to understand in its global form, particularly when Σ is aclosed Riemann surface. Each point of P ∈ BunG(Σ) has an associated dg Lie algebra gPdescribing the formal neighborhood of P. This dg Lie algebra, in the case of the trivialbundle, is precisely the global sections over Σ of gΣ. For a nontrivial bundle P, the Liealgebra gP is also global sections of a natural cosheaf, and we can apply the envelopingconstruction to this cosheaf to obtain a factorization algebras. By studying families of suchbundles, we recognize that our construction C∗gP should recover differential operators onBunG(Σ). When we include a twist κ, we should recover κ-twisted differential operators.When the twist is integral, the twist corresponds to a line bundle on BunG(Σ) and thetwisted differential operators are precisely differential operators for that line bundle.

It is nontrivial to properly define differential operators on the stack BunG(Σ), andmuch work continues on porting the full machinery of D-modules onto this stack. At theformal neighbourhood of a point, however, there are no difficulties and our statementsare rigorous.


5.4.1. The main result. Note that if we take our Riemann surface to be C, the factor-ization algebra F κ is holomorphically translation invariant. This follows from the factthat, on the Lie algebra gκ(U) for any open subset U in C, the derivation ∂

∂z is homotopi-cally trivial, where the homotopy is given by ∂

∂dz . It follows that we are in a situationwhere we can (if certain other properties hold) apply theorem 5.2.2.1. The main result ofthis section is the following.

Theorem. The holomorphically translation invariant factorization algebra F κ on C satisfies theconditions of theorem 5.2.2.1, and so defines a vertex algebra. This vertex algebra is isomorphic tothe affine Kac-Moody vertex algebra.

Before we can prove this statement, we of course need to describe the affine Kac-Moody vertex aigebra.

Recall that the Kac-Moody Lie algebra is the central extension of the loop algebraLg = g[t, t−1],

0→ C · c→ gκ → Lg→ 0.

As vector spaces, we have gκ = g[t, t−1]⊕C · c, and the Lie bracket is given by the formula

[ f (t)⊗ X, g(t)⊗Y]κ := f (t)g(t)⊗ [X, Y] +(∮

f ∂g)

κ(X, Y)

for X, Y ∈ g and f , g ∈ C[t, t−1]. Here c has cohomological degree 0 and is central. Note,that

∮tn∂tm is 2πimδm+n,0.

Note that g[t] is a sub Lie algebra of the Kac-Moody algebra. The vacuum module W forthe Kac-Moody algebra is the induced representation from the trivial rank one represen-tation of g[t]. That there is a natural map C → W of g[t]-modules, where C is the trivialrepresentation. We let |∅〉 ∈W be the image of 1 ∈ C.

As a vector space, we can identify the vacuum representation canonically as

W = U(C · c⊕ t−1g[t−1]).

As, C · c⊕ t−1g[t−1] is a sub-Lie algebra of gκ. Thus, the universal enveloping algebra ofthis sub-algebra acts on W; the action on the vacuum element |∅〉 ∈ W gives rise to thisisomorphism.

The vacuum module W is a C[c] module in a natural way, because C[c] is inside theuniversal enveloping algebra of gκ.

5.4.1.1 Definition. The Kac-Moody vertex algebra is defined as follows. It is a vertex algebrastructure over the base ring C[c] on the vector space W. (Working over the base ring C[c] simplymeans all maps are C[c]-linear). By the reconstruction theorem 5.0.9.2, to specify the vertex algebrastructure it suffices to specify the state-field map on a subset of elements of W which generate all


of W (in the sense of the reconstruction theorem). The following state-field operations define thevertex algebra structure on W.

(1) The vacuum element |∅〉 ∈W is the unit for the vertex algebra, that is, Y(|∅〉 , z) is theidentity.

(2) If X ∈ g ⊂ gκ, we have an element X |∅〉 ∈W. We declare that

Y(X |∅〉 , z) = ∑ Xnz−1−n

where Xn = tnX ∈ gκ, and we are viewing elements of gκ as endomorphisms of W.

5.4.2. Verification of the conditions to define a vertex algebra. We need to verify thatF κ satisfies the conditions listed in theorem ?? guaranteeing that we can construct a vertexalgebra. This is entirely parallel to the corresponding statements for the β− γ system, sowe will be brief. The first thing to check is that the natural S1 action onF κ(D(0, r)) extendsto an action of the algebra D(S1) of distributions on the circle. This is easy to see.

Then, we need to check that, if F κl (D(0, r)) denotes the eigenspace for the S1-action,

then the following properties hold.

(1) The inclusion F κl (D(0, r))→ F κ

l (D(0, s)) for r < s is a quasi-isomorphism.(2) The cohomology H∗(F κ

l (D(0, r)) vanishes (as a sheaf on the site of smooth man-ifolds) for l 0.

(3) The differentiable vector spaces H∗(F κl (D(0, r)) are countable sequential colim-

its (in the category of differentiable vector spaces) of finite-dimensional vectorspaces.

Note that

F κ(D(0, r)) = Sym∗(Ω0,∗

c (D(0, r), g)[1]⊕C · c)

with differential a sum ∂ + dCE where dCE is the Chevalley-Eilenberg differential. GiveF κ(D(0, r) an increasing filtration, by degree of the symmetric power. This filtration iscompatible with the action of S1 and of D(S1). In the associated graded, the differential isjust that from the ∂ differential on Ω0,∗

c (D(0, r)).

It follows that there is a spectral sequence (in the category of sheaves on the site ofsmooth manifolds)

H∗ (Gr∗ F κl (D(0, r)))⇒ H∗ (F κ

l (D(0, r))) .

The analytic results we proved in section ?? concerning compactly supported Dolbeaultcohomology immediately imply that H∗

(Gr∗ F κ

l (D(0, r)))

satisfy properties 1− 3 above.It follows that the same holds for H∗(F κ

l (D(0, r)).


5.4.3. Proof of the theorem. Let us now prove that the vertex algebra associated toFκ is isomorphic to the Kac-Moody vertex algebra. The proof will be a little different thanthe proof of the corresponding result for the β− γ system.

We first prove a statement concerning the behaviour of the factorization algebra Fκ onannuli. Consider the radial projection map

ρ : C× → R>0.

We can define a factorization algebra ρ∗Fκ on R>0 which assigs to any open subset U ⊂R>0 the cochain complex Fκ(ρ−1(U)). In aparticular, this factorization algebra assignsto an interval (a, b) the space Fκ(A(a, b)) where A(a, b) indicates the annulus of those zwith a < |z| < b. The operator product map associated to the inclusion of two disjointintervals in a larger one arises from the operator product map from the inclusion of twodisjoint annuli in a larger one.

Recall ?? that any associative algebra A gives rise to a factorization algebra on R whcihassigns to the interval (a, b) the algebra A, and where the operator product map is themultiplication in A. We let A f act denote the factorization algebra on R associated to A.The first result we will show is the following.

5.4.3.1 Theorem. There is an injective map of C[c]-linear factorization algebras on R>0

U(gκ)f act → H∗(ρ∗F κ)

whose image is a dense subspace.

This map is characterized by the following property. Observe that, for every opensubset U ⊂ C, the space

Ω0,∗c (U, g)[1]⊕C · c ⊂ C∗(Ω0,∗

c (U, g)⊕C · c[1]) = F κ(U)

of linear elements of F κ(U) is in fact a subcomplex. By applying this to U = ρ−1(I) foran interval I ⊂ R>0 and taking cohomology, we obtain a natural cochain map

H1(Ω0,∗c (ρ−1(I)))⊗ g)⊕C · c→ H0(F κ(I)).

We have the natural identification

H1(Ω0,∗c (ρ−1(I))) = Ω1

hol(ρ−1(I))∨.

where ∨ indicates continuous linear dual. If n ∈ Z, then performing a contour integralagainst zn defines a linear function on Ω1

hol(ρ−1(I)), and so an element of H1(Ω0,∗

c (ρ−1(I)))which we call φ(zn).

The map

gκ → H∗(F κ(ρ−1(I)))


constructed by the theorem factors through the map

gκ → H1(Ω0,∗c (ρ−1(I)))⊗ g⊕C · c

znX 7→ φ(zn)Xc 7→ c

PROOF. In Chapter 3 section 3.4, we showed how the universal enveloping algebra ofany Lie algebra a arises as a factorization envelope. Let U f act(a) denote the factorizationalgebra on R which assigns to U the complex

U f act(a)(U) = C∗(Ω∗c (U, a)).

Then, we showed that the cohomology of U f act(a) is locally constant and corresponds tothe ordinary universal enveloping algebra Ua.

Let us apply this construction to a = gκ. To prove the theorem, we need to produce amap of factorization algebras on R>0

U f act(gκ)→ ρ∗F κ.

Since both sides are defined as Chevalley chains of certain Lie algebras, it suffices to pro-duce such a map at the level of dg Lie algebras. We will produce such a map in thehomotopical sense.

Let us introduce some notation to describe the Lie algebras we consider. We let L1 bethe precosheaf of dg Lie algebras on R which assigns to U the dg Lie algebra

L1(U) = Ω∗c ⊗(g[z, z−1]

)⊕C · c[−1].

Thus, L1 is a central extension of Ω∗c ⊗ g[z, z−1]. The cocycle defining the central extensionon the open set U is

αznX⊗ βzmY 7→(∫

Uα ∧ β

)κ(X, Y)

(∮zn∂zzm

)where α, β ∈ Ω∗c (U).

We letL2 = ρ∗(Ω0,∗

c (U)⊗ g⊕C · c[−1])be the precosheaf of dg Lie algebras which assigns to an open subset U the dg Lie algebra

L2(U) = Ω0,∗c (ρ−1(U))⊗ g⊕C · c[−1]

with central extension the one discussed earlier.

Note that both L1 and L2 are prefactorization algebras valued in the category of dgLie algebras equipped with the direct sum symmetric monoidal structure. A precosheafL of dg Lie algebras is such a prefactorization algebra if it has the property that that, if


U, V are disjoint opens in W, then elements in L(W) which are in the image of the mapfrom L(U) commute with those coming from L(V).

Note also that there is a natural map of factorization dg Lie algebras

Ω∗c ⊗ gκ → L1

which is the identity on Ω∗c ⊗ g[z, z−1] and sends

αc 7→(∫

Uα

)c

for α ∈ Ω∗c (U).

This map is clearly a quasi-isomorphism when U is an interval. It follows immediatelythat the map

U f act(gκ) = C∗(Ω∗c ⊗ gκ)→ C∗L1

is a map of prefactorization algebras which is a quasi-isomorphism on intervals. There-fore, the cohomology prefactorization algebra of C∗L1 assigns to an interval U(gκ), andthe factorization product is just the associative product on this algebra.

It suffices to produce a map of precosheaves of dg Lie algebras

L1 → L2.

We do this as follows. First, we define L′1 to be, like L1, a central extension of Ω∗c ⊗g[z, z−1], but where the cocycle defining the central extension is

αznX⊗ βznY 7→(∫

Uα ∧ β

)κ(X, Y)

(∮zn∂zzm

)+ πκ(X, Y)δn+m,0

(∫U

αr∂

∂rβ

)where the vector field r ∂

∂r acts by Lie derivative on the form β ∈ Ω∗c (U).

It is easy to verify that this cochain is closed, and so defines a central extension. In fact,this central extension, as well as the ones defining L1 and L2, are local central extensionsof local dg Lie algebras in the sense of definition 3.6.3.1 of Chapter 3. This concept isstudied in more detail in subsection ?? of Chapter 10.

To prove the result, we will do the following.

(1) Prove that L1 and L′1 are homotopy equivalent prefactorization dg Lie algebras.(2) Construct a map of prefactorization dg Lie algebras L′1 → L2.

For the first point, note that the extra term πκ(X, Y)δn+m,0∫

U αr ∂∂r β in the coycle for the

central extension of Ω∗c ⊗ Lg defining L′1 is an exact cocycle. It is cobounded (in the senseof dg Lie algebras) by the cochain

πκ(X, Y)δn+m,0

∫U

αrι ∂∂r

β


where ι ∂∂r

indicates contraction. The fact that this expression cobounds follows from theCartan homotopy formula. Since the cobounding cochain is also local, L1 and L′1 arehomotopy equivalent as prefactorization dg Lie algebras.

Now we will produce the desired map L′1 → L2. We use the following notation: r willdenote the coordinate on R>0 and the radial coordinate in C×, θ the angular coordinateon C× and X will denote an element of g. We will view gκ as g[z, z−1]⊕C · c.

If U ⊂ R>0 is open, we send

f (r)znX 7→ f (r)znX ∈ L2 if f ∈ Ω0c(U).

f (r)zndr 7→ 12 eiθ f (r)znXdz.

c 7→ c.

We need to verify that this is a map of dg Lie algebras (it is obviously a map of pre-cosheaves). Compatibility with the differential follows from the formula for ∂

∂z in polarcoordinates:

∂

∂z= 1

2 eiθ(

∂

∂r− 1

ir∂

∂θ

)The only possible issue that can arise when checking compatibility with the Lie bracketis from the central extension. Let us denote the map we have constructed by Φ. The factthat the central extension terms match up follows from the following simple identity:

cκ(X, Y)∫

ρ−1(U)f (r)zn∂(zmg(r) 1

2 eiθ)dz

= cκ(X, Y)

2πimδn+m,0

(∫U

f (r)g(r)dr)+ πδn+m,0

∫U

f (r)r∂

∂rg(r)dr

where U ⊂ R>0 is an interval. The expression on the right hand side gives the centralextension term in the Lie bracket on L′1, whereas that on the left is the central extensionterm in the bracket on L2 applied to the elements Φ( f (r)znX) and Φ(g(r)drzmY).

Applying Chevalley chains, we get a map of prefactorization algebras

C∗L′1 → C∗L2 = ρ∗Fκ.

Since C∗L′1 is equivalent to C∗L1, we get the desired map of factorization algebras on R>0

Ugκ → ρ∗Fκ.

The analytical results (concerning compactly supported Dolbeault cohomology) presentedin section 5.3.3 imply immediately this map has dense image.

Finally we will check that this map has the properties stated in the discussion follow-ing the statement of the theorem. Suppose that U ⊂ R>0 is an interval. Then, under theisomorphism

U(gκ) ∼= H∗(C∗(L′1(U))


the element znX (where X ∈ g) is represented by the element

f (r)znXdr ∈ L′1(U)[1]

where f is of compact support and is chosen so that∫

f (r)dr = 1.

To check the desired properties, we need to verify that if α is a holomorphic 1-form onρ−1(U), then ∫

ρ−1(U)α f (r)zn 1

2 eiθdz =∮|z|=1

αzn.

This follows immediately from Stoke’s theorem and the identity

f (r) 12 eiθdz = ∂(h(r))

where, as above, h(r) =∫ r

∞ f (r)dr.

This theorem shows how to relate observables on an annulus to the universal envelop-ing algebra of the affine Kac-Moody algebra. Recall that the Kac-Moody vertex algebra isthe structure of vertex algebra on the vacuum representation. The next result will showthat the vertex algebra associated to the factorization algebra F κ is also a vertex algebrastructure on the vacuum representation.

More precisely, we will show the following.

5.4.3.2 Proposition. LetV = ⊕H∗(F κ

l (D(0, ε))

be the cohomology of the direct sum of the weight spaces of the S1 action on Fκ(D(0, ε)).

Then, the mapU(gκ)→ H∗(F κ(A(r, r′))

constructed in the previous theorem (where A(r, r′) is the annulus) induces an action of U(gκ) onV.

There is a unique isomorphism of U(gκ)-modules from V to the vacuum module W, whichsends the unit observable 1 ∈ V to the vacuum element |∅〉 ∈W.

PROOF. If ε < r < r′, then the factorization product gives a map

F κ(D(0, ε)×F κ(A(r, r′))→ F κ(D(0, r′))

of cochain complexes. Passing to cohomology, and using the relationship between U(gκ)and F κ(A(r, r′)), we get a map

V ⊗U(gκ)→ H∗(F κ(D(0, r′)).


Note that on the left hand side, every element is a finite sum of elements in S1-eigenspaces.This map is S1-equivariant. Therefore, it lands in the subspace of H∗(F (D(0, r′)) whichconsists of finite sums of S1-eigenvectors. This subspace is V. We therefore find a map

(†) U(gκ)⊗V → V.

The fact that the map U(gκ)→ H∗(F κ(A(r, r′)) takes the asosciative product on U(gκ) tothe factorization product map

H∗(F κ(A(r, r′))⊗ H∗(F κ(A(s, s′))→ H∗(F κ(A(r, s′))

(for r < r′ < s < s′), combined with the fact that the following diagram

F κ(D(0, ε))⊗F κ(A(r, r′))⊗F κ(A(s, s′)) //

F κ(D(0, ε))⊗F κ(A(r, s′))

F κ(D(0, r′))⊗F κ(A(s, s′)) // F κ(D(0, s′))

(whose arrows are the factorization product maps) commutes, implies that the map inequation (†) defines an action of the algebra U(gκ) on V.

We need to show that this action identifies V with the vacuum representation W. Theputative map W → V sends the vacuum element |∅〉 to the unit observable 1 ∈ V. Toshow that this map is well-defined, we need to show that 1 ∈ V is annihilated by theelements znX ∈ gκ for n ≥ 0.

The unit axiom for factorization algebras implies that the following diagram com-mutes:

U(gκ) //

H∗(F κ(A(s, s′))

V // H∗(F κ(D(0, s′))

where the left vertical arrow is given by the action of U(gκ) on the unit element 1 ∈ V,and the right vertical arrow is the map arising from the inclusion A(r, r′) ⊂ D(0, r′).The bottom right arrow is the inclusion onto the direct sum of S1-eigenspaces, which isinjective.

The proof of theorem 5.4.3.1 gives an explicit representative for the element znX ∈ gκ

in Fκ(A(s, s′)). Namely, let f (r) be a function which is supported in the interval (s, s′) andwhose integral

∫f (r)dr is one. Then, znX is represented by

12 eiθ f (r)dzznX ∈ Ω0,1

c (A(s, s′)).

We need to show that this is exact when viewed as an element of Ω0,1c (D(0, s′)). If we let

h(r) =∫ r

∞f (t)dt


then h(r) = 0 for r s′ and h(r) = 1 for r < s. Also, ∂∂r h = f . The polar-coordinate

representation of ∂∂z tells us that

∂h(r)zn = 12 eiθ f (r)dzznX.

Thus, we have shown that elements znX ∈ gκ, where n ≥ 0, act by zero on the element1 ∈ V. We thus have a unique map of U(gκ-modules W → V sending |∅〉 → 1.

It remains to show that this is an isomorphism. Note that every object we are dis-cussing is filtered. The universal enveloping algebra U(gκ is filtered by saying that FiU(gκ)is the subspace spanned by products of ≤ i elements of gκ. Similarly, the space

F κ(U) = C∗(Ω0,∗c (U, g)⊕C · c[−1])

is filtered by saying that FiF κ(U) is the subcomplex C≤i. All the maps we have beendiscussing are compatible with these increasing filtrations.

The associated graded of U(gκ) is the symmetric algebra of gκ, and the associatedgraded of F κ(U) is the appropriate completed symmetric algebra on Ω0,∗

c (U, g)[1]⊕C · c.Upon taking associated graded, the maps

Gr U(gκ)→ Gr H∗F κ(A(s, s′))

Gr H∗(F κ(A(s, s′))→ Gr H∗F κ(D(0, s′))

are maps of commutative algebras. It follows that the map

Gr U(gκ)→ Gr V ⊂ H∗ Sym∗(Ω0,∗

c (D(0, s), g)[1]⊕C · c)

is a map of commutative algebras. Now, Gr V is the direct sum of the S1-eigenspaces in thespace on the right of this equation. The direct sum of the S1-eigenspaces in H1(Ω0,1(D(0, s))is naturally identified with z−1C[z−1]. Thus, we find the associated graded of the mapU(gκ) to V is a map of commutative algebras(

Sym∗ g[z, z−1])[c]→

(Sym∗ z−1g[z−1]

)[c].

We have already calculated that on the generators of the commutative algebra, it arisesfrom the natural projection map

g[z, z−1]⊕C · c→ z−1g[z−1]⊕C · c.

It follows immediately that the map Gr W → Gr V is an isomorphism, as desired.

In order to complete the proof that the vertex algebra associated to the factorization al-gebra F κ is isomorphic to the Kac-Moody vertex algebra, we need to identify the operatorproduct expansion map

V ⊗V → V((z)).


Recall that this map is defined, in terms of the factorization algebra F κ, as follows. Con-sider the map

mz,0 : V ⊗V → H∗(F κ(D(0, ∞))

defined by restricting the factorization product map

H∗(F κ(D(z, ε))× H∗(F κ(D(0, ε))→ H∗(F κ(D(0, ∞))

to the subspaceV ⊂ H∗(F κ(D(z, ε)) = H∗(F κ(D(0, ε)).

Composing the map mz,0 with the map

H∗(F κ(D(0, ∞))→ V = ∏k

Vk

where the Vk are the S1 eigenspaces and the map is the product of the projection maps, weget a map

mz,0 : V ⊗V → V.

This depends holomorphically on z. The operator product map is obtained as the Laurentexpansion of mz,0.

Our aim is to calculate the operator product map and identify it with the vertex op-erator map in the Kac-Moody vertex algebra. We use the following notation. If X ∈ g,let Xi = ziX ∈ gκ. We denote the action of gκ on V by the symbol ·. Then we have thefollowing.

5.4.3.3 Proposition. For all v ∈ V,

mz,0(X−1 · 1, v) = ∑i∈Z

z−i−1(Xi · v) ∈ V

where the sum on the right hand side converges.

Before we prove this proposition, let us observe that it proves our main result:

5.4.3.4 Corollary. This isomorphism W → V of U(gκ) from the vacuum representation W to Vis an isomorphism of vertex algebras, where V is given the vertex algebra structure arising fromthe factorization algebra F κ, and W is given the Kac-Moody vertex algebra structure defined in5.4.1.1.

PROOF. This follows immediately from the reconstruction theorem 5.0.9.2.

PROOF OF THE PROPOSITION. As before, we let Xi ∈ gκ denote Xzi if X ∈ g. As weexplained in the discussion following theorem 5.4.3.1, we can view the element

X−1 · 1 ∈ V


as being represented by X times the linear functional

Ω1hol(D(0, s))→ C

which sendsα 7→

∮z−1α.

Cauchy’s theorem tells us that if this linear functional sends h(z)dz (where h is holomor-phic) to 2πih(0).

Let us fix z0 ∈ A(s, s′), and let

ιz0 : V → H∗(F κ(A(s, s′)))

denote the map arising from the restriction to V of the map

H∗(F κ(D(0, ε))) = H∗(F κ(D(z0, ε)))→ H∗(F κ(A(s, s′)))

arising from the inclusion of the disc D(z0, ε) into the annulus A(s, s′). It is clear from thedefinition of mz0,0 and the axioms of a prefactorization algebra that the following diagramcommutes:

V ⊗V ⊗ Id

ιz0

mz0,0 // V

H∗(F κ(A(s, s′))⊗V // H∗(F κ(D(0, ∞)))

OO

where the bottom right arrow is the restriction to V of the factorization structure map

H∗(F κ(A(s, s′))⊗ H∗(F κ(D(0, ε))→ H∗(F κ(D(0, ∞)).

It therefore suffices to show that

ιz0(X−1 · 1) = ∑i∈Z

Xiz−i−i0 ∈ H∗(F κ(A(s, s′)))

where we view Xi ∈ gκ as elements of H0(F κ(A(s, s′))) via the map

U(gκ)→ H0(F κ(A(s, s′)))

constructed in theorem 5.4.3.1.

It is clear from how we construct the factorization algebra F κ that ιz0(X−1 · 1) is in theimage of the natural map

Ω0,∗c (A(s, s′))⊗ g→ F κ(A(s, s′)).

Recall that the cohomology of Ω0,∗c (A(s, s′)) is the linear dual of the space of holomorphic

1-forms on the annulus A(s, s′). The element ιz0(X−1 · 1) can thus be represented by acontinuous linear map

Ω1hol(A(s, s′))→ g.


The map is the one that sends

h(z)dz 7→ 2πih(z0)X.

Similarly, the elements Xi ∈ H∗(F κ(A(s, s′))) are represented by the linear maps whichsend

h(z)dz 7→(∮

zih(z)dz)

X.

It remains to show that, for all holomorphic functions h(z) on the annulus A(s, s′), wehave

2πih(z0) = ∑i∈Z

zi0

(∮|z|=s+ε

z−i−1h(z)dz)

.

This can be proved by a simple argument using Cauchy’s theorem. ?As,

2πih(z0) =∮|z|=s+ε

h(z)z− z0

dz−∮|z|=s′−ε

h(z)z− z0

dz.

Expanding (z − z0)−1 in the regions when |z| < |z0| (relevant for the first integral) andwhen |z| > |z0| (relevant for the second integral) gives the desired expression.

Part 2

Factorization algebras

CHAPTER 6

Factorization algebras: definitions and constructions

Our definition of a prefactorization algebra is closely related to that of a precosheafor of a presheaf. Mathematicians have found it useful to refine the axioms of a presheafto those of a sheaf: a sheaf is a presheaf whose value on a large open set is determined,in a precise way, by values on arbitrarily small subsets. In this chapter we describe asimilar “descent” axiom for prefactorization algebras. We call a prefactorization algebrasatisfying this axiom a factorization algebra.

After defining this axiom, our next task is to verify that the examples we have con-structed so far, such as the observables of a free field theory, satisfy it. This we do insections 6.3 and 6.4.

Philosophically, our descent axiom for factorization algebras is important: a prefac-torization algebra satisfying descent (i.e., a factorization algebra) is built from local data,in a way that a general prefactorization algebra need not be. However, for many practicalpurposes, such as the applications to field theory, this axiom is often not essential.

Thus, a reader with little taste for formal mathematics could skip this part and still beable to follow the rest of this book.

6.1. Factorization algebras

A factorization algebra is a prefactorization algebra that satisfies a local-to-global ax-iom. This axiom is the analog of the gluing axiom for sheaves; it expresses how the valueson big open sets are determined by the values on small open sets. For sheaves, the gluingaxiom says that for any open set U and any cover of that open set, we can determine thevalue of the sheaf on U from the values on the open cover. For factorization algebras, werequire our covers to be fine enough that they capture all the “multiplicative structure” —the structure maps — of a factorization algebra.

We will describe the local-to-global axiom for factorization algebras taking valuesin vector spaces or chain complexes, but the generalization to an arbitrary symmetricmonoidal category is straightforward. In fact, a factorization algebra will be a cosheafwith respect to a modified notion of cover.

149

150 6. FACTORIZATION ALGEBRAS: DEFINITIONS AND CONSTRUCTIONS

6.1.0.5 Definition. Let U be an open set. A collection of open sets U = Ui | i ∈ I is a Weisscover of U if for any finite collection of points x1, . . . , xk in U, there is an open set Ui ∈ U suchthat x1, . . . , xk ⊂ Ui.

The Weiss covers define a Grothendieck topology on Opens(M), the poset category ofopen subsets of a space M. We call it the Weiss topology of M.

Remark: A Weiss cover is certainly a cover in the usual sense, but a Weiss cover typicallycontains an enormous number of opens. It is a kind of “exponentiation” of the usualnotion of cover, because a Weiss cover is well-suited to studying all configuration spacesof finitely many points in U. For instance, given a Weiss cover U of U, the collection

Uni | i ∈ I

provides a cover in the usual sense of Un ⊂ Mn for every positive integer n. ♦

Example: For a smooth n-manifold M, there is a simple way to construct a Weiss cover forM. Fix a Riemannian metric on M, and consider

B = Br(x) : ∀x ∈ M, with 0 < r < InjRad(x),

the collection of open balls, running over each point x ∈ M, whose radii are less than theinjectivity radius at x. We obtain a Weiss cover by taking the collection of all finite tuplesof disjoint balls in B. Another construction is simply to take the collection of open sets inM diffeomorphic to a disjoint union of finitely many copies of the open n-ball. ♦

The examples above suggest the following.

6.1.0.6 Definition. We say that a cover U = Uα of M generates the Weiss cover V if everyopen V ∈ V is given by a finite disjoint union of opens Uα from U.

6.1.1. Strict factorization algebras. The value of a factorization algebra on U is deter-mined by its behavior on a Weiss cover, just as the value of a cosheaf on an open set U isdetermined by its value on any cover of U.

In order to motivate our definition of factorization algebra, let us write briefly recallthe cosheaf axiom. A precosheaf Φ on M is a cosheaf if, for every open cover Ui | i ∈ Iof an open set U ⊂ M, the sequence⊕

i,j

Φ(Ui ∩Uj)→⊕

k

Φ(Uk)→ Φ(U)

is exact on the right. (Alternatively, one can say the map ⊕kΦ(Uk) → Φ(U) coequalizesthe pair of maps ⊕i,jΦ(Ui ∩Uj) ⇒ ⊕kΦ(Uk).)

6.1. FACTORIZATION ALGEBRAS 151

6.1.1.1 Definition. A prefactorization algebra is a lax factorization algebra if it has the prop-erty: For every open subset U ⊂ M and every Weiss cover Ui | i ∈ I of U, the sequence⊕

i,j

F (Ui ∩Uj)→⊕

k

F (Uk)→ F (U)

is exact on the right. That is, F is a lax factorization algebra if it is a cosheaf with respect to theWeiss topology.

A lax factorization algebra is a strict factorization algebra if, in addition, for every pair ofdisjoint open sets U, V ∈ M, the natural map

F (U)⊗F (V)→ F (U tV)

is an isomorphism.

To summarize, a strict factorization algebra satisfies two conditions: a (co)descentaxiom and a factorization axiom. The descent axiom says that it is a cosheaf with respectto the Weiss topology, and the factorization axiom says that its value on finite collectionsof disjoint opens factors into a tensor product of the values on each open.

6.1.2. The Cech complex and homotopy factorization algebras. Now suppose wehave a prefactorization algebra F taking values in complexes. We will define what itmeans for F to be a homotopy factorization algebra. This will happen when F (U) is quasi-isomorphic to the Cech complex constructed from any Weiss cover.

To motivate the definition, let us first recall the definition of a homotopy cosheaf. LetΦ be a pre-cosheaf on M, and let U = Ui | i ∈ I be a cover of some open subset U of M.The Cech complex of U with coefficients in Φ is is defined in the usual way as

C(U, Φ) =∞⊕

k=1

⊕j1,...,jk∈I

ji all distinct

Φ(Uj1 ∩ · · · ∩Ujk)[k− 1]

where the differential is defined in the usual way. Note that this Cech complex is the nor-malized cochain complex arising from a simplicial cochain complex, where we evaluateΦ on the simplicial space U• associated to the cover U.

We say that Φ is a homotopy cosheaf if the natural map from the Cech complex to Φ(U)is a quasi-isomorphism for every open U ⊂ M and every open cover of U.

6.1.2.1 Definition. A lax homotopy factorization algebra on X is a prefactorization algebraF valued in cochain complexes, with the property that for every open set U ⊂ X and Weiss coverU of U, the natural map

C(U,F )→ F (U)


is a quasi-isomorphism. That is, F is a homotopy cosheaf with respect to the Weiss topology.

A lax homotopy factorization algebra is a homotopy factorization algebra if, in addition, forevery pair U, V of disjoint open subsets of X, the natural map

F (U)⊗F (V)→ F (U tV)

is a quasi-isomorphism.

Remark: The notion of strict factorization algebra is not appropriate for the world of cochaincomplexes. Whenever we refer to a factorization algebra in cochain complexes, we willmean a homotopy factorization algebra. ♦

6.1.3. Factorization algebras valued in a multicategory. The factorization algebrasof ultimate interest to us take values not in the symmetric monoidal category of cochaincomplexes, but in the multicategory of differentiable cochain complexes (section B). Inthis setting, the descent axiom continues to make sense but the factorization axiom doesnot, as we cannot speak about tensoring the values on disjoint opens.

Thus, we need to define what it means to be a factorization algebra in a multicategoryC. We will assume that C is equipped with a realization functor from the category C4 ofsimplicial objects of C to the original category C. This will allow us to define the Cechcomplex of an object of C. (In the category of differentiable cochain complexes, the geo-metric realization of a simplicial object is defined in the same way as it is in the categoryof ordinary cochain complexes.)

We will also assume that C is equipped with some notion of weak equivalence (weakequivalences in differentiable cochain complexes are defined in section B).

As before, let DisjM be the multicategory whose objects are open sets in M and whosemulti-morphisms are defined by

Hom(U1, . . . , Un; V) =

∗ if U1 t . . . Un ⊂ V∅ otherwise.

6.1.3.1 Definition. Let C be a multicategory with the structures listed above. A lax factorizationalgebra F with values in C is a functor DisjM → C with the property that, for all open subsetsU ⊂ M and all Weiss covers U of U, the map

C(U,F )→ F (U)

is a weak equivalence.

Remark: Suppose that C is the multicategory underlying a symmetric monoidal categoryC⊗. Then a factorization algebra valued in C, considered as a multicategory, is the sameas a lax factorization algebra valued in C⊗, considered as a symmetric monoidal category.♦

6.1. FACTORIZATION ALGEBRAS 153

6.1.3.2 Definition. A differentiable factorization algebra is a factorization algebra valued inthe multicategory of differentiable cochain complexes.

Remark: Although the factorization axiom does not directly extend to the setting of multi-categories, we can examine the structure map

f ∈ HomC(F (U),F (V);F (U tV))

for two disjoint opens U and V and ask about the image of f . For almost all the examplesconstructed in this book, one can see that the image is dense with respect to the naturaltopology on F (U t V), which is a cousin of the factorization axiom. In practice, though,it is the descent axiom that is important. ♦

6.1.4. Factorization algebras in quantum field theory. We have seen (section 1.4)how prefactorization algebras appear naturally when one thinks about the structure ofobservables of a quantum field theory. It is natural to ask whether the local-to-global ax-iom which distinguishes factorization algebras from prefactorization algebras also has aquantum-field theoretic interpretation.

The local-to-global axiom we posit states, roughly speaking, that all observables onan open set U ⊂ M can be built up as sums of observables supported on arbitrarily smallopen subsets of M. To be concrete, let us consider a Weiss cover Uε of M, built out of allopen balls in M of radius smaller than ε. Applied to this Weiss cover, our local-to-globalaxiom states that any observable O ∈ Obs(U) can be written as a sum of observables ofthe form O1O2 · · ·Ok, where Oi ∈ Obs(Bδi(xi)) and x1, . . . , xk ∈ M.

By taking ε to be very small, we see that our local-to-global axiom implies that allobservables can be written as sums of products of observables which are supported asclose as we like to points in U.

This is a physically reasonable assumption: most of the observables (or operators)that are considered in quantum field theory textbooks are supported at points, so it mightmake sense to restrict attention to observables built from these.

However, more global observables are also considered in the physics literature. Forexample, in a gauge theory, one might consider the observable which measures the mon-odromy of a connection around some loop in the space-time manifold. How would suchobservables fit into the factorization algebra picture?

The answer reveals a key limitation of our axioms: the concept of factorization algebrais only appropriate for perturbative quantum field theories. Indeed, in a perturbative gaugetheory, the gauge field (i.e., the connection) is taken to be an infinitesimally small pertur-bation A0 + δA of a fixed connection A0, which is a solution to the equations of motion.There is a well-known formula (the time-ordered exponential) expressing the holonomy


of A0 + δA as a power series in δA, where the coefficients of the power series are given asintegrals over Lk, where L is the loop which we are considering.

This expression shows that the holonomy of A0 + δA can be built up from observablessupported at points (which happen to lie on the loop L). Thus, the holonomy observablewill form part of our factorization algebra.

However, if we are not working in a perturbative setting, this formula does not apply,and we would not expect (in general) that the prefactorization algebra of observablessatisfies the local-to-global axiom.

6.2. Locally constant factorization algebras

If M is an n-dimensional manifold, then prefactorization algebras locally bear a resem-blance to En algebras. After all, a prefactorization algebra prescribes a way to combine theelements associated to k distinct balls into an element associated to a big ball containingall k balls. In fact, En algebras form a full subcategory of factorization algebras on Rn.

6.2.0.1 Definition. A factorization algebra F on an n-manifold M is locally constant if foreach inclusion of open sets U ⊂ U′ where U is a deformation retraction of U′, then the mapF (U)→ F (U′) is a quasi-isomorphism of cochain complexes.

A central example is the factorization algebra FA on R given by an associative algebraA.

Lurie [Lurb] has shown the following vast extension of this example.

6.2.0.2 Theorem. There is an equivalence of (∞, 1)-categories between En algebras and locallyconstant factorization algebras on Rn.

We remark that Lurie (and others) uses a different gluing axiom than we do. A carefulcomparison of the different axioms and a proof of their equivalence (for locally constantfactorization algebras) can be found in [Mat].

6.2.1. Many examples of En algebras arise naturally from topology, such as labelledconfiguration spaces (as discussed in the work of Segal, McDuff, Bodigheimer, Salvatore,and Lurie). We will discuss an important example, that of mapping spaces.

Recall that (in the appropriate category of spaces) there is an isomorphism Maps(U tV, X) ∼= Maps(U, X) ×Maps(V, X). This fact suggests that we might fix a target spaceX and define a prefactorization algebra by sending an open set U to Maps(U, X). Thisconstruction almost works, but it is not clear how to “extend” a map f : U → X from U

6.2. LOCALLY CONSTANT FACTORIZATION ALGEBRAS 155

to a larger open set V 3 U. By working with “compactly-supported” maps, we solve thisissue.

Fix (X, p) a pointed space. Let F denote the prefactorization algebra on M sendingan open set U to the space of compactly-supported maps from U to (X, p). (Here, “ fis compactly-supported” means that the closure of f−1(X − p) is compact.) Then F is aprefactorization algebra in the category of pointed spaces. (Composing with the singularchains functor gives a prefactorization algebra in abelian groups, but we will work at thelevel of spaces.)

Note that this prefactorization algebra is locally constant: if U → U′ is an inclusion ofopen subsets where U is a deformation retraction of U′, then the map F(U) → F(U′) is aweak homotopy equivalence.

Note also that there is a natural isomorphism

F(U1 tU2) = F(U1)× F(U2)

if U1, U2 are disjoint.

Let’s consider for a moment the case when M = Rn. If D ⊂ Rn is a ball, there is aequivalence

F(D) ' ΩnpX

between the space of compactly supported maps D → X and the n-fold based loop spaceof X, based at the point p. (We are using the topologist’s notation Ωn

pX for the n-fold loopspace; hopefully, this use does not confuse due to the standard notation for the space ofn-forms.)

To see this equivalence, note that a compactly supported map f : D → X extendsuniquely to a map from the closed ball D, sending the boundary ∂D to the base point pof X. Since Ωn

pX is defined to be the space of maps of pairs (D, ∂D) → (X, p), we haveconstructed the desired map from F(D) to Ωn

pX. It is easily verified that this map is ahomotopy equivalence.

If D1, D2 are disjoint balls contained in a ball D3, the prefactorization structure givesus a map

F(D1)× F(D2)→ F(D3).

These maps correspond to the standard En structure on the n-fold loop space.

For a particularly nice example, consider the case where the source manifold is M =R. The structure maps of F then describe the standard concatentation product on thespace ΩpX of based loops in X. At the level of components, we recover the standardproduct on π0ΩX = π1(X, x).


This prefactorization algebra F does not always satisfy the gluing axiom. However,Salvatore [Sal99] and Lurie [Lurb] have shown that if X is sufficiently connected, thisprefactorization algebra is in fact a factorization algebra.

6.2.2. The direct relationship between En algebras and locally constant factorizationalgebras on Rn raises the question of what the local-to-global axiom means from an alge-braic point of view. Evaluating a locally constant factorization algebra on a manifold M isknown as factorization homology or topological chiral homology.

The following result, for n = 1, is striking and helpful.

6.2.2.1 Theorem. For A an E1 algebra (e.g., an associative algebra), there is a weak equivalence

FA(S1) ' HH∗(A),

where FA denotes the locally constant factorization algebra on R associated to Aand HH∗(A)denotes the Hochschild homology of A.

Here HH∗(A) means any cochain complex quasi-isomorphic to the usual bar complex(i.e., we are interested in more than the mere cohomology groups).

This result has several proofs in the literature, depending on choice of gluing axiomand level of generality (for instance, one can work with algebra objects in more general∞-categories). It is one of the primary motivations for the higher dimensional generaliza-tions.

There is a generalization of this result even in dimension 1. Note that there are moregeneral ways to extend FA from the real line to the circle, by allowing “monodromy.”Let σ denote an automorphism of A. Pick an orientation of S1 and fix a point p in S1.Let F σ

A denote the prefactorization algebra on S1 such that on S1 − p it agrees with FAbut where the structure maps across p use the automorphism σ. For instance, if L is asmall interval to the left of p, R is a small interval to the right of p, and M is an intervalcontaining both L and R, then the structure map is

A⊗ A → Aa⊗ b 7→ a⊗ σ(b)

where the leftmost copy of A corresponds to L and so on. It is natural to view the copyof A associated to an interval containing p as the A− A bimodule Aσ where A acts as theleft by multiplication and the right by σ-twisted multiplication.

6.2.2.2 Theorem. There is a weak equivalence F σA(S

1) ' HH∗(A, Aσ).

Beyond the one-dimensional setting, some of the most useful insights into the mean-ing of factorization homology arise from its connection to the cobordism hypothesis and

6.3. FACTORIZATION ALGEBRAS FROM COSHEAVES 157

extended topological field theories. See section 4 of [Lur09b] for an overview of theseideas, and [Sch] for further developments and detailed proofs.

For a deeper discussion of factorization homology than we’ve given, we recommend[Fra] to start, as it combines a clear overview with a wealth of applications. A lovelyexpository account is [Gin]. Locally constant factorization algebras already possess asubstantial literature, as they sit at the nexus of manifold topology and higher algebra.See, for instance, [And], [AFT], [Fra13], [GTZb], [GTZa], [Hora], [Horb], [Lurb], [Mat],and [MW12].

6.3. Factorization algebras from cosheaves

The goal of this section is to describe a natural class of factorization algebras. Thefactorization algebras that we construct from classical and quantum field theory will beclosely related to the factorization algebras discussed here.

The main result of this section is that, given a nice cosheaf of vector spaces or cochaincomplexes F on a manifold M, the functor Sym∗ F : U 7→ Sym(F(U)) is a factorizationalgebra. It is clear how this functor is a prefactorization algebra (see example ??); the hardpart is verifying that it satisfies the local-to-global axiom. The examples in which we areultimately interested arise from cosheaves F that are compactly supported sections of avector bundle, so we focus on cosheaves of this form.

We begin by providing the definitions necessary to state the main result of this section.We then state the main result and explain its role for the rest of the book. Finally, we provethe lemmas that culminate in the proof of the main result.

6.3.1. Preliminary definitions.

6.3.1.1 Definition. A local cochain complex on M is a graded vector bundle E on M (withfinite rank), whose smooth sections will be denoted by E , equipped with a differential operatord : E → E of cohomological degree 1 satisfying d2 = 0.

Recall the notation from section 3.5. For E be a local cochain complex on M and U anopen subset of M, we use E (U) to denote the cochain complex of smooth sections of E onU, and we use Ec(U) to denote the cochain complex of compactly supported sections of Eon U. Similarly, let E (U) denote the distributional sections on U and let E c(U) denote thecompactly supported distributional sections of E on U. In the appendix (B) it is shownthat these four cochain complexes are differentiable cochain complexes in a natural way.


We use E! = E⊗DensM to denote the appropriate dual object. We give E! the differ-ential that is the formal adjoint to d on E. Note that, ignoring the differential, E c(U) is thecontinuous dual to E !(U) and that Ec(U) is the continuous dual to E

!(U).

The factorization algebras we will discuss are constructed from the symmetric algebraon the vector spaces Ec(U) and E

!c(U). Note that, since E

!c(U) is dual to E (U), we can

view Sym E!c(U) as the algebra of formal power series on E (U). Thus, we often write

Sym E!c(U) = O(E (U)),

because we view this algebra as the space of “functions on E (U).”

6.3.2. Note that if U → V is an inclusion of open sets in M, then there are naturalmaps of commutative dg algebras

Sym∗ Ec(U)→ Sym∗ Ec(V)

Sym∗Ec(U)→ Sym

∗Ec(V)

Sym∗ E c(U)→ Sym∗ E c(V)

Sym∗E c(U)→ Sym

∗brEc(V).

Thus, each of these symmetric algebras forms a precosheaf of commutative algebras, andthus a prefactorization algebra. We denote these prefactorization algebras by Sym∗ Ec andso on.

6.3.3. The main result of this section is the following.

6.3.3.1 Theorem. We have the following parallel results for vector bundles and local cochaincomplexes.

(1) Let E be a vector bundle on M. Then(a) Sym∗ Ec and Sym∗ E c are strict (non-homotopical) factorization algebras valued in

the category of differentiable vector spaces, and(b) Sym

∗Ec and Sym

∗E c are strict (non-homotopical) factorization algebras valued in

the category of differentiable pro-vector spaces.(2) Let E be a local cochain complex on M. Then

(a) Sym∗ Ec and Sym∗ Ec are homotopy factorization algebras valued in the category ofdifferentiable cochain complexes, and

(b) Sym∗Ec and Sym

∗Ec are homotopy factorization algebras valued in the category of

differentiable pro-cochain complexes.

6.3. FACTORIZATION ALGEBRAS FROM COSHEAVES 159

PROOF. Let us first prove the strict (non-homotopy) version of the result. To startwith, consider the case of Sym∗ Ec. We need to verify the local-to-global axiom (section6.1).

Let U be an open set in M and let U = Ui | i ∈ I be a Weiss cover of U. We need toprove that Sym∗ Ec(U) is the cokernel of the map⊕

i,j∈I

Sym∗(Ec(Ui ∩Ui))→⊕i∈I

Sym∗(Ec(Ui)).

This map is compatible with the decomposition of Sym∗ Ec(U) into symmetric powers.Thus, it suffices to show that

Symm Ec(U) = coker(⊕i,j∈I Symm(Ec(Ui ∩Uj))→ ⊕i∈I Symm Ec(Ui)

),

for all m.

Now, observe thatEc(U)⊗m = E m

c (Um)

where E mc is the cosheaf on Um of compactly supported smooth sections of the vector

bundle Em, which denotes the external product of E with itself m times.

Thus it is enough to show that

E mc (Um) = coker

⊕i,j∈I

E mc((Ui ∩Uj)

m)→⊕i∈I

E mc (Um

i )

.

Our cover U is a Weiss cover. This means that, for every finite set of points x1, . . . , xk ∈ M,we can find an open Ui in the cover U containing every xj. This implies that the subsets ofUm of the form (Ui)

m, where i ∈ I, cover Um. Further,

(Ui)m ∩ (Uj)

m = (Ui ∩Uj)m.

The desired isomorphism now follows from the fact that E mc is a cosheaf on Mm.

The same argument applies to show that Sym∗ E !c is a factorization algebra. In the

completed case, essentially the same argument applies, with the subtlety (see B) that,when working with pro-cochain complexes, the direct sum is completed.

For the homotopy case, the argument is similar. Let U = Ui | i ∈ I be a Weiss coverof an open subset U of M. Let F = Sym∗ Ec denote the prefactorization algebras we areconsidering (the argument will below will apply when we use the completed symmetricproduct or use E c instead of Ec). We need to show that map

C(U,F )→ F (U)

is an equivalence, where the left hand side is equipped with the standard Cech differential.


Let Fm(U) = Symm Ec. Both sides of the displayed equation above split as a directsum over m, and the map is compatible with this splitting. (If we use the completedsymmetric product, this decomposition is as a product rather than a sum.)

We thus need to show that the map⊕i1,...,in

Symm (Ui1 ∩ · · · ∩Uin) [n− 1]→ Symm (Ec(U))


For i ∈ I, we get an open subset Umi ⊂ Um. Since U is a Weiss cover of U, these open

subsets form a cover of Um. Note that

Umi1 ∩ · · · ∩Um

in= (Ui1 ∩ · · · ∩Uin)

m .

Note that Ec(U)⊗m can be naturally identified with Γc(Um, Em) (where the tensor productis the completed projective tensor product).

Thus, to show that the Cech descent axiom holds, we need to verify that the map⊕i1,...,in∈PI

Γc(Um

i1 ∩ · · · ∩Umin

, Em) [n− 1]→ Γc(Vm, Em)

is a quasi-isomorphism. The left hand side above is the Cech complex for the cosheaf ofcompactly supported sections of Em on Vm. Standard partition-of-unity arguments showthat this map is a weak equivalence.

6.4. Factorization algebras from local Lie algebras

We just showed that for a local cochain complex E, the prefactorization algebra SymEcis, in fact, a factorization algebra. What this construction says is that the “functions onE ,” viewed as a space, satisfy a locality condition on the manifold M over which E lives.We can reconstruct functions about E (M) from knowing about functions on E with verysmall support on M. But E is a simple kind of space, as it is linear in nature. (We shouldremark that E is a simple kind of derived space because it is a cochain complex.) We nowextend to a certain type of nonlinear situation.

In section 3.6, we introduced the notion of a local dg Lie algebra. Since a local cochaincomplex is an Abelian local dg Lie algebra, we might hope that the Chevalley-Eilenbergcochain complex C∗L of a local Lie algebra L also forms a factorization algebra. As weexplain in chapter 10, a local dg Lie algebra can be interpreted as a derived space thatis nonlinear in nature. In this setting, the Chevalley-Eilenberg cochain complex are the“functions” on this space. Hence, if C∗L is a factorization algebra, we would know thatfunctions on this nonlinear space L can also be reconstructed from functions localized onthe manifold M.

6.4. FACTORIZATION ALGEBRAS FROM LOCAL LIE ALGEBRAS 161

In section 3.6, we also constructed an important class of prefactorization algebras:the factorization envelope of a local dg Lie algebra. Again, in the preceding section, weshowed Sym Ec is a factorization algebra, which we can view as the factorization enve-lope of the Abelian local Lie algebra E [−1]. Thus, we might expect that the factorizationenvelope of a local Lie algebra satisfies the local-to-global axiom.

We will now demonstrate that both these Lie-theoretic constructions are factorizationalgebras.

6.4.0.2 Theorem. Let L be a local dg Lie algebra on a manifold M. Then the prefactorizationalgebras

UL :U 7→ C∗(Lc(U))

OL :U 7→ C∗(L(U))

are factorization algebras.

Remark: The argument below applies, with very minor changes, to a local L∞ algebra, amodest generalization we introduce later (see definition ??). ♦

PROOF. The proof is a spectral sequence argument, and we will reuse this idea through-out the book (notably in proving the quantum observables form a factorization algebra).

We start with the factorization envelope. Note that for any dg Lie algebra g, theChevalley-Eilenberg chains C∗g have a natural filtration Fn = Sym≤n(g[1]) compatiblewith the differential. The first page of the associated spectral sequence is simply the co-homology of Sym(g[1]), where g is viewed as a cochain complex rather than a Lie algebra(i.e., we extend the differential on g[1] as a coderivation to the cocommutative coalgebraSym(g[1])).

Consider the Cech complex of UL with respect to some Weiss cover U for an open Uin M. Applying the filtration above to each side of the map

C(U, UL)→ UL(U),

we get a map of spectral sequences. On the first page, we have a quasi-isomorphism bytheorem 6.3.3.1. Hence the original map of filtered complexes is a quasi-isomorphism.

Now we provide the analogous argument for OL. For any dg Lie algebra g, the

Chevalley-Eilenberg cochains C∗g have a natural filtration Fn = Sym≥n

(g∨[−1]) com-patible with the differential. The first page of the associated spectral sequence is simplythe cohomology of Sym(g∨[−1]), where we view g∨[−1] as a cochain complex and extendits differential as a derivation to the completed symmetric algebra.


Consider the Cech complex of OL with respect to some Weiss cover U for an open Uin M. Applying the filtration above to each side of the map

C(U, OL)→ OL(U),

we get a map of spectral sequences. On the first page, we have a quasi-isomorphism bytheorem 6.3.3.1. Hence the original map of filtered complexes is a quasi-isomorphism.

6.5. Some examples of computations

The examples at the heart of this book have an appealing aspect: it is straightforwardto compute the global sections of the factorization algebra – its factorization homology –because we do not need to use the gluing axiom directly. The theorems in the precedingsections allow us to use analysis to compute the colimit of the complicated diagram.

In this section, we compute the global sections of several examples we’ve alreadystudied in the preceding chapters.

6.5.1. Enveloping algebras. Let g be a Lie algebra and let gR denote the local Liealgebra Ω∗R ⊗ g on the real line R. Recall proposition 3.4.0.1 which showed that the fac-torization envelope UgR recovers the universal enveloping algebra Ug.

This factorization algebra UgR is also defined on the circle.

6.5.1.1 Proposition. There is a weak equivalence

UgR(S1) ' C∗(g, Ugad) ' HH∗(Ug),

where in the middle we mean the Lie algebra homology complex for Ug as a g-module via theadjoint action

v · a = va− av,where v ∈ g and a ∈ Ug and va denotes multiplication in Ug.

The second equivalence is a standard fact about Hochschild homology (see, e.g., [Lod98]).It arises by constructing the natural map from the Chevalley-Eilenberg complex to theHochschild complex, which is a quasi-isomorphism because the filtration (inherited fromthe tensor algebra) induces a map of spectral sequences that is an isomorphism on the E1page.

Note one interesting consequence of this theorem: the structure map for an intervalmapping into the whole circle corresponds to the trace map Ug→ Ug/[Ug, Ug].

PROOF. The wonderful fact here is that we do not need to pick a Weiss cover and workwith the Cech complex. Instead, we simply need to examine C∗(Ω∗(S1)⊗ g).

6.5. SOME EXAMPLES OF COMPUTATIONS 163

One approach is to use the natural spectral sequence arising from the filtration Fk =

Sym≤k. The E1 page is C∗(g, Ug), and one must verify that there are no further pages.

Alternatively, recall that the circle is formal, so Ω∗(S1) is quasi-isomorphic to its coho-mology H∗(S1) as a dg algebra. Thus, we get a homotopy equivalence of dg Lie algebras

Ω∗(S1)⊗ g ' g⊕ g[−1],

where, on the right, g acts by the (shifted) adjoint action on g[−1]. Now,

C∗(g⊕ g[−1]) ∼= C∗(g, Sym g),

where Sym g obtains a g-module structure from the Chevalley-Eilenberg homology com-plex. Direct computation verifies this action is precisely the adjoint action of g on Ug (weuse, of course, that Ug and Sym g are isomorphic as vector spaces).

6.5.2. The free scalar field in dimension 1. Recall from section 4.3 that the Weyl alge-bra is recovered from the factorization algebra of quantum observables Obsq for the freescalar field on R. We know that the global sections of Obsq on a circle should thus havesome relationship with the Hochschild homology of the Weyl algebra, but we will see thatthe relationship depends on the ratio of the mass to the radius of the circle.

For simplicity, we restrict attention to the case where S1 has a rotation-invariant met-ric, radius r, and total length L = 2πr. Let S1

L denote this circle R/ZL. We also have ourobservables live over C rather than R.

6.5.2.1 Proposition. If the mass m = 0, then Hk Obsq(S1L) is C[h] for k = 0,−1 and vanishes

for all other k.

If the mass m satisfies mL = in for some integer n, then Hk Obsq(S1L) is C[h] for k = 0,−2

and vanishes for all other k.

For all other values of mass m, H0 Obsq(S1L)∼= C[h] and all other cohomology groups are

zero.

Hochschild homology with monodromy (recall theorem 6.2.2.2) provides an expla-nation for this result. The equations of motion for the free scalar field locally have atwo-dimensional space of solutions, but on a circle the space of solutions depends on therelationship between the mass and the length of the circle. In the massless case, a constantfunction is always a solution, no matter the length. In the massive case, there is either atwo-dimensional space of solutions (for certain imaginary masses, because our conven-tions) or a zero-dimensional space. Viewing the space of local solutions as a local system,we have monodromy around the circle, determined by the Hamiltonian.


When the monodromy is trivial (so mL ∈ iZ), we are simply computing the Hochschildhomology of the Weyl algebra. Otherwise, we have a nontrivial automorphism of theWeyl algebra.

PROOF. We directly compute the global sections, in terms of the analysis of the localLie algebra

L =

(C∞(S1

L)4+m2

−−−→ C∞(S1L)[−1]

)⊕Ch

where h has cohomological degree 1 and the bracket is

[φ0, φ1] = h∫

S1L

φ0φ1

with φk a smooth function with cohomological degree k = 0 or 1. We need to computeC∗(L).

Fourier analysis allows us to make this computation easily. We know that the expo-nentials eikx/L, with k ∈ Z, form a topological basis for smooth functions on S1 and that

(4+ m2)eikx/L =

(k2

L2 + m2)

eikx/L.

Hence, for instance, when m2L2 is not a square integer, it is easy to verify that the complex

C∞(S1L)4+m2

−−−→ C∞(S1L)[−1]

has trivial cohomology. Thus, H∗L = Ch and so H∗(C∗L) = C[h], where in the Chevalley-Eilenberg complex h is shifted into degree 0.

When m = 0, we have

H∗L = Rx⊕Rξ ⊕Rh,

where x represents the constant function in degree 0 and ξ represents the constant func-tion in degree 1. The bracket on this Lie algebra is

[x, ξ] = hL.

Hence H0(C∗L) ∼= C[h] and H−1(C∗L) ∼= C[h], and the remaining cohomology groupsare zero.

When m = in/L for some integer n, we see that the operator 4 + m2 has two-dimensional kernel and cokernel, both spanned by e±inx/L. By a parallel argument to thecase with m = 0, we see H0(C∗L) ∼= C[h] ∼= H−2(C∗L) and all other groups are zero.


6.5.3. The Kac-Moody factorization algebras. Recall from example ?? and section ??that there is a factorization algebra on a Riemann surface Σ associated to every affine Kac-Moody Lie algebra. For g a simple Lie algebra with symmetric invariant pairing 〈−,−〉g,we have a shifted central extension of the local Lie algebra Ω0,∗

Σ ⊗ g with shift

ω(α, β) =∫

Σ〈α, ∂β〉g .

Let UωgΣ denote the twisted factorization envelope for this extended local Lie algebra.

6.5.3.1 Proposition. Let g denote the genus of a closed Riemann surface Σ. Then

H∗(Uσωg(Σ)) ∼= H∗(g, Ug⊗g)[c]

where c denotes a parameter of cohomological degree zero.

PROOF. We need to understand the Lie algebra homology

C∗(Ω0,∗(Σ)⊗ g⊕Cc),

whose differential has the form ∂+dLie. (Note that in the Lie algebra, c has cohomologicaldegree 1.)

Consider the filtration Fk = Sym≤k by polynomial degree. The purely analytic piece∂ preserves polynomial degree while the Lie part dLie lowers polynomial degree by 1.Thus the E1 page of the associated spectral sequence has Sym(H∗(Σ, O)⊗ g⊕ Cc) as itsunderlying graded vector space. The differential now depends purely on the Lie-theoreticaspects. Moreover, the differential on further pages of the spectral sequence are zero.

In the untwisted case (where there is no extension), we find that the E1 page is pre-cisely

H∗(g, Ug⊗g),

by computations directly analogous to those for proposition 6.5.1.1.

In the twisted case, the only subtlety is to understand what happens to the extension.Note that H0(Ω0,∗(Σ) is spanned by the constant functions. The pairing ω vanishes if aconstant function is an input, so we know that the central extension does not contributeto the differential on the E1 page.

An alternative proof is to use the fact the Ω0,∗(Σ) is homotopy equivalent to its coho-mology as a dg algebra. Hence we can compute C∗(H∗(Σ, O)⊗ g⊕Cc) instead.

We can use the ideas of sections 4.5 and ?? to understand the “correlation functions”of this factorization algebra Uωg

Σ. More precisely, given a structure map

UωgΣ(V1)⊗ · · · ⊗Uωg

Σ(Vn)→ UωgΣ(Σ)


for some collection of disjoint opens V1, . . . , Vn ⊂ Σ, we will provide a method for describ-ing the image of this structure map. It has the flavor of a Wick’s formula.

Let [O] denote the cohomology class of an element O of UωgΣ. We want to encode

relations between cohomology classes.

As Σ is closed, Hodge theory lets us construct an operator ∂−1

, which vanishes onthe harmonic functions and (0,1)-forms but provides an inverse to ∂ on the complemen-tary spaces. (We must make a choice of Riemannian metric, of course, to do this.) Nowconsider an element

a1 · · · ak

of cohomological degree 0 in UωgΣ(Σ), where each aj lives in

Ω0,1(Σ)⊗ g⊕Cc.

Then we have

(∂ + dLie)((∂−1

a1)a2 · · · ak

)= ∂(∂

−1a1)a2 · · · ak + dLie

((∂−1

a1)a2 · · · ak

)= ∂(∂

−1a1)a2 · · · ak +

k

∑j=2

[∂−1

a1, aj]a2 · · · aj · · · ak

+k

∑j=2

∫Σ

⟨∂−1

a1, ∂aj

⟩g

c a2 · · · aj · · · ak.

When a1 is in the complementary space to the harmonic forms, we know ∂∂−1

a1 = a1 sowe have the relation

0 = [a1 · · · ak] +k

∑j=2

[[∂−1

a1, aj]a2 · · · aj · · · ak] +k

∑j=2

∫Σ

⟨∂−1

a1, ∂aj

⟩g[c a2 · · · aj · · · ak],

which allows us to iteratively reduce the “polynomial” degree of the original term [a1 · · · ak](from k to k− 1 or less, in this case).

This relation looks somewhat complicated but it is easy to understand for k small. Forinstance, in k = 1, we see that

0 = [a1]

whenever a1 is in the space orthogonal to the harmonic forms. For k = 2, we get

0 = [a1a2] + [[∂−1

a1, a2]] +∫

Σ

⟨∂−1

a1, ∂a2

⟩g[c].


6.5.4. The free βγ system. In section 5.3 we studied the local structure of the freeβγ theory; in other words, we carefully examined the simplest structure maps for thefactorization algebra Obsq of quantum observables on the plane C. It is natural to askabout the global sections on a closed Riemann surface.

There are many ways, however, to extend this theory to a Riemann surface Σ. Let Lbe a holomorphic line bundle on Σ. Then the L-twisted free βγ system has fields

E = Ω0,∗(Σ,L)⊕Ω1,∗(Σ,L∨),where L∨ denotes the dual line bundle. The −1-symplectic pairing on fields is to applypointwise the evaluation pairing between L and L∨ and then to integrate the resultingdensity. The differential is the ∂ operator for these holomorphic line bundles.

Let ObsqL denote the factorization algebra of quantum observables for the L-twisted

βγ system. Locally on Σ, we know how to understand the structure maps: pick a trivial-ization of L and employ our work from section 5.3.

6.5.4.1 Proposition. Let bk = dim Hk(Σ,L). Then H∗ObsqL(Σ) is a rank-one free module over

C[h] and concentrated in degree −b0 − b1.

PROOF. As usual, we use the spectral sequence arising from the filtration Fk = Sym≤k

on Obsq. The first page just depends on the cohomology with respect to ∂, so the under-lying graded vector space is

Sym(

H∗(Σ,L)[1]⊕ H∗(Σ,L∨ ⊗Ω1hol)[1]

)[h],

where Ω1hol denotes the holomorphic cotangent bundle on Σ. By Serre duality, we know

this is the symmetric algebra on a +1-symplectic vector space, concentrated in degrees0 and −1 and with dimension b0 + b1 in each degree. The remaining differential in thespectral sequence is the BV Laplacian for the pairing, and so the cohomology is spannedby the maximal purely odd element, which has degree −b0 − b1.

CHAPTER 7

Formal aspects of factorization algebras

7.1. Pushing forward factorization algebras

A crucial feature of factorization algebras is that they push forward nicely. Let Mand N be topological spaces admitting Weiss covers and let f : M → N be a continuousmap. Given a Weiss cover U = Uα of an open U ⊂ N, let f−1U = f−1Uα denotethe preimage cover of f−1U ⊂ M. Observe that f−1U is Weiss: given a finite collection ofpoints x1, . . . , xn in f−1U, the image points f (x1), . . . , f (xn) are contained in some Uα

in U and hence f−1Uα contains the xj.

7.1.0.2 Definition. Given a factorization algebraF on a space M and a continuous map f : M→N, the pushforward factorization algebra f∗F on N is defined by

f∗F (U) = F ( f−1(U)).

Note that for the map to a point f : M → pt, the pushforward factorization algebraf∗F is simply the global sections of F . We also call this the factorization homology of F onM.

7.2. The category of factorization algebras

In this section, we explain how prefactorization algebras and factorization algebrasform categories. In fact, they naturally form multicategories (or colored operads). We alsoexplain how these multicategories are enriched in simplicial sets when the (pre)factorizationalgebras take values in cochain complexes.

7.2.1. Morphisms and the category structure.

7.2.1.1 Definition. A morphism of prefactorization algebras φ : F → G consists of a mapφU : F(U) → G(U) for each open U ⊂ M, compatible with the structure maps. That is, for anyopen V and any finite collection U1, . . . , Uk of pairwise disjoint open sets, each contained in V, the

169

170 7. FORMAL ASPECTS OF FACTORIZATION ALGEBRAS

following diagram commutes:

F(U1)⊗ · · · ⊗ F(Uk)φU1⊗···⊗φUk−→ G(U1)⊗ · · · ⊗ G(Uk)

↓ ↓F(V)

φV−→ G(V)

.

Likewise, all the obvious associativity relations are respected.

Remark: When our prefactorization algebras take values in cochain complexes, we requirethe φU to be cochain maps, i.e., they each have degree 0 and commute with the differen-tials. When our prefactorization algebras take values in differentiable cochain complexes,we require in addition that the maps φU are smooth. ♦

7.2.1.2 Definition. On a space X, we denote the category of prefactorization algebras on X takingvalues in the multicategory C by PreFA(X, C). The category of factorization algebras, FA(X, C),is the full subcategory whose objects are the factorization algebras.

In practice, C will normally be the multicategory of differentiable cochain complexes.

7.2.2. The multicategory structure. Let SC denote the universal symmetric monoidalcategory containing the multicategory C. Any prefactorization algebra valued in C givesrise to one valued in SC.

There is a natural tensor product on PreFA(X, SC), as follows. Let F, G be prefactor-ization algebras. We define F⊗ G by

F⊗ G(U) = F(U)⊗ G(U),

and we simply define the structure maps as the tensor product of the structure maps. Forinstance, if U ⊂ V, then the structure map is

F(U ⊂ V)⊗ G(U ⊂ V) : F⊗ G(U) = F(U)⊗ G(U)→ F(V)⊗ G(V) = F⊗ G(V).

7.2.2.1 Definition. Let PreFAmc(X, C) denote the multicategory arising from the symmetricmonoidal product on PreFA(X, SC). That is, if Fi, G are prefactorization algebras valued in C,we define the set of multi-morphisms by

PreFAmc(F1, · · · , Fn | G)

to be the set of maps of SC-valued prefactorization algebras

F1 ⊗ · · · ⊗ Fn → G.

Factorization algebras inherit this multicategory structure.

7.2. THE CATEGORY OF FACTORIZATION ALGEBRAS 171

7.2.3. Enrichment over simplicial sets. In this subsection we will explain how themulticategory of factorization algebras with values in an appropriate multicategory issimplicially enriched, in a natural way. In order to define this simplicial enrichment, weneed to introduce some notation.

The first thing to define is the algebra Ω∗(X) of smooth forms on a simplicial set X.An element ω ∈ Ωi(X) consists an i-form f ∗Ωi(4n) for every n-simplex f : 4n → Xsatisfying the condition that for σ : 4m → 4n a face or degeneracy map, we have theequality σ∗ f ∗ω = ( f σ)∗ω.

If V is a differentiable cochain complex, we can define a complex

Ω∗(X, V) = Ω∗(X)⊗C∞(X) C∞(X, V)

where C∞(X, V) refers to the cochain complex of smooth maps from X to V. In this way,we see that the multicategory of differentiable cochain complexes is tensored over theopposite category SSetop to the category of simplicial sets.

This allows us to lift the multicategory DVS of differentiable cochain complexes to asimplicially enriched multicategory, where we define the n-simplices in the simplicial setof multimorphisms by

Hom(V1, . . . , Vn |W)[n] = Hom(V1, . . . , Vn | Ω∗(4n, W)),

where on the right hand side Hom denotes smooth multilinear cochain maps

V1 × · · · ×Vn → Ω∗(4n, W)

which are compatible with differentials.

Now, in general, suppose we have a multicategory C which is tensored over SSetop.Then the multicategory PreFA(X, C) is also tensored over SSetop: the tensor product F Xof a prefactorization algebra F with a simplicial set X is defined by

(F X)(U) = F(U) X.

Then, we can lift our multicategory of C-valued prefactorization algebras to a simpliciallyenriched multicategory, by defining

PreFA4mc(F1, . . . , Fn | G) = PreFA4mc(F1, . . . , Fn | G X).

In particular, we see that the multicategory of prefactorization algebras valued in differ-entiable cochain complexes is simplicially enriched.

7.2.4. Equivalences. In the appendix B, we define a notion of weak equivalence or quasi-isomorphism of differentiable cochain complexes.

7.2.4.1 Definition. Let F, G be factorization algebras valued in cochain complexes. Let φ : F →G be a map. We say that φ is a weak equivalence if, for all open subsets U ⊂ M, the mapF(U)→ G(U) is a quasi-isomorphism of cochain complexes.


Similarly, let F, G be factorization algebras valued in differentiable cochain complexes. Wesay a map φ : F → G is a weak equivalence if, for all U, the map F(U) → G(U) is a weakequivalence of differentiable cochain complexes.

We now provide an explicit criterion for checking weak equivalences, using the notionof a factorizing basis (see definition 7.4.0.2).

7.2.4.2 Lemma. A map F→ G between differentiable factorization algebras is a weak equivalenceif and only if, for every factorizing basis U of X and every U in U, the map

F(U)→ G(U)


PROOF. For any open subset V ⊂ X, let UV denote the Weiss cover of V generated byall open subsets in U that lie in V. By the descent axiom, the map

C(UV , F)→ F(V)

is a weak equivalence, and similarly for G. Thus, it suffices to check that the map

C(UV , F)→ C(UV , G)

is a weak equivalence, for the following reason. Every Cech complex has a natural filtra-tion by number of intersections. We thus obtain of spectral sequences from the map ofCech complexes. The fact that the maps

F(U1 ∩ · · · ∩Uk)→ G(U1 ∩ · · · ∩Uk)

are all weak equivalences implies that we have a quasi-isomorphism on the first page ofthe map of spectral sequences, so our original map is a quasi-isomorphism.

7.3. Descent

Strict factorization algebras – those F for which F (U t V) ' F (U)⊗ F (V) for anypair of disjoint opens – satisfy an a priori different gluing axiom, which is operadic innature. This axiom illuminates the origins and meaning of the Weiss topology, and italso insures that strict factorization algebras satisfy a version of descent with respect tothe usual topology. For the remainder of this section, a factorization algebra will alwaysmean a strict factorization algebra.

We want a factorization algebra to be a prefactorization algebra whose behavior onbig opens is determined by its behavior on small opens. There is a natural operadic wayto phrase this condition, as follows.

7.3. DESCENT 173

Let X be a topological space. Recall from section ?? that DisjX denotes the coloredoperad whose colors are the opens of X and whose operations are

DisjX(V1, . . . , Vn |W) =

∗, if Vj are pairwise disjoint and contained in W∅, else

A prefactorization algebra in a symmetric monoidal category (C,⊗) is an algebra overthis colored operad.

It is straightforward to talk about a prefactorization algebra on “some set of smallopens.” Let U = Uii∈I be a cover with respect to the usual topology (not the Weisstopology). Let DisjU be the colored operad obtained from DisjX by simply restricting tothe opens V such that V is contained in some element Ui of U. (In other words, DisjU is thefull sub-colored operad with objects from the sieve of the cover U.) Then a prefactorizationalgebra subordinate to U is an algebra over the colored operad DisjU.

Let AlgC(DisjX) denote the category of algebras in C for the colored operad DisjX,and let AlgC(DisjU) denote the category of algebras over DisjU. The inclusion of coloredoperads i : DisjU → DisjX induces a pullback (or forgetful) functor

i∗ : AlgC(DisjX)→ AlgC(DisjU)

and, provided C contains the appropriate colimits (we discuss this condition below), a leftadjoint functor

i! : AlgC(DisjU)→ AlgC(DisjX),

which extends a prefactorization algebra subordinate to U to a prefactorization algebra onX. It constructs all the values and structure maps for bigger opens out of the data fromthe opens “smaller than the cover.” Thus, it is an operadic version of gluing.

A natural question is then whether, for a factorization algebra F , the extension i!i∗F isequivalent toF . In other words, we are asking whether we can recoverF just by knowingthe values and structure maps of F on opens subordinate to the cover U. Note that thisoperadic extension a priori has nothing to do with a cosheaf gluing axiom, which onlydepends on the unary operations (i.e., inclusions of opens).

As we are typically interested in the homotopical version of these constructions, wewant to work with derived versions of i∗ and i!. How to provide the necessary derivedreplacements depends on C and one’s taste in homotopical algebra. We momentarilydelay making specific choices and simply denote the derived replacement for i! by Li!.Our goal is thus to verify that

Li!i∗F ' Ffor F a homotopy factorization algebra with values in a good symmetric monoidal cat-egory C. In the remainder of this section, we always mean homotopical factorizationalgebra, unless we say otherwise, although we simply write factorization algebra.


This result states, in particular, that the gluing axiom for the Weiss topology agreeswith the derived operadic left Kan extension. In the course of the proof, it will becomeclear how the Weiss topology is forced to appear by the nature of the structure maps in aprefactorization algebra.

In the next subsection, we will give a precise statement of this result, pinning downconditions on C in particular. In the following subsection we will discuss applications ofthis result, notably how it relates to descent for the category of factorization algebras. Inthe final subsection, we prove the result.

7.3.1. A precise version of the main result. The target symmetric monoidal category(C,⊗) for our prefactorization algebras must have several properties for our proof towork. We want a well-behaved, homotopical version of colimit and clean interaction withthe tensor product, so that it is sensible to talk about a derived or homotopical operadicleft Kan extension Li!F of F ∈ AlgC(DisjU). What we need for the result, in particular,is that this derived operadic extension Li!F is weakly equivalent to the derived left Kanextension L Lani F merely as a functor.

In order to make a precise statement, we must choose some formalism for higher cat-egories, and here we will use model categories. (When we give the proof, we will firstexplain its structure, so that the reader can formulate a version in an alternative formal-ism.) We depend on the machinery developed by Berger and Moerdijk [BM07] for work-ing with algebras over colored operads in a homotopical fashion. We also follow closelythe discussion of these issues in [Hora], which pins down the relevant details.

Following [Hora], we require C to be a cofibrantly generated symmetric monoidalsimplicial model category. We want it to have the following additional properties.

• For any colored operad O, the algebras AlgC(O) obtain a model category struc-ture in which weak equivalences and fibrations are objectwise. That is, for exam-ple, a map of O-algebras F : A → B is a weak equivalence if F(c) : A(c) → B(c)is a weak equivalence for every color c in O.• For any map of colored operads j : O → P , the adjunction

j! : AlgC(O) AlgC(P) : j∗

is a Quillen adjunction.• Let Col(O) denote the discrete category associated to O, where the objects are

simply the colors ofO and the only morphisms are identities. The forgetful func-tor from AlgC(O) to AlgC(Col(O)) preserves cofibrations.

Following Horel, we say such a category C has a good theory of algebras. As Horel shows,C has a good theory of algebras if C is also a left proper model category with a monoidalfibrant replacement functor and a cofibrant unit.

7.3. DESCENT 175

Remark: These conditions insure a good theory of algebras for all colored operads, whereaswe are only working here with a rather restricted class of colored operads and maps. Butthese conditions certainly suffice to obtain our result, and they are fairly natural. ♦

We now state the main result, using the notations from above.

7.3.1.1 Theorem. Let C have a good theory of algebras. Let X be a topological space and U anordinary cover. For F a homotopy factorization algebra on a space X that is not lax, the extensionLi!i∗F is weakly equivalent to F .

In particular, we have Li!i∗F (V) ' F (V) for every open V in X.

In other words, a factorization algebra also satisfies an operadic gluing axiom, in ad-dition to the cosheaf gluing axiom.

7.3.2. Consequences of the theorem. The main application of this theorem is to thequestion of descent. Given an open cover U = Uii∈I of X in the usual sense (i.e., everypoint in X is contained in some Ui), suppose we have a factorization algebra Fi on eachUi, with gluing data on double intersections, coherence data on triple intersections, and soon. In this subsection we will show that we can construct a unique factorization algebraF on X from this descent data.

We will need the following simple construction. If U ⊂ X is an open subset and F isa factorization algebra on X, then the restriction F|U is a factorization algebra on U. Ofcourse, an open embedding iU : U → X of one manifold into another identifies U with anopen subset of X. We will also use i∗UF to denote the “pullback” factorization algebra onU.

The input to our gluing construction is the following. We have, for each finite subsetJ ⊂ I, a factorization algebra FJ on

UJ =⋂j∈J

Uj.

Further, we have weak equivalences

FJ → r∗j FJ\j

of factorization algebras on UJ , for each j ∈ J, where

rj : UJ → UJ\j


is the natural inclusion. Finally, we require that, for every J and every j, j′ ∈ J, the diagramof factorization algebras on UJ

FJ → r∗j FJ\j↓ ↓

r∗j′FJ\j′ → r∗j r∗j′FI\j,j′

commutes.

We can think of this data as defining a factorization algebra Fi for each i ∈ I, togetherwith weak equivalences on double intersections, provided by Fi,j, and with coherencesprovided by the factorization algebras FJ for J ⊂ I of higher cardinality. We call this adescent datum.

For simplicity, we restrict our attention to covers that are locally finite (i.e., each pointin X is contained in only finitely many elements of the cover). Let U = Uii∈I be a locallyfinite cover. For any open W subordinate to the cover, there is then a maximal finite setJ ⊂ I such that

W ⊂ UJ =⋂j∈J

Uj,

by running over all the elements of the cover containing W. We denote that maximalset for W by [W]. Given a descent datum FJ | finite J ⊂ I on a locally finite coverU = Uii∈I , we obtain a factorization algebra F subordinate to U by setting

F (W) = F[W](W).

The structure maps of F use the weak equivalences in the descent datum.

7.3.2.1 Proposition. Given a descent datum FJ | finite J ⊂ I for a factorization algebra on alocally finite cover U, the operadic extension Li!F to a prefactorization algebra on X is a factoriza-tion algebra.

PROOF. Our proof relies on the proofs of theorem 7.3.1.1 and proposition 7.4.0.4, al-though we will try to clearly indicate where we are using ideas from those proofs.

Recall that the sieve of the cover U is the collection of all opens subordinate to U. Weuse U to denote the Weiss cover generated by this sieve: it is the collection of all finitedisjoint unions of opens subordinate to the cover U. Note that it is a factorizing basis (seedefinition 7.4.0.2).

First, observe thatLi!F (U tV) ' F (U)⊗ F (V)

for U and V subordinate to U. This holds because the element [U, V] in S DisjU is homo-topy terminal in the diagram computing the operadic Kan extension. Moreover, following

7.3. DESCENT 177

the argument in the proof of theorem 7.3.1.1, this diagram is equivalent to the Cech gluingdiagram using the Weiss cover for U tV obtained from U.

Thus, Li!F defines a U-factorization algebra, in the sense that it is a factorization alge-bra when restricted to this factorizing basis.

Again, following the proof of theorem 7.3.1.1, we see that the extension of the operadicextension as a U-factorization algebra agrees with the operadic extension Li!F . Hence it is afactorization algebra.

7.3.3. Proof of the theorem. We start by explaining the idea of the proof and thengive a detailed version, using the model category language.

7.3.3.1. As F is a strict factorization algebra, we know that we can recover its valueon any open V from a Weiss cover. Although we have employed Cech diagrams to de-scribe the cosheaf property, it will be convenient here to use the language of sieves instead.Recall that the sieve of a cover V is the partially ordered set of all opens of X subordinate tothe cover. In other words, it is the full subcategory of OpensX over the cover V. As F is acosheaf in the Weiss topology, we know

hocolimSieve(V) F → F (V)

is a weak equivalence for every Weiss cover V of V. (The homotopy colimit can be com-puted by the Cech diagram associated to the cover because the Cech diagram is homotopyterminal in the diagram over the sieve.)

When C is a well-behaved symmetric monoidal category, the derived operadic left Kanextension Li!G(V) can be computed as a homotopy colimit over a diagram built from theinclusion i : DisjU → Disj)X. The key observation is that this diagram is just a fattenedversion of a diagram over a sieve.

To be more precise, let Sieve(U) denote the sieve for the ordinary cover U. The opensin this sieve do not form a Weiss cover, but we can generate a Weiss cover from it. Let Udenote the Weiss cover

V1 t · · · tVk | for each 1 ≤ j ≤ k, Vj ⊂ Ui for some Ui ∈ U.

Let DU denote the diagram such that

Li!i∗F (X) = hocolimDUF .

We will show that the map DU → Sieve(U) is homotopy terminal. Thus, we know the twohomotopy colimits agree. (A parallel argument works for an arbitrary open V rather thanthe whole X. One simply has to work with the natural Weiss cover of V concocted from Uand with the diagram computing Li!i∗F .)


7.3.3.2. We will now explain how to construct the diagram DU. To start, we work atthe 1-categorical level; the derived version comes after.

The first step is to replace colored operads with their symmetric monoidal envelopes.Thus, we can view F as a symmetric monoidal functor F : S DisjX → C, rather than analgebra over a colored operad. This shift of emphasis is just to work with categories ratherthan multicategories.

Note that we will use F for this symmetric monoidal functor, not SF . Likewise, wewill use i for the functor S DisjU → S DisjX, rather than Si.

The functor i! is now the symmetric monoidal left Kan extension. By this we meanthat i! takes as input a symmetric monoidal functor G : DisjU → C and outputs a symmetricmonoidal functor G : DisjX → C together with a monoidal natural transformation

G ⇒ G i,

initial among all such pairs of a symmetric monoidal functor and monoidal natural trans-formation.

Just as the left Kan extension for functors has an objectwise formula if C possessesadequate colimits, the symmetric monoidal left Kan extension also has a nice formula,although we now need the tensor product to play nicely with colimits as well. In fact,Getzler [Get09b] gives conditions under which the left Kan extension as a functor is thesymmetric monoidal left Kan extension:

• C possesses all colimits,• the tensor product preserves colimits on each side (e.g., the functor x ⊗ − pre-

serves colimits for every object x in C), and• the functor i∗ is symmetric monoidal (not just lax symmetric monoidal).

Thus, in many cases, we can simply work with the usual left Kan extension.

For us, the functor i∗ is always symmetric monoidal, since we’re simply restricting toa full subcategory. Hence, we just need C to be a nice symmetric monoidal category, likecochain complexes.

Let us assume that i!G agrees with the usual left Kan extension. Then we know that

i!G(X) = S DisjX(i(−), X)⊗S DisjU G(−)= colim

DU

G

where DU denotes the diagram of all arrows [V1, . . . , Vk]→ X in S DisjX where the source[V1, . . . , Vk] lives in the subcategory S DisjU. In other words, we compute the colimit of Gon the overcategory i ↓ S DisjX.

7.3. DESCENT 179

Remark: This diagram DU looks very similar to the Weiss cover U: in both cases, we areinterested in finite tuples of opens subordinate to the cover U. One could reverse thedirection of our logic in this book and start with the operadic approach and then introducethe Weiss topology, motivated by this colimit diagram. ♦

We want to show that when G = i∗F for F a factorization algebra (in the non-homotopical sense), then there is an equivalence

colimDU

i∗F → colimSieve(U)

F .

This equivalence is the underived version of the theorem, because the colimit on the rightis equivalent to F (X), as F is a cosheaf for the Weiss topology.

First, observe that there is a natural map between the two colimits. An arrow [V1, . . . , Vk]→X in DU consists of a finite tuple of disjoint opens Vj where each is subordinate to the coverU. Hence, we see that the union V1 t · · · tVk is in the Weiss cover U. Denote this map ofdiagrams by t : DU → Sieve(U. Moreover, by hypothesis,

i∗F ([V1, . . . , Vk]) = F (V1)⊗ · · · ⊗ F (Vk) ∼= F (V1 t · · · tVk),

so that we get a map between the colimits.

Second, this map is terminal, so that the colimits agree. To verify this, we need to showthat for any open W in Sieve(U), the undercategory W ↓ t is nonempty and connected.By construction, W is the finite union of opens W1 t · · · tWj that are subordinate to thecover U, so we know that the undercategory is nonempty. Connectedness is also simple:if we have two maps

V1 t · · · tVk ←W → V ′1 t · · · tV ′l ,

then these both factor through ⊔1≤i≤k

1≤j≤l

(Vi ∩V ′j ),

which is also in the image of the map t.

7.3.3.3. The structure of the underived proof carries over easily to the derived setting.Essentially, we replace colimits with homotopy colimits. The technical issues are

(1) to find an analog of Getzler’s result – that the derived left Kan extension is thederived operadic left Kan extension – and

(2) to verify the map of diagrams t is homotopy terminal.

We use model categories to do this, following [Hora].

We use the following result, proposition 2.15, of [Hora].


7.3.3.1 Proposition. Let C have a good theory of algebras and a cofibrant unit. Let G be analgebra over DisjU. Then the derived operadic left Kan extension Li!G is weakly equivalent tothe homotopy left Kan extension of G along i (i.e., viewing G as just a functor, not a symmetricmonoidal functor).

Let QG denote a cofibrant replacement of G as an algebra over DisjU. The value of thehomotopy left Kan extension at X can be computed as

S DisjX(i(−), X)⊗LS DisjU

G(−) = |B•(S DisjX(i(−), X), S DisjU, G)| = |B•(∗, i ↓ X, G)|,

the realization of the usual bar construction. For G = i∗F , this formula is the derivedreplacement for i!i∗F (X).

Similarly, we know that hocolimSieve(U) F , the derived replacement of the other dia-

gram, can be computed using the bar construction |B•(∗, Sieve(U), QF )|.

Recall that there is a natural map t : S DisjU → Sieve(U) sending [V1, . . . , Vk] to V1 t· · · tVk. Once we show it is homotopy terminal, we know that the two homotopy colimitsagree. Given W ∈ Sieve(U), we need to show the undercategory W ↓ t is nonempty andcontractible. We’ve already seen that it is nonempty. Contractibility follows because it iscofiltered: given any finite set of elements in the undercategory, namely

W → V(i)1 t · · · tV(i)

ki

with 1 ≤ i ≤ n, all the maps factor through⊔f∈∏n

i=11,...,ki∩n

i=1V(i)f (i).

Thus the homotopy colimits are equivalent.

7.4. Extension from a basis

Let X be a topological space, and let U be a basis for X, which is closed under takingfinite intersections. It is well-known that there is an equivalence of categories betweensheaves on X and sheaves that are only defined for open sets in the basis U. In this sectionwe will prove a similar statement for lax factorization algebras. (When F is a lax factoriza-tion algebra, the procedure above for the operadic extension does not necessarily apply.The arguments often relied in some way on the factorizing property.)

We begin with a paired set of definitions.

7.4.0.2 Definition. A factorizing basis U = Uii∈I for a space X is a basis for the topology ofX with the following properties:

7.4. EXTENSION FROM A BASIS 181

(1) for every finite set x1, . . . , xn ⊂ X, there exists i ∈ I such that x1, . . . , xn ⊂ Ui;(2) if Ui and Uj are disjoint, then Ui ∪Uj is in U;(3) Ui ∩Uj ∈ U for every Ui and Uj in U.

In particular, a factorizing basis is a Weiss cover of X.

7.4.0.3 Definition. Given a factorizing basis U, a U-prefactorization algebra consists of thefollowing:

(1) for every Ui ∈ U, an object F (Ui);(2) for every finite tuple of pairwise disjoint opens Ui0 , . . . , Uin all contained in Uj — with

all these opens from U — a structure map

F (Ui0)⊗ · · · ⊗ F (Uin)→ F (Uj);

(3) F (∅) ' 1;(4) the structure maps are covariant and associative.

In other words, a U-prefactorization algebra F is like a factorization algebra, except that F (U) isonly defined for sets U in U.

A U-factorization algebra is a U-prefactorization algebra with the property that, for every Uin U and every Weiss cover V of U consisting of open sets in U,

C(V,F ) ' F (U),

where C(V,F ) denotes the Cech complex described earlier (section 6.1).

Note that we have not required the factorizing property, so that we are focused hereon lax factorization algebras. Throughout this section, factorization algebra means laxfactorization algebra.

In this section we will show that any U-factorization algebra on X extends to a factor-ization algebra on X. This extension is unique up to quasi-isomorphism.

Let F be a U-factorization algebra. Let us define a prefactorization algebra ext(F ) onX by

ext(F )(V) = C(UV ,F ),for each open V ⊂ X.

With these definitions in hand, it should be clear why we can recover a factorizationalgebra on X from just a factorization algebra on a factorizing basis U. The first propertyof U insures that the “atomic” structure maps (multiplication out of a finite set of points)factor through the basis. The second property insures that we know how to multiply


within the factorizing basis. In particular, we know

F (Ui0)⊗ · · · ⊗ F (Uin)→ F (Ui0 ∪ · · · ∪Uin),

and associativity insures that this map plus the unary structure maps determine all theother multiplication maps. Finally, the third property insures that we know F on theintersections that appear in the gluing condition.

7.4.0.4 Proposition. With this definition, ext(F ) is a factorization algebra whose restriction toopen sets in the cover U is quasi-isomorphic to F .

7.4.1. The proof. Our goal here is to prove that there is an equivalence of categories

FactAlgX

res++FactAlgU

extkk

where res is the functor of restricting F to the factorizing basis U and ext is a functor ex-tending from that basis. Before we can prove equivalence, we need to explicitly constructext.

Let U be a factorizing basis and F a U-factorization algebra. We will construct a fac-torization algebra ext(F ) on X in several stages:

• we give the value of ext(F ) on every open V in X,• we construct the structure maps and verify associativity and covariance,• we verify that ext(F ) satisfies the gluing axiom.

Finally, it will be manifest from our construction how to extend maps of U-factorizationalgebras to maps of their extensions.

We use the following notations. Given a simplicial cochain complex A• (so each Anis a cochain complex), let C(A•) denote the totalization of the double complex obtainedby taking the unnormalized cochains. Let CN(A•) denote the totalization of the doublecomplex by taking the normalized cochains. Finally, let

shAB : CN(A•)⊗ CN(B•)→ CN(A• ⊗ B•)

denote the Eilenberg-Zilber shuffle map, which is a lax symmetric monoidal functor fromsimplicial cochain complexes to cochain complexes.

7.4.2. Extending values. For an open V ⊂ X, recall UV denotes the Weiss cover of Vgiven by all the opens in U contained inside V. We then define

ext(F )(V) := C(UV ,F ).This construction provides a precosheaf on X.


7.4.3. Extending structure maps. The Cech complex for the factorization gluing ax-iom arises as the normalized cochain complex of a simplicial cochain complex, as shouldbe clear from its construction. We use C(V,F )• to denote this simplicial cochain complex,so that

C(V,F ) = CN(C(V,F )•),with F a factorization algebra and V a Weiss cover.

We construct the structure maps by using the simplicial cochain complexes C(V,F )•,as many properties are manifest at that level. For instance, the unary maps ext(F )(V)→ext(F )(W) arising from inclusions V → W are easy to understand: UV is a subset of UWand thus we get a map between every piece of the simplicial cochain complex.

We now explain in detail the map

mVV′ : ext(F )(V)⊗ ext(F )(V ′)→ ext(F )(V ∪V ′),

where V ∩ V ′ = ∅. Note that knowing this map, we recover every other multiplicationmap

ext(F )(V)⊗ ext(F )(V ′)

mVV′ **

// ext(F )(W)

ext(F )(V ∪V ′)

66

by postcomposing with a unary map.

The n-simplices of C(UV ,F )n are the direct sum of terms

F (Ui0 ∩ · · · ∩Uin)

with the Uik ’s in V. The n-simplices of

C(UV ,F )• ⊗ C(UV′ ,F )•are precisely the levelwise tensor product

C(UV ,F )n ⊗ C(UV′ ,F )n,

which breaks down into a direct sum of terms

F (Ui0 ∩ · · · ∩Uin)⊗F (Uj0 ∩ · · · ∩Ujn)

with the Uik ’s in V and the Ujk ’s in V ′. We need to define a map

mVV′,n : C(UV ,F )n ⊗ C(UV′ ,F )n → C(UV∪V′ ,F )n

for every n, and we will express it in terms of the direct summands.

Now

(Ui0 ∩ · · · ∩Uin) ∪ (Uj0 ∩ · · · ∩Ujn) = (Ui0 ∪Uj0) ∩ · · · ∩ (Uin ∪Ujn).


Thus we have a given map

F (Ui0 ∩ · · · ∩Uin)⊗F (Uj0 ∩ · · · ∩Ujn)→ F((Ui0 ∪Uj0) ∩ · · · ∩ (Uin ∪Ujn)

),

because F is defined on a factorizing basis. The right hand term is one of the directsummands for C(UV∪V′ ,F )n. Summing over the direct summands on the left, we obtainthe desired levelwise map.

Finally, the composition

CNC(U∩V,F )•)⊗ CNC(UV′ ,F )•)

sh

CN(C(U∩V,F )• ⊗ C(UV′ ,F )•

)CNmVV′ ,•)

CNC(UV∪V′ ,F )•)

gives us mVV′ .

A parallel argument works to construct the multiplication maps from n disjoint opensto a bigger open.

The desired associativity and covariance are clear at the level of the simplicial cochaincomplexes C(UV ,F ), since they are inherited from F itself.

7.4.4. Verifying gluing. We have constructed ext(F ) as a prefactorization algebra,but it remains to verify that it is a factorization algebra. Thus, our goal is the following.

7.4.4.1 Proposition. The extension ext(F ) is a cosheaf with respect to the Weiss topology. Inparticular, for every open subset W and every Weiss cover W, the complex C(W, ext(F )) isquasi-isomorphic to ext(F )(W).

As our gluing axiom is simply the axiom for cosheaf — but using a funny class ofcovers — the standard refinement arguments about Cech homology apply. We now spellthis out.

For W an open subset of X and W = Wjj∈J a Weiss cover of W, there are twoassociated covers that we will use:

(a) UW = Ui ⊂W | i ∈ I and(b) UW = Ui | ∃j such that Ui ⊂Wj.


The first is just the factorizing basis of W induced by U, but the second consists of theopens in U subordinate to the cover W. Both are Weiss covers of W.

To prove the proposition, we break the argument into two steps and exploit the inter-mediary Weiss cover UW.

7.4.4.2 Lemma. There is a natural quasi-isomorphism

f : C(W, ext(F )) '→ C(UW,F ),

for every Weiss cover W of an open W.

PROOF OF LEMMA. This argument boils down to combinatorics with the covers. Someextra notation will clarify what’s going on. In the Cech complex for the cover W, for in-stance, we run over n + 1-fold intersections Wj0 ∩ · · · ∩Wjn . We will denote this open byW~j, where ~j = (j0, . . . , jn) ∈ Jn+1, with J the index set for W. Since we are only inter-

ested in intersections for which all the indices are pairwise distinct, we let Jn+1 denotethis subset of Jn+1.

First, we must exhibit the desired map f of cochain complexes. The source complexC(W, ext(F )) is constructed out of F ’s behavior on the opens Ui. Explicitly, we have

C(W, ext(F )) =⊕n≥0

⊕~j∈ Jn+1

C(UW~j,F )[n].

Note that each term F (U~i) appearing in this source complex appears only once in thetarget complex C(UW,F ). Let f send each such term F (U~i) to its unique image in thetarget complex via the identity. This map f is a cochain map: it clearly respects the in-ternal differential of each term F (U~i), and it is compatible with the Cech differential byconstruction.

Second, we need to show f is a quasi-isomorphism. We will show this by imposing afiltration on the map and showing the induced spectral sequence is a quasi-isomorphismon the first page.

We filter the target complex C(UW,F ) by

FnC(UW,F ) :=⊕k≤n

⊕~i∈ Ik+1

F (U~i)[k].

Equip the source complex C(W, ext(F )) with a filtration by pulling this filtration backalong f . In particular, for any U~i = Ui0 ∩ · · · ∩ Uin , the preimage under f consists of adirect sum over all tuples W~j = Wj0 ∩ · · · ∩Wjm such that every Uik is a subset of W~j. Here,m can be any nonnegative integer (in particular, it can be bigger than n).


Consider the associated graded complexes with respect to these filtrations. The sourcecomplex has

Gr C(UW,F ) =⊕n≥0

⊕~i∈ In+1

F (U~i)[n]⊗

⊕~j∈ Jm+1 such that Uik

⊂W~j ∀k

C[m]

.

The rightmost term (after the tensor product) corresponds to the chain complex for a sim-plex — here, the simplex is infinite-dimensional — and hence is contractible. In conse-quence, the map of spectral sequences is a quasi-isomorphism.

We now wish to relate the Cech complex on the intermediary UW to that on UW .

7.4.4.3 Lemma. The complexes C(UW ,F ) and C(UW,F ) are quasi-isomorphic.

PROOF OF LEMMA. We will produce a roof

C(UW ,F ) '← C(UW , extW(F )) '→ C(UW,F ).

Recall that UW is a factorizing basis for W. Then F , restricted to W, is a UW-factorizingbasis. Hence, for any V ∈ UW and any Weiss cover V ⊂ UW of V, we have

C(V,F ) '→ F (V).

For any V ∈ UW , let UW|V = Ui ∈ U |Ui ⊂ V ∩Wj for some j ∈ J. Note that this is aWeiss cover for V. We define

extW(F )(V) := C(UW|V ,F ).

By construction, the natural map

(7.4.4.1) extW(F )(V)→ F (V)


Thus we have a quasi-isomorphism

C(UW , extW(F )) '→ C(UW ,F ),

by using the natural map (7.4.4.1) on each open in the Cech complex for UW .

The map

C(UW , extW(F )) '→ C(UW,F )arises by mimicking the construction in the preceding lemma.

7.6. EQUIVARIANT FACTORIZATION ALGEBRAS AND DESCENT ALONG A TORSOR 187

7.5. Pulling back along an open immersion

Factorization algebras do not pull back along an arbitrary continuous map, at least notin a simple way. Nonetheless, they do pull back along open embeddings (i.e., they restrictto open subsets in an obvious way). In this section we will discuss a generalization of thissituation, namely pulling back along open immersions.

Let f : N → M be an open immersion. Let U f be the cover of N consisting of thoseopen subsets U ⊂ N with the property that

f |U : U → f (U)

is a homeomorphism. (To say that f is an open immersion means that sets of this formcover N.)

For any U ∈ U f , we obtain a factorization algebra f ∗UF on U by defining

f ∗UF (V) = F ( f (V)).

It is simply the pullback of F| f (U) along the embedding f : U → f (U). It is immedi-ate that these f ∗UF satisfy the coherence conditions to glue over the cover U f , followingproposition 7.3.2.1. (One should first take a locally finite refinement of this cover.)

7.5.0.4 Definition. For an open immersion f : N → M and a factorization algebra F on M, thepullback factorization algebra f ∗F is the factorization algebra on N constructed by gluing thefactorization algebras f ∗UF over the cover U f .

7.6. Equivariant factorization algebras and descent along a torsor

Let G be a discrete group acting on a space X.

7.6.0.5 Definition. A G-equivariant factorization algebra on X is a factorization algebra F on Xtogether with isomorphisms

ρg : g∗F ∼= F ,

for each g ∈ G, such that

ρId = Id and ρgh = ρh h∗(ρg) : h∗g∗F → F .

7.6.0.6 Proposition. Let G be a discrete group acting properly discontinuously on X, so that X →X/G is a principal G-bundle. Then there is an equivalence of categories between G-equivariantfactorization algebras on X and factorization algebras on X/G.

PROOF. If F is a factorization algebra on X/G, then f ∗F is a G-equivariant factoriza-tion algebra on X.


Conversely, let F be a G-equivariant factorization algebra on F . Let Ucon be the opencover of X/G consisting of those connected sets where the G-bundle X → X/G admitsa section. Let U denote the factorizing basis for X/G generated by Ucon. We will define aU-factorization algebra FG by defining

FG(U) = F (σ(U)),

where σ is any section of the G-bundle π−1(U)→ U.

Because F is G-equivariant, F (σ(U)) is independent of the section σ chosen. Since U

is a factorizing basis, FG extends canonically to a factorization algebra on X/G.

CHAPTER 8

Structured factorization algebras and quantization

In this chapter we will define what it means to have a factorization algebra endowedwith the structure of an algebra over an operad. Not all operads work for this construc-tion: only operads endowed with an extra structure – that of a Hopf operad – can be used.Add pointer to appendix! The issue is that we need to mix the structure maps of the fac-torization algebra with those of an algebra over an operad P, so we need to know how totensor together P-algebras.

After explaining the relevant machinery, we focus on the cases of interest for us: theP0 and BD operads that appear in the classical and quantum BV formalisms, respectively.These operads play a central role in our quantization theorem, the main result of thisbook, and thus we will have formulated the goal toward which the next two parts of thebook are devoted.

Since, in this book, we are principally concerned with factorization algebras takingvalues in the category of differentiable cochain complexes we will restrict attention to thiscase in the present section.

8.1. Structured factorization algebras

8.1.0.7 Definition. A Hopf operad is an operad in the category of differential graded cocommu-tative coalgebras.

Any Hopf operad P is, in particular, a differential graded operad. In addition, thecochain complexes P(n) are endowed with the structure of differential graded commuta-tive coalgebra. The operadic composition maps

i : P(n)⊗ P(m)→ P(n + m− 1)

are maps of coalgebras, as are the maps arising from the symmetric group action on P(n).

If P is a Hopf operad, then the category of dg P-algebras becomes a symmetric monoidalcategory. If A, B are P-algebras, the tensor product A⊗C B is also a P-algebra. The struc-ture map

PA⊗B : P(n)⊗ (A⊗ B)⊗n → A⊗ B

189

190 8. STRUCTURED FACTORIZATION ALGEBRAS AND QUANTIZATION

is defined to be the composition

P(n)⊗ (A⊗ B)⊗n c(n)−−→ P(n)⊗ P(n)⊗ A⊗n ⊗ B⊗n PA⊗PB−−−→ A⊗ B.

In this diagram, c(n) : P(n)→ P(n)⊗2 is the comultiplication on c(n).

Any dg operad that is the homology operad of an operad in topological spaces is aHopf operad (because topological spaces are automatically cocommutative coalgebras,with comultiplication defined by the diagonal map). For example, the commutative op-erad Com is a Hopf operad, with coproduct defined on the generator ? ∈ Com(2) by

c(?) = ?⊗ ?.

With the comultiplication defined in this way, the tensor product of commutative algebrasis the usual one. If A and B are commutative algebras, the product on A⊗ B is defined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′).

The Poisson operad is also a Hopf operad, with coproduct defined (on the generators?, −,− by

c(?) = ?⊗ ?

c(−,−) = −,−⊗ ?+ ?⊗ −,−.

If A, B are Poisson algebras, then the tensor product A ⊗ B is a Poisson algebra withproduct and bracket defined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′)

a⊗ b, a′ ⊗ b′ = (−1)|a′||b| (a, a′ ⊗ (b ? b′) + (a ? a′)⊗ b, b′

).

8.1.0.8 Definition. Let P be a differential graded Hopf operad. A prefactorization P-algebrais a prefactorization algebra with values in the multicategory of P-algebras. A factorization P-algebra is a prefactorization P-algebra, such that the underlying prefactorization algebra withvalues in cochain complexes is a factorization algebra.

We can unpack this definition as follows. Suppose that F is a factorization P-algebra.Then F is a factorization algebra; and, in addition, for all U ⊂ M, F (U) is a P-algebra.The structure maps

F (U1)× · · · × F (Un)→ F (V)

(defined when U1, . . . , Un are disjoint open subsets of V) are required to be P-algebra mapsin the sense defined above.

The following was floating free elsewhere in the book:If P is a dg Hopf operad, then we can talk about P-algebras in the multicategory of differ-entiable cochain complexes. If F is a differentiable cochain complex, then we can define

8.2. COMMUTATIVE FACTORIZATION ALGEBRAS 191

the endomorphism dg operad End(F) whose nth component is

End(F)(n) = Hom(F, . . . , F | F).

Then, a P-structure on F is a map of dg operads

P→ End(F).

We call such an object a differentiable P-algebra.Such P-algebras themselves form a multicategory, in a natural way. To see this, let S dgDiffdenote the universal symmetric monoidal dg category containing the dg multicategorydgDiff of differentiable cochain complexes. If F1, . . . , Fk are differentiable P-algebras, thenthey are P-algebras in S dgDiff. Since P is a Hopf operad F1 ⊗ · · · ⊗ Fk is then a P-algebrain S dgDiff. A morphism in the multicategory of P-algebras is then a map of P algebras

F1 ⊗ · · · ⊗ Fk → G.

Note that forgetting the the forgetful functor from the multicategory of differentiable P-algebras to that of differentiable cochain complexes is faithful. That is, the map of sets

HomP(F1, . . . , Fk | G)→ HomdgDiff(F1, . . . , Fk | G)

is injective. Further, the image of this map lies in the space of closed degree 0 multimor-phisms; these are the same as multilinear cochain maps.Thus, if Fi, G are differentiable P-algebras, and if

φ : F1 × · · · × Fk → G

is a smooth multilinear cochain map, we can ask whether φ is a map of P-algebras.

8.1.0.9 Definition. Let P be a differential graded Hopf operad. A prefactorization differentiableP-algebra is a prefactorization algebra with values in the multicategory of differentiable P-algebras.A factorization P-algebra is a prefactorization P-algebra, such that the underlying prefactorizationalgebra with values in differentiable cochain complexes is a factorization algebra.

8.2. Commutative factorization algebras

One of the most important examples is when P is the operad Com of commutativealgebras. Then, we find that F (U) is a commutative algebra for each U. Further, ifU1, . . . , Uk ⊂ V are as above, the product map

m : F (U1)× · · · × F (Uk)→ F (V)

is compatible with the commutative algebra structures, in the following sense.

(1) If 1 ∈ F (Ui) is the unit for the commutative product on each F(Ui), then

m(1, . . . , 1) = 1.


(2) If αi, βi ∈ F (Ui), then

m(α1β1, . . . , αkβk) = ±m(α1, . . . , αk)m(β1, . . . , βk)

where ± indicates the usual Koszul rule of signs.

Note that the axioms of a factorization algebra imply that F (∅) is the ground ring k(which we normally take to be R or C for classical theories and R[[h]] or C[[h]] for quan-tum field theories). The axioms above, in the case that k = 1 and U1 = ∅, imply that themap

F (∅)→ F (U)

is a map of unital commutative algebras.

If F is a commutative prefactorization algebra, then we can recover F uniquely fromthe underlying cosheaf of commutative algebras. Indeed, the maps

F (U1)× · · · × F (Uk)→ F (V)

can be described in terms of the commutative product on F (V) and the maps F (Ui) →F (V).

8.3. The P0 operad

Recall that the collection of observables in quantum mechanics form an associativealgebra. The observables of a classical mechanical system form a Poisson algebra. Inthe deformation quantization approach to quantum mechanics, one starts with a Poissonalgebra Acl , and attempts to construct an associative algebra Aq, which is an algebra flatover the ring C[[h]], together with an isomorphism of associative algebras Aq/h ∼= Acl . Inaddition, if a, b ∈ Acl , and a, b are any lifts of a, b to Aq, then

limh→0

1h[a, b] = a, b ∈ Acl .

This book concerns the analog, in quantum field theory, of the deformation quanti-zation picture in quantum mechanics. We have seen that the sheaf of solutions to theEuler-Lagrange equation of a classical field theory can be encoded by a commutative fac-torization algebra. A commutative factorization algebra is the analog, in our setting, of thecommutative algebra appearing in deformation quantization. We have argued (section1.6) that the observables of a quantum field theory should form a factorization algebra.This factorization algebra is the analog of the associative algebra appearing in deforma-tion quantization.

In deformation quantization, the commutative algebra of classical observables has anextra structure – a Poisson bracket – which makes it “want” to deform into an associative

8.3. THE P0 OPERAD 193

algebra. In this section we will explain the analogous structure on a commutative factor-ization algebra which makes it want to deform into a factorization algebra. Later (section12.2) we will see that the commutative factorization algebra associated to a classical fieldtheory has this extra structure.

8.3.1. The E0 operad.

8.3.1.1 Definition. Let E0 be the operad defined by

E0(n) =

0 if n > 0R if n = 0

Thus, an E0 algebra in the category of real vector spaces is a real vector space witha distinguished element in it. More generally, an E0 algebra in a symmetric monoidalcategory C is the same thing as an object A of C together with a map 1C → A

The reason for the terminology E0 is that this operad can be interpreted as the operadof little 0-discs.

The inclusion of the empty set into every open set implies that, for any factorizationalgebra F , there is a unique map from the unit factorization algebra R→ F .

8.3.2. The P0 operad. The Poisson operad is an object interpolating between the com-mutative operad and the associative (or E1) operad. We would like to find an analog of thePoisson operad which interpolates between the commutative operad and the E0 operad.

Let us define the Pk operad to be the operad whose algebras are commutative algebrasequipped with a Poisson bracket of degree 1− k. With this notation, the usual Poissonoperad is the P1 operad.

Recall that the homology of the En operad is the Pn operad, for n > 1. Thus, just as thesemi-classical version of an algebra over the E1 operad is a Poisson algebra in the usualsense (that is, a P1 algebra), the semi-classical version of an En algebra is a Pn algebra.

Thus, we have the following table:

Classical Quantum? E0 operad

P1 operad E1 operadP2 operad E2 operad

......

This immediately suggests that the P0 operad is the semi-classical version of the E0 operad.


Note that the P0 operad is a Hopf operad: the coproduct is defined by

c(?) = ?⊗ ?

c(−,−) = −,−⊗ ?+ ?⊗ −,−.In concrete terms, this means that if A and B are P0 algebras, their tensor product A⊗ Bis again a P0 algebra, with product and bracket defined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′)

a⊗ b, a′ ⊗ b′ = (−1)|a′||b| (a, a′ ⊗ (b ? b′) + (a ? a′)⊗ b, b′

).

8.3.3. P0 factorization algebras. Since the P0 operad is a Hopf operad, it makes senseto talk about P0 factorization algebras. We can give an explicit description of this struc-ture. A P0 factorization algebra is a commutative factorization algebra F , together with aPoisson bracket of cohomological degree 1 on each commutative algebra F (U), with thefollowing additional properties. Firstly, if U ⊂ V, the map

F (U)→ F (V)

must be a homomorphism of P0 algebras.

The second condition is that observables coming from disjoint sets must Poisson com-mute. More precisely, let U1, U2 be disjoint subsets f V. . Let ji : F (Ui) → F (V) be thenatural maps. Let αi ∈ F (Ui), and ji(αi) ∈ F (V). Then, we require that

j1(α1), j2(α2) = 0 ∈ F (V)

where −,− is the Poisson bracket on F (V).

8.3.4. Quantization of P0 algebras. We know what it means to quantize an Poissonalgebra in the ordinary sense (that is, a P1 algebra) into an E1 algebra.

There is a similar notion of quantization for P0 algebras. A quantization is simply anE0 algebra over R[[h]] which, modulo h, is the original P0 algebra, and for which there isa certain compatibility between the Poisson bracket on the P0 algebra and the quantizedE0 algebra.

Let A be a commutative algebra in the category of cochain complexes. Let A1 be an E0

algebra flat over R[[h]]/h2, and suppose that we have an isomorphism of chain complexes

A1 ⊗R[[h]]/h2 R ∼= A.

In this situation, we can define a bracket on A of degree 1, as follows.

We have an exact sequence

0→ hA→ A1 → A→ 0.

8.4. THE BEILINSON-DRINFELD OPERAD 195

The boundary map of this exact sequence is a cochain map

D : A→ A

(well-defined up to homotopy).

Let us define a bracket on A by the formula

a, b = D(ab)− (−1)|a|aDb− (Da)b.

Because D is well-defined up to homotopy, so is this bracket. However, unless D is anorder two differential operator, this bracket is simply a cochain map A⊗ A→ A, and nota Poisson bracket of degree 1.

In particular, this bracket induces one on the cohomology H∗(A) of A. The cohomo-logical bracket is independent of any choices.

8.3.4.1 Definition. Let A be a P0 algebra in the category of cochain complexes. Then a quantiza-tion of A is an E0 algebra A over R[[h]], together with a quasi-isomorphism of E0 algebras

A⊗R[[h]] R ∼= A,

which satisfies the following correspondence principle: the bracket on H∗(A) induced by Amust coincide with that given by the P0 structure on A.

In the next section we will consider a more sophisticated, operadic notion of quanti-zation, which is strictly stronger than this one. To distinguish between the two notions,one could call the definition of quantization presented here a weak quantization, while thedefinition introduced later will be called a strong quantization.

8.4. The Beilinson-Drinfeld operad

Beilinson and Drinfeld [BD04] constructed an operad over the formal disc whichgenerically is equivalent to the E0 operad, but which at 0 is equivalent to the P0 operad.We call this operad the Beilinson-Drinfeld operad.

The operad P0 is generated by a commutative associative product − ?−, of degree 0;and a Poisson bracket −,− of degree +1.

8.4.0.2 Definition. The Beilinson-Drinfeld (or BD) operad is the differential graded operadover the ring R[[h]] which, as a graded operad, is simply

BD = P0 ⊗R[[h]];

but with differential defined byd(− ?−) = h−,−.


If M is a flat differential graded R[[h]] module, then giving M the structure of a BD al-gebra amounts to giving M a commutative associative product, of degree 0, and a Poissonbracket of degree 1, such that the differential on M is a derivation of the Poisson bracket,and the following identity is satisfied:

d(m ? n) = (dm) ? n + (−1)|m|m ? (dn) + (−1)|m|hm, n

8.4.0.3 Lemma. There is an isomorphism of operads,

BD⊗R[[h]] R ∼= P0,

and a quasi-isomorphism of operads over R((h)),

BD⊗R[[h]] R((h)) ' E0 ⊗R((h)).

Thus, the operad BD interpolates between the P0 operad and the E0 operad.

BD is an operad in the category of differential graded R[[h]] modules. Thus, we cantalk about BD algebras in this category, or in any symmetric monoidal category enrichedover the category of differential graded R[[h]] modules.

The BD algebra is, in addition, a Hopf operad, with coproduct defined in the sameway as in the P0 operad. Thus, one can talk about BD factorization algebras.

8.4.1. BD quantization of P0 algebras.

8.4.1.1 Definition. Let A be a P0 algebra (in the category of cochain complexes). A BD quanti-zation of A is a flat R[[h]] module Aq, flat over R[[h]], which is equipped with the structure of aBD algebra, and with an isomorphism of P0 algebras

Aq ⊗R[[h]] R ∼= A.

Similarly, an order k BD quantization of A is a differential graded R[[h]]/hk+1 module Aq,flat over R[[h]]/hk+1, which is equipped with the structure of an algebra over the operad

BD⊗R[[h]] R[[h]]/hk+1,

and with an isomorphism of P0 algebras

Aq ⊗R[[h]]/hk+1 R ∼= A.

This definition applies without any change in the world of factorization algebras.

8.4.1.2 Definition. Let F be a P0 factorization algebra on M. Then a BD quantization of F isa BD factorization algebra F equipped with a quasi-isomorphism

F ⊗R[[h]] R ' F

8.4. THE BEILINSON-DRINFELD OPERAD 197

of P0 factorization algebras on M.

8.4.2. Operadic description of ordinary deformation quantization. We will finishthis section by explaining how the ordinary deformation quantization picture can bephrased in similar operadic terms.

Consider the following operad BD1 over R[[h]]. BD1 is generated by two binary oper-ations, a product ∗ and a bracket [−,−]. The relations are that the product is associative;the bracket is antisymmetric and satisfies the Jacobi identity; the bracket and the productsatisfy a certain Leibniz relation, expressed in the identity

[ab, c] = a[b, c]± [b, c]a

(where ± indicates the Koszul sign rule); and finally the relation

a ∗ b∓ b ∗ a = h[a, b]

holds. This operad was introduced by Ed Segal [Seg10].

Note that, modulo h, BD1 is the ordinary Poisson operad P1. If we set h = 1, we findthat BD1 is the operad E1 of associative algebras. Thus, BD1 interpolates between P1 andE1 in the same way that BD0 interpolates between P0 and E0.

Let A be a P1 algebra. Let us consider possible lifts of A to a BD1 algebra.

8.4.2.1 Lemma. A lift of A to a BD1 algebra, flat over R[[h]], is the same as a deformationquantization of A in the usual sense.

PROOF. We need to describe BD1 structures on A[[h]] compatible with the given Pois-son structure. To give such a BD1 structure is the same as to give an associative producton A[[h]], linear over R[[h]], and which modulo h is the given commutative product on A.Further, the relations in the BD1 operad imply that the Poisson bracket on A is related tothe associative product on A[[h]] by the formula

h−1 (a ∗ b∓ b ∗ a) = a, b mod h.

Part 3

Classical field theory

CHAPTER 9

Introduction to classical field theory

Our goal here is to describe how the observables of a classical field theory naturallyform a factorization algebra (section 6.1). More accurately, we are interested in whatmight be called classical perturbative field theory. “Classical” means that the main ob-ject of interest is the sheaf of solutions to the Euler-Lagrange equations for some localaction functional. “Perturbative” means that we will only consider those solutions whichare infinitesimally close to a given solution. Much of this part of the book is devoted toproviding a precise mathematical definition of these ideas, with inspiration taken fromdeformation theory and derived geometry. In this chapter, then, we will simply sketchthe essential ideas.

9.1. The Euler-Lagrange equations

The fundamental objects of a physical theory are the observables of a theory, that is,the measurements one can make in that theory. In a classical field theory, the fields thatappear “in nature” are constrained to be solutions to the Euler-Lagrange equations (alsocalled the equations of motion). Thus, the measurements one can make are the functionson the space of solutions to the Euler-Lagrange equations.

However, it is essential that we do not take the naive moduli space of solutions. In-stead, we consider the derived moduli space of solutions. Since we are working perturba-tively — that is, infinitesimally close to a given solution — this derived moduli space willbe a “formal moduli problem” [?, Lur11]. In the physics literature, the procedure of tak-ing the derived critical locus of the action functional is implemented by the BV formalism.Thus, the first step (chapter 10.1.3) in our treatment of classical field theory is to develop alanguage to treat formal moduli problems cut out by systems of partial differential equa-tions on a manifold M. Since it is essential that the differential equations we consider areelliptic, we call such an object a formal elliptic moduli problem.

Since one can consider the solutions to a differential equation on any open subsetU ⊂ M, a formal elliptic moduli problem F yields, in particular, a sheaf of formal moduliproblems on M. This sheaf sends U to the formal moduli space F (U) of solutions on U.

201

202 9. INTRODUCTION TO CLASSICAL FIELD THEORY

We will use the notation EL to denote the formal elliptic moduli problem of solutionsto the Euler-Lagrange equation on M; thus, EL(U) will denote the space of solutions onan open subset U ⊂ M.

9.2. Observables

In a field theory, we tend to focus on measurements that are localized in spacetime.Hence, we want a method that associates a set of observables to each region in M. IfU ⊂ M is an open subset, the observables on U are

Obscl(U) = O(EL(U)),

our notation for the algebra of functions on the formal moduli space EL(U) of solutionsto the Euler-Lagrange equations on U. (We will be more precise about which class offunctions we are using later.) As we are working in the derived world, Obscl(U) is adifferential-graded commutative algebra. Using these functions, we can answer any ques-tion we might ask about the behavior of our system in the region U.

The factorization algebra structure arises naturally on the observables in a classicalfield theory. Let U be an open set in M, and V1, . . . , Vk a disjoint collection of open subsetsof U. Then restriction of solutions from U to each Vi induces a natural map

EL(U)→ EL(V1)× · · · × EL(Vk).

Since functions pullback under maps of spaces, we get a natural map

Obscl(V1)⊗ · · · ⊗Obscl(Vk)→ Obscl(U)

so that Obscl forms a prefactorization algebra. To see that Obscl is indeed a factorizationalgebra, it suffices to observe that the functor EL is a sheaf.

Since the space Obscl(U) of observables on a subset U ⊂ M is a commutative algebra,and not just a vector space, we see that the observables of a classical field theory form acommutative factorization algebra (section 8).

9.3. The symplectic structure

Above, we outlined a way to construct, from the elliptic moduli problem associatedto the Euler-Lagrange equations, a commutative factorization algebra. This construction,however, would apply equally well to any system of differential equations. The Euler-Lagrange equations, of course, have the special property that they arise as the criticalpoints of a functional.

In finite dimensions, a formal moduli problem which arises as the derived criticallocus (section 11.1) of a function is equipped with an extra structure: a symplectic form of

9.4. THE P0 STRUCTURE 203

cohomological degree −1. For us, this symplectic form is an intrinsic way of indicatingthat a formal moduli problem arises as the critical locus of a functional. Indeed, anyformal moduli problem with such a symplectic form can be expressed (non-uniquely) inthis way.

We give (section 11.2) a definition of symplectic form on an elliptic moduli prob-lem. We then simply define a classical field theory to be a formal elliptic moduli problemequipped with a symplectic form of cohomological degree −1.

Given a local action functional satisfying certain non-degeneracy properties, we con-struct (section 11.3.1) an elliptic moduli problem describing the corresponding Euler-Lagrange equations, and show that this elliptic moduli problem has a symplectic formof degree −1.

In ordinary symplectic geometry, the simplest construction of a symplectic manifoldis as a cotangent bundle. In our setting, there is a similar construction: given any ellipticmoduli problem F , we construct (section 11.6) a new elliptic moduli problem T∗[−1]Fwhich has a symplectic form of degree −1. It turns out that many examples of field theo-ries of interest in mathematics and physics arise in this way.

9.4. The P0 structure

In finite dimensions, if X is a formal moduli problem with a symplectic form of degree−1, then the dg algebra O(X) of functions on X is equipped with a Poisson bracket ofdegree 1. In other words, O(X) is a P0 algebra (section 8.3).

In infinite dimensions, we show that something similar happens. If F is a classicalfield theory, then we show that on every open U, the commutative algebra O(F (U)) =

Obscl(U) has a P0 structure. We then show that the commutative factorization algebraObscl forms a P0 factorization algebra. This is not quite trivial; it is at this point that weneed the assumption that our Euler-Lagrange equations are elliptic.

CHAPTER 10

Elliptic moduli problems

The essential data of a classical field theory is the moduli space of solutions to theequations of motion of the field theory. For us, it is essential that we take not the naivemoduli space of solutions, but rather the derived moduli space of solutions. In the physicsliterature, the procedure of taking the derived moduli of solutions to the Euler-Lagrangeequations is known as the classical Batalin-Vilkovisky formalism.

The derived moduli space of solutions to the equations of motion of a field theory onX is a sheaf on X. In this chapter we will introduce a general language for discussingsheaves of “derived spaces” on X that are cut out by differential equations.

Our focus in this book is on perturbative field theory, so we sketch the heuristic pic-ture from physics before we introduce a mathematical language that formalizes the pic-ture. Suppose we have a field theory and we have found a solution to the Euler-Lagrangeequations φ0. We want to find the nearby solutions, and a time-honored approach is toconsider a formal series expansion around φ0,

φt = φ0 + tφ1 + t2φ2 + · · · ,

and to solve iteratively the Euler-Lagrange equations for the higher terms φn. Of course,such an expansion is often not convergent in any reasonable sense, but this perturbativemethod has provided insights into many physical problems. In mathematics, particularlythe deformation theory of algebraic geometry, this method has also flourished and ac-quired a systematic geometric interpretation. Here, though, we work in place of t with aparameter ε that is nilpotent, so that there is some integer n such that εn+1 = 0. Let

φ = φ0 + εφ1 + ε2φ2 + · · ·+ εnφn.

Again, the Euler-Lagrange equation applied to φ becomes a system of simpler differentialequations organized by each power of ε. As we let the order of ε go to infinity and findthe nearby solutions, we describe the formal neighborhood of φ0 in the space of all solutionsto the Euler-Lagrange equations. (Although this procedure may seem narrow in scope,its range expands considerably by considering families of solutions, rather a single fixedsolution. Our formalism is built to work in families.)

205

206 10. ELLIPTIC MODULI PROBLEMS

In this chapter we will introduce a mathematical formalism for this procedure, whichincludes derived perturbations (i.e., ε has nonzero cohomological degree). In mathemat-ics, this formalism is part of derived deformation theory or formal derived geometry.Thus, before we discuss the concepts specific to classical field theory, we will explain somegeneral techniques from deformation theory. A key role is played by a deep relationshipbetween Lie algebras and formal moduli spaces.

10.1. Formal moduli problems and Lie algebras

In ordinary algebraic geometry, the fundamental objects are commutative algebras. Inderived algebraic geometry, commutative algebras are replaced by commutative differ-ential graded algebras concentrated in non-positive degrees (or, if one prefers, simplicialcommutative algebras; over Q, there is no difference).

We are interested in formal derived geometry, which is described by nilpotent com-mutative dg algebras.

10.1.0.2 Definition. An Artinian dg algebra over a field K of characteristic zero is a differentialgraded commutative K-algebra R, concentrated in degrees ≤ 0, such that

(1) each graded component Ri is finite dimensional, and Ri = 0 for i 0;(2) R has a unique maximal differential ideal m such that R/m = K, and such that mN = 0

for N 0.

Given the first condition, the second condition is equivalent to the statement thatH0(R) is Artinian in the classical sense.

The category of Artinian dg algebras is simplicially enriched in a natural way. Amap R → S is simply a map of dg algebras taking the maximal ideal mR to that of mS.Equivalently, such a map is a map of non-unital dg algebras mR → mS. An n-simplex inthe space Maps(R, S) of maps from R to S is defined to be a map of non-unital dg algebras

mR → mS ⊗Ω∗(4n)

where Ω∗(4n) is some commutative algebra model for the cochains on the n-simplex.(Normally, we will work over R, and Ω∗(4n) will be the usual de Rham complex.)

We will (temporarily) let Artk denote the simplicially enriched category of Artinian dgalgebras over k.

10.1.0.3 Definition. A formal moduli problem over a field k is a functor (of simplicially enrichedcategories)

F : Artk → sSetsfrom Artk to the category sSets of simplicial sets, with the following additional properties.

10.1. FORMAL MODULI PROBLEMS AND LIE ALGEBRAS 207

(1) F(k) is contractible.(2) F takes surjective maps of dg Artinian algebras to fibrations of simplicial sets.(3) Suppose that A, B, C are dg Artinian algebras, and that B → A, C → A are surjective

maps. Then we can form the fiber product B×A C. We require that the natural map

F(B×A C)→ F(B)×F(A) F(C)

is a weak homotopy equivalence.

We remark that such a moduli problem F is pointed: F assigns to k a point, up tohomotopy, since F(k) is contractible. Since we work mostly with pointed moduli problemsin this book, we will not emphasize this issue. Whenever we work with more generalmoduli problems, we will indicate it explicitly.

Note that, in light of the second property, the fiber product F(B)×F(A) F(C) coincideswith the homotopy fiber product.

The category of formal moduli problems is itself simplicially enriched, in an evidentway. If F, G are formal moduli problems, and φ : F → G is a map, we say that φ is a weakequivalence if for all dg Artinian algebras R, the map

φ(R) : F(R)→ G(R)

is a weak homotopy equivalence of simplicial sets.

10.1.1. Formal moduli problems and L∞ algebras. One very important way in whichformal moduli problems arise is as the solutions to the Maurer-Cartan equation in an L∞algebra. As we will see later, all formal moduli problems are equivalent to formal moduliproblems of this form.

If g is an L∞ algebra, and (R, m) is a dg Artinian algebra, we will let

MC(g⊗m)

denote the simplicial set of solutions to the Maurer-Cartan equation in g⊗ m. Thus, ann-simplex in this simplicial set is an element

α ∈ g⊗m⊗Ω∗(4n)

of cohomological degree 1, which satisfies the Maurer-Cartan equation

dα + ∑n≥2

1n! ln(α, . . . , α) = 0.

It is a well-known result in derived deformation theory that sending R to MC(g ⊗ m)defines a formal moduli problem (see [Get09a], [Hin01]). We will often use the notationBg to denote this formal moduli problem.


If g is finite dimensional, then a Maurer-Cartan element of g⊗m is the same thing asa map of commutative dg algebras

C∗(g)→ R

which takes the maximal ideal of C∗(g) to that of R.

Thus, we can think of the Chevalley-Eilenberg cochain complex C∗(g) as the algebraof functions on Bg.

Under the dictionary between formal moduli problems and L∞ algebras, a dg vectorbundle on Bg is the same thing as a dg module over g. The cotangent complex to Bg cor-responds to the g-module g∨[−1], with the shifted coadjoint action. The tangent complexcorresponds to the g-module g[1], with the shifted adjoint action.

If M is a g-module, then sections of the corresponding vector bundle on Bg is theChevalley-Eilenberg cochains with coefficients in M. Thus, we can define Ω1(Bg) to be

Ω1(Bg) = C∗(g, g∨[−1]).

Similarly, the complex of vector fields on Bg is

Vect(Bg) = C∗(g, g[1]).

Note that, if g is finite dimensional, this is the same as the cochain complex of derivationsof C∗(g). Even if g is not finite dimensional, the complex Vect(Bg) is, up to a shift of one,the Lie algebra controlling deformations of the L∞ structure on g.

10.1.2. The fundamental theorem of deformation theory. The following statementis at the heart of the philosophy of deformation theory:

There is an equivalence of (∞, 1) categories between the category of dif-ferential graded Lie algebras and the category of formal pointed moduliproblems.

In a different guise, this statement goes back to Quillen’s work [Qui69] on rational homo-topy theory. A precise formulation of this theorem has been proved by Hinich [Hin01];more general theorems of this nature are considered in [Lur11], [?] and in [?], which isalso an excellent survey of these ideas.

It would take us too far afield to describe the language in which this statement canbe made precise. We will simply use this statement as motivation: we will only considerformal moduli problems described by L∞ algebras, and this statement asserts that we loseno information in doing so.

10.1. FORMAL MODULI PROBLEMS AND LIE ALGEBRAS 209

10.1.3. Elliptic moduli problems. We are interested in formal moduli problems whichdescribe solutions to differential equations on a manifold M. Since we can discuss solu-tions to a differential equation on any open subset of M, such an object will give a sheafof derived moduli problems on M, described by a sheaf of homotopy Lie algebras. Let usgive a formal definition of such a sheaf.

10.1.3.1 Definition. Let M be a manifold. A local L∞ algebra on M consists of the followingdata.

(1) A graded vector bundle L on M, whose space of smooth sections will be denoted L.(2) A differential operator d : L → L, of cohomological degree 1 and square 0.(3) A collection of poly-differential operators

ln : L⊗n → L

for n ≥ 2, which are alternating, are of cohomological degree 2− n, and endow L withthe structure of L∞ algebra.

10.1.3.2 Definition. An elliptic L∞ algebra is a local L∞ algebra L as above with the propertythat (L, d) is an elliptic complex.

Remark: The reader who is not comfortable with the language of L∞ algebras will loselittle by only considering elliptic dg Lie algebras. Most of our examples of classical fieldtheories will be described using dg Lie algebra rather than L∞ algebras.

If L is a local L∞ algebra on a manifold M, then it yields a presheaf BL of formalmoduli problems on M. This presheaf sends a dg Artinian algebra (R, m) and an opensubset U ⊂ M to the simplicial set

BL(U)(R) = MC(L(U)⊗m)

of Maurer-Cartan elements of the L∞ algebra L(U)⊗m (where L(U) refers to the sectionsof L on U). We will think of this as the R-points of the formal pointed moduli problemassociated to L(U). One can show, using the fact that L is a fine sheaf, that this sheaf offormal moduli problems is actually a homotopy sheaf, i.e. it satisfies Cech descent. Sincethis point plays no role in our work, we will not elaborate further.

10.1.3.3 Definition. A formal pointed elliptic moduli problem (or simply elliptic moduliproblem) is a sheaf of formal moduli problems on M that is represented by an elliptic L∞ algebra.

The basepoint of the moduli problem corresponds, in the setting of field theory, to thedistinguished solution we are expanding around.


10.2. Examples of elliptic moduli problems related to scalar field theories

10.2.1. The free scalar field theory. Let us start with the most basic example of anelliptic moduli problem, that of harmonic functions. Let M be a Riemannian manifold.We want to consider the formal moduli problem describing functions φ on M that areharmonic, namely, functions that satisfy D φ = 0 where D is the Laplacian. The base pointof this formal moduli problem is the zero function.

The elliptic L∞ algebra describing this formal moduli problem is defined by

L = C∞(M)[−1] D−→ C∞(M)[−2].

This complex is thus situated in degrees 1 and 2. The products ln in this L∞ algebra are allzero for n ≥ 2.

In order to justify this definition, let us analyze the Maurer-Cartan functor of this L∞algebra. Let R be an ordinary (not dg) Artinian algebra, and let m be the maximal ideal ofR. The set of 0-simplices of the simplicial set MCL(R) is the set

φ ∈ C∞(M)⊗m | D φ = 0.Indeed, because the L∞ algebra L is Abelian, the set of solutions to the Maurer-Cartanequation is simply the set of closed degree 1 elements of the cochain complex L⊗m. Allhigher simplices in the simplicial set MCL(R) are constant. To see this, note that if φ ∈L⊗m⊗Ω∗(4n) is a closed element in degree 1, then φ must be in C∞(M)⊗m⊗Ω0(4n).The fact that φ is closed amounts to the statement that D φ = 0 and that ddRφ = 0, whereddR is the de Rham differential on Ω∗(4n).

Let us now consider the Maurer-Cartan simplicial set associated to a differential gradedArtinian algebra (R, m) with differential dR. The the set of 0-simplices of MCL(R) is theset

φ ∈ C∞(M)⊗m0, ψ ∈ C∞(M)⊗m−1 | D φ = dRψ.(The superscripts on m indicate the cohomological degree.) Thus, the 0-simplices of oursimplicial set can be identified with the set R-valued smooth functions φ on M that areharmonic up to a homotopy given by ψ and also vanish modulo the maximal ideal m.

Next, let us identify the set of 1-simplices of the Maurer-Cartan simplicial set MCL(R).This is the set of closed degree 1 elements of L ⊗ m⊗Ω∗([0, 1]). Such a closed degree 1element has four terms:

φ0(t) ∈ C∞(M)⊗m0 ⊗Ω0([0, 1])

φ1(t)dt ∈ C∞(M)⊗m−1 ⊗Ω1([0, 1])

ψ0(t) ∈ C∞(M)⊗m−1 ⊗Ω0([0, 1])

ψ1(t)dt ∈ C∞(M)⊗m−2 ⊗Ω1([0, 1]).

10.2. EXAMPLES OF ELLIPTIC MODULI PROBLEMS RELATED TO SCALAR FIELD THEORIES 211

Being closed amounts to satisfying the three equations

D φ0(t) = dRψ0(t)ddt

φ0(t) = dRφ1(t)

D φ1(t) +ddt

ψ0(t) = dRψ1(t).

These equations can be interpreted as follows. We think of φ0(t) as providing a familyof R-valued smooth functions on M, which are harmonic up to a homotopy specified byψ0(t). Further, φ0(t) is independent of t, up to a homotopy specified by φ1(t). Finally, wehave a coherence condition among our two homotopies.

The higher simplices of the simplicial set have a similar interpretation.

10.2.2. Interacting scalar field theories. Next, we will consider an elliptic moduliproblem that arises as the Euler-Lagrange equation for an interacting scalar field theory.Let φ denote a smooth function on the Riemannian manifold M with metric g. The actionfunctional is

S(φ) =∫

M

12 φ D φ + 1

4! φ4 dvolg .

The Euler-Lagrange equation for the action functional S is

D φ + 13! φ

3 = 0,

a nonlinear PDE, whose space of solutions is hard to describe.

Instead of trying to describe the actual space of solutions to this nonlinear PDE, wewill describe the formal moduli problem of solutions to this equation where φ is infinites-imally close to zero.

The formal moduli problem of solutions to this equation can be described as the so-lutions to the Maurer-Cartan equation in a certain elliptic L∞ algebra which continue wecall L. As a cochain complex, L is

L = C∞(M)[−1] D−→ C∞(M)[−2].

Thus, C∞(M) is situated in degrees 1 and 2, and the differential is the Laplacian.

The L∞ brackets ln are all zero except for l3. The cubic bracket l3 is the map

l3 : C∞(M)⊗3 → C∞(M)

φ1 ⊗ φ2 ⊗ φ3 7→ φ1φ2φ3.

Here, the copy of C∞(M) appearing in the source of l3 is the one situated in degree 1,whereas that appearing in the target is the one situated in degree 2.


If R is an ordinary (not dg) Artinian algebra, then the Maurer-Cartan simplicial setMCL(R) associated to R has for 0-simplices the set φ ∈ C∞(M) ⊗ m such that D φ +13! φ

3 = 0. This equation may look as complicated as the full nonlinear PDE, but it issubstantially simpler than the original problem. For example, consider R = R[ε]/(ε2), the“dual numbers.” Then φ = εφ1 and the Maurer-Cartan equation becomes D φ1 = 0. ForR = R[ε]/(ε4), we have φ = εφ1 + ε2φ2 + ε3φ3 and the Maurer-Cartan equation becomesa triple of simpler linear PDE:

D φ1 = 0, D φ2 = 0, and Dφ3 +12 φ3

1 = 0.

We are simply reading off the εk components of the Maurer-Cartan equation. The highersimplices of this simplicial set are constant.

If R is a dg Artinian algebra, then the simplicial set MCL(R) has for 0-simplices theset of pairs φ ∈ C∞(M)⊗m0 and ψ ∈ C∞(M)⊗m−1 such that

D φ + 13! φ

3 = dRψ.

We should interpret this as saying that φ satisfies the Euler-Lagrange equations up to ahomotopy given by ψ.

The higher simplices of this simplicial set have an interpretation similar to that de-scribed for the free theory.

10.3. Examples of elliptic moduli problems related to gauge theories

10.3.1. Flat bundles. Next, let us discuss a more geometric example of an ellipticmoduli problem: the moduli problem describing flat bundles on a manifold M. In thiscase, because flat bundles have automorphisms, it is more difficult to give a direct defini-tion of the formal moduli problem.

Thus, let G be a Lie group, and let P → M be a principal G-bundle equipped with aflat connection ∇0. Let gP be the adjoint bundle (associated to P by the adjoint action ofG on its Lie algebra g). Then gP is a bundle of Lie algebras on M, equipped with a flatconnection that we will also denote ∇0.

For R be an Artinian dg algebra, we want to define the simplicial set DefP(R) of R-families of flat G-bundles on M that deform P. The question is “what local L∞ algebrayields this elliptic moduli problem?”

The answer is L = Ω∗(M, gP), where the differential is d∇0 , the de Rham differentialcoupled to our connection ∇0. But we need to explain how to find this answer so we willprovide the reasoning behind our answer. This reasoning is a model for finding the localL∞ algebras associated to field theories.

10.3. EXAMPLES OF ELLIPTIC MODULI PROBLEMS RELATED TO GAUGE THEORIES 213

As the underlying topological bundle of P is rigid, we can only deform the flat con-nection on P. Let’s consider deformations over a dg Artinian ring R with maximal idealm. A deformation of the connection on P is given by an element

A ∈ Ω1(M, gP)⊗m0.

We would like to ask that A is flat up to homotopy. The curvature F(A) is

F(A) = d∇0 A + 12 [A, A] ∈ Ω2(M, gP)⊗m.

Note that, by the Bianchi identity, d∇0 F(A) + [A, F(A)] = 0.

For A to be flat up to homotopy, we should ask that F(A) is exact in the cochaincomplex Ω2(M, gP) ⊗ m of m-valued 2-forms on M. However, we should also ask thatF(A) is exact in a way compatible with the Bianchi identity.

Thus, as a first approximation, we will define the 0-simplices of the deformation func-tor by

DefprelimP (R)[0] =

A ∈ Ω1(M, gP)⊗m, B ∈ Ω2(M, gP)⊗m | F(A) = dRB, d∇0 B + [A, B] = 0.Here, A is of cohomological degree 1 and B is of cohomological degree 0.

Note that if m is of square zero, then the Bianchi constraint on B just says that d∇B = 0.This leads to a problem: the sheaf of closed 2-forms on M is not fine: it has higher coho-mology groups. Thus, we cannot hope to construct a deformation functor with values inhomotopy sheaves of simplicial sets on M in this way.

Instead, we will ask that B satisfy the Bianchi constraint up a sequence of higher ho-motopies. Thus, the 0-simplices of our simplicial set of deformations are defined by

DefP(R)[0] = A ∈ Ω1(M, gP)⊗m, B ∈ Ω≥2(M, gP)⊗m

| F(A) + d∇0 B + [A, B] + 12 [B, B] = 0..

Here, d refers to the total differential on the tensor product cochain complex Ω≥2(M, gP)⊗m. As before, A is of cohomological degree 1 and B is of cohomological degree 0.

If we let Bi ∈ Ωi(M, gP)⊗m, then the first few constraints on the Bi can be written as

d∇0 B2 + [A, B2] + dRB3 = 0

d∇0 B3 + [A, B3] +12 [B2, B2] + dRB4 = 0.

Thus, B2 satisfies the Bianchi constraint up to a homotopy defined by B3, and so on.

The higher simplices of this simplicial set must relate gauge-equivalent flat connec-tions. If the dg algebra R is concentrated in degree 0 (and so has zero differential), thenwe can define the simplicial set DefP(R) to be the homotopy quotient of DefP(R)[0] by


the nilpotent group associated to the nilpotent Lie algebra Ω0(M, gP)⊗m, which acts onDefP(R)[0] in the standard way (see, for instance, [KS] or [Man09]).

If R is not concentrated in degree 0, however, then the higher simplices of DefP(R)must also involve elements of R of negative cohomological degree. Indeed, degree 0 ele-ments of R should be thought of as homotopies between degree 1 elements of R, and soshould contribute 1-simplices to our simplicial set.

A slick way to define a simplicial set with both desiderata is to set

DefP(R)[n] = A ∈ Ω∗(M, gP)⊗m⊗Ω∗(4n) | d∇0 A + dR A + 12 [A, A] = 0.

Suppose that R is concentrated in degree 0 (so that the differential on R is zero). Then,the higher forms on M don’t play any role, and

DefP(R)[0] = A ∈ Ω1(M, gP)⊗m | d∇0 A + 12 [A, A] = 0.

One can show (see [Get09a]) that in this case, the simplicial set DefP(R) is weakly homo-topy equivalent to the homotopy quotient of DefP(R)[0] by the nilpotent group associ-ated to the nilpotent Lie algebra Ω0(M, gP)⊗m. Indeed, a 1-simplex in the simplicial setDefP(R) is given by a family of the form A0(t) + A1(t)dt, where A0(t) is a smooth familyof elements of Ω1(M, gP) ⊗ m depending on t ∈ [0, 1], and A1(t) is a smooth family ofelements of Ω0(M, gP)⊗m. The Maurer-Cartan equation in this context says that

d∇0 A0(t) + 12 [A0(t), A0(t)] = 0

ddt A0(t) + [A1(t), A0(t)] = 0.

The first equation says that A0(t) defines a family of flat connections. The second equationsays that the gauge equivalence class of A0(t) is independent of t. In this way, gaugeequivalences are represented by 1-simplices in DefP(R).

It is immediate that the formal moduli problem DefP(R) is represented by the ellipticdg Lie algebra

L = Ω∗(M, g).

The differential on L is the de Rham differential d∇0 on M coupled to the flat connectionon g. The only nontrivial bracket is l2, which just arises by extending the bracket of g overthe commutative dg algebra Ω∗(M) in the appropriate way.

10.3.2. Self-dual bundles. Next, we will discuss the formal moduli problem associ-ated to the self-duality equations on a 4-manifold. We won’t go into as much detail as wedid for flat connections; instead, we will simply write down the elliptic L∞ algebra rep-resenting this formal moduli problem. (For a careful explanation, see the original article[AHS78].)

10.4. COCHAINS OF A LOCAL L∞ ALGEBRA 215

Let M be an oriented 4-manifold. Let G be a Lie group, and let P → M be a principalG-bundle, and let gP be the adjoint bundle of Lie algebras. Suppose we have a connectionA on P with anti-self-dual curvature:

F(A)+ = 0 ∈ Ω2+(M, gP)

(here Ω2+(M) denotes the space of self-dual two-forms).

Then, the elliptic Lie algebra controlling deformations of (P, A) is described by thediagram

Ω0(M, gP)d−→ Ω1(M, gP)

d+−→ Ω2+(M, gP).

Here d+ is the composition of the de Rham differential (coupled to the connection on gP)with the projection onto Ω2

+(M, gP).

Note that this elliptic Lie algebra is a quotient of that describing the moduli of flatG-bundles on M.

10.3.3. Holomorphic bundles. In a similar way, if M is a complex manifold and ifP→ M is a holomorphic principal G-bundle, then the elliptic dg Lie algebra Ω0,∗(M, gP),with differential ∂, describes the formal moduli space of holomorphic G-bundles on M.

10.4. Cochains of a local L∞ algebra

Let L be a local L∞ algebra on M. If U ⊂ M is an open subset, then L(U) denotes theL∞ algebra of smooth sections of L on U. Let Lc(U) ⊂ L(U) denote the sub-L∞ algebraof compactly supported sections.

In the appendix (section B.8) we defined the algebra of functions on the space of sec-tions on a vector bundle on a manifold. We are interested in the algebra

O(L(U)[1]) = ∏n≥0

Hom((L(U)[1])⊗n, R

)Sn

where the tensor product is the completed projective tensor product, and Hom denotesthe space of continuous linear maps.

This space is naturally a graded differentiable vector space (that is, we can view it as asheaf of graded vector spaces on the site of smooth manifolds). However, it is importantthat we treat this object as a differentiable pro-vector space. Basic facts about differentiablepro-vector spaces are developed in the Appendix B. The pro-structure comes from thefiltration

FiO(L(U)[1]) = ∏n≥i

Hom((L(U)[1])⊗n, R

)Sn

,

which is the usual filtration on “power series.”


The L∞ algebra structure on L(U) gives, as usual, a differential on O(L(U)[1]), mak-ing O(L(U)[1]) into a differentiable pro-cochain complex.

10.4.0.1 Definition. Define the Lie algebra cochain complex C∗(L(U)) to be

C∗(L(U)) = O(L(U)[1])

equipped with the usual Chevalley-Eilenberg differential. Similarly, define

C∗red(L(U)) ⊂ C∗(L(U))

to be the reduced Chevalley-Eilenberg complex, that is, the kernel of the natural augmentationmap C∗(L(U))→ R. These are both differentiable pro-cochain complexes.

One defines C∗(Lc(U)) in the same way, everywhere substituting Lc for L.

We will think of C∗(L(U)) as the algebra of functions on the formal moduli problemBL(U) associated to the L∞ algebra L(U).

10.4.1. Cochains with coefficients in a module. Let L be a local L∞ algebra on M,and let L denote the smooth sections. Let E be a graded vector bundle on M and equipthe global smooth sections E with a differential that is a differential operator.

10.4.1.1 Definition. A local action of L on E is an action of L on E with the property that thestructure maps

L⊗n ⊗ E → E

(defined for n ≥ 1) are all polydifferential operators.

Note that L has an action on itself, called the adjoint action, where the differential onL is the one coming from the L∞ structure, and the action map

µn : L⊗n ⊗L → L

is the L∞ structure map ln+1.

Let L! = L∨ ⊗C∞M

DensM. Then, L! has a natural local L-action, which we shouldthink of as the coadjoint action. This action is defined by saying that if α1, . . . , αn ∈ L, thedifferential operator

µn(α1, . . . , αn,−) : L! → L!

is the formal adjoint to the corresponding differential operator arising from the action ofL on itself.

has the structure of a local module over L.

10.5. D-MODULES AND LOCAL L∞ ALGEBRAS 217

If E is a local module over L, then, for each U ⊂ M, we can define the Chevalley-Eilenberg cochains

C∗(L(U), E (U))

of L(U) with coefficients in E (U). As above, one needs to take account of the topologieson the vector spaces L(U) and E (U) when defining this Chevalley-Eilenberg cochaincomplex. Thus, as a graded vector space,

C∗(L(U), E (U)) = ∏n≥0

Hom((L(U)[1])⊗n, E (U))Sn

where the tensor product is the completed projective tensor product, and Hom denotesthe space of continuous linear maps. Again, we treat this object as a differentiable pro-cochain complex.

As explained in the section on formal moduli problems (section 10.1), we should thinkof a local module E over L as providing, on each open subset U ⊂ M, a vector bundle onthe formal moduli problem BL(U) associated to L(U). Then the Chevalley-Eilenbergcochain complex C∗(L(U), E (U)) should be thought of as the space of sections of thisvector bundle.

10.5. D-modules and local L∞ algebras

Our definition of a local L∞ algebra is designed to encode the derived moduli spaceof solutions to a system of non-linear differential equations. An alternative language fordescribing differential equations is the theory of D-modules. In this section we will showhow our local L∞ algebras can also be viewed as L∞ algebras in the symmetric monoidalcategory of D-modules.

The main motivation for this extra layer of formalism is that local action functionals— which play a central role in classical field theory — are elegantly described using thelanguage of D-modules.

Let C∞M denote the sheaf of smooth functions on the manifold M, let DensM denote

the sheaf of smooth densities, and let DM the sheaf of differential operators with smoothcoefficients. The ∞-jet bundle Jet(E) of a vector bundle E is the vector bundle whose fiberat a point x ∈ M is the space of jets (or formal germs) at x of sections of E. The sheafof sections of Jet(E), denoted J(E), is equipped with a canonical DM-module structure,i.e., the natural flat connection sometimes known as the Cartan distribution. This flatconnection is characterized by the property that flat sections of J(E) are those sectionswhich arise by taking the jet at every point of a section of the vector bundle E. (Formotivation, observe that a field φ (a section of E) gives a section of Jet(E) that encodes allthe local information about φ.)


The category of DM modules has a symmetric monoidal structure, given by tensoringover C∞

M. The following lemma allows us to translate our definition of local L∞ algebrainto the world of D-modules.

10.5.0.2 Lemma. Let E1, . . . , En, F be vector bundles on M, and let Ei, F denote their spaces ofglobal sections. Then, there is a natural bijection

PolyDiff(E1 × · · · × En, F ) ∼= HomDM(J(E1)⊗ · · · ⊗ J(En), J(F))

where PolyDiff refers to the space of polydifferential operators. On the right hand side, we need toconsider maps which are continuous with respect to the natural adic topology on the bundle of jets.

Further, this bijection is compatible with composition.

A more formal statement of this lemma is that the multi-category of vector bundleson M, with morphisms given by polydifferential operators, is a full subcategory of thesymmetric monoidal category of DM modules. The embedding is given by taking jets.The proof of this lemma (which is straightforward) is presented in [Cos11c], Chapter 5.

This lemma immediately tells us how to interpret a local L∞ algebra in the languageof D-modules.

10.5.0.3 Corollary. Let L be a local L∞ algebra on M. Then J(L) has the structure of L∞ algebrain the category of DM modules.

Indeed, the lemma implies that to give a local L∞ algebra on M is the same as to givea graded vector bundle L on M together with an L∞ structure on the DM module J(L).

We are interested in the Chevalley-Eilenberg cochains of J(L), but taken now in thecategory of DM modules. Because J(L) is an inverse limit of the sheaves of finite-order jets,some care needs to be taken when defining this Chevalley-Eilenberg cochain complex.

In general, if E is a vector bundle, let J(E)∨ denote the sheaf HomC∞M(J(E), C∞

M), whereHomC∞

Mdenotes continuous linear maps of C∞

M-modules. This sheaf is naturally a DM-module. We can form the completed symmetric algebra

Ored(J(E)) = ∏n>0

SymnC∞

M

(J(E)∨

)= ∏

n>0HomC∞

M(J(E)⊗n, C∞

M)Sn .

Note that Ored(J(E) is a DM-algebra, as it is defined by taking the completed symmetricalgebra of J(E)∨ in the symmetric monoidal category of DM-modules where the tensorproduct is taken over C∞

M.


We can equivalently view J(E)∨ as an infinite-rank vector bundle with a flat connec-tion. The symmetric power sheaf Symn

C∞M(J(E)∨) is the sheaf of sections of the infinite-rank

bundle whose fibre at x is the symmetric power of the fibre of J(E)∨ at x.

In the case that E is the trivial bundle R, the sheaf J(R)∨ is naturally isomorphic toDM as a left DM-module. In this case, sections of the sheaf Symn

C∞M(DM) are objects which

in local coordinates are finite sums of expressions like

f (xi)∂I1 . . . ∂In .

where ∂Ij is the partial differentiation operator corresponding to a multi-index.

We should think of an element of Ored(J(E)) as a Lagrangian on the space E of sectionsof E (a Lagrangian in the sense that an action functional is given by a Lagrangian density).Indeed, every element of Ored(J(E)) has a Taylor expansion F = ∑ Fn where each Fn is asection

Fn ∈ HomC∞M(J(E)⊗n, C∞

M)Sn .

Each such Fn is a multilinear map which takes sections φ1, . . . , φn ∈ E and yields a smoothfunction Fn(φ1, . . . , φn) ∈ C∞(M), with the property that Fn(φ1, . . . , φn)(x) only dependson the ∞-jet of φi at x.

In the same way, we can interpret an element F ∈ Ored(J(E)) as something that takesa section φ ∈ E and yields a smooth function

∑ Fn(φ, . . . , φ) ∈ C∞(M),

with the property that F(φ)(x) only depends on the jet of φ at x.

Of course, the functional F is a formal power series in the variable φ. One cannotevaluate most formal power series, since the putative infinite sum makes no sense. In-stead, it only makes sense to evaluate a formal power series on infinitesimal elements. Inparticular, one can always evaluate a formal power series on nilpotent elements of a ring.

Indeed, a formal way to characterize a formal power series is to use the functor ofpoints perspective on Artinian algebras: if R is an auxiliary graded Artinian algebra withmaximal ideal m and if φ ∈ E ⊗m, then F(φ) is an element of C∞(M)⊗m. This assign-ment is functorial with respect to maps of graded Artin algebras.

10.5.1. Local functionals. We have seen that we can interpret Ored(J(E)) as the sheafof Lagrangians on a graded vector bundle E on M. Thus, the sheaf

DensM⊗C∞MOred(J(E))

is the sheaf of Lagrangian densities on M. A section F of this sheaf is something whichtakes as input a section φ ∈ E of E and produces a density F(φ) on M, in such a way that


F(φ)(x) only depends on the jet of φ at x. (As before, F is a formal power series in thevariable φ.)

The sheaf of local action functionals is the sheaf of Lagrangian densities modulo totalderivatives. Two Lagrangian densities that differ by a total derivative define the samelocal functional on (compactly supported) sections because the integral of total derivativevanishes. Thus, we do not want to distinguish them, as they lead to the same physics.The formal definition is as follows.

10.5.1.1 Definition. Let E be a graded vector bundle on M, whose space of global sections is E .Then the space of local action functionals on E is

Oloc(E ) = DensM⊗DMOred(J(E)).

Here, DensM is the right DM-module of densities on M.

Let Ored(Ec) denote the algebra of functionals modulo constants on the space Ec ofcompactly supported sections of E. Integration induces a natural inclusion

ι : Oloc(E )→ Ored(Ec),

where the Lagrangian density S ∈ Oloc(E ) becomes the functional ι(S) : φ 7→∫

M S(φ).(Again, φ must be nilpotent and compactly supported.) From here on, we will use thisinclusion without explicitly mentioning it.

10.5.2. Local Chevalley-Eilenberg complex of a local L∞ algebra. Let L be a local L∞algebra. Then we can form, as above, the reduced Chevalley-Eilenberg cochain complexC∗red(J(L)) of L. This is the DM-algebra Ored(J(L)[1]) equipped with a differential encodingthe L∞ structure on L.

10.5.2.1 Definition. If L is a local L∞-algebra, define the local Chevalley-Eilenberg complex to be

C∗red,loc(L) = DensM⊗DM C∗red(J(L)).

This is the space of local action functionals on L[1], equipped with the Chevalley-Eilenberg differential. In general, if g is an L∞ algebra, we think of the Lie algebra cochaincomplex C∗(g) as being the algebra of functions on Bg. In this spirit, we sometimes usethe notation Oloc(BL) for the complex C∗red,loc(L).

Note that C∗red,loc(L) is not a commutative algebra. Although the DM-module C∗red(J(L))is a commutative DM-module, the functor DensM⊗DM− is not a symmetric monoidalfunctor from DM-modules to cochain complexes, so it does not take commutative alge-bras to commutative algebras.


Note that there’s a natural inclusion of cochain complexes

C∗red,loc(L)→ C∗red(Lc(M)),

where Lc(M) denotes the L∞ algebra of compactly supported sections of L. The complexon the right hand side was defined earlier (see definition 10.4.0.1) and includes nonlocalfunctionals.

10.5.3. Central extensions and local cochains. In this section we will explain howlocal cochains are in bijection with certain central extensions of a local L∞ algebra. Toavoid some minor analytical difficulties, we will only consider central extensions that aresplit as precosheaves of graded vector spaces.

10.5.3.1 Definition. Let L be a local L∞ algebra on M. A k-shifted local central extension ofL is an L∞ structure on the precosheaf Lc ⊕C[k], where C is the constant precosheaf which takesvalue C on any open subset. We use the notation Lc for the precosheaf Lc ⊕C[k]. We require thatthis L∞ structure has the following properties.

(1) The sequence0→ C[k]→ Lc → Lc → 0

is an exact sequence of precosheaves of L∞ algebras, where C[k] is given the abelian struc-ture and Lc is given its original structure.

(2) This implies that the L∞ structure on Lc is determined from that on Lc by L∞ structuremaps

ln : Lc → C[k]for n ≥ 1. We require that these structure maps are given by local action functionals.

Two such central extensions, say Lc and L′c, are isomorphic if there is an L∞-isomorphism

Lc → L′cthat is the identity on C[k] and on the quotientLc. This L∞ isomorphism must satisfy an additionalproperty: the terms in this L∞ -isomorphism, which are given (using the decomposition of Lc andL′c as Lc ⊕C[k]) by functionals

L⊗nc → C[k],

must be local.

This definition refines the definition of central extension given in section 3.6 to includean extra locality property.

Example: Let Σ be a Riemann surface, and let g be a Lie algebra with an invariant pairing.Let L = Ω0,∗

Σ ⊗ g. Consider the Kac-Moody central extension, as defined in section 3.6 of3 We let

Lc = C · c⊕Lc,


where the central parameter c is of degree 1 and the Lie bracket is defined by

[α, β]Lc= [α, β]Lc + c

∫α∂β.

This is a local central extension. As shown in section 5.4 of chapter 5, the factorizationenvelope of this extension recovers the vertex algebra of an associated affine Kac-Moodyalgebra. ♦

10.5.3.2 Lemma. Let L be a local L∞ algebra on a manifold M. There is a bijection betweenisomorphism classes of k-shifted local central extensions of L and classes in Hk+2(Oloc(BL)).

PROOF. This result is almost immediate. Indeed, any closed degree k + 2 element ofOloc(BL) give a local L∞ structure on C[k]⊕Lc, where the L∞ structure maps

ln : Lc(U)→ C[k]

arise from the natural cochain map Oloc(BL)→ C∗red(Lc(U)). The fact that we start with aclosed element of Oloc(BL) corresponds to the fact that the L∞ axioms hold. Isomorphismsof local central extensions correspond to adding an exact cocycle to a closed degree k + 2element in Oloc(BL).

Particularly important is the case when we have a −1-shifted central extension. Asexplained in subsection 3.6.3 in Chapter 3, in this situation we can form the twisted fac-torization envelope, which is a factorization algebra over C[t] (where t is of degree 0)defined by sending an open subset U to the Chevalley-Eilenberg chain complex

U 7→ C∗(Lc(U)).

We think of C[t] as the Chevalley-Eilenberg chains of the Abelian Lie algebra C[−1]. Inthis situation, we can set t to be a particular value, leading to a twisted factorization enve-lope of L. Twisted factorization envelopes will play a central role in our formulation ofNoether’s theorem at the quantum level in chapter 18.

10.5.4. Calculations of local L∞ algebra cohomology play an important role in quan-tum field theory. Indeed, the obstruction-deformation complex describing quantizationsof a classical field theory are local L∞ algebra cohomology groups. Thus, it will be helpfulto be able to compute some examples.

Before we start, let us describe a general result which will facilitate computation.

10.5.4.1 Lemma. Let M be an oriented manifold and let L be a local L∞-algebra on M. Then,there is a natural quasi-isomorphism

Ω∗(M, C∗red(J(L)))[dimM] ∼= C∗red,loc(L).


PROOF. By definition,

O(BL) = DensM⊗DM C∗red J(L)where DM is the sheaf of C∞ differential operators. The DM-module C∗red(J(L)) is flat(this was checked in [Cos11c]), so we can replace the tensor product over DM with theleft-derived tensor product.

Since M is oriented, we can replace DensM by ΩdM where d = dim M. The right DM-

module ΩdM has a free resolution of the form

· · · → Ωd−1M ⊗C∞

MDM → Ωd M⊗C∞

MDM

where ΩiM ⊗C∞

MDM is in cohomological degree −i, and the differential in this complex is

the de Rham differential coupled to the left DM-module structure on DM. (This is some-times called the Spenser resolution).

It follows that we the derived tensor product can be represented as

ΩdM ⊗mbbL

DMC∗red(J(L)) = Ω∗(M, C∗red(J(L)))[d]

as desired.

10.5.4.2 Lemma. Let Σ be a Riemann surface. Let L be the local L∞ algebra on Σ defined byL(U) = Ω0,∗(U, TU). In other words, L is the Dolbeault resolution of the sheaf of holomorphicvector fields on Σ.

Then,Hi(O(BL)) = H∗(Σ)[−1].

Remark: The class in H1(O(BL)) corresponding to the class 1 ∈ H0(Σ) leads to a local cen-tral extension of L. One can check that the corresponding twisted factorization envelopecorresponds to the Virasoro vertex algebra, in the same way that we showed in section 5.4that the Kac-Moody extension above leads to the Kac-Moody vertex algebra. ♦

PROOF. The previous lemma tells us that we need to compute the de Rham cohomol-ogy with coefficients in the DΣ-module C∗red(J(L))[2]. Suppose we want to compute the deRham cohomology with coefficients in any complex M of DΣ-modules. There is a spectralsequence converging to this cohomology, associated to the filtration on Ω∗(Σ, M) by formdegree. The E2 page of this spectral sequence is the de Rham complex Ω∗(Σ,H∗(M)) withcoefficients in the cohomology DΣ-moduleH∗(M).

We will use this spectral sequence in our example. The first step is to compute thecohomology of the DΣ-module C∗red(J(L)). We will compute the cohomology of the fibresof this sheaf at an arbitrary point x ∈ Σ. Let us choose a holomorphic coordinate z at


x. The fibre Jx(L) at x is the dg Lie algebra C[[z, z, dz]]∂z with differential ∂. This dg Liealgebra is quasi-isomorphic to the Lie algebra of formal vector fields C[[z]]∂z.

A calculation performed by Gelfand-Fuchs [] shows that the reduced Lie algebra co-homology of C[[z]]∂z is concentrated in degree 3, where it is one-dimensional. A cochainrepresentative for the unique non-zero cohomology class is ∂∨z (z∂z)∨(z2∂z)∨where (zk∂z)∨

refers to the element in (C[[z]]∂z)∨ in the dual basis.

Thus, we find that the cohomology of C∗red(J(L)) is a rank one local system situatedin cohomological degree 3. Choosing a formal coordinate at a point in a Riemann surfacetrivializes the fibre of this line bundle. The trivialization is independent of the coordinatechoice, and compatible with the flat connection. From this we deduce that

H∗(C∗red(J(L))) = C∞Σ [−3]

is the trivial rank one local system, situated in cohomological degree 3.

Therefore, the cohomology of Oloc(BL) is a shift by−1 of the de Rham cohomology ofthis trivial flat line bundle, completing the result.

10.5.5. Cochains with coefficients in a local module for a local L∞ algebras. Let Lbe a local L∞ algebra on M, and let E be a local module for L. Then J(E) has an action ofthe L∞ algebra J(L), in a way compatible with the DM-module on both J(E) and J(L).

10.5.5.1 Definition. Suppose that E has a local action of L. Then the local cochains C∗loc(L, E )of L with coefficients in E is defined to be the flat sections of the DM-module of cochains of J(L)with coefficients in J(E).

More explicitly, the DM-module C∗(J(L), J(E) is

∏n≥0

HomC∞M

((J(L)[1])⊗n, J(E)

)Sn

,

equipped with the usual Chevalley-Eilenberg differential. The sheaf of flat sections of thisDM module is the subsheaf

∏n≥0

HomDM

((J(L)[1])⊗n, J(E)

)Sn

,

where the maps must be DM-linear. In light of the fact that

HomDM

(J(L)⊗n, J(E)

)= PolyDiff(L⊗n, E ),

we see that C∗loc(L, E ) is precisely the subcomplex of the Chevalley-Eilenberg cochaincomplex

C∗(L, E ) = ∏n≥0

HomR((L[1])⊗n, E )Sn

consisting of those cochains built up from polydifferential operators.

CHAPTER 11

The classical Batalin-Vilkovisky formalism

In the preceding chapter we explained how to encode the formal neighborhood of asolution to the Euler-Lagrange equations — a formal elliptic moduli problem — by anelliptic L∞ algebra. As we explain in this chapter, the elliptic moduli problems arisingfrom action functionals possess even more structure: a shifted symplectic form, so thatthe formal moduli problem is a derived symplectic space.

Our starting point is the finite-dimensional model that motivates the Batalin-Vilkoviskyformalism for classical field theory. With this model in mind, we then develop the rele-vant definitions in the language of elliptic L∞ algebras. The end of the chapter is devotedto several examples of classical BV theories, notably cotangent field theories, which are theanalogs of cotangent bundles in ordinary symplectic geometry.

11.1. The classical BV formalism in finite dimensions

Before we discuss the Batalin-Vilkovisky formalism for classical field theory, we willdiscuss a finite-dimensional toy model (which we can think of as a 0-dimensional classicalfield theory). Our model for the space of fields is a finite-dimensional smooth manifoldmanifold M. The “action functional” is given by a smooth function S ∈ C∞(M). Classicalfield theory is concerned with solutions to the equations of motion. In our setting, theequations of motion are given by the subspace Crit(S) ⊂ M. Our toy model will notchange if M is a smooth algebraic variety or a complex manifold, or indeed a smoothformal scheme. Thus we will write O(M) to indicate whatever class of functions (smooth,polynomial, holomorphic, power series) we are considering on M.

If S is not a nice function, then this critical set can by highly singular. The classi-cal Batalin-Vilkovisky formalism tells us to take, instead the derived critical locus of S.(Of course, this is exactly what a derived algebraic geometer — see [Lur09b], [Toe06] —would tell us to do as well.) We will explain the essential idea without formulating itprecisely inside any particular formalism for derived geometry. For such a treatment, see[Vez11].

The critical locus of S is the intersection of the graph

Γ(dS) ⊂ T∗M

225

226 11. THE CLASSICAL BATALIN-VILKOVISKY FORMALISM

with the zero-section of the cotangent bundle of M. Algebraically, this means that we canwrite the algebra O(Crit(S)) of functions on Crit(S) as a tensor product

O(Crit(S)) = O(Γ(dS))⊗O(T∗M) O(M).

Derived algebraic geometry tells us that the derived critical locus is obtained by replacingthis tensor product with a derived tensor product. Thus, the derived critical locus of S,which we denote Crith(S), is an object whose ring of functions is the commutative dgalgebra

O(Crith(S)) = O(Γ(dS))⊗LO(T∗M) O(M).

In derived algebraic geometry, as in ordinary algebraic geometry, spaces are determinedby their algebras of functions. In derived geometry, however, one allows differential-graded algebras as algebras of functions (normally one restricts attention to differential-graded algebras concentrated in non-positive cohomological degrees).

We will take this derived tensor product as a definition of O(Crith(S)).

11.1.1. An explicit model. It is convenient to consider an explicit model for the de-rived tensor product. By taking a standard Koszul resolution of O(M) as a module overO(T∗M), one sees that O(Crith(S)) can be realized as the complex

O(Crith(S)) ' . . . ∨dS−−→ Γ(M,∧2TM)∨dS−−→ Γ(M, TM)

∨dS−−→ O(M).

In other words, we can identify O(Crith(S)) with functions on the “graded manifold”T∗[−1]M, equipped with the differential given by contracting with the 1-form dS. Thisnotation T∗[−1]M denotes the ordinary smooth manifold M equipped with the graded-commutative algebra SymC∞

M(Γ(M, TM)[1]) as its ring of functions.

Note thatO(T∗[−1]M) = Γ(M,∧∗TM)

has a Poisson bracket of cohomological degree 1, called the Schouten-Nijenhuis bracket.This Poisson bracket is characterized by the fact that if f , g ∈ O(M) and X, Y ∈ Γ(M, TM),then

X, Y = [X, Y]

X, f = X f

f , g = 0

and the Poisson bracket between other elements of O(T∗[−1]M) is inferred from the Leib-niz rule.

The differential on O(T∗[−1]M) corresponding to that on O(Crith(S)) is given by

dφ = S, φfor φ ∈ O(T∗[−1]M).

11.2. THE CLASSICAL BV FORMALISM IN INFINITE DIMENSIONS 227

The derived critical locus of any function thus has a symplectic form of cohomolog-ical degree −1. It is manifest in this model and hence can be found in others. In theBatalin-Vilkovisky formalism, the space of fields always has such a symplectic structure.However, one does not require that the space of fields arises as the derived critical locusof a function.

11.2. The classical BV formalism in infinite dimensions

We would like to consider classical field theories in the BV formalism. We have alreadyexplained how the language of elliptic moduli problems captures the formal geometry ofsolutions to a system of PDE. Now we need to discuss the shifted symplectic structurespossessed by a derived critical locus. For us, a classical field theory will be specified by anelliptic moduli problem equipped with a symplectic form of cohomological degree −1.

We defined the notion of formal elliptic moduli problem on a manifold M using thelanguage of L∞ algebras. Thus, in order to give the definition of a classical field theory,we need to understand the following question: what extra structure on an L∞ algebra gendows the corresponding formal moduli problem with a symplectic form?

In order to answer this question, we first need to understand a little about what itmeans to put a shifted symplectic form on a (formal) derived stack.

In the seminal work of Schwarz [Sch93, AKSZ97], a definition of a shifted symplecticform on a dg manifold is given. Dg manifolds where an early attempt to develop a theoryof derived geometry. It turns out that dg manifolds are sufficient to capture some aspectsof the modern theory of derived geometry, including formal derived geometry.

In the world of dg manifolds, as in any model of derived geometry, all spaces of ten-sors are cochain complexes. In particular, the space of i-forms Ωi(M) on a dg manifold isa cochain complex. The differential on this cochain complex is called the internal differen-tial on i-forms. In addition to the internal differential, there is also a de Rham differentialddR : Ωi(M) → Ωi+1(M) which is a cochain map. Schwarz defined a symplectic formon a dg manifold M to be a two-form ω which is both closed in the differential on thecomplex of two-forms, and which is also closed under the de Rham differential mappingtwo-forms to three-forms. A symplectic form is also required to be non-degenerate. Thesymplectic two-form ω will have some cohomological degree, which for the case relevantto the BV formalsim is −1.

Following these ideas, Pantev et al. [PTVV11] give a definition of (shifted) symplecticstructure in the more modern language of derived stacks. In this approach, instead ofasking that the two-form defining the symplectic structure be closed both in the internaldifferential on two-forms and closed under the de Rham differential, one constructs a


double complex

Ω≥2 = Ω2 → Ω3[−1]→ . . .

as the subcomplex of the de Rham complex consisting of 2 and higher forms. One thenlooks for an element of this double complex which is closed under the total differential(the sum of the de Rham differential and the internal differential on each space of k-forms)and whose 2-form component is non-degenerate in a suitable sense.

However, it turns out that, in the case of formal derived stacks, the definition given bySchwarz and that given by Pantev et al. coincides. One can also show that in this situationthere is a Darboux lemma, showing that we can take the symplectic form to have constantcoefficients. In order to explain what we mean by this, let us explain how to understandforms on a formal derived stack in terms of the associated L∞-algebra.

Given a pointed formal moduli problem M, the associated L∞ algebra gM has theproperty that

gM = TpM[−1].

Further, we can identify geometric objects onM in terms of gM as follows.

C∗(gM) the algebra O(M) of functions onMgM-modules OM-modulesC∗(gM, V) the OM-module corresponding to the gM-module V

the gM-module gM[1] TM

Following this logic, we see that the complex of 2-forms onM is identified with C∗(gM,∧2(g∨M[−1])).

As we have seen, according to Schwarz, a symplectic form onM is a two-form onMwhich is closed for both the internal and de Rham differentials. Any constant-coefficienttwo-form is automatically closed under the de Rham differential. A constant-coefficienttwo-form of degree k is an element of Sym2(gM)∨ of cohomological degree k − 2, i.e.a symmetric pairing on gM of this degree. Such a two-form is closed for the internaldifferential if and only if it is invariant.

To give a formal pointed moduli problem with a symplectic form of cohomologicaldegree k is the same as to give an L∞ algebra with an invariant and non-degenerate pairingof cohomological degree k− 2.

Thus, we find that constant coefficient symplectic two-forms of degree k on M areprecisely the same as non-degenerate symmetric invariant pairings on gM. The relationbetween derived symplectic geometry and invariant pairings on Lie algebras was firstdeveloped by Kontsevich [Kon93].

The following formal Darboux lemma makes this relationship into an equivalence.

11.2. THE CLASSICAL BV FORMALISM IN INFINITE DIMENSIONS 229

11.2.0.1 Lemma. Let g be a finite-dimensional L∞ algebra. Then, k-shifted symplectic structureson the formal derived stack Bg (in the sense of Pantev et al.) are the same as symmetric invariantnon-degenerate pairings on g of cohomological degree k− 2.

The proof is a little technical, and appears in an appendix ??. The proof of a closelyrelated statement in a non-commutative setting was given by Kontsevich and Soibelman[KS06]. In the statement of the lemma, “the same” means that simplicial sets parametriz-ing the two objects are canonically equivalent.

Following this idea, we will define a classical field theory to be an elliptic L∞ algebraequipped with a non-degenerate invariant pairing of cohomological degree −3. Let usfirst define what it means to have an invariant pairing on an elliptic L∞ algebra.

11.2.0.2 Definition. Let M be a manifold, and let E be an elliptic L∞ algebra on M. An invariantpairing on E of cohomological degree k is a symmetric vector bundle map

〈−,−〉E : E⊗ E→ Dens(M)[k]

satisfying some additional conditions:

(1) Non-degeneracy: we require that this pairing induces a vector bundle isomorphism

E→ E∨ ⊗Dens(M)[−3].

(2) Invariance: let Ec denotes the space of compactly supported sections of E. The pairingon E induces an inner product on Ec, defined by

〈−,−〉 : Ec ⊗ Ec → R

α⊗ β→∫

M〈α, β〉 .

We require it to be an invariant pairing on the L∞ algebra Ec.

Recall that a symmetric pairing on an L∞ algebra g is called invariant if, for all n, thelinear map

g⊗n+1 → R

α1 ⊗ · · · ⊗ αn+1 7→ 〈ln(α1, . . . , αn), αn+1〉is graded anti-symmetric in the αi.

11.2.0.3 Definition. A formal pointed elliptic moduli problem with a symplectic form ofcohomological degree k on a manifold M is an elliptic L∞ algebra on M with an invariantpairing of cohomological degree k− 2.

11.2.0.4 Definition. In the BV formalism, a (perturbative) classical field theory on M is aformal pointed elliptic moduli problem on M with a symplectic form of cohomological degree −1.


11.3. The derived critical locus of an action functional

The critical locus of a function f is, of course, the zero locus of the 1-form d f . We areinterested in constructing the derived critical locus of a local functional S ∈ Oloc(BL) onthe formal moduli problem associated to a local L∞ algebra L on a manifold M. Thus,we need to understand what kind of object the exterior derivative dS of such an actionfunctional S is.

If g is an L∞ algebra, then we should think of C∗red(g) as the algebra of functions on theformal moduli problem Bg that vanish at the base point. Similarly, C∗(g, g∨[−1]) shouldbe the thought of as the space of 1-forms on Bg. The exterior derivative is thus a map

d : C∗red(g)→ C∗(g, g∨[−1]),

namely the universal derivation.

We will define a similar exterior derivative for a local Lie algebra L on M. The analogof g∨ is the L-module L!, whose sections are (up to completion) the Verdier dual of thesheaf L. Thus, our exterior derivative will be a map

d : Oloc(BL)→ C∗loc(L,L![−1]).

Recall that Oloc(BL) denotes the subcomplex of C∗red(Lc(M)) consisting of local func-tionals. The exterior derivative for the L∞ algebra Lc(M) is a map

d : C∗red(Lc(M))→ C∗(Lc(M),Lc(M)∨[−1]).

Note that the dual Lc(M)∨ of Lc(M) is the space L!(M) of distributional sections of the

bundle L! on M. Thus, the exterior derivative is a map

d : C∗red(Lc(M))→ C∗(Lc(M),L!(M)[−1]).

Note thatC∗loc(L,L![−1]) ⊂ C∗(Lc(M),L!(M)) ⊂ C∗(Lc(M),L!

(M)).

We will now show that d preserves locality and more.

11.3.0.5 Lemma. The exterior derivative takes the subcomplex Oloc(BL) of C∗red(Lc(M)) to the

subcomplex C∗loc(L,L![−1]) of C∗(Lc(M),L!(M)).

PROOF. The content of this lemma is the familiar statement that the Euler-Lagrangeequations associated to a local action functional are differential equations. We will give aformal proof, but the reader will see that we only use integration by parts.

Any functionalF ∈ Oloc(BL)

11.3. THE DERIVED CRITICAL LOCUS OF AN ACTION FUNCTIONAL 231

can be written as a sum F = ∑ Fn where

Fn ∈ DensM⊗DM HomC∞M

(J(L)⊗n, C∞

M)

Sn.

Any such Fn can be written as a finite sum

Fn = ∑i

ωDi1 . . . Di

n

where ω is a section of DensM and Dij are differential operators from L to C∞

M. (The nota-tion ωDi

1 . . . Din means simply to multiply the density ω by the outputs of the differential

operators, which are smooth functions.)

If we view F ∈ O(Lc(M)), then the nth Taylor component of F is the linear map

Lc(M)⊗n → R

defined by

φ1 ⊗ · · · ⊗ φn →∑i

∫M

ω(Di1φ1) . . . (Di

nφn).

Thus, the (n− 1)th Taylor component of dF is given by the linear map

dFn : Lc(M)⊗n−1 → L!(M) = Lc(M)∨

φ1 ⊗ · · · ⊗ φn−1 ∑i7→ ω(Di

1φ1) . . . (Din−1φn−1)Di

n(−) + symmetric terms

where the right hand side is viewed as a linear map from Lc(M) to R. Now, by integrationby parts, we see that

(dFn)(φ1, . . . , φn−1)

is in the subspace L!(M) ⊂ L!(M) of smooth sections of the bundle L!(M), inside the

space of distributional sections.

It is clear from the explicit expressions that the map

dFn : Lc(M)⊗n−1 → L!(M)

is a polydifferential operator, and so defines an element of C∗loc(L,L![−1]) as desired.

11.3.1. Field theories from action functionals. Physicists normally think of a classicalfield theory as being associated to an action functional. In this section we will show howto construct a classical field theory in our sense from an action functional.

We will work in a very general setting. Recall (section 10.1.3) that we defined a localL∞ algebra on a manifold M to be a sheaf of L∞ algebras where the structure maps aregiven by differential operators. We will think of a local L∞ algebra L on M as defining aformal moduli problem cut out by some differential equations. We will use the notationBL to denote this formal moduli problem.


We want to take the derived critical locus of a local action functional

S ∈ Oloc(BL)of cohomological degree 0. (We also need to assume that S is at least quadratic: thiscondition insures that the base-point of our formal moduli problem BL is a critical pointof S). We have seen (section 11.3) how to apply the exterior derivative to a local actionfunctional S yields an element

dS ∈ C∗loc(L,L![−1]),

which we think of as being a local 1-form on BL.

The critical locus of S is the zero locus of dS. We thus need to explain how to constructa new local L∞ algebra that we interpret as being the derived zero locus of dS.

11.3.2. Finite dimensional model. We will first describe the analogous constructionin finite dimensions. Let g be an L∞ algebra, M be a g-module of finite total dimension,and α be a closed, degree zero element of C∗red(g, M). The subscript red indicates that weare taking the reduced cochain complex, so that α is in the kernel of the augmentationmap C∗(g, M)→ M.

We think of M as a dg vector bundle on the formal moduli problem Bg, and so α isa section of this vector bundle. The condition that α is in the reduced cochain complextranslates into the statement that α vanishes at the basepoint of Bg. We are interested inconstructing the L∞ algebra representing the zero locus of α.

We start by writing down the usual Koszul complex associated to a section of a vectorbundle. In our context, the commutative dg algebra representing this zero locus of α isgiven by the total complex of the double complex

· · · → C∗(g,∧2M∨) ∨α−→ C∗(g, M∨) ∨α−→ C∗(g).

In words, we have written down the symmetric algebra on the dual of g[1]⊕ M[−1]. Itfollows that this commutative dg algebra is the Chevalley-Eilenberg cochain complex ofg⊕M[−2], equipped with an L∞ structure arising from the differential on this complex.

Note that the direct sum g ⊕ M[−2] (without a differential depending on α) has anatural semi-direct product L∞ structure, arising from the L∞ structure on g and the actionof g on M[−2]. This L∞ structure corresponds to the case α = 0.

11.3.2.1 Lemma. The L∞ structure on g⊕M[−2] describing the zero locus of α is a deformationof the semidirect product L∞ structure, obtained by adding to the structure maps ln the maps

Dnα : g⊗n → M

X1 ⊗ · · · ⊗ Xn 7→∂

∂X1. . .

∂

∂Xnα.

11.3. THE DERIVED CRITICAL LOCUS OF AN ACTION FUNCTIONAL 233

This is a curved L∞ algebra unless the section α vanishes at 0 ∈ g.

PROOF. The proof is a straightforward computation.

Note that the maps Dnα in the statement of the lemma are simply the homogeneouscomponents of the cochain α.

We will let Z(α) denote g⊕M[−2], equipped with this L∞ structure arising from α.

Recall that the formal moduli problem Bg is the functor from dg Artin rings (R, m) tosimplicial sets, sending (R, m) to the simplicial set of Maurer-Cartan elements of g⊗ m.In order to check that we have constructed the correct derived zero locus for α, we shoulddescribe the formal moduli problem associated Z(α).

Thus, let (R, m) be a dg Artin ring, and x ∈ g ⊗ m be an element of degree 1, andy ∈ M⊗m be an element of degree −1. Then (x, y) satisfies the Maurer-Cartan equationin Z(α) if and only if

(1) x satisfies the Maurer-Cartan equation in g⊗m and(2) α(x) = dxy ∈ M, where

dx = dy + µ1(x, y) + 12! µ2(x, x, y) + · · · : M→ M

is the differential obtained by deforming the original differential by that arisingfrom the Maurer-Cartan element x. (Here µn : g⊗n ⊗ M → M are the actionmaps.)

In other words, we see that an R-point of BZ(α) is both an R-point x of Bg and a homotopybetween α(x) and 0 in the fiber Mx of the bundle M at x ∈ Bg. The fibre Mx is the cochaincomplex M with differential dx arising from the solution x to the Maurer-Cartan equation.Thus, we are described the homotopy fiber product between the section α and the zerosection in the bundle M, as desired.

Let us make thigs

11.3.3. The derived critical locus of a local functional. Let us now return to the situ-ation where L is a local L∞ algebra on a manifold M and S ∈ O(BL) is a local functionalthat is at least quadratic. Let

dS ∈ C∗loc(L,L![−1])denote the exterior derivative of S. Note that dS is in the reduced cochain complex, i.e.the kernel of the augmentation map C∗loc(L,L![−1])→ L![−1].

LetdnS : L⊗n → L!


be the nth Taylor component of dS. The fact that dS is a local cochain means that dnS is apolydifferential operator.

11.3.3.1 Definition. The derived critical locus of S is the local L∞ algebra obtained by addingthe maps

dnS : L⊗n → L!

to the structure maps ln of the semi-direct product L∞ algebra L ⊕ L![−3]. We denote this localL∞ algebra by Crit(S).

If (R, m) is an auxiliary Artinian dg ring, then a solution to the Maurer-Cartan equa-tion in Crit(S)⊗m consists of the following data:

(1) a Maurer-Cartan element x ∈ L⊗m and(2) an element y ∈ L! ⊗m such that

(dS)(x) = dxy.

Here dxy is the differential on L!⊗m induced by the Maurer-Cartan element x. These twoequations say that x is an R-point of BL that satisfies the Euler-Lagrange equations up toa homotopy specified by y.

11.3.4. Symplectic structure on the derived critical locus. Recall that a classical fieldtheory is given by a local L∞ algebra that is elliptic and has an invariant pairing of degree−3. The pairing on the local L∞ algebra Crit(S) constructed above is evident: it is givenby the natural bundle isomorphism

(L⊕ L![−3])![−3] ∼= L![−3]⊕ L.

In other words, the pairing arises, by a shift, from the natural bundle map L ⊗ L! →DensM .

11.3.4.1 Lemma. This pairing on Crit(S) is invariant.

PROOF. The original L∞ structure on L⊕L![−3] (that is, the L∞ structure not involv-ing S) is easily seen to be invariant. We will verify that the deformation of this structurecoming from S is also invariant.

We need to show that if

α1, . . . , αn+1 ∈ Lc ⊕L!c[−3]

are compactly supported sections of L⊕ L1[−3], then

〈ln(α1, . . . , αn), αn+1〉

11.4. A SUCCINCT DEFINITION OF A CLASSICAL FIELD THEORY 235

is totally antisymmetric in the variables αi. Now, the part of this expression that comesfrom S is just (

∂

∂α1. . .

∂

∂αn+1

)S(0).

The fact that partial derivatives commute — combined with the shift in grading due toC∗(Lc) = O(Lc[1]) — immediately implies that this term is totally antisymmetric.

Note that, although the local L∞ algebra Crit(S) always has a symplectic form, it doesnot always define a classical field theory, in our sense. To be a classical field theory, wealso require that the local L∞ algebra Crit(S) is elliptic.

11.4. A succinct definition of a classical field theory

We defined a classical field theory to be a formal elliptic moduli problem equippedwith a symplectic form of degree −1. In this section we will rewrite this definition in amore concise (but less conceptual) way. This version is included largely for consistencywith [Cos11c] — where the language of elliptic moduli problems is not used — and forease of reference when we discuss the quantum theory.

11.4.0.2 Definition. Let E be a graded vector bundle on a manifold M. A degree−1 symplecticstructure on E is an isomorphism of graded vector bundles

φ : E ∼= E![−1]

that is anti-symmetric, in the sense that φ∗ = −φ where φ∗ is the formal adjoint of φ.

Note that if L is an elliptic L∞ algebra on M with an invariant pairing of degree −3,then the graded vector bundle L[1] on M has a−1 symplectic form. Indeed, by definition,L is equipped with a symmetric isomorphism L ∼= L![−3], which becomes an antisym-metric isomorphism L[1] ∼= (L[1])![−1].

Note also that the tangent space at the basepoint to the formal moduli problem BLassociated to L is L[1] (equipped with the differential induced from that on L). Thus, thealgebra C∗(L) of cochains of L is isomorphic, as a graded algebra without the differential,to the algebra O(L[1]) of functionals on L[1].

Now suppose that E is a graded vector bundle equipped with a −1 symplectic form.Let Oloc(E ) denote the space of local functionals on E , as defined in section 10.5.1.

11.4.0.3 Proposition. For E a graded vector bundle equipped with a −1 symplectic form, letOloc(E ) denote the space of local functionals on E . Then we have the following.

(1) The symplectic form on E induces a Poisson bracket on Oloc(E ), of degree +1.


(2) Equipping E[−1] with a local L∞ algebra structure compatible with the given pairing onE[−1] is equivalent to picking an element S ∈ Oloc(E ) that has cohomological degree 0,is at least quadratic, and satisfies the classical master equation

S, S = 0.

PROOF. Let L = E[−1]. Note that L is a local L∞ algebra, with the zero differential andzero higher brackets (i.e., a totally abelian L∞ algebra). We write Oloc(BL) or C∗red,loc(L) forthe reduced local cochains of L. This is a complex with zero differential which coincideswith Oloc(E ).

We have seen that the exterior derivative (section 11.3) gives a map

d : Oloc(E ) = Oloc(BL)→ C∗loc(L,L![−1]).

Note that the isomorphismL ∼= L![−3]

gives an isomorphismC∗loc(L,L![−1]) ∼= C∗loc(L,L[2]).

Finally, C∗loc(L,L[2]) is the L∞ algebra controlling deformations of L as a local L∞ alge-bra. It thus remains to verify that Oloc(BL) ⊂ C∗loc(L,L[2]) is a sub L∞ algebra, which isstraightforward.

Note that the finite-dimensional analog of this statement is simply the fact that on aformal symplectic manifold, all symplectic derivations (which correspond, after a shift, todeformations of the formal symplectic manifold) are given by Hamiltonian functions, de-fined up to the addition of an additive constant. The additive constant is not mentioned inour formulation because Oloc(E ), by definition, consists of functionals without a constantterm.

Thus, we can make a concise definition of a field theory.

11.4.0.4 Definition. A pre-classical field theory on a manifold M consists of a graded vectorbundle E on M, equipped with a symplectic pairing of degree −1, and a local functional

S ∈ Oloc(Ec(M))

of cohomological degree 0, satisfying the following properties.

(1) S satisfies the classical master equation S, S = 0.(2) S is at least quadratic (so that 0 ∈ Ec(M) is a critical point of S).

In this situation, we can write S as a sum (in a unique way)

S(e) = 〈e, Qe〉+ I(e)

11.5. EXAMPLES OF FIELD THEORIES FROM ACTION FUNCTIONALS 237

where Q : E → E is a skew self-adjoint differential operator of cohomological degree 1and square zero.

11.4.0.5 Definition. A pre-classical field is a classical field theory if the complex (E , Q) iselliptic.

There is one more property we need of a classical field theories in order to be applythe quantization machinery of [Cos11c].

11.4.0.6 Definition. A gauge fixing operator is a map

QGF : E (M)→ E (M)

that is a differential operator of cohomological degree −1 such that (QGF)2 = 0 and

[Q, QGF] : E (M)→ E (M)

is a generalized Laplacian in the sense of [BGV92].

The only classical field theories we will try to quantize are those that admit a gaugefixing operator. Thus, we will only consider classical field theories which have a gaugefixing operator. An important point which will be discussed at length in the chapter onquantum field theory is the fact that the observables of the quantum field theory are inde-pendent (up to homotopy) of the choice of gauge fixing condition.

11.5. Examples of field theories from action functionals

Let us now give some basic examples of field theories arising as the derived criticallocus of an action functional. We will only discuss scalar field theories in this section.

Let (M, g) be a Riemannian manifold. Let R be the trivial line bundle on M and DensMthe density line bundle. Note that the volume form dVolg provides an isomorphism be-tween these line bundles. Let

S(φ) = 12

∫M

φ D φ

denote the action functional for the free massless field theory on M. Here D is the Lapla-cian on M, viewed as a differential operator from C∞(M) to Dens(M), so Dφ = (∆gφ)dVolg.

The derived critical locus of S is described by the elliptic L∞ algebra

L = C∞(M)[−1] D−→ Dens(M)[−2]

where Dens(M) is the global sections of the bundle of densities on M. Thus, C∞(M) issituated in degree 1, and the space Dens(M) is situated in degree 2. The pairing betweenDens(M) and C∞(M) gives the invariant pairing on L, which is symmetric of degree −3as desired.


11.5.1. Interacting scalar field theories. Next, let us write down the derived criticallocus for a basic interacting scalar field theory, given by the action functional

S(φ) = 12

∫M

φ D φ + 14!

∫M

φ4.

The cochain complex underlying our elliptic L∞ algebra is, as before,

L = C∞(M)[−1] D−→ Dens(M)[−2].

The interacting term 14!

∫M φ4 gives rise to a higher bracket l3 on L, defined by the map

C∞(M)⊗3 → Dens(M)

φ1 ⊗ φ2 ⊗ φ3 7→ φ1φ2φ3dVolg.

Let (R, m) be a nilpotent Artinian ring, concentrated in degree 0. Then a section ofφ ∈ C∞(M)⊗m satisfies the Maurer-Cartan equation in this L∞ algebra if and only if

D φ + 13! φ

3dVol = 0.

Note that this is precisely the Euler-Lagrange equation for S. Thus, the formal moduliproblem associated to L is, as desired, the derived version of the moduli of solutions tothe Euler-Lagrange equations for S.

11.6. Cotangent field theories

We have defined a field theory to be a formal elliptic moduli problem equipped witha symplectic form of degree −1. In geometry, cotangent bundles are the basic examplesof symplectic manifolds. We can apply this construction in our setting: given any ellipticmoduli problem, we will produce a new elliptic moduli problem – its shifted cotangentbundle – that has a symplectic form of degree −1. We call the field theories that ariseby this construction cotangent field theories. It turns out that a surprising number of fieldtheories of interest in mathematics and physics arise as cotangent theories, including,for example, both the A- and the B-models of mirror symmetry and their half-twistedversions.

We should regard cotangent field theories as the simplest and most basic class of non-linear field theories, just as cotangent bundles are the simplest class of symplectic man-ifolds. One can show, for example, that the phase space of a cotangent field theory isalways an (infinite-dimensional) cotangent bundle, whose classical Hamiltonian functionis linear on the cotangent fibers.

11.6. COTANGENT FIELD THEORIES 239

11.6.1. The cotangent bundle to an elliptic moduli problem. Let L be an elliptic L∞algebra on a manifold X, and letML be the associated elliptic moduli problem.

Let L! be the bundle L∨ ⊗Dens(X). Note that there is a natural pairing between com-pactly supported sections of L and compactly supported sections of L!.

Recall that we use the notation L to denote the space of sections of L. Likewise, wewill let L! denote the space of sections of L!.

11.6.1.1 Definition. Let T∗[k]BL denote the elliptic moduli problem associated to the elliptic L∞algebra L⊕L![k− 2].

This elliptic L∞ algebra has a pairing of cohomological degree k− 2.

The L∞ structure on the space L ⊕ L![k− 2] of sections of the direct sum bundle L⊕L![k− 2] arises from the natural L-module structure on L!.

11.6.1.2 Definition. LetM = BL be an elliptic moduli problem corresponding to an elliptic L∞algebra L. Then the cotangent field theory associated toM is the −1-symplectic elliptic moduliproblem T∗[−1]M, whose elliptic L∞ algebra is L⊕L![−3].

11.6.2. Examples. In this section we will list some basic examples of cotangent theo-ries, both gauge theories and nonlinear sigma models.

In order to make the discussion more transparent, we will not explicitly describe theelliptic L∞ algebra related to every elliptic moduli problem we describe. Instead, we maysimply define the elliptic moduli problem in terms of the geometric objects it classifies. Inall examples, it is straightforward using the techniques we have discussed so far to writedown the elliptic L∞ algebra describing the formal neighborhood of a point in the ellipticmoduli problems we will consider.

11.6.3. Self-dual Yang-Mills theory. Let X be an oriented 4-manifold equipped witha conformal class of a metric. Let G be a compact Lie group. Let M(X, G) denote theelliptic moduli problem parametrizing principal G-bundles on X with a connection whosecurvature is self-dual.

Then we can consider the cotangent theory T∗[−1]M(X, G). This theory is known inthe physics literature as self-dual Yang-Mills theory.

Let us describe the L∞ algebra of this theory explicitly. Observe that the elliptic L∞algebra describing the completion ofM(X, G) near a point (P,∇) is

Ω0(X, gP)d∇−→ Ω1(X, gP)

d−−→ Ω2−(X, gP)


where gP is the adjoint bundle of Lie algebras associated to the principal G-bundle P. Hered− denotes the connection followed by projection onto the anti-self-dual 2-forms.

Thus, the elliptic L∞ algebra describing T∗[−1]M is given by the diagram

Ω0(X, gP)d∇−→ Ω1(X, gP)

d−−→ Ω2−(X, gP)

⊕ ⊕Ω2−(X, gP)

d∇−→ Ω3(X, gP)d∇−→ Ω4(X, gP)

This is a standard presentation of the fields of self-dual Yang-Mills theory in the BV for-malism (see [CCRF+98] and [Cos11c]). Note that it is, in fact, a dg Lie algebra, so thereare no nontrivial higher brackets.

Ordinary Yang-Mills theory arises as a deformation of the self-dual theory. One sim-ply deforms the differential in the diagram above by including a term that is the identityfrom Ω2

−(X, gP) in degree 1 to the copy of Ω2−(X, gP) situated in degree 2.

11.6.4. The holomorphic σ-model. Let E be an elliptic curve and let X be a com-plex manifold. Let M(E, X) denote the elliptic moduli problem parametrizing holo-morphic maps from E → X. As before, there is an associated cotangent field theoryT∗[−1]M(E, X). (In [Cos11a] it is explained how to describe the formal neighborhood ofany point in this mapping space in terms of an elliptic L∞ algebra on E.)

In [Cos10], this field theory was called a holomorphic Chern-Simons theory, becauseof the formal similarities between the action functional of this theory and that of the holo-morphic Chern-Simons gauge theory. In the physics literature ([Wit05], [Kap05]) this the-ory is known as the twisted (0, 2) supersymmetric sigma model, or as the curved β− γsystem.

This theory has an interesting role in both mathematics and physics. For instance, itwas shown in [Cos10, Cos11a] that the partition function of this theory (at least, the partwhich discards the contributions of non-constant maps to X) is the Witten genus of X.

11.6.5. Twisted supersymmetric gauge theories. Of course, there are many more ex-amples of cotangent theories, as there are very many elliptic moduli problems. In [Cos11b],it is shown how twisted versions of supersymmetric gauge theories can be written ascotangent theories. We will focus on holomorphic (or minimal) twists. Holomorphictwists are richer than the more well-studied topological twists, but contain less informa-tion than the full untwisted supersymmetric theory. As explained in [Cos11b], one canobtain topological twists from holomorphic twists by applying a further twist.

The most basic example is the twisted N = 1 field theory. If X is a complex surfaceand G is a complex Lie group, then the N = 1 twisted theory is simply the cotangenttheory to the elliptic moduli problem of holomorphic principal G-bundles on X. If we fix

11.6. COTANGENT FIELD THEORIES 241

a principal G-bundle P → X, then the elliptic L∞ algebra describing this formal moduliproblem near P is

Ω0,∗(X, gP),where gP is the adjoint bundle of Lie algebras associated to P. It is a classic result ofKodaira and Spencer that this dg Lie algebra describes deformations of the holomorphicprincipal bundle P.

The cotangent theory to this elliptic moduli problem is thus described by the ellipticL∞ algebra

Ω0,∗(X, gP ⊕ g∨P ⊗ KX[−1].).Note that KX denotes the canonical line bundle, which is the appropriate holomorphicsubstitute for the smooth density line bundle.

11.6.6. The twisted N = 2 theory. Twisted versions of gauge theories with moresupersymmetry have similar descriptions, as is explained in [Cos11b]. The N = 2 theoryis the cotangent theory to the elliptic moduli problem for holomorphic G-bundles P→ Xtogether with a holomorphic section of the adjoint bundle gP. The underlying elliptic L∞algebra describing this moduli problem is

Ω0,∗(X, gP + gP[−1]).

Thus, the cotangent theory has

Ω0,∗(X, gP + gP[−1]⊕ g∨P ⊗ KX ⊕ g∨P ⊗ KX[−1])

for its elliptic L∞ algebra.

11.6.7. The twisted N = 4 theory. Finally, we will describe the twisted N = 4theory. There are two versions of this twisted theory: one used in the work of Vafa-Witten[VW94] on S-duality, and another by Kapustin-Witten [KW06] in their work on geometricLanglands. Here we will describe only the latter.

Let X again be a complex surface and G a complex Lie group. Then the twisted N =4 theory is the cotangent theory to the elliptic moduli problem describing principal G-bundles P→ X, together with a holomorphic section φ ∈ H0(X, T∗X⊗ gP) satisfying

[φ, φ] = 0 ∈ H0(X, KX ⊗ gP).

Here T∗X is the holomorphic cotangent bundle of X.

The elliptic L∞ algebra describing this is

Ω0,∗(X, gP ⊕ T∗X⊗ gP[−1]⊕ KX ⊗ gP[−2]).

Of course, this elliptic L∞ algebra can be rewritten as

(Ω∗,∗(X, gP), ∂),


where the differential is just ∂ and does not involve ∂. The Lie bracket arises from ex-tending the Lie bracket on gP by tensoring with the commutative algebra structure on thealgebra Ω∗,∗(X) of forms on X.

Thus, the corresponding cotangent theory has

Ω∗,∗(X, gP)⊕Ω∗,∗(X, gP)[1]

for its elliptic Lie algebra.

CHAPTER 12

The observables of a classical field theory

So far we have given a definition of a classical field theory, combining the ideas ofderived deformation theory and the classical BV formalism. Our goal in this chapter isto show that the observables for such a theory do indeed form a commutative factor-ization algebra, denoted Obscl , and to explain how to equip it with a shifted Poissonbracket. The first part is straightforward — implicitly, we have already done it! — but thePoisson bracket is somewhat subtle, due to complications that arise when working with

infinite-dimensional vector spaces. We will exhibit a sub-factorization algebra Obscl

ofObscl which is equipped with a commutative product and Poisson bracket, and such that

the inclusion map Obscl→ Obscl is a quasi-isomorphism.

12.1. The factorization algebra of classical observables

We have given two descriptions of a classical field theory, and so we provide the twodescriptions of the associated observables.

Let L be the elliptic L∞ algebra of a classical field theory on a manifold M. Thus, theassociated elliptic moduli problem is equipped with a symplectic form of cohomologicaldegree −1.

12.1.0.1 Definition. The observables with support in the open subset U is the commutativedg algebra

Obscl(U) = C∗(L(U)).

The factorization algebra of observables for this classical field theory, denoted Obscl , assignsthe cochain complex Obscl(U) to the open U.

The interpretation of this definition should be clear from the preceding chapters. Theelliptic L∞ algebra L encodes the space of solutions to the Euler-Lagrange equations forthe theory (more accurately, the formal neighborhood of the solution given by the base-point of the formal moduli problem). Its Chevalley-Eilenberg cochains C∗(L(U)) on theopen U are interpreted as the algebra of functions on the space of solutions over the openU.

243

244 12. THE OBSERVABLES OF A CLASSICAL FIELD THEORY

By the results of section 6.4, we know that this construction is in fact a factorizationalgebra.

We often call Obscl simply the classical observables, in contrast to the factorization alge-bras of some quantization, which we will call the quantum observables.

Alternatively, let E be a graded vector bundle on M, equipped with a symplecticpairing of degree −1 and a local action functional S which satisfies the classical masterequation. As we explained in section 11.4 this data is an alternative way of describing aclassicla field theory. The bundle L whose sections are the local L∞ algebra L is E[−1].

12.1.0.2 Definition. The observables with support in the open subset U is the commutativedg algebra

Obscl(U) = O(E (U)),

equipped with the differential S,−.

The factorization algebra of observables for this classical field theory, denoted Obscl , as-signs the cochain complex Obscl(U) to the open U.

Recall that the operator S,− is well-defined because the bracket with the local func-tional is always well-defined.

The underlying graded-commutative algebra of Obscl(U) is manifestly the functionson the fields E (U) over the open set U. The differential imposes the relations betweenobservables arising from the Euler-Lagrange equations for S. In physical language, weare giving a cochain complex whose cohomology is the “functions on the fields that areon-shell.”

It is easy to check that this definition of classical observables coincides with the one interms of cochains of the sheaf of L∞-algebras L(U).

12.2. The graded Poisson structure on classical observables

Recall the following definition.

12.2.0.3 Definition. A P0 algebra (in the category of cochain complexes) is a commutative dif-ferential graded algebra together with a Poisson bracket −,− of cohomological degree 1, whichsatisfies the Jacobi identity and the Leibniz rule.

The main result of this chapter is the following.

12.3. THE POISSON STRUCTURE FOR FREE FIELD THEORIES 245

12.2.0.4 Theorem. For any classical field theory (section 11.4) on M, there is a P0 factorization

algebra Obscl

, together with a weak equivalence of commutative factorization algebras.

Obscl ∼= Obscl .

Concretely, Obscl(U) is built from functionals on the space of solutions to the Euler-

Lagrange equations that have more regularity than the functionals in Obscl(U).

The idea of the definition of the P0 structure is very simple. Let us start with a finite-dimensional model. Let g be an L∞ algebra equipped with an invariant antisymmetricelement P ∈ g⊗ g of cohomological degree 3. This element can be viewed (according tothe correspondence between formal moduli problems and Lie algebras given in section10.1) as a bivector on Bg, and so it defines a Poisson bracket on O(Bg) = C∗(g). Con-cretely, this Poisson bracket is defined, on the generators g∨[−1] of C∗(g), as the map

g∨ ⊗ g∨ → R

determined by the tensor P.

Now let L be an elliptic L∞ algebra describing a classical field theory. Then the ker-nel for the isomorphism L(U) ∼= L!(U)[−3] is an element P ∈ L(U) ⊗ L(U), which issymmetric, invariant, and of degree 3.

We would like to use this idea to define the Poisson bracket on


As in the finite dimensional case, in order to define such a Poisson bracket, we would needan invariant tensor in L(U)⊗2. The tensor representing our pairing is instead in L(U)⊗2,which contains L(U)⊗2 as a dense subspace. In other words, we run into a standardproblem in analysis: our construction in finite-dimensional vector spaces does not portimmediately to infinite-dimensional vector spaces.

We solve this problem by finding a subcomplex

Obscl(U) ⊂ Obscl(U)

such that the Poisson bracket is well-defined on the subcomplex and the inclusion is aweak equivalence. Up to quasi-isomorphism, then, we have the desired Poisson structure.

12.3. The Poisson structure for free field theories

In this section, we will construct a P0 structure on the factorization algebra of observ-ables of a free field theory. More precisely, we will construct for every open subset U, a


subcomplex

Obscl(U) ⊂ Obscl(U)

of the complex of classical observables such that

(1) Obscl

forms a sub-commutative factorization algebra of Obscl ;

(2) the inclusion Obscl(U) ⊂ Obscl(U) is a weak equivalence of differentiable pro-

cochain complexes for every open set U; and

(3) Obscl

has the structure of P0 factorization algebra.

The complex Obscl(U) consists of a product over all n of certain distributional sections

of a vector bundle on Un. The complex Obscl

is defined by considering instead smoothsections on Un of the same vector bundle.

Let us now make this definition more precise. Recall that a free field theory is a clas-sical field theory associated to an elliptic L∞ algebra L that is abelian, i.e., where all thebrackets ln | n ≥ 2 vanish.

Thus, let L be the graded vector bundle associated to an abelian elliptic L∞ algebra,and let L(U) be the elliptic complex of sections of L on U. To say that L defines a fieldtheory means we have a symmetric isomorphism L ∼= L![−3].

Recall (section B.2) that we use the notation L(U) to denote the space of distributionalsections of L on U. A lemma of Atiyah-Bott (section B.10) shows that the inclusion

L(U) → L(U)

is a continuous homotopy equivalence of topological cochain complexes.

It follows that the natural map

C∗(L(U)) → C∗(L(U))

is a cochain homotopy equivalence. Indeed, because we are dealing with an abelian L∞algebra, the Chevalley-Eilenberg cochains become quite simple:

C∗(L(U)) = Sym(L(U)∨[−1]),

C∗(L(U)) = Sym(L(U)∨[−1]),

where, as always, the symmetric algebra is defined using the completed tensor product.The differential is simply the differential on, for instance, L(U)∨ extended as a derivation,so that we are simply taking the completed symmetric algebra of a complex. The complexC∗(L(U)) is built from distributional sections of the bundle (L!)n[−n] on Un, and thecomplex C∗(L(U)) is built from smooth sections of the same bundle.

12.3. THE POISSON STRUCTURE FOR FREE FIELD THEORIES 247

Note thatL(U)∨ = L!

c(U) = Lc(U)[3].

Thus,

C∗(L(U)) = Sym(Lc(U)[2]),

C∗(L(U)) = Sym(Lc(U)[2]).

We can define a Poisson bracket of degree 1 on C∗(L(U)) as follows. On the generatorsLc(U)[2], it is defined to be the given pairing

〈−,−〉 : Lc(U)×Lc(U)→ R,

since we can pair smooth sections. This pairing extends uniquely, by the Leibniz rule, tocontinuous bilinear map

C∗(L(U))× C∗(L(U))→ C∗(L(U)).

In particular, we see that C∗(L(U)) has the structure of a P0 algebra in the multicategoryof differentiable cochain complexes.

Let us define the modified observables in this theory by


We have seen that Obscl(U) is homotopy equivalent to Obscl(U) and that Obs

cl(U) has a

P0 structure.

12.3.0.5 Lemma. Obscl(U) has the structure of a P0 factorization algebra.

PROOF. It remains to verify that if U1, . . . , Un are disjoint open subsets of M, eachcontained in an open subset W, then the map

Obscl(U1)× · · · × Obs

cl(Un)→ Obs

cl(W)

is compatible with the P0 structures. This map automatically respects the commutative

structure, so it suffices to verify that for α ∈ Obscl(Ui) and β ∈ Obs

cl(Uj), where i 6= j,

then

α, β = 0 ∈ Obscl(W).

That this bracket vanishes follows from the fact that if two “linear observables” φ, ψ ∈Lc(W) have disjoint support, then

〈φ, ψ〉 = 0.

Every Poisson bracket reduces to a sum of brackets between linear terms by applying theLeibniz rule repeatedly.


12.4. The Poisson structure for a general classical field theory

In this section we will prove the following.

12.4.0.6 Theorem. For any classical field theory (section 11.4) on M, there is a P0 factorization

algebra Obscl

, together with a quasi-isomorphism of commutative factorization algebras

Obscl ∼= Obscl .

12.4.1. Functionals with smooth first derivative. For a free field theory, we defined a

subcomplex Obscl

of observables which are built from smooth sections of a vector bundleon Un, instead of distributional sections as in the definition of Obscl . It turns out that,for an interacting field theory, this subcomplex of Obscl is not preserved by the differ-ential. Instead, we have to find a subcomplex built from distributions on Un which arenot smooth but which satisfy a mild regularity condition. We will call also this complex

Obscl

(thus introducing a conflict with the terminology introduced in the case of free fieldtheories).

Let L be an elliptic L∞ algebra on M that defines a classical field theory. Recall thatthe cochain complex of observables is

Obscl(U) = C∗(L(U)),

where L(U) is the L∞ algebra of sections of L on U.

Recall that as a graded vector space, C∗(L(U)) is the algebra of functionals O(L(U)[1])on the graded vector space L(U)[1]. In the appendix (section B.8), given any graded vec-tor bundle E on M, we define a subspace

O sm(E (U)) ⊂ O(E (U))

of functionals that have “smooth first derivative”. A function Φ ∈ O(E (U)) is in O sm(E (U))precisely if

dΦ ∈ O(E (U))⊗ E !c (U).

(The exterior derivative of a general function in O(E (U)) will lie a priori in the larger spaceO(E (U))⊗ E

!c(U).) The space O sm(E (U)) is a differentiable pro-vector space.

Recall that if g is an L∞ algebra, the exterior derivative maps C∗(g) to C∗(g, g∨[−1]).The complex C∗sm(L(U)) of cochains with smooth first derivative is thus defined to bethe subcomplex of C∗(L(U)) consisting of those cochains whose first derivative lies inC∗(L(U),L!

c(U)[−1]), which is a subcomplex of C∗(L(U),L(U)∨[−1]).

12.4. THE POISSON STRUCTURE FOR A GENERAL CLASSICAL FIELD THEORY 249

In other words, C∗sm(L(U)) is defined by the fiber diagram

C∗sm(L(U))d−→ C∗(L(U),L!

c(U)[−1])↓ ↓

C∗(L(U))d−→ C∗(L(U),Lc

!(U)[−1]).

(Note that differentiable pro-cochain complexes are closed under taking limits, so that thisfiber product is again a differentiable pro-cochain complex; more details are provided inthe appendix B.8).

Note thatC∗sm(L(U)) ⊂ C∗(L(U))

is a sub-commutative dg algebra for every open U. Furthermore, as U varies, C∗sm(L(U))defines a sub-commutative prefactorization algebra of the prefactorization algebra de-fined by C∗(L(U)).

We define

Obscl(U) = C∗sm(L(U)) ⊂ C∗(L(U)) = Obscl(U).

The next step is to construct the Poisson bracket.

12.4.2. The Poisson bracket. Because the elliptic L∞ algebra L defines a classical fieldtheory, it is equipped with an isomorphism L ∼= L![−3]. Thus, we have an isomorphism

Φ : C∗(L(U),L!c(U)[−1]) ∼= C∗(L(U),Lc(U)[2]).

In the appendix (section B.9), we show that C∗(L(U),L(U)[1]) — which we think of asvector fields on the formal manifold BL(U) — has a natural structure of a dg Lie algebrain the multicategory of differentiable pro-cochain complexes. The bracket is, of course, aversion of the bracket of vector fields. Further, C∗(L(U),L(U)[1]) acts on C∗(L(U)) byderivations. This action is in the multicategory of differentiable pro-cochain complexes:the map

C∗(L(U),L(U)[1])× C∗(L(U))→ C∗(L(U))

is a smooth bilinear cochain map. We will write Der(C∗(L(U))) for this dg Lie algebraC∗(L(U),L(U)[1]).

Thus, composing the map Φ above with the exterior derivative d and with the inclu-sion Lc(U) → L(U), we find a cochain map

C∗sm(L(U))→ C∗(L(U),Lc(U)[2])→ Der(C∗(L(U)))[1].

If f ∈ C∗sm(L(U)), we will let X f ∈ Der(C∗(L(U))) denote the corresponding derivation.If f has cohomological degree k, then X f has cohomological degree k + 1.


If f , g ∈ C∗sm(L(U)) = Obscl(U), we define

f , g = X f g ∈ Obscl(U).

This bracket defines a bilinear map

Obscl(U)× Obs

cl(U)→ Obs

cl(U).

Note that we are simply adopting the usual formulas to our setting.

12.4.2.1 Lemma. This map is smooth, i.e., a bilinear map in the multicategory of differentiablepro-cochain complexes.

PROOF. This follows from the fact that the map

d : Obscl(U)→ Der(C∗(L(U)))[1]

is smooth, which is immediate from the definitions, and from the fact that the map

Der(C∗(L(U))× C∗(L(U))→ C∗(L(U))

is smooth (which is proved in the appendix B.9).

12.4.2.2 Lemma. This bracket satisfies the Jacobi rule and the Leibniz rule. Further, for U, V

disjoint subsets of M, both contained in W, and for any f ∈ Obscl(U), g ∈ Obs

cl(V), we have

f , g = 0 ∈ Obscl(W).

PROOF. The proof is straightforward.

Following the argument for lemma 12.3.0.5, we obtain a P0 factorization algebra.

12.4.2.3 Corollary. Obscl

defines a P0 factorization algebra in the valued in the multicategory ofdifferentiable pro-cochain complexes.

The final thing we need to verify is the following.

12.4.2.4 Proposition. For all open subset U ⊂ M, the map

Obscl(U)→ Obscl(U)


12.4. THE POISSON STRUCTURE FOR A GENERAL CLASSICAL FIELD THEORY 251

PROOF. It suffices to show that it is a weak equivalence on the associated graded for

the natural filtration on both sides. Now, Grn Obscl(U) fits into a fiber diagram

Grn Obscl(U) //

Symn(L!c(U)[−1])⊗L!

c(U)

Grn Obscl(U) // Symn(L!c(U)[−1])⊗L!

c(U).

Note also thatGrn Obscl(U) = Symn L!

c(U).The Atiyah-Bott lemma B.10 shows that the inclusion

L!c(U) → L!

c(U)

is a continuous cochain homotopy equivalence. We can thus choose a homotopy inverse

P : L!c(U)→ L!

c(U)

and a homotopy

H : L!c(U)→ L!

c(U)

such that [d, H] = P− Id and such that H preserves the subspace L!c(U).

Now,

Symn L!c(U) ⊂ Grn Obs

cl(U) ⊂ Symn L!

c(U).Using the projector P and the homotopy H, one can construct a projector

Pn = P⊗n : L!c(U)⊗n → L!

c(U)⊗n.

We can also construct a homotopy

Hn : L!c(U)⊗n → L!

c(U)⊗n.

The homotopy Hn is defined inductively by the formula

Hn = H ⊗ Pn−1 + 1⊗ Hn−1.

This formula defines a homotopy because

[d, Hn] = P⊗ Pn−1 − 1⊗ Pn−1 + 1⊗ Pn−1 − 1⊗ 1.

Notice that the homotopy Hn preserves all the subspaces of the form

L!c(U)⊗k ⊗L!

c(U)⊗L!c(U)⊗n−k−1.

This will be important momentarily.

Next, letπ : L!

c(U)⊗n[−n]→ Symn(L!c(U)[−1])


be the projection, and let

Γn = π−1 Grn Obscl(U).

Then Γn is acted on by the symmetric group Sn, and the Sn invariants are Obscl(U).

Thus, it suffices to show that the inclusion

Γn → Lc(U)⊗n

is a weak equivalence of differentiable spaces. We will show that it is continuous homo-topy equivalence.

The definition of Obscl(U) allows one to identify

Γn = ∩n−1k=0L

!c(U)⊗k ⊗L!

c(U)⊗L!c(U)⊗n−k−1.

The homotopy Hn preserves Γn, and the projector Pn maps

L!c(U)⊗n → Lc(U)⊗n ⊂ Γn.

Thus, Pn and Hn provide a continuous homotopy equivalence between L!c(U)⊗n and Γn,

as desired.

Part 4

Quantum field theory

CHAPTER 13

Introduction to quantum field theory

As explained in the introduction, this book develops a version of deformation quan-tization for field theories, rather than mechanics. In the chapters on classical field theory,we showed that the observables of a classical BV theory naturally form a commutativefactorization algebra, with a homotopical P0 structure. In the following chapters, we willshow that every quantization of a classical BV theory produces a factorization algebra (inBeilinson-Drinfeld algebras) that we call the quantum observables of the quantum fieldtheory. To be precise, the main theorem of this part is the following.

13.0.2.5 Theorem. Any quantum field theory on a manifold M, in the sense of [Cos11c], givesrise to a factorization algebra Obsq on M of quantum observables. This is a factorization algebraover C[[h]], valued in differentiable pro-cochain complexes, and it quantizes (in the weak sense of1.7) the P0 factorization algebra of classical observables of the corresponding classical field theory.

For free field theories, this factorization algebra of quantum observables is essentiallythe same as the one discussed in Chapter 4. (The only difference is that, when discussingfree field theories, we normally set h = 1 and took our observables to be polynomialfunctions of the fields. When we discuss interacting theories, we take our observables tobe power series on the space of fields, and we take h to be a formal parameter).

Chapter 14 is thus devoted to reviewing the formalism of [Cos11c], stated in a formmost suitable to our purposes here. It’s important to note that, in contrast to the de-formation quantization of Poisson manifolds, a classical BV theory may not possess anyquantizations (i.e., quantization may be obstructed) or it may have many quantizations. Acentral result of [Cos11c], stated in section 14.5, is that there is a space of BV quantiza-tions. Moreover, this space can be constructed as a tower of fibrations, where the fiberbetween any pair of successive layers is described by certain cohomology groups of localfunctionals. These cohomology groups can be computed just from the classical theory.

The machinery of [Cos11c] allows one to construct many examples of quantum fieldtheories, by calculating the appropriate cohomology groups. For example, in [Cos11c],the quantum Yang-Mills gauge theory is constructed. Theorem 13.0.2.5, together with theresults of [Cos11c], thus produces many interesting examples of factorization algebras.

255

256 13. INTRODUCTION TO QUANTUM FIELD THEORY

Remark: We forewarn the reader that our definitions and constructions involve a heavyuse of functional analysis and (perhaps more surprisingly) simplicial sets, which is ourpreferred way of describing a space of field theories. Making a quantum field theorytypically requires many choices, and as mathematicians, we wish to pin down preciselyhow the quantum field theory depends on these choices. The machinery we use gives usvery precise statements, but statements that can be forbidding at first sight. We encouragethe reader, on a first pass through this material, to simply make all necessary choices (suchas a parametrix) and focus on the output of our machine, namely the factorization algebraof quantum observables. Keeping track of the dependence on choices requires carefulbookkeeping (aided by the machinery of simplicial sets) but is straightforward once theprimary construction is understood. ♦

The remainder of this chapter consists of an introduction to the quantum BV formal-ism, building on our motivation for the classical BV formalism in section 11.1.

13.1. The quantum BV formalism in finite dimensions

In section 11.1, we motivated the classical BV formalism with a finite-dimensional toymodel. To summarize, we described the derived critical locus of a function S on a smoothmanifold M of dimension n. The functions on this derived space O(Crith(S)) form acommutative dg algebra,

Γ(M,∧nTM)∨dS−−→ . . . ∨dS−−→ Γ(M,∧2TM)

∨dS−−→ Γ(M, TM)∨dS−−→ C∞(M),

the polyvector fields PV(M) on M with the differential given by contraction with dS. Thiscomplex remembers how dS vanishes and not just where it vanishes.

The quantum BV formalism uses a deformation of this classical BV complex to encode,in a homological way, oscillating integrals.

In finite dimensions, there already exists a homological approach to integration: thede Rham complex. For instance, on a compact, oriented n-manifold without boundary,M, we have the commuting diagram

Ωn(M)

∫M //

[−] %%

R

Hn(M)〈[M],−〉

<<

where [µ] denotes the cohomology class of the top form µ and 〈[M],−〉 denotes pairingthe class with the fundamental class of M. Thus, integration factors through the de Rhamcohomology.

13.1. THE QUANTUM BV FORMALISM IN FINITE DIMENSIONS 257

Suppose µ is a smooth probability measure, so that∫

M µ = 1 and µ is everywherenonnegative (which depends on the choice of orientation). Then we can interpret theexpected value of a function f on M — an “observable on the space of fields M” — as thecohomology class [ f µ] ∈ Hn(M).

The BV formalism in finite dimensions secretly exploits this use of the de Rham com-plex, as we explain momentarily. For an infinite-dimensional manifold, though, the deRham complex ceases to encode integration over the whole manifold because there are notop forms. In contrast, the BV version scales to the infinite-dimensional setting. Infinitedimensions, of course, introduces extra difficulties to do with the fact that integration ininfinite dimensions is not well-defined. These difficulties manifest themselves as ultra-violte divergences of quantum field theory, and we deal with them using the techniquesdeveloped in [Cos11c].

In the classical BV formalism, we work with the polyvector fields rather than de Rhamforms. A choice of probability measure µ, however, produces a map between these gradedvector spaces

Γ(M,∧nTM)

∨µ

. . . Γ(M,∧2TM)

∨µ

Γ(M, TM)

∨µ

C∞(M)

∨µ

C∞(M) . . . Ωn−2(M) Ωn−1(M) Ωn(M)

where ∨µ simply contracts a k-polyvector field with µ to get a n − k-form. When µ isnowhere-vanishing (i.e., when µ is a volume form), this map is an isomorphism and so wecan “pull back” the exterior derivative to equip the polyvector fields with a differential.This differential is usually called the divergence operator for µ, so we denote it divµ.

By the divergence complex for µ, we mean the polyvector fields (concentrated in non-positive degrees) with differential divµ. Its cohomology is isomorphic, by construction, toH∗dR(M)[n]. In particular, given a function f on M, viewed as living in degree zero andproviding an “observable,” we see that its cohomology class [ f ] in the divergence com-plex corresponds to the expected value of f against µ. More precisely, we can define theratio [ f ]/[1] as the expected value of f . Under the map ∨µ, it goes to the usual expectedvalue.

What we’ve done above is provide an alternative homological approach to integration.More accurately, we’ve shown how “integration against a volume form” can be encodedby an appropriate choice of differential on the polyvector fields. Cohomology classes inthis divergence complex encode the expected values of functions against this measure.Of course, this is what we want from the path integral! The divergence complex is themotivating example for the quantum BV formalism, and so it is also called a quantum BVcomplex.


We can now explain why this approach to homological integration is more suitableto extension to infinite dimensions than the usual de Rham picture. Even for an infinite-dimensional manifold M, the polyvector fields are well-defined (although one must makechoices in how to define them, depending on one’s preferences with functional analysis).One can still try to construct a “divergence-type operator” and view it as the effectivereplacement for the probability measure. By taking cohomology classes, we computethe expected values of observables. The difficult part is making sense of the divergenceoperator; this is achieved through renormalization.

This vein of thought leads to a question: how to characterize, in an abstract fashion,the nature of a divergence operator? An answer leads, as we’ve shown, to a process fordefining a homological path integral. Below, we’ll describe one approach, but first weexamine a simple case.

Remark: The cohomology of the complex (both in the finite and infinite dimensional set-tings) always makes sense, but H0 is not always one-dimensional. For example, on amanifold X that is not closed, the de Rham cohomology often vanishes at the top. If themanifold is disconnected but closed, the top de Rham cohomology has dimension equalto the number of components of the manifold. In general, one must choose what classof functions to integrate against the volume form, and the cohomology depends on thischoice (e.g., consider compactly supported de Rham cohomology).

Instead of computing expected values, the cohomology provides relations betweenexpected values of observables. We will see how the cohomology encodes relations in theexample below. In the setting of conformal field theory, for instance, one often uses suchrelations to obtain formulas for the operator product expansion. ♦

13.2. The “free scalar field” in finite dimensions

A concrete example is in order. We will work with a simple manifold, the real line,equipped with the Gaussian measure and recover the baby case of Wick’s lemma. Thegeneralization to a finite-dimensional vector space will be clear.

Remark: This example is especially pertinent to us because in this book we are workingwith perturbative quantum field theories. Hence, for us, there is always a free field theory— whose space of fields is a vector space equipped with some kind of Gaussian measure— that we’ve modified by adding an interaction to the action functional. The underlyingvector space is equipped with a linear pairing that yields the BV Laplacian, as we workwith it. As we will see in this example, the usual BV formalism relies upon the underlying“manifold” being linear in nature. To extend to a global nonlinear situation, on e needs todevelop new techniques (see, for instance, [Cos11a]). ♦

13.2. THE “FREE SCALAR FIELD” IN FINITE DIMENSIONS 259

Before we undertake the Gaussian measure, let’s begin with the Lebesgue measure dxon R. This is not a probability measure, but it is nowhere-vanishing, which is the onlyproperty necessary to construct a divergence operator. In this case, we compute

divLeb : f∂

∂x7→ ∂ f

∂x.

In one popular notion, we use ξ to denote the vector field ∂/∂x, and the polyvector fieldsare then C∞(R)[ξ], where ξ has cohomological degree −1. The divergence operator be-comes

divLeb =∂

∂x∂

∂ξ,

which is also the standard example of the BV Laplacian4. (In short, the usual BV Lapla-cian on Rn is simply the divergence operator for the Lebesgue measure.) We will use 4for it, as this notation will continue throughout the book.

It is easy to see, by direct computation or the Poincare lemma, that the cohomology ofthe divergence complex for the Lebesgue measure is simply H−1 ∼= R and H0 ∼= R.

Let µb be the usual Gaussian probability measure on R with variance b:

µb =

√1

2πbe−x2/2bdx.

As µ is a nowhere-vanishing probability measure, we obtain a divergence operator

divb : f∂

∂x7→ ∂ f

∂x− x

bf .

We havedivb = 4+ ∨dS

where S = −x2/2b. Note that this complex is a deformation of the classical BV complexfor S by adding the BV Laplacian4.

This divergence operator preserves the subcomplex of polynomial polyvector fields.That is, a vector field with polynomial coefficient goes to a polynomial function.

Explicitly, we see

divb

(xn ∂

∂x

)= nxn−1 − 1

bxn+1.

Hence, at the level of cohomology, we see [xn+1] = bn[xn−1]. We have just obtained thefollowing, by a purely cohomological process.

13.2.0.6 Lemma (Baby case of Wick’s lemma). The expected value of xn with respect to theGaussian measure is zero if n odd and bk(2k− 1)(2k− 3) · · · 5 · 3 if n = 2k.


Since Wick’s lemma appears by this method, it should be clear that one can recover theusual Feynman diagrammatic expansion. Indeed, the usual arguments with integrationby parts are encoded here by the relations between cohomology classes.

Note that for any function S : R→ R, the volume form eSdx has divergence operator

divS = 4+∂S∂x

∂

∂x,

and using the Schouten bracket −,− on polyvector fields, we can write it as

divS = 4+ S,−.

The quantum master equation (QME) is the equation div2S = 0. The classical master equation

(CME) is the equation S, S = 0, which just encodes the fact that the differential of theclassical BV complex is square-zero. (In the examples we’ve discussed so far, this propertyis immediate, but in many contexts, such as gauge theories, finding such a function S canbe a nontrivial process.)

13.3. An operadic description

Before we provide abstract properties that characterize a divergence operator, weshould recall properties that characterize the classical BV complex. Of course, functionson the derived critical locus are a commutative dg algebra. Polyvector fields, however,also have the Schouten bracket — the natural extension of the Lie bracket of vector fieldsand functions — which is a Poisson bracket of cohomological degree 1 and which is com-patible with the differential ∨S = S,−. Thus, we introduced the notion of a P0 algebra,where P0 stands for “Poisson-zero,” in section 8.3. In chapter 12, we showed that thefactorization algebra of observables for a classical BV theory have a lax P0 structure.

Examining the divergence complex for a measure of the form eSdx in the precedingsection, we saw that the divergence operator was a deformation of S,−, the differentialfor the classical BV complex. Moreover, a simple computation shows that a divergenceoperator satisfies

div(ab) = (div a)b + (−1)|a|a(div b) + (−1)|a|a, b

for any polyvector fields a and b. (This relation follows, under the polyvector-de Rhamisomorphism given by the measure, from the fact that the exterior derivative is a deriva-tion for the wedge product.) Axiomatizing these two properties, we obtain the notionof a Beilinson-Drinfeld algebra, discussed in section 8.4. The differential of a BD algebrapossesses many of the essential properties of a divergence operator, and so we view a BDalgebra as a homological way to encode integration on (a certain class of) derived spaces.

13.4. EQUIVARIANT BD QUANTIZATION AND VOLUME FORMS 261

In short, the quantum BV formalism aims to find, for a P0 algebra Acl , a BD algebra Aq

such that Acl = Aq ⊗R[[h]] R[[h]]/(h). We view it as moving from studying functions onthe derived critical locus of some action functional S to the divergence complex for eSDφ.

This motivation for the definition of a BD algebra is complementary to our earliermotivation, which emphasizes the idea that we simply want to deform from a commu-tative factorization algebra to a “plain,” or E0, factorization algebra. It grows out of thepath integral approach to quantum field theory, rather than extending to field theory thedeformation quantization approach to mechanics.

For us, the basic situation is a formal moduli spaceMwith−1-symplectic pairing. Itsalgebra of functions is a P0 algebra. By a version of the Darboux lemma for formal modulispaces, we can identifyM with an L∞ algebra g equipped with an invariant symmetricpairing. Geometrically, this means the symplectic pairing is translation-invariant and allthe nonlinearity is pushed into the brackets. As the differential d on O(M) respects thePoisson bracket, we view it as a symplectic vector field of cohomological degree 1, and inthis formal situation, we can find a Hamiltonian function S such that d = S,−.

Comparing to our finite-dimensional example above, we are seeing the analog of thefact that any nowhere-vanishing volume form on Rn can be written as eSdx1 · · ·dxn. Theassociated divergence operator looks like 4 + S,−, where the BV Laplacian 4 is thedivergence operator for Lebesgue measure.

The translation-invariant Poisson bracket on O(M) also produces a translation-invariantBV Laplacian4. Quantizing then amounts to finding a function I ∈ hO(M)[[h]] such that

S,−+ I,−+ h4is square-zero. In the BV formalism, we call I a “solution to the quantum master equationfor the action S.” As shown in chapter 6 of [Cos11c], we have the following relationship.

13.3.0.7 Proposition. LetM be a formal moduli space with −1-symplectic structure. There isan equivalence of spaces

solutions of the QME ' BD quantizations.

13.4. Equivariant BD quantization and volume forms

We now return to our discussion of volume forms and formulate a precise relationshipwith BD quantization. This relationship, first noted by Koszul [Kos85], generalizes natu-rally to the setting of cotangent field theories. In section 16.4, we explain how cotangentquantizations provide volume forms on elliptic moduli problems.

For a smooth manifold M, there is a special feature of a divergence complex that wehave not yet discussed. Polyvector fields have a natural action of the multiplicative group


Gm, where functions have weight zero, vector fields have weight −1, and k-vector fieldshave weight −k. This action arises because polyvector fields are functions on the shiftedcotangent bundle T∗[−1]M, and there is always a scaling action on the cotangent fibers.

We can make the classical BV complex into a Gm-equivariant P0 algebra, as follows.Simply equip the Schouten bracket with weight 1 and the commutative product withweight zero. We now ask for a Gm-equivariant BD quantization.

To make this question precise, we rephrase our observations operadically. Equip theoperad P0 with the Gm action where the commutative product is weight zero and thePoisson bracket is weight 1. An equivariant P0 algebra is then a P0 algebra with a Gmaction such that the bracket has weight 1 and the product has weight zero. Similarly,equip the operad BD with the Gm action where h has weight −1, the product has weightzero, and the bracket has weight 1. A filtered BD algebra is a BD algebra with a Gm actionwith the same weights.

Given a volume form µ on M, the h-weighted divergence complex

(PV(M)[[h]], h divµ)

is a filtered BD algebra.

On an smooth manifold, we saw that each volume form µ produced a divergenceoperator divµ, via “conjugating” the exterior derivative d by the isomorphism ∨µ. In fact,any rescaling cµ, with c ∈ R×, produces the same divergence operator. Since we wantto work with probability measures, this fact meshes well with our objectives: we wouldalways divide by the integral

∫X µ anyway. In fact, one can show that every filtered BD

quantization of the P0 algebra PV(M) arises in this way.

13.4.0.8 Proposition. There is a bijection between projective volume forms on M, and filtered BVquantizations of PV(M).

See [Cos11a] for more details on this.

13.5. How renormalization group flow interlocks with the BV formalism

So far, we have introduced the quantum BV formalism in the finite dimensional set-ting and extracted the essential algebraic structures. Applying these ideas in the settingof field theories requires nontrivial work. Much of this work is similar in flavor to ourconstruction of a lax P0 structure on Obscl : issues with functional analysis block the mostnaive approach, but there are alternative approaches, often well-known in physics, thataccomplish our goal, once suitably reinterpreted.

13.6. OVERVIEW OF THE REST OF THIS PART 263

Here, we build on the approach of [Cos11c]. The book uses exact renormalizationgroup flow to define the notion of effective field theory and develops an effective versionof the BV formalism. In chapter 14, we review these ideas in detail. We will sketch howto apply the BV formalism to formal elliptic moduli problems M with −1-symplecticpairing.

The main problem here is the same as in defining a shifted Poisson structure on theclassical observables: the putative Poisson bracket −,−, arising from the symplecticstructure, is well-defined only on a subspace of all observables. As a result, the associatedBV Laplacian4 is also only partially-defined.

To work around this problem, we use the fact that every parametrix Φ for the ellipticcomplex underlying M yields a mollified version 4Φ of the BV Laplacian, and hencea mollified bracket −,−Φ. An effective field theory consists of a BD algebra ObsΦ forevery parametrix and a homotopy equivalence for any two parametrices, ObsΦ ' ObsΨ, satisfying coherence relations. In other words, we get a family of BD algebras overthe space of parametrices. The renormalization group (RG) flow provides the homotopyequivalences for any pair of parametrices. Modulo h, we also get a family Obscl

Φ of P0algebras over the space of parametrices. The tree-level RG flow produces the homotopyequivalences modulo h.

An effective field theory is a quantization ofM if, in the limit as4Φ goes to4, the P0algebra goes to the functions O(M) on the formal moduli problem.

The space of parametrices is contractible, so an effective field theory describes just oneBD algebra, up to homotopy equivalence. From the perspective developed thus far, weinterpret this BD algebra as encoding integration overM.

There is another way to interpret this definition, though, that may be attractive. TheRG flow amounts to a Feynman diagram expansion, and hence we can see it as a definitionof functional integration (in particular, flowing from energy scale Λ to Λ′ integrates overthe space of functions with energies between those scales). In [Cos11c], the RG flow isextended to the setting where the underlying free theory is an elliptic complex, not justgiven by an elliptic operator.

13.6. Overview of the rest of this Part

Here is a detailed summary of the chapters on quantum field theory.

(1) In section 15.1 we recall the definition of a free theory in the BV formalism andconstruct the factorization algebra of quantum observables of a general free the-ory, using the factorization envelope construction of section 3.6 of Chapter 3. Thisgeneralizes the discussion in chapter 4.


(2) In sections ?? to 14.5 we give an overview of the definition of QFT developed in[Cos11c].

(3) In section 15.2 we show how the definition of a QFT leads immediately to a con-struction of a BD algebra of “global observables” on the manifold M, which wedenote Obsq

P(M).(4) In section 15.3 we start the construction of the factorization algebra associated to

a QFT. We construct a cochain complex Obsq(M) of global observables, which isquasi-isomorphic to (but much smaller than) the BD algebra Obsq

P(M).(5) In section 15.5 we construct, for every open subset U ⊂ M, the subspace Obsq(U) ⊂

Obsq(M) of observables supported on U.(6) Section 15.6 accomplishes the primary aim of the chapter. In it, we prove that the

cochain complexes Obsq(U) form a factorization algebra. The proof of this resultis the most technical part of the chapter.

(7) In section 16.1 we show that translation-invariant theories have translation-invariantfactorization algebras of observables, and we treat the holomorphic situation aswell.

(8) In section 16.4 we explain how to interpret our definition of a QFT in the specialcase of a cotangent theory: roughly speaking, a quantization of the cotangenttheory to an elliptic moduli problem yields a locally-defined volume form on themoduli problem we start with.

CHAPTER 14

Effective field theories and Batalin-Vilkovisky quantization

In this chapter, we will give a summary of the definition of a QFT as developed in[Cos11c]. We will emphasize the aspects used in our construction of the factorizationalgebra associated to a QFT. This means that important aspects of the story there — suchas the concept of renormalizability — will not be mentioned. The introductory chapterof [Cos11c] is a leisurely exposition of the main physical and mathematical ideas, and weencourage the reader to examine it before delving into what follows. The approach thereis perturbative and hence has the flavor of formal geometry (that is, geometry with formalmanifolds).

A perturbative field theory is defined to be a family of effective field theories parametrizedby some notion of “scale.” The notion of scale can be quite flexible; the simplest versionis where the scale is a positive real number, the length. In this case, the effective theoryat a length scale L is obtained from the effective theory at scale ε by integrating out overfields with length scale between ε and L. In order to construct factorization algebras, weneed a more refined notion of “scale,” where there is a scale for every parametrix Φ ofa certain elliptic operator. We denote such a family of effective field theories by I[Φ],where I[Φ] is the “interaction term” in the action functional S[Φ] at “scale” Φ. We alwaysstudy families with respect to a fixed free theory.

A local action functional (see section 14.1) S is a real-valued function on the space offields such that S(φ) is given by integrating some function of the field and its derivativesover the base manifold (the “spacetime”). The main result of [Cos11c] states that the spaceof perturbative QFTs is the “same size” as the space of local action functionals. Moreprecisely, the space of perturbative QFTs defined modulo hn+1 is a torsor over the space ofQFTs defined modulo hn for the abelian group of local action functionals. In consequence,the space of perturbative QFTs is non-canonically isomorphic to local action functionalswith values in R[[h]] (where the choice of isomorphism amounts to choosing a way toconstruct counterterms).

The starting point for many physical constructions — such as the path integral —is a local action functional. However, a naive application of these constructions to suchan action functional yields a nonsensical answer. Many of these constructions do work if,instead of applying them to a local action functional, they are applied to a family I[Φ] ofeffective action functionals. Thus, one can view the family of effective action functionals

265

266 14. EFFECTIVE FIELD THEORIES AND BATALIN-VILKOVISKY QUANTIZATION

I[Φ] as a quantum version of the local action functional defining classical field theory.The results of [Cos11c] allow one to construct such families of action functionals. Manyformal manipulations with path integrals in the physics literature apply rigorously tofamilies I[Φ] of effective actions. Our strategy for constructing the factorization algebraof observables is to mimic path-integral definitions of observables one can find in thephysics literature, but replacing local functionals by families of effective actions.

14.1. Local action functionals

In studying field theory, there is a special class of functions on the fields, known aslocal action functionals, that parametrize the possible classical physical systems. Let Mbe a smooth manifold. Let E = C∞(M, E) denote the smooth sections of a Z-gradedsuper vector bundle E on M, which has finite rank when all the graded components areincluded. We call E the fields.

Various spaces of functions on the space of fields are defined in the appendix B.8.

14.1.0.9 Definition. A functional F is an element of

O(E ) =∞

∏n=0

HomDVS(E×n, R)Sn .

This is also the completed symmetric algebra of E ∨, where the tensor product is the completedprojective one.

Let Ored(E ) = O(E )/C be the space of functionals on E modulo constants.

Note that every element of O(E ) has a Taylor expansion whose terms are smoothmultilinear maps

E ×n → C.Such smooth mulitilinear maps are the same as compactly-supported distributional sec-tions of the bundle (E!)n on Mn. Concretely, a functional is then an infinite sequence ofvector-valued distributions on powers of M.

The local functionals depend only on the local behavior of a field, so that at each pointof M, a local functional should only depend on the jet of the field at that point. In theLagrangian formalism for field theory, their role is to describe the permitted actions, sowe call them local action functionals. A local action functional is the essential datum of aclassical field theory.

14.1.0.10 Definition. A functional F is local if each homogeneous component Fn is a finite sumof terms of the form

Fn(φ) =∫

M(D1φ) · · · (Dnφ) dµ,

14.2. THE DEFINITION OF A QUANTUM FIELD THEORY 267

where each Di is a differential operator from E to C∞(M) and dµ is a density on M.

We letOloc(E ) ⊂ Ored(E )

denote denote the space of local action functionals modulo constants.

As explained in section 11.4, a classical BV theory is a choice of local action functionalS of cohomological degree 0 such that S, S = 0. That is, S must satisfy the classicalmaster equation.

14.2. The definition of a quantum field theory

In this section, we will give the formal definition of a quantum field theory. The defi-nition is a little long and somewhat technical. The reader should consult the first chapterof [Cos11c] for physical motivations for this definition. We will provide some justificationfor the definition from the point of view of homological algebra shortly (section 15.2).

14.2.1.

14.2.1.1 Definition. A free BV theory on a manifold M consists of the following data:

(1) a Z-graded super vector bundle π : E→ M that is of finite rank;(2) a graded antisymmetric map of vector bundles 〈−,−〉loc : E⊗ E → Dens(M) of coho-

mological degree −1 that is fiberwise nondegenerate. It induces a graded antisymmetricpairing of degree −1 on compactly supported smooth sections Ec of E:

〈φ, ψ〉 =∫

x∈M〈φ(x), ψ(x)〉loc;

(3) a square-zero differential operator Q : E → E of cohomological degree 1 that is skew selfadjoint for the symplectic pairing.

In our constructions, we require the existence of a gauge-fixing operator QGF : E → Ewith the following properties:

(1) it is a square-zero differential operator of cohomological degree −1 ;(2) it is self adjoint for the symplectic pairing;(3) D = [Q, QGF] is a generalized Laplacian on M, in the sense of [BGV92]. This

means that D is an order 2 differential operator whose symbol σ(D), which is anendomorphism of the pullback bundle p∗E on the cotangent bundle p : T∗M →M, is

σ(D) = g Idp∗E


where g is some Riemannian metric on M, viewed as a function on T∗M.

All our constructions vary homotopically with the choice of gauge fixing operator.In practice, there is a natural contractible space of gauge fixing operators, so that ourconstructions are independent (up to contractible choice) of the choice of gauge fixing op-erator. (As an example of contractibility, if the complex E is simply the de Rham complex,each metric gives a gauge fixing operator d∗. The space of metrics is contractible.)

14.2.2. Operators and kernels. Let us recall the relationship between kernels and op-erators on E . Any continuous linear map F : Ec → E can be represented by a kernel

KF ∈ D(M2, E E!).

Here D(M,−) denotes distributional sections. We can also identify this space as

D(M2, E E!) = HomDVS(E!

c × Ec, C)

= HomDVS(Ec, E )

= E ⊗πE!.

Here ⊗π denotes the completed projective tensor product.

The symplectic pairing on E gives an isomorphism between E and E![−1]. This allows

us to view the kernel for any continuous linear map F as an element

KF ∈ E ⊗πE = HomDVS(E!

c × E !c , C)

. If F is of cohomological degree k, then the kernel KF is of cohomological degree k + 1.

If the map F : Ec → E has image in E c and extends to a continuous linear map E → E c,then the kernel KF has compact support. If F has image in E and extends to a continuouslinear map E c → E , then the kernel KF is smooth.

Our conventions are such that the following hold.

(1) K[Q,F] = QKF, where Q is the total differential on E ⊗πE .(2) Suppose that F : Ec → Ec is skew-symmetric with respect to the degree −1 pair-

ing on Ec. Then KF is symmetric. Similarly, if F is symmetric, then KF is anti-symmetric.

14.2.3. The heat kernel. In this section we will discuss heat kernels associated to thegeneralized Laplacian D = [Q, QGF]. These generalized heat kernels will not be essen-tial to our story; most of our constructions will work with a general parametrix for theoperator D, and the heat kernel simply provides a convenient example.


Suppose that we have a free BV theory with a gauge fixing operator QGF. As above,let D = [Q, QGF]. If our manifold M is compact, then this leads to a heat operator e−tD

acting on sections E . The heat kernel Kt is the corresponding kernel, which is an elementof E ⊗πE ⊗πC∞(R≥0). Further, if t > 0, the operator e−tD is a smoothing operator, so thatthe kernel Kt is in E ⊗πE . Since the operator e−tD is skew symmetric for the symplecticpairing on E , the kernel Kt is symmetric.

The kernel Kt is uniquely characterized by the following properties:

(1) The heat equation:ddt

Kt + (D⊗ 1)Kt = 0.

(2) The initial condition that K0 ∈ E ⊗πE is the kernel for the identity operator.

On a non-compact manifold M, there is more than one heat kernel satisfying these prop-erties.

14.2.4. Parametrices. In [Cos11c], two equivalent definitions of a field theory are given:one based on the heat kernel, and one based on a general parametrix. We will use exclu-sively the parametrix version in this book.

Before we define the notion of parametrix, we need a technical definition.

14.2.4.1 Definition. If M is a manifold, a subset V ⊂ Mn is proper if all of the projection mapsπ1, . . . , πn : V → M are proper. We say that a function, distribution, etc. on Mn has propersupport if its support is a proper subset of Mn.

14.2.4.2 Definition. A parametrix Φ is a distributional section

Φ ∈ E (M)⊗πE (M)

of the bundle E E on M×M with the following properties.

(1) Φ is symmetric under the natural Z/2 action on E (M)⊗πE (M).(2) Φ is of cohomological degree 1.(3) Φ has proper support.(4) Let QGF : E → E be the gauge fixing operator. We require that

([Q, QGF]⊗ 1)Φ− KId

is a smooth section of E E on M×M. Thus,

([Q, QGF]⊗ 1)Φ− KId ∈ E (M)⊗πE (M).

(Here KId is the kernel corresponding to the identity operator).


Remark: For clarity’s sake, note that our definition depends on a choice of QGF. Thus, weare defining here parametrices for the generalized Laplacian [Q, QGF], not general para-metrices for the elliptic complex E . ♦

Note that the parametrix Φ can be viewed (using the correspondence between ker-nels and operators described above) as a linear map AΦ : E → E . This operator is ofcohomological degree 0, and has the property that

AΦ[Q, QGF] = Id+ a smoothing operator

[Q, QGF]AΦ = Id+ a smoothing operator.

This property – being both a left and right inverse to the operator [Q, QGF], up to a smooth-ing operator – is the standard definition of a parametrix.

An example of a parametrix is the following. For M compact, let Kt ∈ E ⊗πE be theheat kernel. Then, the kernel

∫ L0 Ktdt is a parametrix, for any L > 0.

It is a standard result in the theory of pseudodifferential operators (see e.g. [Tar87])that every elliptic operator admits a parametrix. Normally a parametrix is not assumedto have proper support; however, if Φ is a parametrix satisfying all conditions except thatof proper support, and if f ∈ C∞(M×M) is a smooth function with proper support thatis 1 in a neighborhood of the diagonal, then f Φ is a parametrix with proper support. Thisshows that parametrices with proper support always exist.

Let us now list some key properties of parametrices, all of which are consequences ofelliptic regularity.

14.2.4.3 Lemma. (1) If Φ, Ψ are parametrices, then the section Φ−Ψ of the bundle E Eon M×M is smooth.

(2) Any parametrix Φ is smooth away from the diagonal in M×M.(3) Any parametrix Φ is such that (Q⊗ 1 + 1⊗ Q)Φ is smooth on all of M × M. (Note

that Q⊗ 1 + 1⊗Q is the natural differential on the space E ⊗βE ).

PROOF. We will let Q denote Q⊗ 1 + 1⊗Q, and similarly QGF = QGF ⊗ 1 + 1⊗QGF,acting on the space E ⊗βE . Note that

[Q, QGF] = [Q, QGF]⊗ 1 + 1⊗ [Q, QGF].

(1) Since [Q, QGF](Φ−Ψ) is smooth, and the operator [Q, QGF] is elliptic, this followsfrom elliptic regularity.

(2) Away from the diagonal, Φ is annihilated by the elliptic operator [Q, QGF], andso is smooth.

(3) Note that[Q, QGF]QΦ = Q[Q, QGF]Φ


and that [Q, QGF]Φ − 2KId is smooth, where KId is the kernel for the identityoperator. Since QKId = 0, the statement follows.

If Φ, Ψ are parametrices, we say that Φ < Ψ if the support of Φ is contained in thesupport of Ψ. In this way, parametrices acquire a partial order.

14.2.5. The propagator for a parametrix. In what follows, we will use the notationQ, QGF, [Q, QGF] for the operators Q⊗ 1 + 1⊗Q, etc.

If Φ is a parametrix, we let

P(Φ) = 12 QGFΦ ∈ E ⊗πE .

This is the propagator associated to Φ. We let

KΦ = KId −QP(Φ)..

Note that

QP(Φ)) = 12 [Q, QGF]Φ−QΦ

= Kid + smooth kernels .

Thus, KΦ is smooth.

An important identity we will often use is that

KΦ − KΨ = QP(Ψ)−QP(Φ).

To relate to section 14.2.3 and [Cos11c], we note that if M is a compact manifold and if

Φ =∫ L

0Ktdt

is the parametrix associated to the heat kernel, then

P(Φ) = P(0, L) =∫ L

0(QGF ⊗ 1)Ktdt

and

KΦ = KL.


14.2.6. Classes of functionals. In the appendix B.8 we define various classes of func-tions on the space Ec of compactly-supported fields. Here we give an overview of thoseclasses. Many of the conditions seem somewhat technical at first, but they arise naturallyas one attempts both to discuss the support of an observable and to extend the algebraicideas of the BV formalism in this infinite-dimensional setting.

We are interested, firstly, in functions modulo constants, which we call Ored(Ec). Everyfunctional F ∈ Ored(Ec) has a Taylor expansion in terms of symmetric smooth linear maps

Fk : E ×kc → C

(for k > 0). Such linear maps are the same as distributional sections of the bundle (E!)k

on Mk. We say that F has proper support if the support of each Fk (as defined above) isa proper subset of Mk. The space of functionals with proper support is denoted OP(Ec)(as always in this section, we work with functionals modulo constants). This conditionequivalently means that, when we think of Fk as an operator

E ×k−1c → E

!,

it extends to a smooth multilinear map

Fk : E ×k−1 → E!.

At various points in this book, we will need to consider functionals with smooth firstderivative, which are functionals satisfying a certain technical regularity constraint. Func-tionals with smooth first derivative are needed in two places in the text: when we definethe Poisson bracket on classical observables, and when we give the definition of a quan-tum field theory. In terms of the Taylor components Fk, viewed as multilinear operatorsE ×k−1

c → E!, this condition means that the Fk has image in E !. (For more detail, see

Appendix B, section B.8.)

We are interested in the functionals with smooth first derivative and with proper sup-port. We denote this space by OP,sm(E ). These are the functionals with the property thatthe Taylor components Fk, when viewed as operators, give continuous linear maps

E ×k−1 → E !.

14.2.7. The renormalization group flow. Let Φ and Ψ be parametrices. Then P(Φ)−P(Ψ) is a smooth kernel with proper support.

Given any elementα ∈ E ⊗πE = C∞(M×M, E E)

of cohomological degree 0, we define an operator

∂α : O(E )→ O(E ).


This map is an order 2 differential operator, which, on components, is the map given bycontraction with α:

α ∨− : Symn E ∨ → Symn−2 E ∨.

The operator ∂α is the unique order 2 differential operator that is given by pairing with α

on Sym2 E ∨ and that is zero on Sym≤1 E ∨.

We define a map

W (α,−) : O+(E )[[h]]→ O+(E )[[h]]

F 7→ h log(

eh∂α eF/h)

,

known as the renormalization group flow with respect to α. (When α = P(Φ)− P(Ψ), wecall it the RG flow from Ψ to Φ.) This formula is a succinct way of summarizing a Feyn-man diagram expansion. In particular, W (α, F) can be written as a sum over Feynmandiagrams with the Taylor components Fk of F labelling vertices of valence k, and with αas propagator. (All of this, and indeed everything else in this section, is explained in fargreater detail in chapter 2 of [Cos11c].) For this map to be well-defined, the functional Fmust have only cubic and higher terms modulo h. The notation O+(E )[[h]] denotes thisrestricted class of functionals.

If α ∈ E ⊗πE has proper support, then the operator W (α,−) extends (uniquely, ofcourse) to a continuous (or equivalently, smooth) operator

W (α,−) : O+P,sm(Ec)[[h]]→ O+

P,sm(Ec)[[h]].

Our philosophy is that a parametrix Φ is like a choice of “scale” for our field the-ory. The renormalization group flow relating the scale given by Φ and that given by Ψ isW (P(Φ)− P(Ψ),−).

Because P(Φ) is not a smooth kernel, the operator W (P(Φ),−) is not well-defined.This is just because the definition of W (P(Φ),−) involves multiplying distributions. Inphysics terms, the singularities that appear when one tries to define W (P(Φ),−) arecalled ultraviolet divergences.

However, if I ∈ O+P,sm(E ), the tree level part

W0 (P(Φ), I) = W ((P(Φ), I) mod h

is a well-defined element of O+P,sm(E ). The h → 0 limit of W (P(Φ), I) is called the tree-

level part because, whereas the whole object W (P(Φ), I) is defined as a sum over graphs,the h → 0 limit W0 (P(Φ), I) is defined as a sum over trees. It is straightforward to seethat W0 (P(Φ), I) only involves multiplication of distributions with transverse singularsupport, and so is well defined.


14.2.8. The BD algebra structure associated to a parametrix. A parametrix also leadsto a BV operator

4Φ = ∂KΦ : O(E )→ O(E ).Again, this operator preserves the subspace OP,sm(E ) of functions with proper supportand smooth first derivative. The operator4Φ commutes with Q, and it satisfies (4Φ)

2 =0. In a standard way, we can use the BV operator 4Φ to define a bracket on the spaceO(E ), by

I, JΦ = 4Φ(I J)− (4Φ I)J − (−1)|I| I4Φ J.

This bracket is a Poisson bracket of cohomological degree 1. If we give the graded-commutative algebra O(E )[[h]] the standard product, the Poisson bracket −,−Φ, andthe differential Q + h4Φ, then it becomes a BD algebra.

The bracket −,−Φ extends uniquely to a continuous linear map

OP(E )×O(E )→ O(E ).

Further, the space OP,sm(E ) is closed under this bracket. (Note, however, that OP,sm(E )is not a commutative algebra if M is not compact: the product of two functionals withproper support no longer has proper support.)

A functional F ∈ O(E )[[h]] is said to satisfy the Φ-quantum master equation if

QF + h4ΦF + 12F, FΦ = 0.

It is shown in [Cos11c] that if F satisfies the Φ-QME, and if Ψ is another parametrix, thenW (P(Ψ)− P(Φ), F) satisfies the Ψ-QME. This follows from the identity

[Q, ∂P(Φ) − ∂P(Ψ)] = 4Ψ −4Φ

of order 2 differential operators on O(E ). This relationship between the renormalizationgroup flow and the quantum master equation is a key part of the approach to QFT of[Cos11c].

14.2.9. The definition of a field theory. Our definition of a field theory is as follows.

14.2.9.1 Definition. Let (E , Q, 〈−,−〉) be a free BV theory. Fix a gauge fixing condition QGF.Then a quantum field theory (with this space of fields) consists of the following data.

(1) For all parametrices Φ, a functional

I[Φ] ∈ O+P,sm(Ec)[[h]]

that we call the scale Φ effective interaction. As we explained above, the subscriptsindicate that I[Φ] must have smooth first derivative and proper support. The superscript+ indicates that, modulo h, I[Φ] must be at least cubic. Note that we work with functionsmodulo constants.


(2) For two parametrices Φ, Ψ, I[Φ] must be related by the renormalization group flow:

I[Φ] = W (P(Φ)− P(Ψ), I[Ψ]) .

(3) Each I[Φ] must satisfy the Φ-quantum master equation

(Q + h4Φ)eI[Φ]/h = 0.

Equivalently,

QI[Φ] + h4Φ I[Φ] + 12I[Φ], I[Φ]Φ.

(4) Finally, we require that I[Φ] satisfies a locality axiom. Let

Ii,k[Φ] : E ×kc → C

be the kth Taylor component of the coefficient of hi in I[Φ]. We can view this as a distri-butional section of the bundle (E!)k on Mk. Our locality axiom says that, as Φ tends tozero, the support of

Ii,k[Φ]

becomes closer and closer to the small diagonal in Mk.For the constructions in this book, it turns out to be useful to have precise bounds on

the support of Ii,k[Φ]. To give these bounds, we need some notation. Let Supp(Φ) ⊂ M2

be the support of the parametrix Φ, and let Supp(Φ)n ⊂ M2 be the subset obtained byconvolving Supp(Φ) with itself n times. (Thus, (x, y) ∈ Supp(Φ)n if there exists asequence x = x0, x1, . . . , xn = y such that (xi, xi+1) ∈ Supp(Φ).)

Our support condition is that, if ej ∈ Ec, then

Ii,k(e1, . . . , ek) = 0

unless, for all 1 ≤ r < s ≤ k,

Supp(er)× Supp(es) ⊂ Supp(Φ)3i+k.

Remark: (1) The locality axiom condition as presented here is a little unappealing.An equivalent axiom is that for all open subsets U ⊂ Mk containing the smalldiagonal M ⊂ Mk, there exists a parametrix ΦU such that

Supp Ii,k[Φ] ⊂ U for all Φ < ΦU .

In other words, by choosing a small parametrix Φ, we can make the support ofIi,k[Φ] as close as we like to the small diagonal on Mk.

We present the definition with a precise bound on the size of the support ofIi,k[Φ] because this bound will be important later in the construction of the factor-ization algebra. Note, however, that the precise exponent 3i + k which appearsin the definition (in Supp(Φ)3i+k) is not important. What is important is that wehave some bound of this form.


(2) It is important to emphasize that the notion of quantum field theory is only de-fined once we have chosen a gauge fixing operator. Later, we will explain indetail how to understand the dependence on this choice. More precisely, we willconstruct a simplicial set of QFTs and show how this simplicial set only dependson the homotopy class of gauge fixing operator (in most examples, the space ofnatural gauge fixing operators is contractible).

♦

Let I0 ∈ Oloc(E ) be a local functional (defined modulo constants) that satisfies theclassical master equation

QI0 +12I0, I0 = 0.

Suppose that I0 is at least cubic.

Then, as we have seen above, we can define a family of functionals

I0[Φ] = W0 (P(Φ), I0) ∈ OP,sm(E )

as the tree-level part of the renormalization group flow operator from scale 0 to the scalegiven by the parametrix Φ. The compatibility between this classical renormalizationgroup flow and the classical master equation tells us that I0[Φ] satisfies the Φ-classicalmaster equation

QI0[Φ] + 12I0[Φ], I0[Φ]Φ = 0.

14.2.9.2 Definition. Let I[Φ] ∈ O+P,sm(E )[[h]] be the collection of effective interactions defining

a quantum field theory. Let I0 ∈ Oloc(E ) be a local functional satisfying the classical masterequation, and so defining a classical field theory. We say that the quantum field theory I[Φ] is aquantization of the classical field theory defined by I0 if

I[Φ] = I0[Φ] mod h,

or, equivalently, iflimΦ→0

I[Φ]− I0 mod h = 0.

14.3. Families of theories over nilpotent dg manifolds

Before discussing the interpretation of these axioms and also explaining the results of[Cos11c] that allow one to construct such quantum field theories, we will explain howto define families of quantum field theories over some base dg algebra. The fact that wecan work in families in this way means that the moduli space of quantum field theoriesis something like a derived stack. For instance, by considering families over the base dgalgebra of forms on the n-simplex, we see that the set of quantizations of a given classicalfield theory is a simplicial set.

14.3. FAMILIES OF THEORIES OVER NILPOTENT DG MANIFOLDS 277

One particularly important use of the families version of the theory is that it allowsus to show that our constructions and results are independent, up to homotopy, of thechoice of gauge fixing condition (provided one has a contractible — or at least connected— space of gauge fixing conditions, which happens in most examples).

In later sections, we will work implicitly over some base dg ring in the sense describedhere, although we will normally not mention this base ring explicitly.

14.3.0.3 Definition. A nilpotent dg manifold is a manifold X (possibly with corners), equippedwith a sheaf A of commutative differential graded algebras over the sheaf Ω∗X, with the followingproperties.

(1) A is concentrated in finitely many degrees.(2) Each A i is a locally free sheaf of Ω0

X-modules of finite rank. This means that A i is thesheaf of sections of some finite rank vector bundle Ai on X.

(3) We are given a map of dg Ω∗X-algebras A → C∞X .

We will let I ⊂ A be the ideal which is the kernel of the map A → C∞X : we require

that I , its powers I k, and each A /I k are locally free sheaves of C∞X -modules. Also,

we require that I k = 0 for k sufficiently large.

Note that the differential d on A is necessary a differential operator.

We will use the notation A ] to refer to the bundle of graded algebras on X whose smoothsections are A ], the graded algebra underlying the dg algebra A .

If (X, A ) and (Y, B) are nilpotent dg manifolds, a map (Y, B) → (X, A ) is a smooth mapf : Y → X together with a map of dg Ω∗(X)-algebras A → B.

Here are some basic examples.

(1) A = C∞(X) and I = 0. This describes the smooth manifold X.(2) A = Ω∗(X) and I = Ω>0(X). This equips X with its de Rham complex as

a structure sheaf. (Informally, we can say that “constant functions are the onlyfunctions on a small open” so that this dg manifold is sensitive to topologicalrather than smooth structure.)

(3) If R is a dg Artinian C-algebra with maximal ideal m, then R can be viewed asgiving the structure of nilpotent graded manifold on a point.

(4) If again R is a dg Artinian algebra, then for any manifold (X, R ⊗ Ω∗(X)) is anilpotent dg manifold.

(5) If X is a complex manifold, then A = (Ω0,∗(X), ∂) is a nilpotent dg manifold.

Remark: We study field theories in families over nilpotent dg manifolds for both practicaland structural reasons. First, we certainly wish to discuss familes of field theories over


smooth manifolds. However, we would also like to access a “derived moduli space” offield theories.

In derived algebraic geometry, one says that a derived stack is a functor from the cate-gory of non-positively graded dg rings to that of simplicial sets. Thus, such non-positivelygraded dg rings are the “test objects” one uses to define derived algebraic geometry. Ouruse of nilpotent dg manifolds mimics this story: we could say that a C∞ derived stackis a functor from nilpotent dg manifolds to simplicial sets. The nilpotence hypothesis isnot a great restriction, as the test objects used in derived algebraic geometry are naturallypro-nilpotent, where the pro-nilpotent ideal consists of the elements in degrees < 0.

Second, from a practical point of view, our arguments are tractable when workingover nilpotent dg manifolds. This is related to the fact that we choose to encode theanalytic structure on the vector spaces we consider using the language of differentiablevector spaces. Differentiable vector spaces are, by definition, objects where one can talkabout smooth families of maps depending on a smooth manifold. In fact, the definitionof differentiable vector space is strong enough that one can talk about smooth families ofmaps depending on nilpotent dg manifolds. ♦

We can now give a precise notion of “family of field theories.” We will start with thecase of a family of field theories parameterized by the nilpotent dg manifold X = (X, C∞

X ),i.e. the sheaf of dg rings on X is just the sheaf of smooth functions.

14.3.0.4 Definition. Let M be a manifold and let (X, A ) be a nilpotent dg manifold. A familyover (X, A ) of free BV theories is the following data.

(1) A graded bundle E on M× X of locally free A]-modules. We will refer to global sectionsof E as E . The space of those sections s ∈ Γ(M× X, E) with the property that the mapSupp s→ X is proper will be denoted Ec. Similarly, we let E denote the space of sectionswhich are distributional on M and smooth on X, that is,

E = E ⊗C∞(M×X)

(D(M)⊗πC∞(X)

).

(This is just the algebraic tensor product, which is reasonable as E is a finitely generatedprojective C∞(M× X)-module).

As above, we let

E! = HomA](E, A])⊗DensM

denote the “dual” bundle. There is a natural A ]-valued pairing between E and E !c .

(2) A differential operator Q : E → E , of cohomological degree 1 and square-zero, making Einto a dg module over the dg algebra A .

(3) A map

E⊗A] E→ DensM⊗A]


which is of degree −1, anti-symmetric, and leads to an isomorphism

HomA](E, A])⊗DensM → E

of sheaves of A]-modules on M× X.This pairing leads to a degree −1 anti-symmetric A -linear pairing

〈−,−〉 : Ec⊗πEc → A .

We require it to be a cochain map. In other words, if e, e′ ∈ Ec,

dA

⟨e, e′⟩=⟨

Qe, e′⟩+ (−1)|e|

⟨e, Qe′

⟩.

14.3.0.5 Definition. Let (E, Q, 〈−,−〉) be a family of free BV theories on M parameterized byA . A gauge fixing condition on E is an A -linear differential operator

QGF : E → E

such thatD = [Q, QGF] : E → E

is a generalized Laplacian, in the following sense.

Note that D is an A -linear cochain map. Thus, we can form

D0 : E ⊗A C∞(X)→ E ⊗A C∞(X)

by reducing modulo the maximal ideal I of A .

Let E0 = E/I be the bundle on M× X obtained by reducing modulo the ideal I in the bundleof algebras A. Let

σ(D0) : π∗E0 → π∗E0

be the symbol of the C∞(X)-linear operator D0. Thus, σ(D0) is an endomorphism of the bundle ofπ∗E0 on (T∗M)× X.

We require that σ(D0) is the product of the identity on E0 with a smooth family of metrics onM parameterized by X.

Throughout this section, we will fix a family of free theories on M, parameterized byA . We will take A to be our base ring throughout, so that everything will be A -linear.We would also like to take tensor products over A . Since A is a topological dg ring andwe are dealing with topological modules, the issue of tensor products is a little fraught.Instead of trying to define such things, we will use the following shorthand notations:

(1) E ⊗A E is defined to be sections of the bundle

E A] E = π∗1 E⊗A] π∗2 E

on M × M × X, with its natural differential which is a differential operator in-duced from the differentials on each copy of E .


(2) E is the space of sections of the bundle E on M × X which are smooth in theX-direction and distributional in the M-direction. Similarly for E c, E

!, etc.

(3) E ⊗A E is defined to be sections of the bundle E A] E on M×M× X, which aredistributions in the M-directions and smooth as functions of X.

(4) If x ∈ X, let Ex denote the sections on M of the restriction of the bundle E onM× X to M× x. Note that Ex is an A]

x-module. Then, we define O(E ) to be thespace of smooth sections of the bundle of topological (or differentiable) vectorspaces on X whose fibre at x is

O(E )x = ∏n

HomDVS/A]x(E ×n

x , A ]x )Sn .

That is an element of O(Ex) is something whose Taylor expansion is given bysmooth A]

x-multilinear maps to A]x.

If F ∈ O(E ) is a smooth section of this bundle, then the Taylor terms of Fare sections of the bundle (E!)A]n on Mn×X which are distributional in the Mn-directions, smooth in the X-directions, and whose support maps properly to X.

In other words: when we want to discuss spaces of functionals on E , or tensor powersof E or its distributional completions, we just to everything we did before fibrewise on Xand linear over the bundle of algebras A]. Then, we take sections of this bundle on X.

14.3.1. Now that we have defined free theories over a base ring A , the definition ofan interacting theory over A is very similar to the definition given when A = C. First,one defines a parametrix to be an element

Φ ∈ E ⊗A E

with the same properties as before, but where now we take all tensor products (and so on)over A . More precisely,

(1) Φ is symmetric under the natural Z/2 action on E ⊗ E .(2) Φ is of cohomological degree 1.(3) Φ is closed under the differential on E ⊗ E .(4) Φ has proper support: this means that the map Supp Φ→ M× X is proper.(5) Let QGF : E → E be the gauge fixing operator. We require that

([Q, QGF]⊗ 1)Φ− KId

is an element of E ⊗ E (where, as before, KId ∈ E ⊗ E is the kernel for the identitymap).

An interacting field theory is then defined to be a family of A -linear functionals

I[Φ] ∈ Ored(E )[[h]] = ∏n≥1

HomA (E ⊗A n, A )Sn [[h]]


satisfying the renormalization group flow equation, quantum master equation, and lo-cality condition, just as before. In order for the RG flow to make sense, we require thateach I[Φ] has proper support and smooth first derivative. In this context, this means thefollowing. Let Ii,k[Φ] : E ⊗k → A be the kth Taylor component of the coefficient of hi inIi,k[Φ]. Proper support means that any projection map

Supp Ii,k[Φ] ⊂ Mk × X → M× X

is proper. Smooth first derivative means, as usual, that when we think of Ii,k[Φ] as anoperator E ⊗k−1 → E , the image lies in E .

If we have a family of theories over (X, A ), and a map

f : (Y, B)→ (X, A )

of dg manifolds, then we can base change to get a family over (Y, B). The bundle on Y ofB]

x-modules of fields is defined, fibre by fibre, by

( f ∗E )y = E f (y) ⊗A]f (y)

B]y.

The gauge fixing operatorQGF : f ∗E → f ∗E

is the B-linear extension of the gauge fixing condition for the family of theories over A .

IfΦ ∈ E ⊗A E ⊂ f ∗E ⊗B f ∗E

is a parametrix for the family of free theories E over A , then it defines a parametrix f ∗Φfor the family of free theories f ∗E over B. For parametrices of this form, the effectiveaction functionals

f ∗ I[ f ∗Φ] ∈ O+sm,P( f ∗E )[[h]] = O+

sm,P(E )[[h]]⊗A B

is simply the image of the original effective action functional

I[Φ] ∈ O+sm,P(E )[[h]] ⊂ O+

sm,P( f ∗E )[[h]].

For a general parametrix Ψ for f ∗E , the effective action functional is defined by the renor-malization group equation

f ∗ I[Ψ] = W (P(Ψ)− P( f ∗Φ), f ∗ I[ f ∗Φ]) .

This is well-defined because

P(Ψ)− P( f ∗Φ) ∈ f ∗E ⊗B f ∗E

has no singularities.

The compatibility between the renormalization group equation and the quantum mas-ter equation guarantees that the effective action functionals f ∗ I[Ψ] satisfy the QME for ev-ery parametrix Ψ. The locality axiom for the original family of effective action functionals


I[Φ] guarantees that the pulled-back family f ∗ I[Ψ] satisfy the locality axiom necessary todefine a family of theories over B.

14.4. The simplicial set of theories

One of the main reasons for introducing theories over a nilpotent dg manifold (X, A )is that this allows us to talk about the simplicial set of theories. This is essential, becausethe main result we will use from [Cos11c] is homotopical in nature: it relates the simplicialset of theories to the simplicial set of local functionals.

We introduce some useful notation. Let us fix a family of classical field theories on amanifold M over a nilpotent dg manifold (X, A ). As above, the fields of such a theoryare a dg A -module E equipped with an A -linear local functional I ∈ Oloc(E ) satisfyingthe classical master equation QI + 1

2I, I = 0.

By pulling back along the projection map

(X×4n, A ⊗ C∞(4n))→ (X, A ),

we get a new family of classical theories over the dg base ring A ⊗ C∞(4n), whose fieldsare E ⊗ C∞(4n). We can then ask for a gauge fixing operator

QGF : E ⊗ C∞(4n)→ E ⊗ C∞(4n).

for this family of theories. This is the same thing as a smooth family of gauge fixingoperators for the original theory depending on a point in the n-simplex.

14.4.0.1 Definition. Let (E , I) denote the classical theory we start with over A . Let G F (E , I)denote the simplicial set whose n-simplices are such families of gauge fixing operators over A ⊗C∞(4n). If there is no ambiguity as to what classical theory we are considering, we will denotethis simplicial set by G F .

Any such gauge fixing operator extends, by Ω∗(4n)-linearity, to a linear map E ⊗Ω∗(4n) → E ⊗ Ω∗(4n), which thus defines a gauge fixing operator for the family oftheories over A ⊗Ω∗(4n) pulled back via the projection

(X×4n, A ⊗Ω∗(4n))→ (X, A ).

(Note that Ω∗(4n) is equipped with the de Rham differential.)

Example: Suppose that A = C, and the classical theory we are considering is Chern-Simons theory on a 3-manifold M, where we perturb around the trivial bundle. Then,the space of fields is E = Ω∗(M)⊗ g[1] and Q = ddR. For every Riemannian metric on M,we find a gauge fixing operator QGF = d∗. More generally, if we have a smooth family

gσ | σ ∈ 4n

14.4. THE SIMPLICIAL SET OF THEORIES 283

of Riemannian metrics on M, depending on the point σ in the n-simplex, we get an n-simplex of the simplicial set G F of gauge fixing operators.

Thus, if Met(M) denotes the simplicial set whose n-simplices are the set of Riemann-ian metrics on the fibers of the submersion M×4n → 4n, then we have a map of sim-plicial sets

Met(M)→ G F .

Note that the simplicial set Met(M) is (weakly) contractible (which follows from the fa-miliar fact that, as a topological space, the space of metrics on M is contractible).

A similar remark holds for almost all theories we consider. For example, suppose wehave a theory where the space of fields

E = Ω0,∗(M, V)

is the Dolbeault complex on some complex manifold M with coefficients in some holo-morphic vector bundle V. Suppose that the linear operator Q : E → E is the ∂-operator.The natural gauge fixing operators are of the form ∂

∗. Thus, we get a gauge fixing oper-

ator for each choice of Hermitian metric on M together with a Hermitian metric on thefibers of V. This simplicial set is again contractible.

It is in this sense that we mean that, in most examples, there is a natural contractiblespace of gauge fixing operators. ♦

14.4.1. We will use the shorthand notation (E , I) to denote the classical field theoryover A that we start with; and we will use the notation (E4n , I4n) to refer to the familyof classical field theories over A ⊗Ω∗(4n) obtained by base-change along the projection(X×4n, A ⊗Ω∗(4n))→ (X, A ).

14.4.1.1 Definition. We let T (n) denote the simplicial set whose k-simplices consist of the follow-ing data.

(1) A k-simplex QGF4k ∈ G F [k], defining a gauge-fixing operator for the family of theories

(E4k , I4k) over A ⊗Ω∗(4k).(2) A quantization of the family of classical theories with gauge fixing operator (E4k , I4k , QGF

4k ),

defined modulo hn+1.

We let T (∞) denote the corresponding simplicial set where the quantizations are defined to allorders in h.

Note that there are natural maps of simplicial sets T (n) → T (m), and that T (∞) = lim←−T (n).Further, there are natural maps T (n) → G F .


Note further that T (0) = G F .

This definition describes the most sophisticated version of the set of theories we willconsider. Let us briefly explain how to interpret this simplicial set of theories.

Suppose for simplicity that our base ring A is just C. Then, a 0-simplex of T (0) issimply a gauge-fixing operator for our theory. A 0-simplex of T (n) is a gauge fixingoperator, together with a quantization (defined with respect to that gauge-fixing operator)to order n in h.

A 1-simplex of T (0) is a homotopy between two gauge fixing operators. Supposethat we fix a 0-simplex of T (0), and consider a 1-simplex of T (∞) in the fiber over this0-simplex. Such a 1-simplex is given by a collection of effective action functionals

I[Φ] ∈ O+P,sm(E )⊗Ω∗([0, 1])[[h]]

one for each parametrix Φ, which satisfy a version of the QME and the RG flow, as ex-plained above.

We explain in some more detail how one should interpret such a 1-simplex in the spaceof theories. Let us fix a parametrix Φ on E and extend it to a parametrix for the family oftheories over Ω∗([0, 1]). We can then expand our effective interaction I[Φ] as

I[Φ] = J[Φ](t) + J′[Φ](t)dt

where J[Φ](t), J′[Φ](t) are elements

J[Φ](t), J′[Φ](t) ∈ O+P,sm(E )⊗ C∞([0, 1])[[h]].

Here t is the coordinate on the interval [0, 1].

The quantum master equation implies that the following two equations hold, for eachvalue of t ∈ [0, 1],

QJ[Φ](t) + 12J[Φ](t), J[Φ](t)Φ + h4Φ J[Φ](t) = 0,

∂

∂tJ[Φ](t) + QJ′[Φ](t) + J[Φ](t), J′[Φ](t)Φ + h4Φ J′[Φ](t) = 0.

The first equation tells us that for each value of t, J[Φ](t) is a solution of the quantum mas-ter equation. The second equation tells us that the t-derivative of J[Φ](t) is homotopicallytrivial as a deformation of the solution to the QME J[Φ](t).

In general, if I is a solution to some quantum master equation, a transformation of theform

I 7→ I + εJ = I + εQI′ + I, I′+ h4I′

is often called a “BV canonical transformation” in the physics literature. In the physicsliterature, solutions of the QME related by a canonical transformation are regarded as

14.5. THE THEOREM ON QUANTIZATION 285

equivalent: the canonical transformation can be viewed as a change of coordinates on thespace of fields.

For us, this interpretation is not so important. If we have a family of theories overΩ∗([0, 1]), given by a 1-simplex in T (∞), then the factorization algebra we will constructfrom this family of theories will be defined over the dg base ring Ω∗([0, 1]). This impliesthat the factorization algebras obtained by restricting to 0 and 1 are quasi-isomorphic.

14.4.2. Generalizations. We will shortly state the theorem which allows us to con-struct such quantum field theories. Let us first, however, briefly introduce a slightly moregeneral notion of “theory.”

We work over a nilpotent dg manifold (X, A ). Recall that part of the data of such amanifold is a differential ideal I ⊂ A whose quotient is C∞(X). In the above discussion,we assumed that our classical action functional S was at least quadratic; we then split Sas

S = 〈e, Qe〉+ I(e)into kinetic and interacting terms.

We can generalize this to the situation where S contains linear terms, as long as theyare accompanied by elements of the ideal I ⊂ A . In this situation, we also have somefreedom in the splitting of S into kinetic and interacting terms; we require only that linearand quadratic terms in the interaction I are weighted by elements of the nilpotent idealI .

In this more general situation, the classical master equation S, S = 0 does not implythat Q2 = 0, only that Q2 = 0 modulo the ideal I . However, this does not lead to anyproblems; the definition of quantum theory given above can be easily modified to dealwith this more general situation.

In the L∞ language used in Chapter 10, this more general situation describes a familyof curved L∞ algebras over the base dg ring A with the property that the curving vanishesmodulo the nilpotent ideal I .

Recall that ordinary (not curved) L∞ algebras correspond to formal pointed moduliproblems. These curved L∞ algebras correspond to families of formal moduli problemsover A which are pointed modulo I .

14.5. The theorem on quantization

Let M be a manifold, and suppose we have a family of classical BV theories on M overa nilpotent dg manifold (X, A ). Suppose that the space of fields on M is the A -moduleE . Let Oloc(E ) be the dg A -module of local functionals with differential Q + I,−.


Given a cochain complex C, we denote the Dold-Kan simplicial set associated to C byDK(C). Its n-simplices are the closed, degree 0 elements of C⊗Ω∗(4n).

14.5.0.1 Theorem. All of the simplicial sets T (n)(E , I) are Kan complexes and T (∞)(E , I). Themaps p : T (n+1)(E , I)→ T (n)(E , I) are Kan fibrations.

Further, there is a homotopy fiber diagram of simplicial sets

T (n+1)(E , I)

p

// 0

T (n)(E , I) O // DK(Oloc(E )[1], Q + I,−)

where O is the “obstruction map.”

In more prosaic terms, the second part of the theorem says the following. If α ∈T (n)(E , I)[0] is a zero-simplex of T (n)(E , I), then there is an obstruction O(α) ∈ Oloc(E ).This obstruction is a closed degree 1 element. The simplicial set p−1(α) ∈ T (n+1)(E , I) ofextensions of α to the next order in h is homotopy equivalent to the simplicial set of waysof making O(α) exact. In particular, if the cohomology class [O(α)] ∈ H1(Oloc(E), Q +I,−) is non-zero, then α does not admit a lift to the next order in h. If this cohomologyclass is zero, then the simplicial set of possible lifts is a torsor for the simplicial Abeliangroup DK(Oloc(E ))[1].

Note also that a first order deformation of the classical field theory (E , Q, I) is givenby a closed degree 0 element of Oloc(E ). Further, two such first order deformations areequivalent if they are cohomologous. Thus, this theorem tells us that the moduli space ofQFTs is “the same size” as the moduli space of classical field theories: at each order in h,the data needed to describe a QFT is a local action functional.

The first part of the theorem says can be interpreted as follows. A Kan simplicial setcan be thought of as an “infinity-groupoid.” Since we can consider families of theoriesover arbitrary nilpotent dg manifolds, we can consider T ∞(E , I) as a functor from thecategory of nilpotent dg manifolds to that of Kan complexes, or infinity-groupoids. Thus,the space of theories forms something like a “derived stack” [Toe06, Lur11].

This theorem also tells us in what sense the notion of “theory” is independent of thechoice of gauge fixing operator. The simplicial set T (0)(E , I) is the simplicial set G F ofgauge fixing operators. Since the map

T (∞)(E , I)→ T (0)(E , I) = G F

is a fibration, a path between two gauge fixing conditions QGF0 and QGF

1 leads to a ho-motopy between the corresponding fibers, and thus to an equivalence between the ∞-groupoids of theories defined using QGF

0 and QGF1 .

14.5. THE THEOREM ON QUANTIZATION 287

As we mentioned several times, there is often a natural contractible simplicial set map-ping to the simplicial set G F of gauge fixing operators. Thus, G F often has a canonical“homotopy point”. From the homotopical point of view, having a homotopy point is justas good as having an actual point: if S→ G F is a map out of a contractible simplicial set,then the fibers in T (∞) above any point in S are canonically homotopy equivalent.

CHAPTER 15

The observables of a quantum field theory

15.1. Free fields

Before we give our general construction of the factorization algebra associated to aquantum field theory, we will give the much easier construction of the factorization alge-bra for a free field theory.

Let us recall the definition of a free BV theory.

15.1.0.2 Definition. A free BV theory on a manifold M consists of the following data:

(1) a Z-graded super vector bundle π : E→ M that has finite rank;(2) an antisymmetric map of vector bundles 〈−,−〉loc : E⊗ E → Dens(M) of degree −1

that is fiberwise nondegenerate. It induces a symplectic pairing on compactly supportedsmooth sections Ec of E:

〈φ, ψ〉 =∫

x∈M〈φ(x), ψ(x)〉loc;

(3) a square-zero differential operator Q : E → E of cohomological degree 1 that is skew selfadjoint for the symplectic pairing.

Remark: When we consider deforming free theories into interacting theories, we will needto assume the existence of a “gauge fixing operator”: this is a degree −1 operator QGF :E → E such that [Q, QGF] is a generalized Laplacian in the sense of [BGV92]. ♦

On any open set U ⊂ M, the commutative dg algebra of classical observables sup-ported in U is

Obscl(U) = (Sym(E ∨(U)), Q),

where

E ∨(U) = E!c(U)

denotes the distributions dual to E with compact support in U and Q is the derivationgiven by extending the natural action of Q on the distributions.

289

290 15. THE OBSERVABLES OF A QUANTUM FIELD THEORY

In section 12.3 we constructed a sub-factorization algebra

Obscl(U) = (Sym(E !

c (U)), Q)

defined as the symmetric algebra on the compactly-supported smooth (rather than distri-

butional) sections of the bundle E!. We showed that the inclusion Obscl(U) → Obscl(U)

is a weak equivalence of factorization algebras. Further, Obscl(U) has a Poisson bracket

of cohomological degree 1, defined on the generators by the natural pairing

E !c (U)⊗πE !

c (U)→ R,

which arises from the dual pairing on Ec(U). In this section we will show how to construct

a quantization of the P0 factorization algebra Obscl

.

15.1.1. The Heisenberg algebra construction. Our quantum observables on an openset U will be built from a certain Heisenberg Lie algebra.

Recall the usual construction of a Heisenberg algebra. If V is a symplectic vectorspace, viewed as an abelian Lie algebra, then the Heisenberg algebra Heis(V) is the centralextension

0→ C · h→ Heis(V)→ V

whose bracket is [x, y] = h〈x, y〉.

Since the element h ∈ Heis(V) is central, the algebra U(Heis(V)) is an algebra overC[[h]], the completed universal enveloping algebra of the Abelian Lie algebra C · h.

In quantum mechanics, this Heisenberg construction typically appears in the study ofsystems with quadratic Hamiltonians. In this context, the space V can be viewed in twoways. Either it is the space of solutions to the equations of motion, which is a linear spacebecause we are dealing with a free field theory; or it is the space of linear observables dualto the space of solutions to the equations of motion. The natural symplectic pairing onV gives an isomorphism between these descriptions. The algebra U(Heis(V)) is then thealgebra of non-linear observables.

Our construction of the quantum observables of a free field theory will be formallyvery similar. We will start with a space of linear observables, which (after a shift) is acochain complex with a symplectic pairing of cohomological degree 1. Then, instead ofapplying the usual universal enveloping algebra construction, we will take Chevalley-Eilenberg chain complex, whose cohomology is the Lie algebra homology.1 This fits withour operadic philosophy: Chevalley-Eilenberg chains are the E0 analog of the universalenveloping algebra.

1As usual, we always use gradings such that the differential has degree +1.

15.1. FREE FIELDS 291

15.1.2. The basic homological construction. Let us start with a 0-dimensional freefield theory. Thus, let V be a cochain complex equipped with a symplectic pairing ofcohomological degree −1. We will think of V as the space of fields of our theory. Thespace of linear observables of our theory is V∨; the Poisson bracket on O(V) induces asymmetric pairing of degree 1 on V∨. We will construct the space of all observables froma Heisenberg Lie algebra built on V∨[−1], which has a symplectic pairing 〈−,−〉 of degree−1. Note that there is an isomorphism V ∼= V∨[−1] compatible with the pairings on bothsides.

15.1.2.1 Definition. The Heisenberg algebra Heis(V) is the Lie algebra central extension

0→ C · h[−1]→ Heis(V)→ V∨[−1]→ 0

whose bracket is[v + ha, w + hb] = h 〈v, w〉

The element h labels the basis element of the center C[−1].

Putting the center in degree 1 may look strange, but it is necessary to do this in orderto get a Lie bracket of cohomological degree 0.

Let C∗(Heis(V)) denote the completion2 of the Lie algebra chain complex of Heis(V),defined by the product of the spaces Symn Heis(V), instead of their sum.

In this zero-dimensional toy model, the classical observables are

Obscl = O(V) = ∏n

Symn(V∨).

This is a commutative dg algebra equipped with the Poisson bracket of degree 1 arisingfrom the pairing on V. Thus, O(V) is a P0 algebra.

15.1.2.2 Lemma. The completed Chevalley-Eilenberg chain complex C∗(Heis(V)) is a BD alge-bra (section 8.4) which is a quantization of the P0 algebra O(V).

PROOF. The completed Chevalley-Eilenberg complex for Heis(V) has the completedsymmetric algebra Sym(Heis(V)[1]) as its underlying graded vector space. Note that

Sym(Heis(V)[1]) = Sym(V∨ ⊕C · h) = Sym(V∨)[[h]],

so that C∗(Heis(V)) is a flat C[[h]] module which reduces to Sym(V∨) modulo h. TheChevalley-Eilenberg chain complex C∗(Heis(V)) inherits a product, corresponding to thenatural product on the symmetric algebra Sym(Heis(V)[1]). Further, it has a natural Pois-son bracket of cohomological degree 1 arising from the Lie bracket on Heis(V), extended

2One doesn’t need to take the completed Lie algebra chain complex. We do this to be consistent with ourdiscussion of the observables of interacting field theories, where it is essential to complete.


to be a derivation of C∗(Heis(V)). Note that, since C · h[−1] is central in Heis(V), thisPoisson bracket reduces to the given Poisson bracket on Sym(V∨) modulo h.

In order to prove that we have a BD quantization, it remains to verify that, althoughthe commutative product on C∗(Heis(V)) is not compatible with the product, it satisfiesthe BD axiom:

d(a · b) = (da) · b + (−1)|a|a · (db) + ha, b.This follows by definition.

15.1.3. Cosheaves of Heisenberg algebras. Next, let us give the analog of this con-struction for a general free BV theory E on a manifold M. As above, our classical observ-ables are defined by

Obscl(U) = Sym E !

c (U)

which has a Poisson bracket arising from the pairing on E !c (U). Recall that this is a factor-

ization algebra.

To construct the quantum theory, we define, as above, a Heisenberg algebra Heis(U)as a central extension

0→ C[−1] · h→ Heis(U)→ E !c (U)[−1]→ 0.

Note that Heis(U) is a pre-cosheaf of Lie algebras. The bracket in this Heisenberg algebraarises from the pairing on E !

c (U).

We then define the quantum observables by

Obsq(U) = C∗(Heis(U)).

The underlying cochain complex is, as before,

Sym(Heis(U)[1])

where the completed symmetric algebra is defined (as always) using the completed tensorproduct.

15.1.3.1 Proposition. Sending U to Obsq(U) defines a BD factorization algebra in the categoryof differentiable pro-cochain complexes over R[[h]], which quantizes Obscl(U).

PROOF. First, we need to define the filtration on Obsq(U) making it into a differen-tiable pro-cochain complex. The filtration is defined, in the identification

Obsq(U) = Sym E !c (U)[[h]]

by sayingFn Obsq(U) = ∏

khk Sym≥n−2k E !

c (U).

This filtration is engineered so that the Fn Obsq(U) is a subcomplex of Obsq(U).

15.2. THE BD ALGEBRA OF GLOBAL OBSERVABLES 293

It is immediate that Obsq is a BD pre-factorization algebra quantizing Obscl(U). Thefact that it is a factorization algebra follows from the fact that Obscl(U) is a factorizationalgebra, and then a simple spectral sequence argument. (A more sophisticated version ofthis spectral sequence argument, for interacting theories, is given in section 15.6.)

15.2. The BD algebra of global observables

In this section, we will try to motivate our definition of a quantum field theory fromthe point of view of homological algebra. All of the constructions we will explain willwork over an arbitrary nilpotent dg manifold (X, A ), but to keep the notation simple wewill not normally mention the base ring A .

Thus, suppose that (E , I, Q, 〈−,−〉) is a classical field theory on a manifold M. Wehave seen (Chapter 12, section 12.2) how such a classical field theory gives immediately acommutative factorization algebra whose value on an open subset is

Obscl(U) = (O(E (U)), Q + I,−) .

Further, we saw that there is a P0 sub-factorization algebra

Obscl(U) = (Osm(E (U)), Q + I,−) .

In particular, we have a P0 algebra Obscl(M) of global sections of this P0 algebra. We

can think of Obscl(M) as the algebra of functions on the derived space of solutions to the

Euler-Lagrange equations.

In this section we will explain how a quantization of this classical field theory will

give a quantization (in a homotopical sense) of the P0 algebra Obscl(M) into a BD algebra

Obsq(M) of global observables. This BD algebra has some locality properties, which wewill exploit later to show that Obsq(M) is indeed the global sections of a factorizationalgebra of quantum observables.

In the case when the classical theory is the cotangent theory to some formal ellipticmoduli problem BL on M (encoded in an elliptic L∞ algebra L on M), there is a par-ticularly nice class of quantizations, which we call cotangent quantizations. Cotangentquantizations have a very clear geometric interpretation: they are locally-defined volumeforms on the sheaf of formal moduli problems defined by L.

15.2.1. The BD algebra associated to a parametrix. Suppose we have a quantizationof our classical field theory (defined with respect to some gauge fixing condition, or familyof gauge fixing conditions). Then, for every parametrix Φ, we have seen how to constructa cohomological degree 1 operator

4Φ : O(E )→ O(E )


and a Poisson bracket

−,−Φ : O(E )×O(E )→ O(E )

such that O(E )[[h]], with the usual product, with bracket −,−Φ and with differentialQ + h4Φ, forms a BD algebra.

Further, since the effective interaction I[Φ] satisfies the quantum master equation, wecan form a new BD algebra by adding I[Φ],−Φ to the differential of O(E )[[h]].

15.2.1.1 Definition. Let ObsqΦ(M) denote the BD algebra

ObsqΦ(M) = (O(E )[[h]], Q + h4Φ + I[Φ],−Φ) ,

with bracket −,−Φ and the usual product.

Remark: Note that I[Φ] is not in O(E )[[h]], but rather in O+P,sm(E )[[h]]. However, as we

remarked earlier in 14.2.8, the bracket

I[Φ],−Φ : O(E )[[h]]→ O(E )[[h]]

is well-defined. ♦

Remark: Note that we consider ObsqΦ(M) as a BD algebra valued in the multicategory of

differentiable pro-cochain complexes (see Appendix B). This structure includes a filtrationon Obsq

Φ(M) = O(E )[[h]]. The filtration is defined by saying that

FnO(E )[[h]] = ∏i

hi Sym≥(n−2i)(E ∨);

it is easily seen that the differential Q + h4Φ + I[Φ],−Φ preserves this filtration. ♦

We will show that for varying Φ, the BD algebras ObsqΦ(M) are canonically weakly

equivalent. Moreover, we will show that there is a canonical weak equivalence of P0algebras

ObsqΦ(M)⊗C[[h]] C ' Obs

cl(M).

To show this, we will construct a family of BD algebras over the dg base ring of forms on acertain contractible simplicial set of parametrices that restricts to Obsq

Φ(M) at each vertex.

Before we get into the details of the construction, however, let us say something abouthow this result allows us to interpret the definition of a quantum field theory.

A quantum field theory gives a BD algebra for each parametrix. These BD algebrasare all canonically equivalent. Thus, at first glance, one might think that the data of a QFTis entirely encoded in the BD algebra for a single parametrix. However, this does not takeaccount of a key part of our definition of a field theory, that of locality.


The BD algebra associated to a parametrix Φ has underlying commutative algebraO(E )[[h]], equipped with a differential which we temporarily denote

dΦ = Q + h4Φ + I[Φ],−Φ.

If K ⊂ M is a closed subset, we have a restriction map

E = E (M)→ E (K),

where E (K) denotes germs of smooth sections of the bundle E on K. There is a dual mapon functionals O(E (K))→ O(E ). We say a functional f ∈ O(E )[[h]] is supported on K if itis in the image of this map.

As Φ → 0, the effective interaction I[Φ] and the BV Laplacian 4Φ become more andmore local (i.e., their support gets closer to the small diagonal). This tells us that, for verysmall Φ, the operator dΦ only increases the support of a functional in O(E )[[h]] by a smallamount. Further, by choosing Φ to be small enough, we can increase the support by anarbitrarily small amount.

Thus, a quantum field theory is

(1) A family of BD algebra structures on O(E )[[h]], one for each parametrix, whichare all homotopic (and which all have the same underlying graded commutativealgebra).

(2) The differential dΦ defining the BD structure for a parametrix Φ increases supportby a small amount if Φ is small.

This property of dΦ for small Φ is what will allow us to construct a factorization al-gebra of quantum observables. If dΦ did not increase the support of a functional f ∈O(E )[[h]] at all, the factorization algebra would be easy to define: we would just setObsq(U) = O(E (U))[[h]], with differential dΦ. However, because dΦ does increase sup-port by some amount (which we can take to be arbitrarily small), it takes a little work topush this idea through.

Remark: The precise meaning of the statement that dΦ increases support by an arbitrarilysmall amount is a little delicate. Let us explain what we mean. A functional f ∈ O(E )[[h]]has an infinite Taylor expansion of the form f = ∑ hi fi,k, where fi,k : E ⊗πk → C is asymmetric linear map. We let Supp≤(i,k) f be the unions of the supports of fr,s where(r, s) ≤ (i, k) in the lexicographical ordering. If K ⊂ M is a subset, let Φn(K) denotethe subset obtained by convolving n times with Supp Φ ⊂ M2. The differential dΦ hasthe following property: there are constants ci,k ∈ Z>0 of a purely combinatorial nature(independent of the theory we are considering) such that, for all f ∈ O(E )[[h]],

Supp≤(i,k) dΦ f ⊂ Φci,k(Supp≤(i,k) f ).

Thus, we could say that dΦ increase support by an amount linear in Supp Φ. We will usethis concept in the main theorem of this chapter. ♦


15.2.2. Let us now turn to the construction of the equivalences between ObsqΦ(M) for

varying parametrices Φ. The first step is to construct the simplicial set P of parametrices;we will then construct a BD algebra Obsq

P(M) over the base dg ring Ω∗(P), which wedefine below.

LetV ⊂ C∞(M×M, E E) = E ⊗πE

denote the subspace of those elements which are cohomologically closed and of degree 1,symmetric, and have proper support.

Note that the set of parametrices has the structure of an affine space for V: if Φ, Ψ areparametrices, then

Φ−Ψ ∈ V

and, conversely, if Φ is a parametrix and A ∈ V, then Φ + A is a new parametrix.

Let P denote the simplicial set whose n-simplices are affine-linear maps from 4n tothe affine space of parametrices. It is clear that P is contractible.

For any vector space V, let V4 denote the simplicial set whose k-simplices are affinelinear maps4k → V. For any convex subset U ⊂ V, there is a sub-simplicial set U4 ⊂ V4whose k-simplices are affine linear maps 4k → U. Note that P is a sub-simplicial set of

E⊗π24 , corresponding to the convex subset of parametrices inside E

⊗π2.

Let C P [0] ⊂ E⊗π2

denote the cone on the affine subspace of parametrices, with vertex

the origin 0. An element of C P [0] is an element of E⊗π2

of the form tΦ, where Φ is aparametrix and t ∈ [0, 1]. Let C P denote the simplicial set whose k-simplices are affinelinear maps to C P [0].

Recall that the simplicial de Rham algebra Ω∗4(S) of a simplicial set S is defined asfollows. Any element ω ∈ Ωi

4(S) consists of an i-form

ω(φ) ∈ Ωi(4k)

for each k-simplex φ : 4k → S. If f : 4k → 4l is a face or degeneracy map, then werequire that

f ∗ω(φ) = ω(φ f ).

The main results of this section are as follows.

15.2.2.1 Theorem. There is a BD algebra ObsqP(M) over Ω∗(P) which, at each 0-simplex Φ,

is the BD algebra ObsqΦ(M) discussed above.

The underlying graded commutative algebra of ObsqP(M) is O(E )⊗Ω∗(P)[[h]].


For every open subset U ⊂ M × M, let PU denote the parametrices whose support is inU. Let Obsq

PU(M) denote the restriction of Obsq

P(M) to U. The differential on ObsqPU

(M)increases support by an amount linear in U (in the sense explained precisely in the remark above).

The bracket −,−PU on ObsqPU

(M) is also approximately local, in the following sense. IfO1, O2 ∈ Obsq

PU(M) have the property that

Supp O1 × Supp O2 ∩U = ∅ ∈ M×M,

then O1, O2PU = 0.

Further, there is a P0 algebra ObsclC P(M) over Ω∗(C P) equipped with a quasi-isomorphism

of P0 algebras over Ω∗(P),

ObsclC P(M)

∣∣∣P' Obsq

P(M) modulo h,

and with an isomorphism of P0 algebras,

ObsclC P(M)

∣∣∣0∼= Obs

cl(M),

where Obscl(M) is the P0 algebra constructed in Chapter 12.

The underlying commutative algebra of ObsclC P(M) is Obs

cl(M)⊗Ω∗(C P), the differen-

tial on ObsclC P(M) increases support by an arbitrarily small amount, and the Poisson bracket on

ObsclC P(M) is approximately local in the same sense as above.

PROOF. We need to construct, for each k-simplex φ : 4k →P , a BD algebra Obsqφ(M)

over Ω∗(4k). We view the k-simplex as a subset of Rk+1 by

4k :=

(λ0, . . . , λk) ⊂ [0, 1]k+1 : ∑

iλi = 1

.

Since simplices in P are affine linear maps to the space of parametrices, the simplex φ isdetermined by k + 1 parametrices Φ0, . . . , Φk, with

φ(λ0, . . . , λk) = ∑i

λiΦi

for λi ∈ [0, 1] and ∑ λi = 1.

The graded vector space underlying our BD algebra is

Obsqφ(M) = O(E )[[h]]⊗Ω∗(4k).


The structure as a BD algebra will be encoded by an order two, Ω∗(4k)-linear differentialoperator

4φ : Obsqφ(M)→ Obsq

φ(M).

We need to recall some notation in order to define this operator. Each parametrix Φprovides an order two differential operator4Φ on O(E ), the BV Laplacian correspondingto Φ. Further, if Φ, Ψ are two parametrices, then the difference between the propagatorsP(Φ) − P(Ψ) is an element of E ⊗ E , so that contracting with P(Φ) − P(Ψ) defines anorder two differential operator ∂P(Φ) − ∂P(Ψ) on O(E ). (This operator defines the infini-tesimal version of the renormalization group flow from Ψ to Φ.) We have the equation

[Q, ∂P(Φ) − ∂P(Ψ)] = −4Φ +4Ψ.

Note that although the operator ∂P(Φ) is only defined on the smaller subspace O(E ), be-cause P(Φ) ∈ E ⊗ E , the difference ∂P(Φ) and ∂P(Ψ) is nonetheless well-defined on O(E )because P(Φ)− P(Ψ) ∈ E ⊗ E .

The BV Laplacian 4φ associated to the k-simplex φ : 4k → P is defined by theformula

4φ =k

∑i=0

λi4Φi −k

∑i=0

dλi∂P(Φi),

where the λi ∈ [0, 1] are the coordinates on the simplex 4k and, as above, the Φi are theparametrices associated to the vertices of the simplex φ.

It is not entirely obvious that this operator makes sense as a linear map O(E ) →O(E ) ⊗ Ω∗(4k), because the operators ∂P(Φ) are only defined on the smaller subspaceO(E ). However, since ∑ dλi = 0, we have

∑ dλi∂P(Φi) = ∑ dλi(∂P(Φi) − ∂P(Φ0)),

and the right hand side is well defined.

It is immediate that42φ = 0. If we denote the differential on the classical observables

O(E )⊗Ω∗(4n) by Q + ddR, we have

[Q + ddR,4φ] = 0.

To see this, note that

[Q + ddR,4φ] = ∑ dλi4Φi + ∑ dλi[Q, ∂Φi − ∂Φ0 ]

= ∑ dλi4Φi −∑ dλi(4Φi −4Φ0)

= ∑ dλi4Φ0

= 0,

where we use various identities from earlier.


The operator4φ defines, in the usual way, an Ω∗(4k)-linear Poisson bracket −,−φ

on O(E )⊗Ω∗(4k).

We have effective action functionals I[Ψ] ∈ O+sm,P(E )[[h]] for each parametrix Ψ. Let

I[φ] = I[∑ λiΦi] ∈ O+sm,P(E )[[h]]⊗ C∞(4k).

The renormalization group equation tells us that I[∑ λiΦi] is smooth (actually polyno-mial) in the λi.

We define the structure of BD algebra on the graded vector space

Obsqφ(M) = O(E )[[h]]⊗Ω∗(4k)

as follows. The product is the usual one; the bracket is −,−φ, as above; and the differ-ential is

Q + ddR + h4φ + I[φ],−φ.We need to check that this differential squares to zero. This is equivalent to the quantummaster equation

(Q + ddR + h4φ)eI[φ]/h = 0.This holds as a consequence of the quantum master equation and renormalization groupequation satisfied by I[φ]. Indeed, the renormalization group equation tells us that

eI[φ]/h = exp(

h ∑ λi

(∂PΦi) − ∂P(Φ0)

))eI[Φ0]/h.

Thus,ddReI[φ]/h = h ∑ dλi∂P(Φi)e

I[φ]/h

The QME for each I[∑ λiΦi] tells us that

(Q + h ∑ λi4Φi)eI[φ]/h = 0.

Putting these equations together with the definition of 4φ shows that I[φ] satisfies theQME.

Thus, we have constructed a BD algebra Obsqφ(M) over Ω∗(4k) for every simplex

φ : 4k →P . It is evident that these BD algebras are compatible with face and degeneracymaps, and so glue together to define a BD algebra over the simplicial de Rham complexΩ∗4(P) of P .

Let φ be a k-simplex of P , and let

Supp(φ) = ∪λ∈4k Supp(∑ λiΦi).

We need to check that the bracket O1, O2φ vanishes for observables O1, O2 such that(Supp O1 × Supp)O2 ∩ Supp φ = ∅. This is immediate, because the bracket is defined bycontracting with tensors in E ⊗ E whose supports sit inside Supp φ.


Next, we need to verify that, on a k-simplex φ of P , the differential Q + I[φ],−φ in-creases support by an amount linear in Supp(φ). This follows from the support propertiessatisfied by I[Φ] (which are detailed in the definition of a quantum field theory, definition14.2.9.1).

It remains to construct the P0 algebra over Ω∗(C P). The construction is almost iden-tical, so we will not give all details. A zero-simplex of C P is an element of E ⊗ E of theform Ψ = tΦ, where Φ is a parametrix. We can use the same formulae we used for para-metrices to construct a propagator P(Ψ) and Poisson bracket −,−Ψ for each Ψ ∈ C P .The kernel defining the Poisson bracket −,−Ψ need not be smooth. This means thatthe bracket −,−Ψ is only defined on the subspace Osm(E ) of functionals with smoothfirst derivative. In particular, if Ψ = 0 is the vertex of the cone C P , then −,−0 is the

Poisson bracket defined in Chapter 12 on Obscl(M) = Osm(E ).

For each Ψ ∈ C P , we can form a tree-level effective interaction

I0[Ψ] = W0 (P(Ψ), I) ∈ Osm,P(E ),

where I ∈ Oloc(E ) is the classical action functional we start with. There are no difficultiesdefining this expression because we are working at tree-level and using functionals withsmooth first derivative. If Ψ = 0, then I0[0] = I.

The P0 algebra over Ω∗(C P) is defined in almost exactly the same way as we definedthe BD algebra over Ω∗P . The underlying commutative algebra is Osm(E )⊗Ω∗(C P). Ona k-simplex ψ with vertices Ψ0, . . . , Ψk, the Poisson bracket is

−,−ψ = ∑ λi−,−Ψi + ∑ dλi−,−P(Ψi),

where −,−P(Ψi) is the Poisson bracket of cohomological degree 0 defined using thepropagator P(Ψi) ∈ E ⊗πE as a kernel. If we let I0[ψ] = I0[∑ λiΨi], then the differential is

dψ = Q + I0[ψ],−ψ.

The renormalization group equation and classical master equation satisfied by the I0[Ψ]

imply that d2ψ = 0. If Ψ = 0, this P0 algebra is clearly the P0 algebra Obs

cl(M) constructed

in Chapter 12. When restricted to P ⊂ C P , this P0 algebra is the sub P0 algebra ofObsq

P(M)/h obtained by restricting to functionals with smooth first derivative; the inclu-sion

ObsclC P(M) |P → Obsq

P(M)/his thus a quasi-isomorphism, using proposition 12.4.2.4 of Chapter 12.

15.3. Global observables

In the next few sections, we will prove the first version (section 1.7) of our quantiza-tion theorem. Our proof is by construction, associating a factorization algebra on M to a

15.3. GLOBAL OBSERVABLES 301

quantum field theory on M, in the sense of [Cos11c]. This is a quantization (in the weaksense) of the P0 factorization algebra associated to the corresponding classical field theory.

More precisely, we will show the following.

15.3.0.2 Theorem. For any quantum field theory on a manifold M over a nilpotent dg manifold(X, A ), there is a factorization algebra Obsq on M, valued in the multicategory of differentiablepro-cochain complexes flat over A [[h]].

There is an isomorphism of factorization algebras

Obsq⊗A [[h]]A∼= Obscl

between Obsq modulo h and the commutative factorization algebra Obscl .

Further, Obsq is a weak quantization (in the sense of Chapter 1, section 1.7) of the P0 factor-ization algebra Obscl of classical observables.

15.3.1. So far we have constructed a BD algebra ObsqΦ(M) for each parametrix Φ;

these BD algebras are all weakly equivalent to each other. In this section we will definea cochain complex Obsq(M) of global observables which is independent of the choice ofparametrix. For every open subset U ⊂ M, we will construct a subcomplex Obsq(U) ⊂Obsq(M) of observables supported on U. The complexes Obsq(U) will form our factor-ization algebra.

Thus, suppose we have a quantum field theory on M, with space of fields E andeffective action functionals I[Φ], one for each parametrix (as explained in section 14.2).

An observable for a quantum field theory (that is, an element of the cochain complexObsq(M)) is simply a first-order deformation I[Φ] + δO[Φ] of the family of effectiveaction functionals I[Φ], which satisfies a renormalization group equation but does notnecessarily satisfy the locality axiom in the definition of a quantum field theory. Definition15.3.1.3 makes this idea precise.

Remark: This definition is motivated by a formal argument with the path integral. LetS(φ) be the action functional for a field φ, and let O(φ) be another function of the field,describing a measurement that one could make. Heuristically, the expectation value ofthe observable is

〈O〉 = 1ZS

∫O(φ)e−S(φ)/h Dφ,

where ZS denotes the partition function, simply the integral without O. A formal manip-ulation shows that

〈O〉 = ddδ

1ZS

∫e(−S(φ)+hδO(φ))/h Dφ.

In other words, we can view O as a first-order deformation of the action functional Sand compute the expectation value as the change in the partition function. Because the


book [Cos11c] gives an approach to the path integral that incorporates the BV formalism,we can define and compute expectation values of observables by exploiting the seconddescription of 〈O〉 given above. ♦

Earlier we defined cochain complexes ObsqΦ(M) for each parametrix. The underlying

graded vector space of ObsqΦ(M) is O(E )[[h]]; the differential on Obsq

Φ(M) is

QΦ = Q + I[Φ],−Φ + h4Φ.

15.3.1.1 Definition. Define a linear map

WΦΨ : O(E )[[h]]→ O(E )[[h]]

by requiring that, for an element f ∈ O(E )[[h]] of cohomological degree i,

I[Φ] + δWΦΨ ( f ) = W (P(Φ)− P(Ψ), I[Ψ] + δ f )

where δ is a square-zero parameter of cohomological degree −i.

15.3.1.2 Lemma. The linear map

WΦΨ : Obsq

Ψ(M)→ ObsqΦ(M)

is an isomorphism of differentiable pro-cochain complexes.

PROOF. The fact that WΦΨ intertwines the differentials QΦ and QΨ follows from the

compatibility between the quantum master equation and the renormalization group equa-tion described in [Cos11c], Chapter 5 and summarized in section 14.2. It is not hard toverify that WΦ

Ψ is a map of differentiable pro-cochain complexes. The inverse to WΦΨ is

WΨΦ .

15.3.1.3 Definition. A global observable O of cohomological degree i is an assignment to everyparametrix Φ of an element

O[Φ] ∈ ObsqΦ(M) = O(E)[[h]]

of cohomological degree i such that

WΦΨ O[Ψ] = O[Φ].

If O is an observable of cohomological degree i, we let QO be defined by

Q(O)[Φ] = QΦ(O[Φ]) = QO[Φ] + I[Φ], O[Φ]Φ + h4ΦO[Φ].

This makes the space of observables into a differentiable pro-cochain complex, which we call Obsq(M).

Thus, if O ∈ Obsq(M) is an observable of cohomological degree i, and if δ is a square-zero parameter of cohomological degree −i, then the collection of effective interactionsI[Φ] + δO[Φ] satisfy most of the axioms needed to define a family of quantum field

15.4. LOCAL OBSERVABLES 303

theories over the base ring C[δ]/δ2. The only axiom which is not satisfied is the localityaxiom: we have not imposed any constraints on the behavior of the O[Φ] as Φ→ 0.

15.4. Local observables

So far, we have defined a cochain complex Obsq(M) of global observables on thewhole manifold M. If U ⊂ M is an open subset of M, we would like to isolate thoseobservables which are “supported on U”.

The idea is to say that an observable O ∈ Obsq(M) is supported on U if, for sufficientlysmall parametrices, O[Φ] is supported on U. The precise definition is as follows.

15.4.0.4 Definition. An observable O ∈ Obsq(M) is supported on U if, for each (i, k) ∈ Z≥0 ×Z≥0, there exists a compact subset K ⊂ Uk and a parametrix Φ, such that for all parametricesΨ ≤ Φ

Supp Oi,k[Ψ] ⊂ K.

Remark: Recall that Oi,k[Φ] : E ⊗k → C is the kth term in the Taylor expansion of thecoefficient of hi of the functional O[Φ] ∈ O(E )[[h]]. ♦

Remark: As always, the definition works over an arbitrary nilpotent dg manifold (X, A ),even though we suppress this from the notation. In this generality, instead of a compactsubset K ⊂ Uk we require K ⊂ Uk × X to be a set such that the map K → X is proper. ♦

We let Obsq(U) ⊂ Obsq(M) be the sub-graded vector space of observables supportedon U.

15.4.0.5 Lemma. Obsq(U) is a sub-cochain complex of Obsq(M). In other words, if O ∈Obsq(U), then so is QO.

PROOF. The only thing that needs to be checked is the support condition. We need tocheck that, for each (i, k), there exists a compact subset K of Uk such that, for all sufficientlysmall Φ, QOi,k[Φ] is supported on K.

Note that we can write

QOi,k[Φ] = QOi,k[Φ] + ∑a+b=i

r+s=k+2

Ia,r[Φ], Ob,s[Φ]Φ + ∆ΦOi−1,k+2[Φ].

We now find a compact subset K for QOi,k[Φ]. We know that, for each (i, k) and for allsufficiently small Φ, Oi,k[Φ] is supported on K, where K is some compact subset of Uk. Itfollows that QOi,k[Φ] is supported on K.


By making K bigger, we can assume that for sufficiently small Φ, Oi−1,k+2[Φ] is sup-ported on L, where L is a compact subset of Uk+2 whose image in Uk, under every projec-tion map, is in K. This implies that ∆ΦOi−1,k+2[Φ] is supported on K.

The locality condition for the effective actions I[Φ] implies that, by choosing Φ tobe sufficiently small, we can make Ii,k[Φ] supported as close as we like to the small di-agonal in Mk. It follows that, by choosing Φ to be sufficiently small, the support ofIa,r[Φ], Ob,s[Φ]Φ can be taken to be a compact subset of Uk. Since there are only a finitenumber of terms like Ia,r[Φ], Ob,s[Φ]Φ in the expression for (QO)i,k[Φ], we see that forΦ sufficiently small, (QO)i,k[Φ] is supported on a compact subset K of Uk, as desired.

15.4.0.6 Lemma. Obsq(U) has a natural structure of differentiable pro-cochain complex space.

PROOF. Our general strategy for showing that something is a differentiable vectorspace is to ensure that everything works in families over an arbitrary nilpotent dg mani-fold (X, A ). Thus, suppose that the theory we are working with is defined over (X, A ).If Y is a smooth manifold, we say a smooth map Y → Obsq(U) is an observable forthe family of theories over (X × Y, A ⊗πC∞(Y)) obtained by base-change along the mapX×Y → X (so this family of theories is constant over Y).

The filtration on Obsq(U) (giving it the structure of pro-differentiable vector space) isinherited from that on Obsq(M). Precisely, an observable O ∈ Obsq(U) is in Fk Obsq(U)if, for all parametrices Φ,

O[Φ] ∈∏ hi Sym≥(2k−i) E ∨.

The renormalization group flow WΨΦ preserves this filtration.

So far we have verified that Obsq(U) is a pro-object in the category of pre-differentiablecochain complexes. The remaining structure we need is a flat connection

∇ : C∞(Y, Obsq(U))→ Ω1(Y, Obsq(U))

for each manifold Y, where C∞(Y, Obsq(U)) is the space of smooth maps Y → Obsq(U).

This flat connection is equivalent to giving a differential on

Ω∗(Y, Obsq(U)) = C∞(Y, Obsq(U))⊗C∞(Y) Ω∗(Y)

making it into a dg module for the dg algebra Ω∗(Y). Such a differential is provided byconsidering observables for the family of theories over the nilpotent dg manifold (X ×Y, A ⊗πΩ∗(Y)), pulled back via the projection map X×Y → Y.

15.5. LOCAL OBSERVABLES FORM A PREFACTORIZATION ALGEBRA 305

15.5. Local observables form a prefactorization algebra

At this point, we have constructed the cochain complex Obsq(M) of global observ-ables of our factorization algebra. We have also constructed, for every open subset U ⊂M, a sub-cochain complex Obsq(U) of observables supported on U.

In this section we will see that the local quantum observables Obsq(U) of a quantumfield on a manifold M form a prefactorization algebra.

The definition of local observables makes it clear that they form a pre-cosheaf: thereare natural injective maps of cochain complexes

Obsq(U)→ Obsq(U′)

if U ⊂ U′ is an open subset.

Let U, V be disjoint open subsets of M. The structure of prefactorization algebra onthe local observables is specified by the pre-cosheaf structure mentioned above, and abilinear cochain map

Obsq(U)×Obsq(V)→ Obsq(U qV).

These product maps need to be maps of cochain complexes which are compatible withthe pre-cosheaf structure and with reordering of the disjoint opens. Further, they need tosatisfy a certain associativity condition which we will verify.

15.5.1. Defining the product map. Suppose that O ∈ Obsq(U) and O′ ∈ Obsq(V)are observables on U and V respectively. Note that O[Φ] and O′[Φ] are elements of thecochain complex

ObsqΦ(M) =

(O(E )[[h]], QΦ

)which is a BD algebra and so a commutative algebra (ignoring the differential, of course).(The commutative product is simply the usual product of functions on E .) In the defini-tion of the prefactorization product, we will use the product of O[Φ] and O′[Φ] taken inthe commutative algebra O(E). This product will be denoted O[Φ] ∗O′[Φ] ∈ O(E).

Recall (see definition 15.3.1.1) that we defined a linear renormalization group flowoperator WΨ

Φ , which is an isomorphism between the cochain complexes ObsqΦ(M) and

ObsqΨ(M).

The main result of this section is the following.

15.5.1.1 Theorem. For all observables O ∈ Obsq(U), O′ ∈ Obsq(V), where U and V aredisjoint, the limit

limΨ→0

WΦΨ(O[Ψ] ∗O′[Ψ]

)∈ O(E )[[h]]


exists. Further, this limit satisfies the renormalization group equation, so that we can define anobservable m(O, O′) by

m(O, O′)[Φ] = limΨ→0

WΦΨ(O[Ψ] ∗O′[Ψ]

).

The map

Obsq(U)×Obsq(V) 7→ Obsq(U qV)

O×O′ 7→ m(O, O′)

is a smooth bilinear cochain map, and it makes Obsq into a prefactorization algebra in the multi-category of differentiable pro-cochain complexes.

PROOF. We will show that, for each i, k, the Taylor term

WΨΦ (O[Φ] ∗O′[Φ])i,k : E ⊗k → C

is independent of Φ for Φ sufficiently small. This will show that the limit exists.

Note thatWΨ

Γ

(WΓ

Φ(O[Φ] ∗O′[Φ]

))= WΨ

Φ(O[Φ] ∗O′[Φ]

).

Thus, to show that the limit limΦ→0 WΨΦ (O[Φ] ∗O′[Φ]) is eventually constant, it suffices

to show that, for all sufficiently small Φ, Γ satisfying Φ < Γ,

WΓΦ(O[Φ] ∗O′[Φ])i,k = (O[Γ] ∗O′[Γ])i,k.

This turns out to be an exercise in the manipulation of Feynman diagrams. In order toprove this, we need to recall a little about the Feynman diagram expansion of WΓ

Φ(O[Φ]).(Feynman diagram expansions of the renormalization group flow are discussed exten-sively in [Cos11c].)

We have a sum of the form

WΓΦ(O[Φ])i,k = ∑

G

1|Aut(G)|wG (O[Φ]; I[Φ]; P(Γ)− P(Φ)) .

The sum is over all connected graphs G with the following decorations and properties.

(1) The vertices v of G are labelled by an integer g(v) ∈ Z≥0, which we call the genusof the vertex.

(2) The first Betti number of G, plus the sum of over all vertices of the genus g(v),must be i (the “total genus”).

(3) G has one special vertex.(4) G has k tails (or external edges).

The weight wG (O[Φ]; I[Φ]; P(Γ)− P(Φ)) is computed by the contraction of a collection ofsymmetric tensors. One places O[Φ]r,s at the special vertex, when that vertex has genus r

15.5. LOCAL OBSERVABLES FORM A PREFACTORIZATION ALGEBRA 307

and valency s; places I[Φ]g,v at every other vertex of genus g and valency v; and puts thepropagator P(Γ)− P(Φ) on each edge.

Let us now consider WΓΦ(O[Φ] ∗O′[Φ]). Here, we a sum over graphs with one special

vertex, labelled by O[Φ] ∗ O′[Φ]. This is the same as having two special vertices, oneof which is labelled by O[Φ] and the other by O′[Φ]. Diagrammatically, it looks like wehave split the special vertex into two pieces. When we make this maneuver, we introducepossibly disconnected graphs; however, each connected component must contain at leastone of the two special vertices.

Let us now compare this to the graphical expansion of

O[Γ] ∗O′[Γ] = WΓΦ(O[Φ]) ∗WΓ

Φ(O′[Φ]).

The Feynman diagram expansion of the right hand side of this expression consists ofgraphs with two special vertices, labelled by O[Φ] and O′[Φ] respectively (and, of course,any number of other vertices, labelled by I[Φ], and the propagator P(Γ)− P(Φ) labellingeach edge). Further, the relevant graphs have precisely two connected components, eachof which contains one of the special vertices.

Thus, we see that

WΓΦ(O[Φ] ∗O′[Φ])−WΓ

Φ(O[Φ]) ∗WΓΦ(O

′[Φ]).

is a sum over connected graphs, with two special vertices, one labelled by O[Φ] and theother by O′[Φ]. We need to show that the weight of such graphs vanish for Φ, Γ sufficientlysmall, with Φ < Γ.

Graphs with one connected component must have a chain of edges connecting thetwo special vertices. (A chain is a path in the graph with no repeated vertices or edges.)For a graph G with “total genus” i and k tails, the length of any such chain is bounded by2i + k.

It is important to note here that we require a non-special vertex of genus zero to havevalence at least three and a vertex of genus one to have valence at least one. See [Cos11c]for more discussion. If we are considering a family of theories over some dg ring, wedo allow bivalent vertices to be accompanied by nilpotent parameters in the base ring;nilpotence of the parameter forces there to be a global upper bound on the number ofbivalent vertices that can appear. The argument we are presenting works with minormodifications in this case too.

Each step along a chain of edges involves a tensor with some support that dependson the choice of parametrices Phi and Γ. As we move from the special vertex O towardthe other O′, we extend the support, and our aim is to show that we can choose Φ and Γto be small enough so that the support of the chain, excluding O′[Φ], is disjoint from the


support of O′[Φ]. The contraction of a distribution and function with disjoint supports iszero, so that the weight will vanish. We now make this idea precise.

Let us choose arbitrarily a metric on M. By taking Φ and Γ to be sufficiently small, wecan assume that the support of the propagator on each edge is within ε of the diagonal inthis metric, and ε can be taken to be as small as we like. Similarly, the support of the Ir,s[Γ]labelling a vertex of genus r and valency s can be taken to be within cr,sε of the diagonal,where cr,s is a combinatorial constant depending only on r and s. In addition, by choosingΦ to be small enough we can ensure that the supports of O[Φ] and O′[Φ] are disjoint.

Now let G′ denote the graph G with the special vertex for O′ removed. This graph cor-responds to a symmetric tensor whose support is within some distance CGε of the smalldiagonal, where CG is a combinatorial constant depending on the graph G′. As the sup-ports K and K′ (of O and O′, respectively) have a finite distance d between them, we canchoose ε small enough that CGε < d. It follows that, by choosing Φ and Γ to be sufficientlysmall, the weight of any connected graph is obtained by contracting a distribution and afunction which have disjoint support. The graph hence has weight zero.

As there are finitely many such graphs with total genus i and k tails, we see that wecan choose Γ small enough that for any Φ < Γ, the weight of all such graphs vanishes.

Thus we have proved the first part of the theorem and have produced a bilinear map

Obsq(U)×Obsq(V)→ Obsq(U qV).

It is a straightforward to show that this is a cochain map and satisfies the associativityand commutativity properties necessary to define a prefactorization algebra. The factthat this is a smooth map of differentiable pro-vector spaces follows from the fact thatthis construction works for families of theories over an arbitrary nilpotent dg manifold(X, A ).

15.6. Local observables form a factorization algebra

We have seen how to define a prefactorization algebra Obsq of observables for ourquantum field theory. In this section we will show that this prefactorization algebra isin fact a factorization algebra. In the course of the proof, we show that modulo h, thisfactorization algebra is isomorphic to Obscl .

15.6.0.2 Theorem. (1) The prefactorization algebra Obsq of quantum observables is, in fact,a factorization algebra.

(2) Further, there is an isomorphism

Obsq⊗C[[h]]C∼= Obscl

15.6. LOCAL OBSERVABLES FORM A FACTORIZATION ALGEBRA 309

between the reduction of the factorization algebra of quantum observables modulo h, andthe factorization algebra of classical observables.

15.6.1. Proof of the theorem. This theorem will be a corollary of a more technicalproposition.

15.6.1.1 Proposition. For any open subset U ⊂ M, filter Obsq(U) by saying that the k-th filteredpiece Gk Obsq(U) is the sub C[[h]]-module consisting of those observables which are zero modulohk. Note that this is a filtration by sub prefactorization algebras over the ring C[[h]].

Then, there is an isomorphism of prefactorization algebras (in differentiable pro-cochain com-plexes)

Gr Obsq ' Obscl ⊗CC[[h]].This isomorphism makes Gr Obsq into a factorization algebra.

Remark: We can give Gk Obsq(U) the structure of a pro-differentiable cochain complex,as follows. The filtration on Gk Obsq(U) that defines the pro-structure is obtained byintersecting Gk Obsq(U) with the filtration on Obsq(U) defining the pro-structure. Thenthe inclusion Gk Obsq(U) → Obsq(U) is a cofibration of differentiable pro-vector spaces(see definition B.6.0.7). ♦

PROOF OF THE THEOREM, ASSUMING THE PROPOSITION. We need to show that forevery open U and for every Weiss cover U, the natural map

(†) C(U, Obsq)→ Obsq(U)

is a quasi-isomorphism of differentiable pro-cochain complexes.

The basic idea is that the filtration induces a spectral sequence for both C(U, Obsq) andObsq(U), and we will show that the induced map of spectral sequences is an isomorphismon the first page. Because we are working with differentiable pro-cochain complexes,this is a little subtle. The relevant statements about spectral sequences in this context aredeveloped in this context in Appendix B.

Note that C(U, Obsq) is filtered by C(U, Gk Obsq). The map (†) preserves the filtrations.Thus, we have a maps of inverse systems

C(U, Obsq /Gk Obsq)→ Obsq(U)/Gk Obsq(U).

These inverse systems satisfy the properties of Appedix B, lemma B.6.0.11. Further, it isclear that

Obsq(U) = lim←−Obsq(U)/Gk Obsq(U).

We also haveC(U, Obsq) = lim←− C(U, Obsq /Gk Obsq).


This equality is less obvious, and uses the fact that the Cech complex is defined using thecompleted direct sum as described in Appendix B, section B.6.

Using lemma B.6.0.11, we need to verify that the map

C(U, Gr Obsq)→ Gr Obsq(U)

is an equivalence. This follows from the proposition because Gr Obsq is a factorizationalgebra.

PROOF OF THE PROPOSITION. The first step in the proof of the proposition is the fol-lowing lemma.

15.6.1.2 Lemma. Let Obsq(0) denote the prefactorization algebra of observables which are only

defined modulo h. Then there is an isomorphism

Obsq(0) ' Obscl

of differential graded prefactorization algebras.

PROOF OF LEMMA. Let O ∈ Obscl(U) be a classical observable. Thus, O is an elementof the cochain complex O(E (U)) of functionals on the space of fields on U. We need toproduce an element of Obsq

(0) from O. An element of Obsq(0) is a collection of functionals

O[Φ] ∈ O(E ), one for every parametrix Φ, satisfying a classical version of the renormal-ization group equation and an axiom saying that O[Φ] is supported on U for sufficientlysmall Φ.

Given an elementO ∈ Obscl(U) = O(E (U)),

we define an elementO[Φ] ∈ Obsq

(0)

by the formulaO[Φ] = lim

Γ→0WΦ

Γ (O) modulo h.

The Feynman diagram expansion of the right hand side only involves trees, since we areworking modulo h. As we are only using trees, the limit exists. The limit is defined bya sum over trees with one special vertex, where each edge is labelled by the propagatorP(Φ), the special vertex is labelled by O, and every other vertex is labelled by the classicalinteraction I0 ∈ Oloc(E ) of our theory.

The mapObscl(U)→ Obsq

(0)(U)

we have constructed is easily seen to be a map of cochain complexes, compatible with thestructure of prefactorization algebra present on both sides. (The proof is a variation on


the argument in section 11, chapter 5 of [Cos11c], about the scale 0 limit of a deformationof the effective interaction I modulo h.)

A simple inductive argument on the degree shows this map is an isomorphism.

Because the construction works over an arbitrary nilpotent dg manifold, it is clear thatthese maps are maps of differentiable cochain complexes.

The next (and most difficult) step in the proof of the proposition is the followinglemma. We use it to work inductively with the filtration of quantum observables.

Let Obsq(k) denote the prefactorization algebra of observables defined modulo hk+1.

15.6.1.3 Lemma. For all open subsets U ⊂ M, the natural quotient map of differentiable pro-cochain complexes

Obsq(k+1)(U)→ Obsq

(k)(U)

is a fibration of differentiable pro-cochain complexes (see Appendix B, Definition B.6.0.7 for thedefinition of a fibration). The fiber is isomorphic to Obscl(U).

PROOF OF LEMMA. We give the set (i, k) ∈ Z≥0 ×Z≥0 the lexicographical ordering,so that (i, k) > (r, s) if i > r or if i = r and k > s.

We will let Obsq≤(i,k)(U) be the quotient of Obsq

(i) consisting of functionals

O[Φ] = ∑(r,s)≤(i,k)

hrO(r,s)[Φ]

satisfying the renormalization group equation and locality axiom as before, but whereO(r,s)[Φ] is only defined for (r, s) ≤ (i, k). Similarly, we will let Obsq

<(i,k)(U) be the quotientwhere the O(r,s)[Φ] are only defined for (r, s) < (i, k).

We will show that the quotient map

q : Obsq≤(i,k)(U)→ Obsq

<(i,k)(U)

is a fibration. The result will follow.

Recall what it means for a map f : V → W of differentiable cochain complexes to bea fibration. For X a manifold, let C∞

X (V) denote the sheaf of cochain complexes on X ofsmooth maps to V. We say f is a fibration if for every manifold X, the induced map ofsheaves C∞

X (V) → C∞X (W) is surjective in each degree. Equivalently, we require that for

all smooth manifolds X, every smooth map X →W lifts locally on X to a map to V.

Now, by definition, a smooth map from X to Obsq(U) is an observable for the constantfamily of theories over the nilpotent dg manifold (X, C∞(X)). Thus, in order to show q is


a fibration, it suffices to show the following. For any family of theories over a nilpotentdg manifold (X, A ), any open subset U ⊂ M, and any observable α in the A -moduleObsq

<(i,k)(U), we can lift α to an element of Obsq≤(i,k)(U) locally on X.

To prove this, we will first define, for every parametrix Φ, a map

LΦ : Obsq<(i,k)(U)→ Obsq

≤(i,k)(M)

with the property that the composed map

Obsq<(i,k)(U)

LΦ−→ Obsq≤(i,k)(M)→ Obsq

<(i,k)(M)

is the natural inclusion map. Then, for every observable O ∈ Obsq<(i,k)(U), we will show

that LΦ(O) is supported on U, for sufficiently small parametrices Φ, so that LΦ(O) pro-vides the desired lift.

ForO ∈ Obsq

<(i,k)(U),

we defineLΦ(O) ∈ Obsq

≤(i,k)(M)

by

LΦ(O)r,s[Φ] =

Or,s[Φ] if (r, s) < (i, k)0 if (r, s) = (i, k)

.

For Ψ 6= Φ, we obtain LΦ(O)r,s[Ψ] by the renormalization group flow from LΦ(O)r,s[Φ].The RG flow equation tells us that if (r, s) < (i, k), then

LΦ(O)r,s[Ψ] = Or,s[Ψ].

However, the RG equation for LΦ(O)r,s is non-trivial and tells us that

Ii,k[Ψ] + δ (LΦ(O)i,k[Ψ]) = Wi,k (P(Ψ)− P(Φ), I[Φ] + δO[Φ])

for δ a square-zero parameter of cohomological degree opposite to that of O.

To complete the proof of this lemma, we prove the required local lifting property inthe sublemma below.

15.6.1.4 Sub-lemma. For each O ∈ Obsq<(i,k)(U), we can find a parametrix Φ — locally over

the parametrizing manifold X — so that LΦO lies in Obsq≤(i,k)(U) ⊂ Obsq

≤(i,k)(M).

PROOF. Although the observables Obsq form a factorization algebra on the manifoldM, they also form a sheaf on the parametrizing base manifold X. That is, for every opensubset V ⊂ X, let Obsq(U) |V denote the observables for our family of theories restrictedto V. In other words, Obsq(U) |V denotes the sections of this sheaf Obsq(U) on V.


The map LΦ constructed above is then a map of sheaves on X.

For every observable O ∈ Obsq<(i,k)(U), we need to find an open cover

X =⋃α

Yα

of X, and on each Yα a parametrix Φα (for the restriction of the family of theories to Yα)such that

LΦα(O |Yα) ∈ Obsq

≤(i,k)(U) |Yα.

More informally, we need to show that locally in X, we can find a parametrix Φ such thatfor all sufficiently small Ψ, the support of LΦ(O)(i,k)[Ψ] is in a subset of Uk × X whichmaps properly to X.

This argument resembles previous support arguments (e.g., the product lemma fromsection 15.5). The proof involves an analysis of the Feynman diagrams appearing in theexpression

(?) LΦ(O)i,k[Ψ] = ∑γ

1|Aut(γ)|wγ (O[Φ]; I[Φ]; P(Ψ)− P(Φ)) .

The sum is over all connected Feynman diagrams of genus i with k tails. The edges arelabelled by P(Ψ)− P(Φ). Each graph has one special vertex, where O[Φ] appears. Moreexplicitly, if this vertex is of genus r and valency s, it is labelled by Or,s[Φ]. Each non-special vertex is labelled by Ia,b[Φ], where a is the genus and b the valency of the vertex.Note that only a finite number of graphs appear in this sum.

By assumption, O is supported on U. This means that there exists some parametrixΦ0 and a subset K ⊂ U × X mapping properly to X such that for all Φ < Φ0, Or,s[Φ] issupported on Ks. (Here by Ks ⊂ Us × X we mean the fibre product over X.)

Further, each Ia,b[Φ] is supported as close as we like to the small diagonal M × X inMk × X. We can find precise bounds on the support of Ia,b[Φ], as explained in section14.2. To describe these bounds, let us choose metrics for X and M. For a parametrix Φsupported within ε of the diagonal M× X in M×M× X, the effective interaction Ia,b[Φ]is supported within (2a + b)ε of the diagonal.

(In general, if A ⊂ Mn × X, the ball of radius ε around A is defined to be the union ofthe balls of radius ε around each fibre Ax of A → X. It is in this sense that we mean thatIa,b[Φ] is supported within (2a + b)ε of the diagonal.)

Similarly, for every parametrix Ψ with Ψ < Φ, the propagator P(Ψ) − P(Φ) is sup-ported within ε of the diagonal.

In sum, there exists a set K ⊂ U × X, mapping properly to X, such that for all ε > 0,there exists a parametrix Φε, such that


(1) O[Φε]r,s is supported on Ks for all (r, s) < (i, k).(2) Ia,b[Φε] is supported within (2a + b)ε of the small diagonal.(3) For all Ψ < Φε, P(Ψ)− P(Φε) is supported within ε of the small diagonal.

The weight wγ of a graph in the graphical expansion of the expression (?) above (usingthe parametrices Φε and any Ψ < Φε) is thus supported in the ball of radius cε aroundKk (where c is some combinatorial constant, depending on the number of edges and ver-tices in γ). There are a finite number of such graphs in the sum, so we can choose thecombinatorial constant c uniformly over the graphs.

Since K ⊂ U × X maps properly to X, locally on X, we can find an ε so that the closedball of radius cε is still inside Uk × X. This completes the proof.

15.7. The map from theories to factorization algebras is a map of presheaves

In [Cos11c], it is shown how to restrict a quantum field theory on a manifold M to anyopen subset U of M. Factorization algebras also form a presheaf in an obvious way. Inthis section, we will prove the following result.

15.7.0.5 Theorem. The map from the simplicial set of theories on a manifold M to the ∞-groupoidof factorization algebras on M extends to a map of simplicial presheaves.

The proof of this will rely on the results we have already proved, and in particular onthefact that observables form a factorization algebra.

As a corollary, we have the following very useful result.

15.7.0.6 Corollary. For every open subset U ⊂ M, there is an isomorphism of graded differen-tiable vector spaces

Obsq(U) ∼= Obscl(U)[[h]].

Note that what we have proved already is that there is a filtration on Obsq(U) whoseassociated graded is Obscl(U)[[h]]. This result shows that this filtration is split as a filtra-tion of differentiable vector spaces.

PROOF. By the theorem, Obsq(U) can be viewed as global observables for the fieldtheory obtained by restricting our field theory on M to one on U. Choosing a parametrixon U allows one to identify global observables with Obscl(U)[[h]], with differential d +I[Φ],−Φ + h4Φ. This is an isomorphism of differentiable vector spaces.

15.7. THE MAP FROM THEORIES TO FACTORIZATION ALGEBRAS IS A MAP OF PRESHEAVES 315

The proof of this theorem is a little technical, and uses the same techniques we havediscussed so far. Before we explain the proof of the theorem, we need to explain how torestrict theories to open subsets.

Let E (M) denote the space of fields for a field theory on M. In order to relate fieldtheories on U and on M, we need to relate parametrices on U and on M. If

Φ ∈ E (M)⊗βE (M)

is a parametrix on M (with proper support as always), then the restriction

Φ |U∈ E (U)⊗βE (U)

of Φ to U may no longer be a parametrix. It will satisfy all the conditions required to be aparametrix except that it will typically not have proper support.

We can modify Φ |U so that it has proper support, as follows. Let K ⊂ U be a compactset, and let f be a smooth function on U ×U with the following properties:

(1) f is 1 on K× K.(2) f is 1 on a neighbourhood of the diagonal.(3) f has proper support.

Then, f Φ |U does have proper support, and further, f Φ |U is equal to Φ on K× K.

Conversely, given any parametrix Φ on U, there exists a parametrix Φ on M suchthat Φ and Φ agree on K. One can construct Φ by taking any parametrix Ψ on M, andobserving that, when restricted to U, Ψ and Φ differ by a smooth section of the bundleE E on U ×U.

We can then choose a smooth section σ of this bundle on U ×U such that f has com-pact support and σ = Ψ−Φ on K× K. Then, we let Φ = Ψ− f .

Let us now explain what it means to restrict a theory on M to one on U. Then we willstate the theorem that there exists a unique such restriction.

Fix a parametrix Φ on U. Let K ⊂ U be a compact set, and consider the compact set

Ln = (Supp Φ∗)nK ⊂ U.

Here we are using the convolution construction discussed earlier, whereby the collectionof proper subsets of U ×U acts on that of compact sets in U by convolution. Thus, Ln isthe set of those x ∈ U such that there exists a sequence (y0, . . . , yn) where (yi, yi+1) is inSupp Φ, yn ∈ K and y0 = x.

15.7.0.7 Definition. Fix a theory on M, specified by a collection I[Ψ] of effective interactions.Then a restriction of I[Ψ] to U consists of a collection of effective interactions IU [Φ] with the


following propery. For every parametrix Φ on U, and for all compact sets K ⊂ U, let Ln ⊂ U beas above.

Let Φn be a parametrix on M with the property that

Φn = Φ on Ln × Ln.

Then we require that

IUi,k[Φ](e1, . . . , ek) = Ii,k[Φn](e1, . . . , ek)

where ei ∈ Ec(U) have support on K, and where n ≥ 2i + k.

This definition makes sense in families with obvious modifications.

15.7.0.8 Theorem. Any theory I[Ψ] on M has a unique restriction on U.

This restriction map works in families, and so defines a map of simplicial sets from the simpli-cial set of theories on M to that on U.

In this way, we have a simplicial presheaf T on M whose value on U is the simplicial set oftheories on U (quantizing a given classical theory). This simplicial presheaf is a homotopy sheaf,meaning that it satisfies Cech descent.

PROOF. It is obvious that the restriction, it if exists, is unique. Indeed, we have speci-fied each IU

i,k[Φ] for every Φ and for every compact subset K ⊂ U. Since each IUi,k[Φ] must

have compact support on Uk, it is determined by its behaviour on compact sets of theform Kk.

In [Cos11c], a different definition of restriction was given, defined not in terms ofgeneral parametrices but in terms of those defined by the heat kernel. One therefore needsto check that the notion of restriction defined in [Cos11c] coincides with the one discussedin this theorem. This is easy to see by a Feynman diagram argument similar to the oneswe discussed earlier. The statement that the simplicial presheaf of theories satisfies Cechdescent is proved in [Cos11c].

Now here is the main theorem in this section.

15.7.0.9 Theorem. The map which assigns to a field theory the corresponding factorization alge-bra is a map of presheaves. Further, the map which assigns to an n-simplex in the simplicial set oftheories, a factorization algebra over Ω∗(4n), is also a map of presheaves.

Let us explain what this means concretely. Consider a theory on M and let ObsqM

denote the corresponding factorization algebra. Let ObsqU denote the factorization algebra


for the theory restricted to U, and let ObsqM |U denote the factorization algebra Obsq

Mrestricted to U (that is, we only consider open subsets contained in M). Then there is acanonical isomorphism of factorization algebras on U,

ObsqU∼= Obsq

M |U .

In addition, this construction works in families, and in particular in families over Ω∗(4n).

PROOF. Let V ⊂ U be an open set whose closure in U is compact. We will first con-struct an isomorphism of differentiable cochain complexes

ObsqM(V) ∼= Obsq

U(V).

Later we will check that this isomorphism is compatible with the product structures. Fi-nally, we will use the codescent properties for factorization algebras to extend to an iso-morphism of factorization algebras defined on all open subsets V ⊂ U, and not just thosewhose closure is compact.

Thus, let V ⊂ U have compact closure, and let O ∈ ObsqM(V). Thus, O is something

which assigns to every parametrix Φ on M a collection of functionals Oi,k[Φ] satisfyingthe renormalization group equation and a locality axiom stating that for each i, k, thereexists a parametrix Φ0 such that Oi,k[Φ] is supported on V for Φ ≤ Φ0.

We want to construct from such an observable a collection of functionals ρ(O)i,k[Ψ],one for each parametrix Ψ on U, satisfying the RG flow on U and the same locality axiom.It suffices to do this for a collection of parametrices which include parametrices which arearbitrarily small (that is, with support contained in an arbitrarily small neighbourhood ofthe diagonal in U ×U).

Let L ⊂ U be a compact subset with the property that V ⊂ Int L. Choose a functionf on U ×U which is 1 on a neighbourhood of the diagonal, 1 on L× L, and has propersupport. If Ψ is a parametrix on M, we let Ψ f be the parametrix on U obtained by multi-plying the restriction of Ψ to U×U by f . Note that the support of Ψ f is a subset of that ofΨ.

The construction is as follows. Choose (i, k). We define

ρ(O)r,s[Ψ f ] = Or,s[Ψ]

for all (r, s) ≤ (i, k) and all Ψ sufficiently small. We will not spell out what we meanby sufficiently small, except that it in particular means it is small enough so that Or,s[Ψ]is supported on V for all (r, s) ≤ (i, k). The value of ρ(O)r,s for other parametrices isdetermined by the RG flow.

To check that this construction is well-defined, we need to check that if we take someparametrix Ψ on M which is also sufficiently small, then the ρ(O)r,s[Ψ f ] and ρ(O)r,s[Ψ f ]


are related by the RG flow for observables for the theory on U. This RG flow equa-tion relating these two quantities is a sum over connected graphs, with one vertex la-belled by ρ(O)[Ψ f ], all other vertices labelled by IU [Ψ f ], and all internal edges labelledby P(Ψ f )− P(Ψ f ). Since we are only considering (r, s) ≤ (i, k) only finitely many graphscan appear, and the number of internal edges of these graphs is bounded by 2i + k. Weare assuming that both Ψ and Ψ are sufficiently small so that Or,s[Ψ] and Or,s[Ψ] havecompact support on V. Also, by taking Ψ sufficiently small, we can assume that IU [Ψ]

has support arbitrarily close to the diagonal. It follows that, if we choose both Ψ and Ψ tobe sufficiently small, there is a compact set L′ ⊂ U containing V such that the weight ofeach graph appearing in the RG flow is zero if one of the inputs (attached to the tails) hassupport on the complement of L′. Further, by taking Ψ and Ψ sufficiently small, we canarrange so that L′ is as small as we like, and in particular, we can assume that L′ ⊂ Int L(where L is the compact set chosen above).

Recall that the weight of a Feynman diagram involves pairing quantities attached toedges with multilinear functionals attached to vertices. A similar combinatorial analy-sis tells us that, for each vertex in each graph appearing in this sum, the inputs to themultilinear functional attached to the vertex are all supported in L′.

Now, for Ψ sufficiently small, we have

IUr,s[Ψ

f ](e1, . . . , es) = Ir,s[Ψ](e1, . . . , es)

if all of the ei are supported in L′. (This follows from the definition of the restriction of atheory. Recall that IU indicates the theory on U and I indicates the theory on M).

It follows that, in the sum over diagrams computing the RG flow, we get the sameanswer if we label the vertices by I[Ψ] instead of IU [Ψ f ]. The RG flow equation nowfollows from that for the original observable O[Ψ] on M.

The same kind of argument tells us that if we change the choice of compact set L ⊂ Uwith V ⊂ Int L, and if we change the bumb function f we chose, the map

ρ : ObsqM(V)→ Obsq

U(V)

does not change.

A very similar argument also tells us that this map is a cochain map. It is immediatethat ρ is an isomorphism, and that it commutes with the maps arising from inclusionsV ⊂ V ′.

We next need to verify that this map respects the product structure. Recall that theproduct of two observables O, O′ in V, V ′ is defined by saying that ([Ψ]O′[Ψ])r,s is simplythe naive product in the symmetric algebra Sym∗ E !

c (V qV ′) for (r, s) ≤ (i, k) (some fixed(i, k)) and for Ψ sufficiently small.


Since, for (r, s) ≤ (i, k) and for Ψ sufficiently small, we defined

ρ(O)r,s[Ψ f ] = Or,s[Ψ],

we see immediately that ρ respects products.

Thus, we have constructed an isomorphism

ObsqM |U∼= Obsq

U

of prefactorization algebras on U, where we consider open subsets in U with compactclosure. We need to extend this to an isomorphism of factorization algebras. To do this,we use the following property: for any open subset W ⊂ U,

ObsqU(W) = colim

V⊂WObsq

U(V)

where the colimit is over all open subsets with compact closure. (The colimit is taken,of course, in the category of filtered differentiable cochain complexes, and is simply thenaive and not homotopy colimit). The same holds if we replace Obsq

U by ObsqM. Thus we

have constructed an isomorphism

ObsqU(W) ∼= Obsq

M(W)

for all open subsets W. The associativity axioms of prefactorization algebras, combinedwith the fact that Obsq(W) is a colimit of Obsq(V) for V with compact closure and the factthat the isomorphisms we have constructed respect the product structure for such opensubsets V, implies that we have constructed an isomorphism of factorization algebras onU.

CHAPTER 16

Further aspects of quantum observables

16.1. Translation-invariant factorization algebras from translation-invariant quantumfield theories

In this section, we will show that a translation-invariant quantum field theory on Rn

gives rise to a smoothly translation-invariant factorization algebra on Rn (see section 4.7).We will also show that a holomorphically translation-invariant field theory on Cn givesrise to a holomorphically translation-invariant factorization algebra.

16.1.1. First, we need to define what it means for a field theory to be translation-invariant. Let us consider a classical field theory on Rn. Recall that this is given by

(1) A graded vector bundle E whose sections are E ;(2) An antisymmetric pairing E⊗ E→ DensRn ;(3) A differential operator Q : E → E making E into an elliptic complex, which is

skew-self adjoint;(4) A local action functional I ∈ Oloc(E ) satisfying the classical master equation.

A classical field theory is translation-invariant if

(1) The graded bundle E is translation-invariant, so that we are given an isomor-phism between E and the trivial bundle with fibre E0.

(2) The pairing, differential Q, and local functional I are all translation-invariant.

It takes a little more work to say what it means for a quantum field theory to betranslation-invariant. Suppose we have a translation-invariant classical field theory, equippedwith a translation-invariant gauge fixing operator QGF. As before, a quantization of sucha field theory is given by a family of interactions I[Φ] ∈ Osm,P(E ), one for each parametrixΦ.

16.1.1.1 Definition. A translation-invariant quantization of a translation-invariant classical fieldtheory is a quantization with the property that, for all translation-invariant parametrices Φ, I[Φ]is translation-invariant.

321

322 16. FURTHER ASPECTS OF QUANTUM OBSERVABLES

Remark: In general, in order to give a quantum field theory on a manifold M, we do notneed to give an effective interaction I[Φ] for all parametrices. We only need to specifyI[Φ] for a collection of parametrices such that the intersection of the supports of Φ is thesmall diagonal M ⊂ M2. The functional I[Ψ] for all other parametrices Ψ is defined bythe renormalization group flow. It is easy to construct a collection of translation-invariantparametrices satisfying this condition. ♦

16.1.1.2 Proposition. The factorization algebra associated to a translation-invariant quantumfield theory is smoothly translation-invariant (see section 4.7 in Chapter 4 for the definition).

PROOF. Let Obsq denote the factorization algebra of quantum observables for ourtranslation-invariant theory. An observable supported on U ⊂ Rn is defined by a familyO[Φ] ∈ O(E )[[h]], one for each translation-invariant parametrix, which satisfiese the RGflow and (in the sense we explained in section 15.4) is supported on U for sufficientlysmall parametrices. The renormalization group flow

WΦΨ : O(E )[[h]]→ O(E )[[h]]

for translation-invariant parametrices Ψ, Φ commutes with the action of Rn on O(E ) bytranslation, and therefore acts on Obsq(Rn). For x ∈ Rn and U ⊂ Rn, let TxU denote thex-translate of U. It is immediate that the action of x ∈ Rn on Obsq(Rn) takes Obsq(U) ⊂Obsq(Rn) to Obsq(TxU). It is not difficult to verify that the resulting map

Obsq(U)→ Obsq(TxU)

is an isomorphism of differentiable pro-cochain complexes and that it is compatible withthe structure of a factorization algebra.

We need to verify the smoothness hypothesis of a smoothly translation-invariant fac-torization algebra. This is the following. Suppose that U1, . . . , Uk are disjoint open subsetsof Rn, all contained in an open subset V. Let A′ ⊂ Rnk be the subset consisting of thosex1, . . . , xk such that the closures of Txi Ui remain disjoint and in V. Let A be the connectedcomponent of 0 in A′. We need only examine the case where A is non-empty.

We need to show that the composed map

mx1,...,xk : Obsq(U1)× · · · ×Obsq(Uk)→Obsq(Tx1U1)× · · · ×Obsq(Txk Uk)→ Obsq(V)

varies smoothly with (x1, . . . , xk) ∈ A. In this diagram, the first map is the product of thetranslation isomorphisms Obsq(Ui) → Obsq(Txi Ui), and the second map is the productmap of the factorization algebra.

The smoothness property we need to check says that the map mx1,...,xk lifts to a multi-linear map of differentiable pro-cochain complexes

Obsq(U1)× · · · ×Obsq(Uk)→ C∞(A, Obsq(V)),

16.1. TRANSLATION INVARIANCE 323

where on the right hand side the notation C∞(A, Obsq(V)) refers to the smooth maps fromA to Obsq(V).

This property is local on A, so we can replace A by a smaller open subset if necessary.

Let us assume (replacing A by a smaller subset if necessary) that there exist open sub-sets U′i containing Ui, which are disjoint and contained in V and which have the propertythat for each (x1, . . . , xk) ∈ A, Txi Ui ⊂ U′i .

Then, we can factor the map mx1,...,xk as a composition

(†) Obsq(U1) × · · · × Obsq(Uk)ix1×···×ixk−−−−−−→ Obsq(U′1) × · · · × Obsq(U′k) → Obsq(V).

Here, the map ixi : Obsq(Ui)→ Obsq(U′i ) is the composition

Obsq(Ui)→ Obsq(Txi Ui)→ Obsq(U′i )

of the translation isomorphism with the natural inclusion map Obsq(Txi Ui) → Obsq(U′i ).The second map in equation (†) is the product map associated to the disjoint subsetsU′1, . . . , U′k ⊂ V.

By possibly replacing A by a smaller open subset, let us assume that A = A1 × · · · ×Ak, where the Ai are open subsets of Rn containing the origin. It remains to show that themap

ixi : Obsq(Ui)→ Obsq(U′i )

is smooth in xi, that is, extends to a smooth map

Obsq(Ui)→ C∞(Ai, Obsq(U′i )).

Indeed, the fact that the product map

m : Obsq(U′1)× · · · ×Obsq(U′k)→ Obsq(V)

is a smooth multilinear map implies that, for every collection of smooth maps αi : Yi →Obsq(U′i ) from smooth manifolds Yi, the resulting map

Y1 × · · · ×Yk → Obsq(V)

(y1, . . . yk) 7→ m(α1(y), . . . , αk(y))

is smooth.

Thus, we have reduced the result to the following statement: for all open subsets A ⊂Rn and for all U ⊂ U′ such that TxU ⊂ U′ for all x ∈ A, the map ix : Obsq(U)→ Obsq(U′)is smooth in x ∈ A.

But this statement is tractable. Let

O ∈ Obsq(U) ⊂ Obsq(U′) ⊂ Obsq(Rn)


be an observable. It is obvious that the family of observables TxO, when viewed as ele-ments of Obsq(Rn), depends smoothly on x. We need to verify that it depends smoothlyon x when viewed as an element of Obsq(U′).

This amounts to showing that the support conditions which ensure an observable isin Obsq(U′) hold uniformly for x in compact sets in A.

The fact that O is in Obsq(U) means the following. For each (i, k), there exists a com-pact subset K ⊂ U and ε > 0 such that for all translation-invariant parametrices Φ sup-ported within ε of the diagonal and for all (r, s) ≤ (i, k) in the lexicographical ordering,the Taylor coefficient Or,s[Φ] is supported on Ks.

We need to enlarge K to a subset L ⊂ U′ × A, mapping properly to A, such that TxOis supported on L in this sense (again, for (r, s) ≤ (i, k)). Taking L = K× A, embedded inU′ × A by

(k, x) 7→ (Txk, x)suffices.

Remark: Essentially the same proof will give us the somewhat stronger result that for anymanifold M with a smooth action of a Lie group G, the factorization algebra correspond-ing to a G-equivariant field theory on M is smoothly G-equivariant. ♦

16.2. Holomorphically translation-invariant theories and their factorization algebras

Similarly, we can talk about holomorphically translation-invariant classical and quan-tum field theories on Cn. In this context, we will take our space of fields to be Ω0,∗(Cn, V),where V is some translation-invariant holomorphic vector bundle on Cn. The pairingmust arise from a translation-invariant map of holomorphic vector bundles

V ⊗V → KCn

of cohomological degree n− 1, where KCn denotes the canonical bundle. This means thatthe composed map

Ω0,∗c (Cn, V)⊗2 → Ω0,∗

c (Cn, KCn)

∫−→ C

is of cohomological degree −1.

Let

ηi =∂

∂zi∨− : Ω0,k(Cn, V)→ Ω0,k−1(Cn, V)

be the contraction operator. The cohomological differential operator Q on Ω0,∗(V) mustbe of the form

Q = ∂ + Q0

where Q0 is translation-invariant and satisfies the following conditions:

16.2. HOLOMORPHICALLY TRANSLATION-INVARIANT THEORIES AND THEIR FACTORIZATION ALGEBRAS325

(1)(2) Q0 (and hence Q) must be skew self-adjoint with respect to the pairing on Ω0,∗

c (Cn, V).(3) We assume that Q0 is a purely holomorphic differential operator, so that we can

write Q0 as a finite sum

Q0 = ∑∂

∂zI µI

where µI : V → V are linear maps of cohomological degree 1. (Here we are usingmulti-index notation). Note that this implies that

[Q0, ηi] = 0,

for i = 1, . . . , n. In terms of the µI , the adjointness condition says that µI is skew-symmetric if |I| is even and symmetric if |I| is odd.

The other piece of data of a classical field theory is the local action functional I ∈ Oloc(Ω0,∗(Cn, V)).We assume that I is translation-invariant, of course, but also that

ηi I = 0

for i = 1 . . . n, where the linear map ηi on Ω0,∗(Cn, V) is extended in the natural way to aderivation of the algebra O(Ω0,∗

c (Cn, V)) preserving the subspace of local functionals.

Any local functional I on Ω0,∗(Cn, V) can be written as a sum of functionals of theform

φ 7→∫

Cndz1 . . . dzn A(D1φ . . . Dkφ)

where A : V⊗k → C is a linear map, and each Di is in the space

C

[dzi, ηi,

∂

∂zi,

∂

∂zi

].

(Recall that ηi indicates ∂∂dzi

). The condition that ηi I = 0 for each i means that we onlyconsider those Di which are in the subspace

C

[ηi,

∂

∂zi,

∂

∂zi

].

In other words, as a differential operator on the graded algebra Ω0,∗(Cn), each Di hasconstant coefficients.

It turns out that, under some mild hypothesis, any such action functional I is equiva-lent (in the sense of the BV formalism) to one which has only zi derivatives, and no zi ordzi derivatives.

16.2.0.3 Lemma. Suppose that Q = ∂, so that Q0 = 0. Then, any interaction I satisfying theclassical master equation and the condition that ηi I = 0 for i = 1, . . . , n is equivalent to one onlyinvolving derivatives in the zi.


PROOF. Let E = Ω0,∗(Cn, V) denote the space of fields of our theory, and let Oloc(E )denote the space of local functionals on E . Let Oloc(E )hol denote those functions which aretranslation-invariant and are in the kernel of the operators ηi, and let Oloc(E )hol′ denotethose which in addition have only zi derivatives. We will show that the inclusion map

Oloc(E )hol′ → Oloc(E )hol

is a quasi-isomorphism, where both are equipped with just the ∂ differential. Both sidesare graded by polynomial degree of the local functional, so it suffices to show this for localfunctionals of a fixed degree.

Note that the space V is filtered, by saying that Fi consists of those elements of degrees≥ i. This induces a filtration on E by the subspaces Ω0,∗(Cn, FiV). After passing to theassociated graded, the operator Q becomes ∂. By considering a spectral sequence withrespect to this filtration, we see that it suffices to show we have a quasi-isomorphism inthe case Q = ∂.

But this follows immediately from the fact that the inclusion

C

[∂

∂zi

]→ C

[∂

∂zi,

∂

∂zi, ηi

]

is a quasi-isomorphism, where the right hand side is equipped with the differential [∂,−].To see that this map is a quasi-isomorphism, note that the ∂ operator sends ηi to ∂

∂zi.

Recall that the action functional I induces the structure of L∞ algebra on Ω0,∗(Cn, V)[−1]whose differential is Q, and whose L∞ structure maps are encoded by the Taylor compo-nents of I. Under the hypothesis of the previous lemma, this L∞ algebra is L∞ equivalentto one which is the Dolbeault complex with coefficients in a translation-invariant local L∞algebra whose structure maps only involve zi derivatives.

There are many natural examples of holomorphically translation-invariant classicalfield theories. Geometrically, they arise from holomorphic moduli problems. For instance,one could take the cotangent theory to the derived moduli of holomorphic G bundles onCn, or the cotangent theory to the derived moduli space of such bundles equipped withholomorphic sections of some associated bundles, or the cotangent theory to the moduliof holomorphic maps from Cn to some complex manifold.

As is explained in great detail in [], holomorphically translation-invariant field theo-ries arise very naturally in physics as holomorphic (or minimal) twists of supersymmetricfield theories in even dimensions.


16.2.1. A holomorphically translation invariant classical theory on Cn has a naturalgauge fixing operator, namely

∂∗= −∑ ηi

∂

∂zi.

Since [ηi, Q0] = 0, we see that [Q, ∂∗] = [∂, ∂

∗] is the Laplacian. (More generally, we can

consider a family of gauge fixing operators coming from the ∂∗

operator for a family offlat Hermitian metrics on Cn. Since the space of such metrics is GL(n, C)/U(n) and thuscontractible, we see that everything is independent up to homotopy of the choice of gaugefixing operator.)

We say a translation-invariant parametrix

Φ ∈ Ω0,∗(Cn, V)⊗2

is holomorphically translation-invariant if

(ηi ⊗ 1 + 1⊗ ηi)Φ = 0

for i = 1, . . . , n. For example, if Φ0 is a parametrix for the scalar Laplacian

4 = −∑∂

∂zi

∂

∂zi

then

Φ0

n

∏i=1

d(zi − wi)c

defines such a parametrix. Here zi and wi indicate the coordinates on the two copies ofCn, and c ∈ V ⊗V is the inverse of the pairing on v. Clearly, we can find holomorphicallytranslation-invariant parametrices which are supported arbitrarily close to the diagonal.This means that we can define a field theory by only considering I[Φ] for holomorphicallytranslation-invariant parametrices Φ.

16.2.1.1 Definition. A holomorphically translation-invariant quantization of a holomorphicallytranslation-invariant classical field theory as above is a translation-invariant quantization suchthat for each holomorphically translation-invariant parametrix Φ, the effective interaction I[Φ]satisfies

ηi I[Φ] = 0

for i = 1, . . . , n. Here ηi abusively denotes the natural extension of the contraction ηi to a deriva-tion on O(Ω0,∗

c (Cn, V)).

The usual obstruction theory arguments hold for constructing holomorphically-translationinvariant quantizations. At each order in h, the obstruction-deformation complex is thesubcomplex of the complex Oloc(E )Cn

of translation-invariant local functionals which arealso in the kernel of the operators ηi.


16.2.1.2 Proposition. A holomorphically translation-invariant quantum field theory on Cn leadsto a holomorphically translation-invariant factorization algebra.

PROOF. This follows immediately from proposition 16.1.1.2. Indeed, quantum ob-servables form a smoothly translation-invariant factorization algebra. Such an observ-able O on U is specified by a family O[Φ] ∈ O(Ω0,∗(Cn, V)) of functionals defined foreach holomorphically translation-invariant parametrix Φ, which are supported on U forΦ sufficiently small. The operators ∂

∂zi, ∂

∂zi, ηi act in a natural way on O(Ω0,∗(Cn, V)) by

derivations, and each commutes with the renormalization group flow WΦΨ for holomor-

phically translation-invariant parametrices Ψ, Φ. Thus, ∂∂zi

, ∂∂zi

and ηi define derivations ofthe factorization algebra Obsq. Explicitly, if O ∈ Obsq(U) is an observable, then for eachholomorphically translation-invariant parametrix Φ,(

∂

∂ziO)[Φ] =

∂

∂zi(O[Φ]),

and similarly for ∂∂zi

and ηi.

By definition (Definition 5.1.1.1), a holomorphically translation-invariant factorizationalgebra is a translation-invariant factorization algebra where the derivation operator ∂

∂zion observables is homotopically trivialized.

Note that, for a holomorphically translation-invariant parametrix Φ, [ηi,4Φ] = 0 andηi is a derivation for the Poisson bracket −,−Φ. It follows that

[Q + I[Φ],−Φ + h4Φ, ηi] = [Q, ηi]

as operators on O(Ω0,∗(Cn, V)). Since we wrote Q = ∂+Q0 and required that [Q0, ηi] = 0,we have

[Q, ηi] = [∂, ηi] =∂

∂zi.

Since the differential on Obsq(U) is defined by

(QO)[Φ] = QO[Φ] + I[Φ], O[Φ]Φ + h4ΦO[Φ],

we see that [Q, ηi] =∂

∂zi, as desired.

As we showed in Chapter 5, a holomorphically translation invariant factorization al-gebra in one complex dimension, with some mild additional conditions, gives rise to avertex algebra. Let us verify that these conditions hold in the examples of interest. Wefirst need a definition.

16.2.1.3 Definition. A holomorphically translation-invariant field theory on C is S1-invariant ifthe following holds. First, we have an S1 action on the vector space V, inducing an action of S1

on the space E = Ω0,∗(C, V) of fields, by combining the S1 action on V with the natural one


on Ω0,∗(C) coming from rotation on C. . We suppose that all the structures of the field theoryare S1-invariant. More precisely, the symplectic pairing on E and the differential Q on E mustbe S1-invariant. Further, for every S1-invariant parametrix Φ, the effective interaction I[Φ] isS1-invariant.

16.2.1.4 Lemma. Suppose we have a holomorphically translation invariant field theory on C

which is also S1-invariant. Then, the corresponding factorization algebra satisfies the conditionsstated in theorem 5.2.2.1 of Chapter 5 allowing us to construct a vertex algebra structure on thecohomology.

PROOF. Let F denote the factorization algebra of observables of our theory. Note thatif U ⊂ C is an S1-invariant subset, then S1 acts on F (U).

Recall that F is equipped with a complete decreasing filtration, and is viewed as afactorization algebra valued in pro-differentiable cochain complexes. Recall that we needto check the following properties.

(1) The S1 action on F (D(0, r)) extends to a smooth action of the algebra D(S1) ofdistributions on S1.

(2) Let Grm F (D(0, r)) denote the associated graded with respect to the filtration onF (D(0, r)). Let Grm

k F (D(0, r)) refer to the kth S1-eigenspace in Grm F (D(0, r)).Then, we require that the map

Grmk F (D(0, r))→ Grm

k F (D(0, r′))

is a quasi-isomorphism of differentiable vector spaces.(3) The differentiable vector space H∗(Grm

k F (D(0, r))) is finite-dimensional for all kand is zero for k 0.

Let us first check that the S1 action extends to a D(S1)-action. If λ ∈ S1 let ρ∗λ denotethis action. We need to check that for any observable O[Φ] and for every distributionD(λ) on S1 the expression ∫

λ∈S1D(λ)ρ∗λO[Φ]

makes sense and defines another observable. Further, this construction must be smoothin both D(λ) and the observable O[Φ], meaning that it must work families.

For fixed Φ, each Oi,k[Φ] is simply a distribution on Ck with some coefficients. Forany distribution α on Ck, the expression

∫λ D(λ)ρ∗λα makes sense and is continuous in

both α and the distributionD. Indeed,∫

λ D(λ)ρ∗λα is simply the push-forward map indistributions applied to the action map S1 ×Ck → Ck.


It follows that, for each distribution D on S1, we can define

D ∗Oi,k[Φ] :=∫

λ∈S1D(λ)ρ∗λOi,k[Φ].

As a function of D and Oi,k[Φ], this construction is smooth. Further, sending an observableO[Φ] to D ∗O[Φ] commutes with the renormalization group flow (between S1-equivariantparametrices). It follows that we can define a new observable D ∗O by

(D ∗O)i,k[Φ] = D ∗ (Oi,k[Φ]).

Now, a family of observables Ox (parametrized by x ∈ M, a smooth manifold) is smoothif and only if the family of functionals Ox

i,k[Φ] are smooth for all i, k and all Φ. In factone need not check this for all Φ, but for any collection of parametrices which includesarbitrarily small parametrices. If follows that the map sending D and O to D ∗O is smooth,that is, takes smooth families to smooth families.

Let us now check the remaining assumptions of theorem 5.2.2.1. Let F denote thefactorization algebra of quantum observables of the theory and let Fk denote the ktheigenspace of the S1 action. We first need to check that the inclusion

Grmk F (D(0, r))→ Grm

k F (D(0, r′))

is a quasi-isomorphism for r < r′. We need it to be a quasi-isomorphism of completedfiltered differentiable vector spaces. The space Grm F (D(0, r)) is a finite direct sum ofspaces of the form

Ω0,∗c (D(0, r)l , Vl)Sl .

It thus suffices to check that for the map

Ω0,∗c (D(0, r)m)→ Ω

0,∗c (D(0, r′)m)

is a quasi-isomorphism on each S1-eigenspace. This is immediate.

The same holds to check that the cohomology of Grmk F (D(0, r)) is zero for k 0 and

that it is finite-dimensional as a differentiable vector space.

We have seen that any S1-equivariant and holmorphically translation-invariant factor-ization algebra on C gives rise to a vertex algebra. We have also seen that the obstruction-theory method applies in this situation to construct holomorphically translation invariantfactorization algebras from appropriate Lagrangians. In this way, we have a very generalmethod for constructing vertex algebras.

16.3. Renormalizability and factorization algebras

A central concept in field theory is that of renormalizability. This is discussed in detailin [Cos11c]. The basic idea is the following.

16.3. RENORMALIZABILITY AND FACTORIZATION ALGEBRAS 331

The group R>0 acts on the collection of field theories on Rn, where the action is in-duced from the scaling action of R>0 on Rn. This action is implemented differently in dif-ferent models for field theories. In the language if factorization algebras it is very simple,because any factorization algebra on Rn can be pushed forward under any diffeomor-phism of Rn to yield a new factorization algebra on Rn. Push-forward of factorizationalgebras under the map x 7→ λ−1x (for λ ∈ R>0) defines the renormalization group flowon factorization algebra.

We will discuss how to implement this rescaling in the definition of field theory givenin [Cos11c] shortly. The main result of this section is the statement that the map which as-signs to a field theory the corresponding factorization algebra of observables intertwinesthe action of R>0.

Acting by elements λ ∈ R>0 on a fixed quantum field theory produces a one-parameterfamily of theories, depending on λ. Let F denote a fixed theory, either in the language offactorization algebras, the language of [Cos11c], or any other approach to quantum fieldtheory. We will call this family of theories ρλ(F). We will view the theory ρλ(F) as be-ing obtained from F by “zooming in” on Rn by an amount dicated by λ, if λ < 1, or byzooming out if λ > 1.

We should imagine the theory F as having some number of continuous parameters,called coupling constants. Classically, the coupling constants are simply constants ap-pearing next to various terms in the Lagrangian. At the quantum level, we could thinkof the structure constants of the factorization algebra as being functions of the couplingconstants (we will discuss this more precisely below).

Roughly speaking, a theory is renormalizable if, as λ → 0, the family of theories ρλ(F)converges to a limit. While this definition is a good one non-pertubatively, in perturbationtheory it is not ideal. The reason is that often the coupling constants depend on the scalethrough quantities like λh. If h was an actual real number, we could analyze the behaviourof λh for λ small. In perturbation theory, however, h is a formal parameter, and we mustexpand λh in a series of the form 1 + h log λ + . . . . The coefficients of this series alwaysgrow as λ→ 0.

In other words, from a perturbative point of view, one can’t tell the difference betweena theory that has a limit as λ→ 0 and a theory whose coupling constants have logarithmicgrowth in λ.

This motivates us to define a theory to be perturbatively renormalizable if it has logarith-mic growth as λ → 0. We will introduce a formal definition of perturbative renormaliz-ability shortly. Let us first indicate why this definition is important.

It is commonly stated (especially in older books) that perturbative renormalizabilityis a necessary condition for a theory to exist (in perturbation theory) at the quantum level.


This is not the case. Instead, renormalizability is a criterion which allows one to select afinite-dimensional space of well-behaved quantizations of a given classical field theory,from a possibly infinite dimensional space of all possible quantizations.

There are other criteria which one wants to impose on a quantum theory and whichalso help select a small space of quantizations: for instance, symmetry criteria. (In ad-dition, one also requires that the quantum master equation holds, which is a strong con-straint. This, however, is part of the definition of a field theory that we use). There areexamples of non-renormalizable field theories for which nevertheless a unique quantiza-tion can be selected by other criteria. (An example of this nature is BCOV theory).

16.3.1. The renormalization group action on factorization algebras. Let us now dis-cuss the concept of renormalizability more formally. We will define the action of the groupR>0 on the set of theories in the definition used in [Cos11c], and on the set of factorizationalgebras on Rn. We will see that the map which assigns a factorization algebra to a theoryis R>0-equivariant.

Let us first define the action of R>0 on the set of factorization algebras on Rn.

16.3.1.1 Definition. If F is a factorization algebra on Rn, and λ ∈ R>0, let ρλ(F ) denote thefactorization algebra on Rn which is the push-forward ofF under the diffeomorphism λ−1 : Rn →Rn given by multiplying by λ−1. Thus,

ρλ(F )(U) = F (λ(U))

and the product maps in ρλ(F ) arise from those in F . We will call this action of R>0 on thecollection of factorization algebras on Rn the local renormalization group action.

Thus, the action of R>0 on factorization algebras on Rn is simply the obvious actionof diffeomorphisms on Rn on factorization algebras on Rn.

16.3.2. The renormalization group flow on classical theories. The action on field the-ories as defined in [Cos11c] is more subtle. Let us start by describing the action of R>0 onclassical field theories. Suppose we have a translation-invariant classical field on Rn, withspace of fields E . The space E is the space of sections of a trivial vector bundle on Rn withfibre E0. The vector space E0 is equipped with a degree −1 symplectic pairing valued inthe line ω0, the fibre of the bundle of top forms on Rn at 0. We also, of course, have atranslation-invariant local functional I ∈ Oloc(E ) satisfying the classical master equation.

Let us choose an action ρ0λ of the group R>0 on the vector space E0 with the property

that the symplectic pairing on E0 is R>0-equivariant, where the action of R>0 acts on theline ω0 with weight −n. Let us further assume that this action is diagonalizable, andthat the eigenvalues of ρ0

λ are rational integer powers of λ. (In practise, only integer orhalf-integer powers appear).


The choice of such an action, together with the action of R>0 on Rn by rescaling,induces an action of R>0 on

E = C∞(Rn)⊗ E0

which sends

φ⊗ e0 7→ φ(λ−1x)ρ0λ(e0),

where φ ∈ C∞(Rn) and e0 ∈ E0. The convention that x 7→ λ−1x means that for small λ,we are looking at small scales (for instance, as λ→ 0 the metric becomes large).

This action therefore induces an action on spaces associated to E , such as the spacesO(E ) of functionals and Oloc(E ) of local functionals. The compatibility between the actionof R>0 and the symplectic pairing on E0 implies that the Poisson bracket on the spaceOloc(E ) of local functionals on E is preserved by the R>0 action. Let us denote the actionof R>0 on Oloc(E ) by ρλ.

16.3.2.1 Definition. The local renormalization group flow on the space of translation-invariantclassical field theories sends a classical action functional I ∈ Oloc(E ) to ρλ(I).

This definition makes sense, because ρλ preserves the Poisson bracket on Oloc(E ).Note also that, if the action of ρ0

λ on E0 has eigenvalues in 1n Z, then the action of ρλ on the

space Oloc(E ) is diagonal and has eigenvalues again in 1n Z.

The action of R>0 on the space of classical field theories up to isomorphism is inde-pendent of the choice of action of R>0 on E0. If we choose a different action, inducinga different action ρ′λ of R>0 on everything, then ρλ I and ρ′λ I are related by a linear andsymplectic change of coordinates on the space of fields which covers the identity on Rn.Field theories related by such a change of coordinates are equivalent.

It is often convenient to choose the action of R>0 on the space E0 so that the quadraticpart of the action is invariant. When we can do this, the local renormalization group flowacts only on the interactions (and on any small deformations of the quadratic part thatone considers). Let us give some examples of the local renormalization group flow onclassical field theories. Many more details are given in [Cos11c].

Consider the free massless scalar field theory on Rn. The complex of fields is

C∞(Rn)D−→ C∞(Rn).

We would like to choose an action of R>0 so that both the symplectic pairing and theaction functional

∫φ D φ are invariant. This action must, of course, cover the action of

R>0 on Rn by rescaling. If φ, ψ denote fields in the copies of C∞(Rn) in degrees 0 and 1


respectively, the desired action sends

ρλ(φ(x)) = λ2−n

2 φ(λ−1x)

ρλ(ψ(x)) = λ−n−2

2 ψ(λ−1x).

Let us then consider how ρλ acts on possible interactions. We find, for example, that if

Ik(φ) =∫

φk

thenρλ(Ik) = λn− k(n−2)

2 Ik.

16.3.2.2 Definition. A classical theory is renormalizable if, as λ → 0, it flows to a fixed pointunder the local renormalization group flow.

For instance, we see that in dimension 4, the most general renormalizable classicalaction for a scalar field theory which is invariant under the symmetry φ 7→ −φ is∫

φ D φ + m2φ2 + cφ4.

Indeed, the φ4 term is fixed by the local renormalization group flow, whereas the φ2 termis sent to zero as λ→ 0.

16.3.2.3 Definition. A classical theory is strictly renormalizable if it is a fixed point under thelocal renormalization group flow.

A theory which is renormalizable has good small-scale behaviour, in that the couplingconstants (classically) become small at small scales. (At the quantum level there may alsobe logaritmic terms which we will discuss shortly). A renormalizable theory may, how-ever, have bad large-scale behaviour: for instance, in four dimensions, a mass term

∫φ2

becomes large at large scales. A strictly renormalizable theory is one which is classicallyscale invariant. At the quantum level, we will define a strictly renormalizable theory tobe one which is scale invariant up to logarithmic corrections.

Again in four dimensions, the only strictly renormalizable interaction for the scalarfield theory which is invariant under φ 7→ −φ is the φ4 interaction. In six dimensions,the φ3 interaction is strictly renormalizable, and in three dimensions the φ6 interaction(together with finitely many other interactions involving derivatives) are strictly renor-malizable.

As another example, recall that the graded vector space of fields of pure Yang-Millstheory (in the first order formalism) is(

Ω0[1]⊕Ω1 ⊕Ω2+ ⊕Ω2

+[−1]⊕Ω3[−1]⊕Ω4[−2])⊗ g.


(Here Ωi indicates forms on R4). The action of R>0 is the natural one, coming from pull-back of forms under the map x 7→ λ−1x. The Yang-Mills action functional

S(A, B) =∫

F(A) ∧ B + B ∧ B

is obviously invariant under the action of R>0, since it only involves wedge product andintegration, as well as projection to Ω2

+. (Here A ∈ Ω1 ⊗ g) and B ∈ Ω2+ ⊗ g). The other

terms in the full BV action functional are also invariant, because the symplectic pairing onthe space of fields and the action of the gauge group are both scale-invariant.

Something similar holds for Chern-Simons theory on R3, where the space of fieldsis Ω∗(R3) ⊗ g[1]. The action of R> 0 is by pull-back by the map x 7→ λ−1x, and theChern-Simons functional is obviously invariant.

16.3.2.4 Lemma. The map which assigns to a translation-invariant classical field theory on Rn

the associated P0 factorization algebra commutes with the action of the local renormalization groupflow.

PROOF. The action of R>0 on the space of fields of the theory induces an action on thespace Obscl(Rn) of classical observables on Rn, by sending an observable O (which is afuncton on the space E (Rn) of fields) to the observable

ρλO : φ 7→ O(ρλ(φ)).

This preserves the Poisson bracket on the subspace Obscl(Rn) of functionals with smooth

first derivative, because by assumption the symplectic pairing on the space of fields isscale invariant. Further, it is immediate from the definition of the local renormalizationgroup flow on classical field theories that

ρλS, O = ρλ(I), ρλ(O)where S ∈ Oloc(E ) is a translation-invariant solution of the classical master equation(whose quadratic part is elliptic).

Let Obsclλ denote the factorization algebra on Rn coming from the theory ρλ(S) (where

S is some fixed classical action). Then, we see that we have an isomorphism of cochaincomplexes

ρλ : Obscl(Rn) ∼= Obsclλ (R

n).We next need to check what this isomorphism does to the support conditions. Let U ⊂ Rn

and let O ∈ Obscl(U) be an observable supported on U. Then, one can check easily thatρλ(O) is supported on λ−1(U). Thus, ρλ gives an isomorphism

Obscl(U) ∼= Obsclλ (λ

−1(U)).

and so,Obscl(λU) ∼= Obscl

λ (U).


The factorization algebraρλ Obscl = λ∗Obscl

assigns to an open set U ⊂ Rn the value of Obscl on λ(U). Thus, we have constructed anisomorphism of precosheaves on Rn,

ρλ Obscl ∼= Obsclλ .

This isomorphism compatible with the commutative product and the (homotopy) Poissonbracket on both side, as well as the factorization product maps.

16.3.3. The renormalization group flow on quantum field theories. The most inter-esting version of the renormalization group flow is, of course, that on quantum fieldtheories. Let us fix a classical field theory on Rn, with space of fields as above E =C∞(Rn)⊗ E0 where E0 is a graded vector space. In this section we will define an action ofthe group R>0 on the simplicial set of quantum field theories with space of fields E , quan-tizing the action on classical field theories that we constructed above. We will show thatthe map which assigns to a quantum field theory the corresponding factorization algebracommutes with this action.

Let us assume, for simplicity, that we have chosen the linear action of R>0 on E0 so thatit leaves invariant a quadratic action functional on E defining a free theory. Let Q : E →E be the corresponding cohomological differential, which, by assumption, is invariantunder the R>0 action. (This step is not necessary, but will make the exposition simpler).

Let us also assume (again for simplicity) that there exists a gauge fixing operator QGF :E → E with the property that

ρλQGFρ−λ = λkQGF

for some k ∈ Q. For example, for a massless scalar field theory on Rn, we have seen that

the action of R>0 on the space C∞(Rn)⊕ C∞(Rn)[−1] of fields sends φ to λ2−n

2 φ(λ−1x)

and ψ to λ−2−n

2 ψ(λ−1x) (where φ is the field of cohomological degree 0 and ψ is thefield of cohomological degree 1). The gauge fixing operator is the identity operator fromC∞(Rn)[−1] to C∞(Rn)[0]. In this case, we have ρλQGFρ−λ = λ2QGF.

As another example, consider pure Yang-Mills theory on R4. The fields, as we havedescribed above, are built from forms on R4, equipped with the natural action of R>0.The gauge fixing operator is d∗. It is easy to see that ρλd∗ρ−λ = λ2d∗. The same holds forChern-Simons theory, which also has a gauge fixing operator defined by d∗ on forms.

A translation-invariant quantum field theory is defined by a family

I[Φ] ∈ O+P,sm(E )Rn

[[h]] | Φ a translation-invariant parametrixwhich satisfies the renormalization group equation, quantum master equation, and thelocality condition. We need to explain how scaling of Rn by R>0 acts on the (simplicial)


set of quantum field theories. To do this, we first need to explain how this scaling actionacts on the set of parametrices.

16.3.3.1 Lemma. If Φ is a translation-invariant parametrix, then λkρλ(Φ) is also a parametrix,where as above k measures the failure of QGF to commute with ρλ.

PROOF. All of the axioms characterizing a parametrix are scale invariant, except thestatement that

([Q, QGF]⊗ 1)Φ = Kid − something smooth.

We need to check that λkρλΦ also satisfies this. Note that

ρλ([Q, QGF]⊗ 1)Φ = λk([Q, QGF]⊗ 1)ρλ(Φ)

since ρλ commutes with Q but not with QGF. Also, ρλ preserves Kid and smooth kernels,so the desired identity holds.

This lemma suggests a way to define the action of the group R>0 on the set of quantumfield theories.

16.3.3.2 Lemma. If I[Φ] is a theory, define Iλ[Φ] by

Iλ[Φ] = ρλ(I[λ−kρ−λ(Φ)]).

Then, the collection of functionals Iλ[Φ] define a new theory.

On the right hand side of the equation in the lemma, we are using the natural actionof ρλ on all spaces associated to E , such as the space E ⊗πE (to define ρ−λ(Φ)) and thespace of functions on E (to define how ρλ acts on the function I[λ−kρ−λ(Φ)]).

Note that this lemma, as well as most things we discuss about renormalizability offield theories which do not involve factorization algebras, is discussed in more detail in[Cos11c], except that there the language of heat kernels is used. We will prove the lemmahere anyway, because the proof is quite simple.

PROOF. We need to check that Iλ[Φ] satisfies the renormalization group equation, lo-cality action, and quantum master equation. Let us first check the renormalization groupflow. As a shorthand notation, let us write Φλ for the parametrix λkρλ(Φ). Then, note thatthe propagator P(Φλ) is

P(Φλ) = ρλP(Φ).

Indeed,

ρλ12 (Q

GF ⊗ 1 + 1⊗QGF)Φ = λk 12 (Q

GF ⊗ 1 + 1⊗QGF)ρλ(Φ)

= P(Φλ).


It follows from this that, for all functionals I ∈ O+P (E )[[h]],

ρλ(W (P(Φ)− P(Ψ), I) = W (P(Φλ)− P(Ψλ), ρλ(I).)

We need to verify the renormalization group equation, which states that

W (P(Φ)− P(Ψ), Iλ[Ψ]) = Iλ[Φ].

Because Iλ[Φ] = ρλ I[Φ−λ], this is equivalent to

ρ−λW (P(Φ)− P(Ψ), ρλ(I[Ψ−λ]) = I[Φ−λ].

Bringing ρ−λ inside the W reduces us to proving the identity

W (P(Φ−λ − P(Ψ−λ, I[Ψ−λ]) = I[Φ−λ]

which is the renormalization group identity for the functionals I[Φ].

The fact that Iλ[Φ] satisfies the quantum master equation is proved in a similar way,using the fact that

ρλ(4Φ I) = 4Φλρλ(I)

where4Φ denotes the BV Laplacian associated to Φ and I is any functional.

Finally, the locality axiom is an immediate consequence of that for the original func-tionals I[Φ].

16.3.3.3 Definition. Define the local renormalization group flow to be the action of R>0 onthe set of theories which sends, as in the previous lemma, a theory I[Φ] to the theory

Iλ[Φ] = ρλ(I[λ−kρ−λΦ]).

Note that this works in families, and so defines an action of R>0 on the simplicial set of theories.

Note that this definition simply means that we act by R>0 on everything involved inthe definition of a theory, including the parametrices.

Let us now quote some results from [Cos11c], concerning the behaviour of this action.Let us recall that to begin with, we chose an action of R>0 on the space E = C∞(R4)⊗ E0of fields, which arose from the natural rescaling action on C∞(R4) and an action on thefinite-dimensional vector space E0. We assumed that the action on E0 is diagonalizable,where on each eigenspace ρλ acts by λa for some a ∈ Q. Let m ∈ Z be such that theexponents of each eigenvalue are in 1

m Z.

16.3.3.4 Theorem. For any theory I[Φ] and any parametrix Φ, the family of functionals Iλ[Φ]depending on λ live in

O+sm,P(E )

[log λ, λ

1m , λ−

1m

][[h]].


In other words, the functionals Iλ[Φ] depend on λ only through polynomials in log λ

and λ±1m . (More precisely, each functional Iλ,i,k[Φ] in the Taylor expansion of Iλ[Φ] has

such polynomial dependence, but as we quantify over all i and k the degree of the poly-nomials may be arbitrarily large).

In [Cos11c], this result is only stated under the hypothesis that m = 2, which is thecase that arises in most examples, but the proof in [Cos11c] works in general.

16.3.3.5 Lemma. The action of R>0 on quantum field theories lifts that on classical field theoriesdescribed earlier.

This basic point is also discussed in [Cos11c]; it follows from the fact that at the clas-sical level, the limit of I[Φ] as Φ→ 0 exists and is the original classical interaction.

16.3.3.6 Definition. A quantum theory is renormalizable if the functionals Iλ[Φ] depend on

λ only by polynomials in log λ and λ1m (where we assume that m > 0). A quantum theory is

strictly renormalizable if it only depends on λ through polynomials in log λ.

Note that at the classical level, a strictly renormalizable theory must be scale-invariant,because logarithmic contributions to the dependence on λ only arise at the quantum level.

16.3.4. Quantization of renormalizable and strictly renormalizable theories. Let usdecompose Oloc(E )R4

, the space of translation-invariant local functionals on E , into eigenspacesfor the action of R>0. For k ∈ 1

m Z, we let O(k)loc (E )R4

be the subspace on which ρλ acts by

λk. Let O(≥0)loc (E )R4

denote the direct sum of all the non-negative eigenspaces.

Let us suppose that we are interested in quantizing a classical theory, given by aninteraction I, which is either strictly renormalizable or renormalizable. In the first case, Iis in O

(0)loc (E )R4

, and in the second, it is in O(≥0)loc (E )R4

.

By our initial assumptions, the Lie bracket on Oloc(E )R4commutes with the action of

R>0. Thus, if we have a strictly renormalizable classical theory, then O(0)loc (E )R4

is a cochaincomplex with differential Q + I−, . This is the cochain complex controlling first-orderdeformations of our classical theory as a strictly renormalizable theory. In physics termi-nology, this is the cochain complex of marginal deformations.

If we start with a classical theory which is simply renormalizable, then the spaceO

(≥0)loc (E )R4

is a cochain complex under the differential Q + I−, . This is the cochaincomplex of renormalizable deformations.


Typically, the cochain complexes of marginal and renormalizable deformations arefinite-dimensional. (This happens, for instance, for scalar field theories in dimensionsgreater than 2).

Here is the quantization theorem for renormalizable and strictly renormalizalble quan-tizations.

16.3.4.1 Theorem. Fix a classical theory on Rn which is renormalizable with classical interactionI. Let R(n) denote the set of renormalizable quantizations defined modulo hn+1. Then, given anyelement inR(n), there is an obstruction to quantizing to the next order, which is an element

On+1 ∈ H1(O

(≥0)loc (E )R4

, Q + I,−)

.

If this obstruction vanishes, then the set of quantizations to the next order is a torsor for H0(O

(≥0)loc (E )R4

).

This statement holds in the simplicial sense too: if R(n)4 denotes the simplicial set of renor-

malizable theories defined modulo hn+1 and quantizing a given classical theory, then there is ahomotopy fibre diagram of simplicial sets

R(n+1)4

//

R(n)4

0 // DK(O

(≥0)loc (E )R4

[1], Q + I,−)

On the bottom right DK indicates the Dold-Kan functor from cochain complexes to simplicial sets.

All of these statements hold for the (simplicial) sets of strictly renormalizable theories quantiz-ing a given strictly renormalizable classical theory, except that we should replace O

(≥0)loc by O

(0)loc

everywhere. Further, all these results hold in families iwth evident modifications.

Finally, if G F denotes the simplicial set of translation-invariant gauge fixing conditions forour fixed classical theory (where we only consider gauge-fixing conditions which scale well withrespect to ρλ as discussed earlier), then the simplicial sets of (strictly) renormalizable theories witha fixed gauge fixing condition are fibres of a simplicial set fibred over GF . As before, this means thatthe simplicial set of theories is independent up to homotopy of the choice of gauge fixing condition.

This theorem is proved in [Cos11c], and is the analog of the quantization theorem fortheories without the renormalizability criterion.

Let us give some examples of how this theorem allows us to construct small-dimensionalfamilies of quantizations of theories where without the renormalizability criterion therewould be an infinite dimensional space of quantizations.


Consider, as above, the massless φ4 theory on R4, with interaction∫

φ D φ + φ4. Atthe classical level this theory is scale-invariant, and so strictly renormalizable. We havethe following.

16.3.4.2 Lemma. The space of strictly-renormalizable quantizations of the massless φ4 theory in4 dimensions which are also invariant under the Z/2 action φ 7→ −φ is isomorphic to hR[[h]].That is, there is a single h-dependent coupling constant.

PROOF. We need to check that the obstruction group for this problem is zero, and thedeformation group is one-dimensional. The obstruction group is zero for degree reasons,because for a theory without gauge symmetry the complex of local functionals is concen-trated in degrees ≤ 0. To compute the deformation group, note that the space of localfunctionals which are scale invariant and invariant under φ 7→ −φ is two-dimensional,spanned by

∫φ4 and

∫φ D φ. The quotient of this space by the image of the differential

Q + I,− is one dimensional, because we can eliminate one of the two possible terms bya change of coordinates in φ.

Let us give another, and more difficult, example.

16.3.4.3 Theorem. The space of renormalizable (or strictly renormalizable) quantizations of pureYang-Mills theory on R4 with simple gauge Lie algebra g is isomorphic to hR[[h]]. That is, thereis a single h-dependent coupling constant.

PROOF. The relevant cohomology groups were computed in [Cos11c], where it wasshown that the deformation group is one dimensional and that the obstruction group isH5(g). The obstruction group is zero unless g = sln and n ≥ 3. By considering theouter automorphisms of sln, it was argued in [Cos11c] that the obstruction must alwaysvanish.

This theorem then tells us that we have an essentially canonical quantization of pureYang-Mills theory on R4, and hence a corresonding factorization algebra.

The following is the main new result of this section.

16.3.4.4 Theorem. The map from translation-invariant quantum theories on Rn to factorizationalgebras on Rn commutes with the local renormalization group flow.

PROOF. Suppose we have a translation-invariant quantum theory on Rn with space offields E and family of effective actions I[Φ]. Recall that the RG flow on theories sendsthis theory to the theory defined by

Iλ[Φ] = ρλ(I[λ−kρ−λ(Φ)]).


We let Φλ = λkρλΦ. As we have seen in the proof of lemma 16.3.3.2, we have

P(Φλ) = ρλ(P(Φ))

4Φλ= ρλ(4Φ).

Suppose that O[Φ] is an observable for the theory I[Φ]. First, we need to show that

Oλ[Φ] = ρλ(O[Φ−λ])

is an observable for the theory Oλ[Φ]. The fact that Oλ[Φ] satisfies the renormalizationgroup flow equation is proved along the same lines as the proof that Iλ[Φ] satisfies therenormalization group flow equation in lemma 16.3.3.2.

If Obsqλ denotes the factorization algebra for the theory Iλ, then we have constructed

a map

Obsq(Rn)→ Obsqλ(R

n)

O[Φ] 7→ Oλ[Φ].The fact that4Φλ

= ρλ(4Φ) implies that this is a cochain map. Further, it is clear that thisis a smooth map, and so a map of differentiable cochain complexes.

Next we need to check is the support condition. We need to show that if O[Φ] isin Obsq(U), where U ⊂ Rn is open, then Oλ[Φ] is in Obsq(λ−1(U)). Recall that thesupport condition states that, for all i, k, there is some parametrix Φ0 and a compact setK ⊂ U such that Oi,k[Φ] is supported in K for all Φ ≤ Φ0.

By making Φ0 smaller if necessary, we can assume that Oi,k[Φλ] is supported on K forΦ ≤ Φ0. (If Φ is supported within ε of the diagonal, then Φλ is supported within λ−1ε.)Then, ρλOi,k[Φλ] will be supported on λ−1K for all Φ ≤ Φ0. This says that Oλ is supportedon λ−1K as desired.

Thus, we have constructed an isomorphism

Obsq(U) ∼= Obsqλ(λ

−1(U)).

This isomorphism is compatible with inclusion maps and with the factorization product.Therefore, we have an isomorphism of factorization algebras

(λ−1)∗Obsq ∼= Obsqλ

where (λ−1)∗ indicates pushforward under the map given by multiplication by λ−1. Sincethe action of the local renormalization group flow on factorization algebras on Rn sendsF to (λ−1)∗F , this proves the result.

The advantage of the factorization algebra formulation of the local renormalizationgroup flow is that it is very easy to define; it captures precisely the intuition that therenormalization group flow arises the action of R>0 on Rn. This theorem shows that

16.4. COTANGENT THEORIES AND VOLUME FORMS 343

the less-obvious definition of the renormalization group flow on theories, as defined in[Cos11c], coincides with the very clear definition in the language of factorization algebras.The advantage of the definition presented in [Cos11c] is that it is possible to computewith this definition, and that the relationship between this definition and how physicistsdefine the β-function is more or less clear. For example, the one-loop β-function (one-loopcontribution to the renormalization group flow) is calculated explicitly for the φ4 theoryin [Cos11c].

16.4. Cotangent theories and volume forms

In this section we will examine the case of a cotangent theory, in which our defini-tion of a quantization of a classical field theory acquires a particularly nice interpretation.Suppose that L is an elliptic L∞ algebra on a manifold M describing an elliptic moduliproblem, which we denote by BL. As we explained in Chapter ??, section 11.6, we canconstruct a classical field theory from L, whose space of fields is E = L[1]⊕L![−2]. Themain observation of this section is that a quantization of this classical field theory can beinterpreted as a kind of “volume form” on the elliptic moduli problem BL. This point ofview was developed in [Cos11b], and used in [Cos11a] to provide a geometric interpreta-tion of the Witten genus.

The relationship between quantization of field theories and volume forms was dis-cussed already at the very beginning of this book, in Chapter 2. There, we explainedhow to interpret (heuristically) the BV operator for a free field theory as the divergenceoperator for a volume form.

While this heuristic interpretation holds for many field theories, cotangent theoriesare a class of theories where this relationship becomes very clean. If we have a cotan-gent theory to an elliptic moduli problem L on a compact manifold, then the L∞ algebraL(M) has finite-dimensional cohomology. Therefore, the formal moduli problem BL(M)is an honest finite-dimensional formal derived stack. We will find that a quantization of acotangent theory leads to a volume form on BL(M) which is of a “local” nature.

Morally speaking, the partition function of a cotangent theory should be the volume ofBL(M) with respect to this volume form. If, as we’ve been doing, we work in perturbationtheory, then the integral giving this volume often does not converge. One has to replaceBL(M) by a global derived moduli space of solutions to the equations of motion to have achance at defining the volume. The volume form on a global moduli space is obtained bydoing perturbation theory near every point and then gluing together the formal volumeforms so obtained near each point.

This program has been successfully carried out in a number of examples, such as[?, GG11, ?]. For example, in [Cos11a], the cotangent theory to the space of holomorphic


maps from an elliptic curve to a complex manifold was studied, and it was shown thatthe partition function (defined in the way we sketched above) is the Witten elliptic genus.

16.4.1. A finite dimensional model. We first need to explain an algebraic interpreta-tion of a volume form in finite dimensions. Let X be a manifold (or complex manifold orsmooth algebraic variety; nothing we will say will depend on which geometric categorywe work in). Let O(X) denote the smooth functions on X, and let Vect(X) denote thevector fields on X.

If ω is a volume form on X, then it gives a divergence map

Divω : Vect(X)→ O(X)

defined via the Lie derivative:Divω(V)ω = LVω

for V ∈ Vect(X). Note that the divergence operator Divω satisfies the equations

(†)Divω( f V) = f Divω V + V( f ).

Divω([V, W]) = V Divω W −W Divω V.

The volume form ω is determined up to a constant by the divergence operator Divω.

Conversely, to give an operator Div : Vect(X) → O(X) satisfying equations (†) is thesame as to give a flat connection on the canonical bundle KX of X, or, equivalently, to givea right D-module structure on the structure sheaf O(X).

16.4.1.1 Definition. A projective volume form on a space X is an operator Div : Vect(X) →O(X) satisfying equations (†).

The advantage of this definition is that it makes sense in many contexts where morestandard definitions of a volume form are hard to define. For example, if A is a quasi-freedifferential graded commutative algebra, then we can define a projective volume form onthe dg scheme Spec A to be a cochain map Der(A)→ A satisfying equations (†). Similarly,if g is a dg Lie or L∞ algebra, then a projective volume form on the formal moduli problemBg is a cochain map C∗(g, g[1])→ C∗(g) satisfying equations (†).

16.4.2. There is a generalization of this notion that we will use where, instead ofvector fields, we take any Lie algebroid.

16.4.2.1 Definition. Let A be a commutative differential graded algebra over a base ring k. A Liealgebroid L over A is a dg A-module with the following extra data.

(1) A Lie bracket on L making it into a dg Lie algebra over k. This Lie bracket will be typicallynot A-linear.

(2) A homomorphism of dg Lie algebras α : L→ Der∗(A), called the anchor map.


(3) These structures are related by the Leibniz rule

[l1, f l2] = (α(l1)( f )) l2 + (−1)|l1|| f | f [l1, l2]

for f ∈ A, li ∈ L.

In general, we should think of L as providing the derived version of a foliation. Inordinary as opposed to derived algebraic geometry, a foliation on a smooth affine schemewith algebra of functions A consists of a Lie algebroid L on A which is projective as anA-module and whose anchor map is fibrewise injective.

16.4.2.2 Definition. If A is a commutative dg algebra and L is a Lie algebroid over A, then anL-projective volume form on A is a cochain map

Div : L→ A

satisfying

Div(al) = a Div l + (−1)|l||a|α(l)a.

Div([l1, l2]) = l1 Div l2 − (−1)|l1||l2|l2 Div l1.

Of course, if the anchor map is an isomorphism, then this structure is the same as aprojective volume form on A. In the more general case, we should think of an L-projectivevolume form as giving a projective volume form on the leaves of the derived foliation.

16.4.3. Let us explain how this definition relates to the notion of quantization of P0algebras.

16.4.3.1 Definition. Give the operad P0 a C× action where the product has weight 0 and thePoisson bracket has weight 1. A graded P0 algebra is a C×-equivariant differential graded algebraover this dg operad.

Note that, if X is a manifold, O(T∗[−1]X) has the structure of graded P0 algebra,where the C× action on O(T∗[−1]X) is given by rescaling the cotangent fibers.

Similarly, if L is a Lie algebroid over a commutative dg algebra A, then Sym∗A L[1] is aC×-equivariant P0 algebra. The P0 bracket is defined by the bracket on L and the L-actionon A; the C× action gives Symk L[1] weight −k.

16.4.3.2 Definition. Give the operad BD over C[[h]] a C× action, covering the C× action onC[[h]], where h has weight −1, the product has weight 0, and the Poisson bracket has weight 1.


Note that this C× action respects the differential on the operad BD, which is definedon generators by

d(− ∗−) = h−,−.Note also that by describing the operad BD as a C×-equivariant family of operads overA1, we have presented BD as a filtered operad whose associated graded operad is P0.

16.4.3.3 Definition. A filtered BD algebra is a BD algebra A with a C× action compatible withthe C× action on the ground ring C[[h]], where h has weight −1, and compatible with the C×

action on BD.

16.4.3.4 Lemma. If L is Lie algebroid over a dg commutative algebra A, then every L-projectivevolume form yields a filtered BD algebra structure on Sym∗A(L[1])[[h]], quantizing the graded P0algebra Sym∗A(L[1]).

PROOF. If Div : L→ A is an L-projective volume form, then it extends uniquely to anorder two differential operator4 on Sym∗A(L[1]) which maps

SymiA(L[1])→ Symi−1

A (L[1]).

Then Sym∗A L[1][[h]], with differential d + h4, gives the desired filtered BD algebra.

16.4.4. Let BL be an elliptic moduli problem on a compact manifold M. The mainresult of this section is that there exists a special kind of quantization of the cotangentfield theory for BL that gives a projective volume on this formal moduli problem BL.Projective volume forms arising in this way have a special “locality” property, reflectingthe locality appearing in our definition of a field theory.

Thus, let L be an elliptic L∞ algebra on M. This gives rise to a classical field theorywhose space of fields is E = L[1]⊕ L![−2], as described in Chapter ??, section 11.6. Letus give the space E a C×-action where L[1] has weight 0 and L![−1] has weight 1. Thisinduces a C× action on all associated spaces, such as O(E ) and Oloc(E ).

This C× action preserves the differential Q + I,− on O(E ), as well as the commu-tative product. Recall (Chapter ??, section 12.2) that the subspace

Obscl(M) = Osm(E ) ⊂ O(E )

of functionals with smooth first derivative has a Poisson bracket of cohomological degree1, making it into a P0 algebra. This Poisson bracket is of weight 1 with respect to the C×

action on Obscl(M), so Obs

cl(M) is a graded P0 algebra.

We are interested in quantizations of our field theory where the BD algebra ObsqΦ(M)

of (global) quantum observables (defined using a parametrix Φ) is a filtered BD algebra.


16.4.4.1 Definition. A cotangent quantization of a cotangent theory is a quantization, given byeffective interaction functionals I[Φ] ∈ O+

sm,P(E )[[h]] for each parametrix Φ, such that I[Φ] is ofweight −1 under the C× action on the space O+

sm,P(E )[[h]] of functionals.

This C× action gives h weight −1. Thus, this condition means that if we expand

I[Φ] = ∑ hi Ii[Φ],

then the functionals Ii[Φ] are of weight i− 1.

Since the fields E = L[1]⊕ L![−2] decompose into spaces of weights 0 and 1 under the C×

action, we see that I0[Φ] is linear as a function of L![−2], that I1[Φ] is a function only of L[1],and that Ii[Φ] = 0 for i > 1.

Remark: (1) The quantization I[Φ] is a cotangent quantization if and only if thedifferential Q+ I[Φ],−Φ + h4Φ preserves the C× action on the space O(E )[[h]]of functionals. Thus, I[Φ] is a cotangent quantization if and only if the BDalgebra Obsq

Φ(M) is a filtered BD algebra for each parametrix Φ.(2) The condition that I0[Φ] is of weight −1 is automatic.(3) It is easy to see that the renormalization group flow

W (P(Φ)− P(Ψ),−)

commutes with the C× action on the space O+sm,P(E )[[h]].

♦

16.4.5. Let us now explain the volume-form interpretation of cotangent quantization.Let L be an elliptic L∞ algebra on M, and let O(BL) = C∗(L) be the Chevalley-Eilenbergcochain complex of M. The cochain complexes O(BL(U)) for open subsets U ⊂ M definea commutative factorization algebra on M.

As we have seen in Chapter ??, section ??, we should interpret modules for an L∞algebra g as sheaves on the formal moduli problem Bg. The g-module g[1] correspondsto the tangent bundle of Bg, and so vector fields on g correspond to the O(Bg)-moduleC∗(g, g[1]).

Thus, we use the notation

Vect(BL) = C∗(L,L[1]);

this is a dg Lie algebra and acts on C∗(L) by derivations (see Appendix B, section B.9, fordetails).


For any open subset U ⊂ M, the L(U)-module L(U)[1] has a sub-module Lc(U)[1]given by compactly supported elements of L(U)[1]. Thus, we have a sub-O(BL(U))-module

Vectc(BL(U)) = C∗(L(U),Lc(U)[1]) ⊂ Vect(BL(U)).

This is in fact a sub-dg Lie algebra, and hence a Lie algebroid over the dg commutativealgebra O(BL(U)). Thus, we should view the subspaceLc(U)[1] ⊂ Lc(U)[1] as providinga foliation of the formal moduli problem BL(U), where two points of BL(U) are in thesame leaf if they coincide outside a compact subset of U.

If U ⊂ V are open subsets of M, there is a restriction map of L∞ algebras L(V) →L(U). The natural extension map Lc(U)[1]→ Lc(V)[1] is a map of L(V)-modules. Thus,by taking cochains, we find a map

Vectc(BL(U))→ Vectc(BL(V)).

Geometrically, we should think of this map as follows. If we have an R-point α of BL(V)for some dg Artinian ring R, then any compactly-supported deformation of the restrictionα |U of α to U extends to a compactly supported deformation of α.

We want to say that a cotangent quantization of L leads to a “local” projective vol-ume form on the formal moduli problem BL(M) if M is compact. If M is compact, thenVectc(BL(M)) coincides with Vect(BL(M)). A local projective volume form on BL(M)should be something like a divergence operator

Div : Vect(BL(M))→ O(BL(M))

satisfying the equations (†), with the locality property that Div maps the subspace

Vectc(BL(U)) ⊂ Vect(BL(M))

to the subspace O(BL(U)) ⊂ O(BL(M)).

Note that a projective volume form for the Lie algebroid Vectc(BL(U)) over O(BL(U))is a projective volume form on the leaves of the foliation of BL(U) given by Vectc(BL(U)).The leaf space for this foliation is described by the L∞ algebra

L∞(U) = L(U)/Lc(U) = colimK⊂U

L(U \ K).

(Here the colimit is taken over all compact subsets of U.) Consider the one-point com-pactification U∞ of U. Then the formal moduli problem L∞(U) describes the germs at ∞on U∞ of sections of the sheaf on U of formal moduli problems given by L.

Thus, the structure we’re looking for is a projective volume form on the fibers of themaps BL(U) → BL∞(U) for every open subset U ⊂ M, where the divergence operatorsdescribing these projective volume forms are all compatible in the sense described above.


What we actually find is something a little weaker. To state the result, recall (section15.2) that we use the notation P for the contractible simplicial set of parametrices, andC P for the cone on P . The vertex of the cone C P will denoted 0.

16.4.5.1 Theorem. A cotangent quantization of the cotangent theory associated to the elliptic L∞algebra L leads to the following data.

(1) A commutative dg algebra OC P(BL) over Ω∗(C P). The underlying graded algebra ofthis commutative dg algebra is O(BL)⊗Ω∗(C P). The restriction of this commutativedg algebra to the vertex 0 of C P is the commutative dg algebra O(BL).

(2) A dg Lie algebroid VectC Pc (BL) over OC P(BL), whose underlying graded OC P(BL)-

module is Vectc(BL)⊗Ω∗(C P). At the vertex 0 of C P , the dg Lie algebroid VectC Pc (BL)

coincides with the dg Lie algebroid Vectc(BL).(3) We let OP(BL) and VectPc (BL) be the restrictions of OC P(BL) and VectC P

c (BL) toP ⊂ C P . Then we have a divergence operator

DivP : VectPc (BL)→ OP(BL)

defining the structure of a VectPc (BL) projective volume form on OP(BL) and VectPc (BL).

Further, when restricted to the sub-simplicial set PU ⊂ P of parametrices with support in asmall neighborhood of the diagonal U ⊂ M×M, all structures increase support by an arbitrarilysmall amount (more precisely, by an amount linear in U, in the sense explained in section 15.2).

PROOF. This follows almost immediately from theorem 15.2.2.1. Indeed, because wehave a cotangent theory, we have a filtered BD algebra

ObsqP(M) =

(O(E )[[h]]⊗Ω∗(P), QP , −,−P

).

Let us consider the sub-BD algebra ObsqP(M), which, as a graded vector space, is Osm(E )[[h]]⊗

Ω∗(P) (as usual, Osm(E ) indicates the space of functionals with smooth first derivative).

Because we have a filtered BD algebra, there is a C×-action on this complex ObsqP(M).

We letOP(BL) = Obs

qP(M)0

be the weight 0 subspace. This is a commutative differential graded algebra over Ω∗(P),whose underlying graded algebra is O(BL); further, it extends (using again the results of15.2.2.1) to a commutative dg algebra OC P(BL) over Ω∗(C P), which when restricted tothe vertex is O(BL).

Next, consider the weight −1 subspace. As a graded vector space, this is

ObsqP(M)−1 = Vectc(BL)⊗Ω∗(P)⊕ hOP(BL).


We thus letVectPc (BL) = Obs

qP(M)−1/hOP(BL).

The Poisson bracket on ObsqP(M) is of weight 1, and it makes the space Obs

qP(M)−1 into

a sub Lie algebra.

We have a natural decomposition of graded vector spaces

ObsqP(M)−1 = VectPc (BL)⊕ hOP(BL).

The dg Lie algebra structure on ObsqP(M)−1 gives us

(1) The structure of a dg Lie algebra on VectPc (BL) (as the quotient of ObsqP(M)−1

by the differential Lie algebra ideal hOP(BL)).(2) An action of VectPc (BL) on OP(BL) by derivations; this defines the anchor map

for the Lie algebroid structure on VectPc (BL).(3) A cochain map

VectPc (BL)→ hOP(BL).This defines the divergence operator.

It is easy to verify from the construction of theorem 15.2.2.1 that all the desired propertieshold.

16.4.6. The general results about quantization of [Cos11c] thus apply to this situa-tion, to show that the following.

16.4.6.1 Theorem. Consider the cotangent theory E = L[1]⊕L![−2] to an elliptic moduli prob-lem described by an elliptic L∞ algebra L on a manifold M.

The obstruction to constructing a cotangent quantization is an element in

H1(Oloc(E )C×) = H1(Oloc(BL)).If this obstruction vanishes, then the simplicial set of cotangent quantizations is a torsor for thesimplicial Abelian group arising from the cochain complex Oloc(BL) by the Dold-Kan correspon-dence.

As in Chapter ??, section 10.5, we are using the notation Oloc(BL) to refer to a “local”Chevalley-Eilenberg cochain for the elliptic L∞ algebra L. If L is the vector bundle whosesections are L, then as we explained in [Cos11c], the jet bundle J(L) is a DM L∞ algebraand

Oloc(BL) = DensM⊗DM C∗red(J(L)).There is a de Rham differential (see section 11.3) mapping Oloc(BL) to the complex of local1-forms,

Ω1loc(BL) = C∗loc(L,L![−1]).


The de Rham differential maps Oloc(BL) isomorphically to the subcomplex of Ω1locBL) of

closed local one-forms. Thus, the obstruction is a local closed 1-form on BL of cohomol-ogy degree 1: it is in

H1(Ω1loc(BL).

Since the obstruction to quantizing the theory is the obstruction to finding a locally-defined volume form on BL, we should view this obstruction as being the local first Chernclass of BL.


So far in this chapter, we have proved the quantization theorem showing that froma field theory we can construct a factorization algebra. We like to think that this factor-ization algebra encodes most things one would want to with a quantum field theory inperturbation theory. To illustrate this, in this section, we will explain how to constructcorrelation frunctions form the factorization algebra, under certain additional hypothesis.

Suppose we have a field theory on a compact manifold M, with space of fields E andlinearized differential Q on the space of fields. Let us suppose that

H∗(E (M), Q) = 0.

This means the following: the complex (E (M), Q) is tangent complex to the formal mod-uli space of solutions to the equation of motion to our field theory, at the base point aroundwhich we are doing perturbation theory. The statement that this tangent complex has nocohomology means that there the trivial solution of the equation of motion has no defor-mations (up to whatever gauge symmetry we have). In other words, we are working withan isolated point in the moduli of solutions to the equations of motion.

As an example, consider a massive interacting scalar field theory on a compact mani-fold M, with action functional for example∫

Mφ(D+m2)φ + φ4

where φ ∈ C∞(M) and m > 0. Then, the complex E (M) of fields is the complex

C∞(M)D+m2

−−−→ C∞(M).

Hodge theory tells us that this complex has no cohomology.

Let Obsq denote the factorization algebra of quantum observables of a quantum fieldtheory which satisfies this (classical) condition.

16.5.0.2 Lemma. In this situation, there is a canonical isomorphism

H∗(Obsq(M)) = C[[h]].


(Note that we usually work, for simplicity, with complex vector spaces; this resultholds where everything is real too, in which case we find R[[h]] on the right hand side).

PROOF. There’s a spectral sequence

H∗(Obscl(M))[[h]] H∗(Obsq(M)).

Further, Obscl(M) has a complete decreasing filtration whose associated graded is thecomplex

Gr Obscl(M) = ∏n

Symn(E (M)∨)

with differential arising from the linear differential Q on E (M). The condition that H∗(E (M), Q) =

0 implies that the cohomology of Symn(E (M)∨) is also zero, so that H∗(Obscl(M)) =C. This shows that there is an isomorphism of C[[h]]-modules from H∗(Obsq(M)) toC[[h]]. To make this isomorphism canonical, we declare that the vacuum observable |∅〉 ∈H0(Obsq(M)) (that is, the unit in the factorization algebra) gets sent to 1 ∈ C[[h]].

16.5.0.3 Definition. As above, let Obsq denote the factorization algebra of obsevables of a QFTon M which satisfies H∗(E (M), Q) = 0.

Let U1, . . . , Un ⊂ M be disjoint open sets, and let Oi ∈ Obsq(Ui). Define the expectationvalue (or correlation function) of the obsevables Oi, denoted by

〈O1, . . . , On〉 ∈ C[[h]],

to be the image of the product observable

O1 ∗ · · · ∗On ∈ H∗(Obsq(M))

under the canonical isomorphism between H∗(Obsq(M)) and C[[h]].

We have already encountered this definition when we discussed free theories (see def-inition 4.6.0.2 in Chapter 4). There we saw that this definition reproduced usual physicsdefinitions of correlation functions for free field theories.

Part 5

Using the machine

CHAPTER 17

Noether’s theorem in classical field theory

Noether’s theorem is a central result in field theory, which states that there is a bijec-tion between symmetries of a field theory and conserved currents. In this chapter we willdevelop a very general version of Noether’s theorem for classical field theories in the lan-guage of factorization algebras. In the following chapter, we will develop the analogoustheorem for quantum field theories.

The statement for classical field theories is the following. Suppose we have a classical

field theory on a manifold M, and let Obscl

denote the P0 factorization algebra of observ-ables of the theory. Suppose that L is a local L∞ algebra on M which acts on our classicalfield theory (we will define precisely what we mean by an action shortly). Let Lc denotethe precosheaf of L∞ algebras on M given by compactly supported section of L. Note that

the P0 structure on Obscl

means that Obscl[−1] is a precosheaf of dg Lie algebras.

The formulation of Noether’s theorem we will prove involves shifted central exten-sions of hte cosheaf Lc of L∞ algebras on M. Such central extensions were discussed in-section 3.6.3; we are interested in −1-shifted central extensions, which fit into short exactsequences

0→ C[−1]→ Lc → Lc → 0,

where C is the constant precosheaf.

The theorem is the following.

Theorem. Suppose that a local L∞ algebra L acts on a classical field theory with observablesObscl . Then, there is a −1-shifted central extension Lc of the precosheaf Lc of L∞ algebras on M,and a map of precosheaves of L∞ algebras

Lc → Obscl[−1]

which, for every open subset U, sends the central element of Lc to the observable 1 ∈ Obscl(U)[−1].

This map is not arbitrary. Rather, it is compatible with the action of the cosheaf Lc onObscl arising from the action Lc on the field theory. Let us explain the form this compati-bility takes.

355

356 17. NOETHER’S THEOREM IN CLASSICAL FIELD THEORY

Note that the dg Lie algebra Obscl(U)[−1] acts on Obscl(U) by the Poisson bracket,

in such a way that the subspace spanned by the observable 1 acts by zero. The L∞ mapwe just discussed therefore gives an action of Lc(U) on Obscl(U), which descends to anaction of Lc(U) because the central element acts by zero.

Theorem. In this situation, the action of Lc(U) coming from the L∞-map Lc → Obscl

and theaction coming from the action of L on the classical field theory coincide up to a homotopy.

. Let us relate this formulation of Noether’s theorem to familiar statements in classicalfield theory. Suppose we have a symplectic manifold X with an action of a Lie algebra gby symplectic vector fields. Let us work locally on X, so that we can assume H1(X) = 0.Then, there is a short exact sequence of Lie algebras

0→ R→ C∞(X)→ SympVect(X)→ 0

where SympVect(X) is the Lie algebra of symplectic vector fields on X, and C∞(X) is aLie algebra under the Poisson bracket.

We can pull back this central extension under the Lie algebra homomorphism g →SympVect(X) to obtain a central extension g of g. This is the analog of the central extensionLc that appeared in our formulation of Noether’s theorem.

The Poisson algebra C∞(X) is observables of the classical field theory. The map g →C∞(X) sends the central element of g to 1 ∈ C∞(X). Further, the action of g on C∞(X)arising from the homomorphism g → C∞(X) coincides with the one arising from theoriginal homomorphism g→ SympVect(X).

Thus, the map g → C∞(X) is entirely analogous to the map that appears in our for-mulation of classical Noether’s theorem. Indeed, we defined a field theory to be a sheaf offormal moduli problems with a−1-shifted symplectic form. The P0 Poisson bracket on theobservables of a classical field theory is analogous to the Poisson bracket on observablesin classical mechanics. Our formulation of Noether’s theorem can be rephrased as sayingthat, after passing to a central extension, an action of a sheaf of Lie algebras by symplecticsymmetries on a sheaf of formal moduli probblems is Hamiltonian.

This similarity is more than just an analogy. After some non-trivial work, one canshow that our formulation of Noether’s theorem, when applied to classical mechanics,yields the statement discussed above about actions of a Lie algebra on a symplectic man-ifold. The key result one needs in order to translate is a result of Nick Rozenblyum [].Observables of classical mechanics form a locally-constant P0 factorization algebra on thereal line, and so (by a theorem of Lurie discussed in section 6.2) an E1 algebra in P0 alge-bras. Rozenblyum shows E1 algebras in P0 algebras are the same as P1, that is ordinaryPoisson, algebras. This allows us to translate the shifted Poisson bracket on the factoriza-tion algebra on R of observable of classical mechanics into the ordinary unshifted Poisson

17.1. SYMMETRIES OF A CLASSICAL FIELD THEORY 357

bracket that is more familiar in classical mechanics, and to translate our formulation ofNoether’s theorem into the statement about Lie algebra actions on symplectic manifoldsdiscussed above.

17.0.1.

17.0.1.1 Theorem. Suppose we have a quantum field theory on a manifold M, which is acted onby a local dg Lie (or L∞) algebra on M. Then, there is a map of factorization algebras on M fromthe twisted factorization envelope (3.6.3) of L to observables of the field theory.

Of course, this is not an arbitrary map; rather, the action of L on observables can berecovered from this map together with the factorization product.

This theorem may seem quite different from Noether’s theorem as it is usually stated.We explain the link between this result and the standard formulation in section ??.

For us, the power of this result is that it gives us a very general method for under-standing quantum observables. The factorization envelope of a local L∞ algebra is a veryexplicit and easily-understood object. By contrast, the factorization algebra of quantumobservables of an interacting field theory is a complicated object which resists explicit de-scription. Our formulation of Noether’s theorem shows us that, if we have a field theorywhich has many symmetries, we can understand explicitly a large part of the factorizationalgebra of quantum observables.

17.1. Symmetries of a classical field theory

We will start our discussion of Noether’s theorem by examining what it means fora homotopy Lie algebra to act on a field theory. We are particularly interested in what itmeans for a local L∞ algebra to act on a classical field theory. Recall ?? that a local L∞ algebraL is a sheaf of L∞ algebras which is the sheaf of sections of a graded vector bundle L, andwhere the L∞-structure maps are poly-differential operators.

We know from chapter ?? that a perturbative classical field theory is described byan elliptic moduli problem on M with a degree −1 symplectic form. Equivalently, it isdescribed by a local L∞ algebra M on M equipped with an invariant pairing of degree−3. Therefore, an action of L on M should be an L∞ action of L on M. Thus, the firstthing we need to understand is what it means for one L∞ algebra to act on the other.

17.1.1. Actions of L∞ algebras. If g, h are ordinary Lie algebras, then it is straightfor-ward to say what it means for g to act on h. If g does act on h, then we can define thesemi-direct product gn h. This semi-direct product lives in a short exact sequence of Lie


algebras0→ h→ gn h→ g→ 0.

Further, we can recover the action of g on h from the data of a short exact sequence of Liealgebras like this.

We will take this as a model for the action of one L∞ algebra g on another L∞ algebrah.

17.1.1.1 Definition. An action of an L∞ algebra g on an L∞ algebra h is, by definition, an L∞-algebra structure on g⊕ h with the property that the (linear) maps in the exact sequence

0→ h→ g⊕ h→ g→ 0

are maps of L∞ algebras.

Remark: (1) The set of actions of g on h enriches to a simplicial set, whose n-simplicesare families of actions over the dg algebra Ω∗(4n).

(2) There are other possible notions of action of g on h which might seem more nat-ural to some readers. For instance, an abstract notion is to say that an action of gon h is an L∞ algebra h with a map φ : h→ g and an isomorphism of L∞ algebrasbetween the homotopy fibre φ−1(0) and h. One can show that this more fancydefinition is equivalent to the concrete one proposed above, in the sense that thetwo ∞-groupoids of possible actions are equivalent.

If h is finite dimensional, then we can identify the dg Lie algebra of derivations ofC∗(h) with C∗(h, h[1]) with a certain dg Lie bracket. We can thus view C∗(h, h[1]) as thedg Lie algebra of vector fields on the formal moduli problem Bh.

17.1.1.2 Lemma. Actions of g on h are the same as L∞-algebra maps g→ C∗(h, h[1]).

PROOF. This is straightforward.

This lemma shows that an action of g on h is the same as an action of g on the formalmoduli problem Bh which may not preserve the base-point of Bh.

17.1.2. Actions of local L∞ algebras. Now let us return to the setting of local L∞ al-gebras, and define what it means for one local L∞ algebra to act on another.

17.1.2.1 Definition. Let L,M be local L∞ algebras on M. Then an L action onM is given by alocal L∞ structure on L⊕M, such that the exact sequence

0→M→ L⊕M→ L → 0

is a sequence of L∞ algebras.


More explicitly, this says thatM (with its original L∞ structure) is a sub-L∞-algebraofM⊕L, and is also an L∞-ideal: all operations which take as input at least one elementofM land inM. We will refer to the L∞ algebra L ⊕M with the L∞ structure definingthe action as LnM.

17.1.2.2 Definition. Suppose thatM has an invariant pairing. An action of L onM preservesthe pairing if, for local compactly supported sections αi, β j of L andM the tensor

〈lr+s(α1, . . . , αr, β1, . . . , βs), βs+1〉

is totally symmetric if s + 1 is even (or antisymmetric if s + 1 is odd) under permutation of the βi.

17.1.2.3 Definition. An action of a local L∞ algebra L on a classical field theory defined by a localL∞ algebraM with an invariant pairing of degree−3 is, as above, an L∞ action of L onM whichpreserves the pairing.

As an example, we have the following.

17.1.2.4 Lemma. Suppose that L acts on an elliptic L∞ algebraM. Then L acts on the cotangenttheory forM.

PROOF. This is immediate by naturality, but we can also write down explicitly thesemi-direct product L∞ algebra describing the action. Note that LnM acts linearly on

(LnM)! [−3] = L![−3]⊕M![−3].

Further, L![−3] is a submodule for this action, so that we can form the quotientM![−3].Then,

(LnM)nM![−3]

is the desired semi-direct product.

Remark: Note that this construction is simply giving the−1-shifted relative cotangent bun-dle to the map

B(LnM)→ BL.

The definition we gave above of an action of a local L∞ algebra on a classical fieldtheory is a little abstract. We can make it more concrete as follows.

Recall that the space of fields of the classical field theory associated toM isM[1], andthat the L∞ structure onM is entirely encoded in the action functional

S ∈ Oloc(M[1])

which satisfies the classical master equation S, S = 0. (The notation Oloc always indi-cates local functionals modulo constants).


An action of a local L∞ algebra L onM can also be encoded in a certain local func-tional, which depends on L. We need to describe the precise space of functionals that arisein this interpretation.

If X denotes the space-time manifold on which L andM are sheaves, then L(X) is anL∞ algebra. Thus, we can form the Chevalley-Eilenberg cochain complex

C∗(L(X)) = (O(L(X)[1]), dL)

as well as it’s reduced version C∗red(L(X)).

We can form the completed tensor product of this dg algebra with the shifted Liealgebra Oloc(M[1]), to form a new shifted dg Lie algebra C∗red(L(X))⊗Oloc(M[1]).

Inside this is the subspace

Oloc(L[1]⊕M[1])/ (Oloc(L[1])⊕Oloc(M[1])) ⊂ C∗red(L(X))⊗Oloc(M[1])

of functionals which are local as a function of L[1]. Note that we are working with func-tionals which must depend on both L[1] and M[1]: we discard those functionals whichdepend only on one or the other.

One can check that this graded subspace is preserved both by the Lie bracket −,−,the differential dL and the differential dM (coming from the L∞ structure on L andM).This space thus becomes a shifted dg Lie algebra, with the differential dL ⊕ dM and withthe degree +1 bracket −,−.

17.1.2.5 Lemma. To give an action of a local L∞ algebraL on a classical field theory correspondingto a local L∞ algebraM with invariant pairing, is the same as to give an action functional

SL ∈ Oloc(L[1]⊕M[1])/ (Oloc(L[1])⊕Oloc(M[1]))

which is of cohomological degree 0, and satisfied the Maurer-Cartan equation

(dL + dM)SL + 12S

L, SL = 0.

PROOF. Given such an SL, then

dL + dM + SL,−

defines a differential on O(L(X)[1] ⊕M(X)[1]). The classical master equation impliesthat this differential is of square zero, so that it defines an L∞ structure on L(X)⊕M(X).The locality condition on SL guarantees that this is a local L∞ algebra structure. A simpleanalysis shows that this L∞ structure respects the exact sequence

0→M→ L⊕M→ L → 0

and the invariant pairing onM.


This lemma suggests that we should look at a classical field theory with an action of Las a family of classical field theories over the sheaf of formal moduli problems BL. Furtherjustification for this idea will be offered in proposition ??.

17.1.3. Let g be an ordinary L∞ algebra (not a sheaf of such), which we assume to befinite-dimensional for simplicity. Let C∗(g) be it’s Chevalley-Eilenberg cochain algebra,viewed as a pro-nilpotent commutative dga. Suppose we have a classical field theory,represented as an elliptic L∞ algebra M with an invariant pairing. Then we can definethe notion of a g-action onM as follows.

17.1.3.1 Definition. A g-action onM is any of the following equivalent data.

(1) An L∞ structure on g⊕M(X) such that the exact sequence

0→M(X)→ g⊕M(X)→ g

is a sequence of L∞-algebras, and such that the structure maps

g⊗n ⊗M(X)⊗m →M(X)

are poly-differential operators in theM-variables.(2) An L∞-homomorphism

g→ Oloc(BM)[−1](the shift is so that Oloc(BM)[−1] is an ordinary, and not shifted, dg Lie algebra).

(3) An elementSg ∈ C∗red(g)⊗Oloc(BM)

which satisfies the Maurer-Cartan

dgSgdMSg + 12S

g, Sg = 0.

It is straightforward to verify that these three notions are identical. The third versionof the definition can be viewed as saying that a g-action on a classical field theory is a fam-ily of classical field theories over the dg ring C∗(g) which reduces to the original classicalfield theory modulo the maximal ideal C>0(g). This version of the definition generalizesto the quantum level.

Our formulation of Noether’s theorem will be phrased in terms of the action of a localL∞ algebra on a field theory. However, we are often presented with the action of an or-dinary, finite-dimensional L∞-algebra on a theory, and we would like to apply Noether’stheorem to this situation. Thus, we need to be able to formulate this kind of action as anaction of a local L∞ algebra.

The following lemma shows that we can do this.

17.1.3.2 Lemma. Let g be an L∞-algebra. Then, the simplicial sets describing the following arecanonically homotopy equivalent:


(1) Actions of g on a fixed classical field theory on a space-time manifold X.(2) Actions of the local L∞ algebra Ω∗X ⊗ g on the same classical field theory.

Note that the sheaf Ω∗X⊗ g is a fine resolution of the constant sheaf of L∞ algebras withvalue g. The lemma can be generalized to show that, given any locally-constant sheaf ofL∞ algebras g, an action of g on a theory is the same thing as an action of a fine resolutionof g.

PROOF. Suppose thatM is a classical field theory, and suppose that we have an actionof the local L∞ algebra Ω∗X ⊗ g onM.

Actions of g onM are Maurer-Cartan elements of the pro-nilpotent dg Lie algebra

Act(g,M)def= C∗red(g)⊗Oloc(M[1])

, with dg Lie structure described above. Actions of Ω∗X ⊗ g are Maurer-Cartan elements ofthe pro-nilpotent dg Lie algebra

Act(Ω∗X ⊗ g,M)def= Oloc(Ω∗X ⊗ g[1]⊕M[1])/ (Oloc(Ω∗X ⊗ g[1])⊕Oloc(M[1])) ,

again with the dg Lie algebra structure defined above.

Recall that there is an inclusion of dg Lie algebras

Oloc(Ω∗X ⊗ g[1]⊕M[1])/Oloc(Ω∗X ⊗ g[1]) ⊂ C∗(Ω∗(X)⊗ g)⊗Oloc(M[1]).

Further, there is an inclusion of L∞ algebras

g → Ω∗(X)⊗ g

(by tensoring with the constant functions). Composing these maps gives a map of dg Liealgebras

Act(Ω∗X ⊗ g,M)→ Act(g,M).It suffices to show that this map is an equivalence.

We will do this by using the DX-module interpretation of the left hand side. Let J(M)and J(Ω∗X) refer to the DX-modules of jets of sections ofM and of the de Rham complex,respectively. Note that the natural map of DX-modules

C∞X → J(Ω∗X)

is a quasi-isomorphism (this is the Poincare lemma.).

Recall thatOloc(M[1]) = ωX ⊗DX C∗red(J(M)).

The cochain complex underlying the dg Lie algebra Act(Ω∗X ⊗ g,M) has the followinginterpretation in the language of DX-modules:

Act(Ω∗X ⊗ g,M) = ωX ⊗DX

(C∗red(J(Ω∗X)⊗C g)⊗C∞

XC∗red(J(M))

).

17.2. EXAMPLES OF CLASSICAL FIELD THEORIES WITH AN ACTION OF A LOCAL L∞ ALGEBRA 363

Under the other hand, the complex Act(g,M) has the DX-module interpretation

Act(g,M) = ωX ⊗DX

(C∗red(g⊗ C∞

X )⊗C∞X

C∗red(J(M)))

.

Because the map C∞X → J(Ω∗X) is a quasi-isomorphism of DX-modules, the natural map

C∗red(J(Ω∗X)⊗C g)⊗C∞X

C∗red(J(M))

→ C∗red(g⊗ C∞X )⊗C∞

XC∗red(J(M))

is a quasi-isomorphism of DX-modules. Now, both sides of this equation are flat as leftDX-modules; this follows from the fact that C∗red(J(M)) is a flat DX-module. If followsthat this map is still a quasi-isomorphism after tensoring over DX with ωX.

17.2. Examples of classical field theories with an action of a local L∞ algebra

One is often interested in particular classes of field theories: for example, conformalfield theories, holomorphic field theories, or field theories defined on Riemannian mani-folds. It turns out that these ideas can be formalized by saying that a theory is acted onby a particular local L∞ algebra, corresponding to holomorphic, Riemannian, or confor-mal geometry. This generalizes to any geometric structure on a manifold which can bedescribed by a combination of differential equations and symmetries.

In this section, we will describe the local L∞ algebras corresponding to holomorphic,conformal, and Riemannian geometry, and give examples of classical field theories actedon by these L∞ algebras.

We will first discuss the holomorphic case. Let X be a complex manifold, and define alocal dg Lie algebra algebra Lhol by setting

Lhol(X) = Ω0,∗(X, TX),

equipped with the Dolbeault differential and the Lie bracket of vector fields. A holomor-phic classical field theory will be acted on by Lhol(X).

Remark: A stronger notion of holomorphicity might require the field theory to be actedon by the group of holomorphic symmetries of X, and that the derivative of this actionextends to an action of the local dg Lie algebra Lhol .

Let us now give some examples of field theories acted on by Lhol .

Example: Let X be a complex manifold of complex dimension d, and let g be a finite-dimensional Lie algebra with Lie group G. Then, Ω0,∗(X, g) describes the formal modulispace of principle G-bundles on X. We can form the cotangent theory to this, which is aclassical field theory, by letting

M = Ω0,∗(X, g)⊕Ωd,∗(X, g∨)[d− 3].


As discussed in [Cos11b], this example is important in physics. If d = 2 it describes aholomorphic twist of N = 1 supersymmetric gauge theory. In addition, one can use theformalism of L∞ spaces [Cos11a, Cos11b] to write twisted supersymmetric σ-models inthese terms (when d = 1 and g is a certain sheaf of L∞ algebras on the target space).

The dg Lie algebra Lhol(X) acts by Lie derivative on Ωk,∗(X) for any k. One can makethis action explicit as follows: the contraction map

Ω0,∗(X, TX)×Ωk,∗(X)→ Ωk−1,∗(X)

(V, ω) 7→ ιVω

is Ω0,∗(X)-linear and defined on Ω0,0(X, TX) in the standard way. The Lie derivative isdefined by the Cartan homotopy formula

LVω = [ιV , ∂]ω.

In this way, L acts onM. This action preserves the invariant pairing.

We can write this in terms of an L-dependent action functional, as follows. If α ∈Ω0,∗(X, g)[1], β ∈ Ωd,∗(X, g∨)[d− 2] and V ∈ Ω0,∗(X, TX)[1], we define

SL(α, β, V) =∫ ⟨

β, (∂ + LV)α⟩+ 1

2 〈β, [α, α]〉 .

(The fields α, β, V can be of mixed degree).

Note that if V ∈ Ω0,∗(X, TX) is of cohomological degree 1, it defines a deformation ofcomplex structure of X, and the ∂-operator for this deformed complex structure is ∂ +LV .The action functional SL therefore describes the variation of the original action functinalS as we vary the complex structure on X. Other terms in SL encode the fact that S isinvariant under holomorphic symmetries of X.

We will return to this example throughout our discussion of Noether’s theorem. Wewill see that, in dimension d = 1 and with g Abelian, it leads to a version of the Segal-Sugawara construction: a map from the Virasoro vertex algebra to the vertex algebra as-sociated to a free β− γ system.

Example: Next, let us discuss the situation of field theories defined on a complex mani-fold X together with a holomorphic principal G-bundle. In the case that X is a Riemannsurface, field theories of this form play an important role in the mathematics of chiralconformal field theory.

For this example, we define a local dg Lie algebra L on a complex manifold X by

L(X) = Ω0,∗(X, TX)n Ω0,∗(X, g)

so that L(X) is the semi-direct product of the Dolbeault resolution of holomorphic vectorfields with the Dolbeault complex with coefficients in g. Thus, L(X) is the dg Lie algebra


controlling deformations of X as a complex manifold equipped with a holomorphic G-bundle, near the trivial bundle. (The dg Lie algebra controlling deformations of the pair(X, P) where P is a non-trivial principal G-bundle on X is Ω0,∗(X, AtP) where AtP is theAtiyah algebroid of P, and everything that follows works in the more general case whenP is non-trivial and we use Ω0,∗(X, AtP) in place of L).

Let V be a representation of G. We can form the cotangent theory to the elliptic moduliproblem of sections of V, defined by the Abelian elliptic L∞ algebra

M(X) = Ω0,∗(X, V)[−1]⊕Ω0,∗(X, V∨)[d− 2].

This is acted on by the local L∞ algebra L we described above.

More generally, we could replace V by a complex manifold M with a G-action andconsider the cotangent theory to the moduli of holomorphic maps to M.

Example: In this example we will introduce the local dg Lie algebra LRiem on a Riemannianmanifold X which controls deformations of X as a Riemannian manifold. This local dgLie algebra acts on field theories which are defined on Riemannian manifolds; we willshow this explicitly in the case of scalar field theories.

Let (X, g0) be a Riemannian manifold of dimension d, which for simplicity we assumeto be oriented.

Consider the local dg Lie algebra

LRiem(X) = Vect(X)⊕ Γ(X, Sym2 TX)[−1].

The differential is dV = LV g0 where LV indicates Lie derivative. The Lie bracket isdefined by saying that the bracket of a vector field V with anything is given by Lie deriv-ative.

Note that LRiem(X) is the dg Lie algebra describing the formal neighbourhood of X inthe moduli space of Riemannian manifolds.

Consider the free scalar field theory on X, defined by the abelian elliptic dg Lie algebra

M(X) = C∞(X)[−1]4g0−−→ Ωd(X)[−2]

where the superscript indicates cohomological degree, and

4g0 = d ∗ d

is the Laplacian for the metric g0, landing in top-forms.


We define the action of LRiem(X) onM(X) by defining an action functional SL whichcouples the fields in LRiem(X) to those inM(X). If φ, ψ ∈ M(X)[1] are fields of cohomo-logical degree 0 and 1, and V ∈ Vect(X), α ∈ Γ(X, Sym2 TX), then we define SL by

SL(φ, ψ, V, α) =∫

φ(4g0+α −4g0)φ +∫(Vφ)ψ.

On the right hand side we interpret 4g0+α as a formal power series in the field α. Thefact that this satisfies the master equation follows from the fact that the Laplacian 4g0+α

is covariant under infinitesimal diffeomorphisms:

4g0+α + ε[V,4g0+α] = 4g0+α+εLV g0+εLV α.

One can rewrite this in the language of L∞ algebras by Taylor expanding 4g0+α inpowers of α. The resulting semi-direct product L∞-algebra LRiem(X)nM(X) is the de-scribes the formal moduli space of Riemannian manifolds together with a harmonic func-tion φ.

Example: Let us modify the previous example by considering a scalar field theory with apolynomial interaction, so that the action functional is of the form∫

φ4g0 φ + ∑n≥2

λn1n! φ

ndVolg0 .

In this case,M is deformed into a non-abelian L∞ algebra, with maps ln defined by

ln : C∞(X)⊗n → Ωd(X)

ln(φ1, . . . , φn) = λnφ1 · · · φndVolg0 .

The action of LRiem onM is defined, as above, by declaring that the action functional SLcoupling the two theories is

SL(φ, ψ, V, α) + S(φ, ψ) =∫

φ4g0+αφ + ∑n≥2

λn1n! φ

ndVolg0+α +∫(Vφ)ψ.

Example: Next let us discuss the classical conformal field theories.

As above, let (X, g0) be a Riemannian manifold. Define a local dg Lie algebra Lcon f onX by setting

Lcon f (X) = Vect(X)⊕ C∞(X)⊕ Γ(X, Sym2 TX)[−1].The copy of C∞(X) corresponds to Weyl rescalings.

The differential on Lcon f (X) is

d(V, f ) = LV g0 + f g0

where V ∈ Vect(X) and f ∈ C∞(X). The Lie bracket is defined by saying that Vect(X)acts on everything by Lie derivative, and that if f ∈ C∞(X) and α ∈ Γ(X, Sym2 TX),[ f , α] = f α.


It is easy to verify that H0(Lcon f (X)) is the Lie algebra of conformal symmetries of X,and that H1(Lcon f (X)) is the space of first-order conformal deformations of X. The localdg Lie algebra Lcon f will act on any classical conformal field theory.

We will see this explicitly in the case of the free scalar field theory in dimension 2.Let M be the elliptic dg Lie algebra corresponding to the free scalar field theory on aRiemannian 2-manifold X, as described in the previous example.

The action of Lcon f onM is defined such that the sub-algebra LRiem acts in the sameway as before, and that C∞(X) acts by zero.

This does not define an action for the two-dimensional theory with polynomial inter-action, because the polynomial interaction is not conformally invariant.

There are many other, more complicated, examples of this nature. If X is a conformal4-manifold, then Yang-Mills theory on X is conformally invariant at the classical level.The same goes for self-dual Yang-Mills theory. One can explicitly write an action of Lcon f

on the elliptic L∞-algebra on X describing either self-dual or full Yang-Mills theory.

Example: In this example, we will see how we can describe sources for local operators inthe language of local dg Lie algebras.

Let X be a Riemannian manifold, and consider a scalar field theory on X with a φ3

interaction, whose associated elliptic L∞ algebraM has been described above. Recall thatM(X) consists of C∞(X) in degree 1 and of Ωd(X) in degree 2, where d = dim X.

The action functional encoding the L∞ structure onM is the functional onM[1] de-fined by

S(φ, ψ) =∫

φ4φ +∫

φ3.

where φ is a degree 0 element ofM(X)[1], so that φ is a smooth function.

Let us view the sheaf C∞X [−1] as an Abelian local L∞-algebra on X, situated in degree

1 with zero differential and bracket.

Let us define an action of L onM by giving an action functional

SL(α, φ, ψ) =∫

φ4φ +∫

φα.

Here,

α ∈ L[1] = C∞(M)


and φ, ψ are elements of degrees 0 and 1 ofM(X)[1]. This gives a semi-direct product L∞algebra LnM, whose underlying cochain complex is

C∞(X)1

Id

%%C∞(X)1 4 // C∞(X)2

where the first row is L and the second row isM. The only non-trivial Lie bracket is theoriginal Lie bracket onM.

17.3. The factorization algebra of equivariant observable

17.3.0.3 Proposition. Suppose thatM is a classical field theory with an action of L. Then, thereis a P0 factorization algebra Obscl

L of equivariant observables, which is a factorization algebra inmodules for the factorization algebra in commutative dg algebras C∗(L), which assigns to an opensubset U the commutative dga C∗(L(U)).

PROOF. Since L acts onM, we can construct the semi-direct product local L∞ algebraLnM. We define the equivariant classical observables

ObsclL = C∗(LnM)

to be the Chevalley-Eilenberg cochain factorization algebra associated to this semi-directproduct.

As in section 12.4, we will construct a sub-factorization algebra on which the Poissonbracket is defined and which is quasi-isomorphic. We simply let

ObsclL(U) ⊂ Obscl

L(U)

be the subcomplex consisting of those functionals which have smooth first derivative butonly in the M-directions. As in section 12.4, there is a P0 structure on this subcomplex,which on generators is defined by the dual of the non-degenerate invariant pairing onM.Those functionals which lie in C∗(L(U)) are central for this Poisson bracket.

It is clear that this constructs a P0-factorization algebra over the factorization algebraC∗(L).

17.4. Inner actions

A stronger notion of action of a local L∞ algebra on a classical field theory will beimportant for Noeother’s theorem. We will call this stronger notion an inner action of a

17.4. INNER ACTIONS 369

local L∞ algebra on a classical field theory. For classical field theories, every action can belifted canonically to an inner action, but at the quantum level this is no longer the case.

We defined an action of a local L∞ algebra L on a field theoryM (both on a manifoldX) to be a Maurer-Cartan element in the differential graded Lie algebra Act(L,M) whoseunderlying cochain complex is

Oloc(L[1]⊕M[1])/ (Oloc(L[1])⊕Oloc(M[1]))

with the Chevalley-Eilenberg differential for the direct sum L∞ algebra L⊕M.

An inner action will be defined as a Maurer-Cartan element in a larger dg Lie algebrawhich is a central extension of Act(L,M) by Oloc(L[1]). Note that

Oloc(L[1]⊕M[1]) ⊂ C∗red(L(X)⊕M)

has the structure of dg Lie algebra, where the differential is the Chevalley-Eilenberg differ-ential for the direct sum dg Lie algebra, and the bracket arises, as usual, from the invariantpairing onM.

Further, there’s a natural map of dg Lie algebras from this to Oloc(M[1]), which arisesby applying the functor of Lie algebra cochains to the inclusion M → M⊕ L of L∞algebras.

We letInnerAct(L,M) ⊂ Oloc(L[1]⊕M[1])

be the kernel of this map. Thus, as a cochain complex,

InnerAct(L,M) = Oloc(L[1]⊕M[1])/Oloc(M[1])

with differential the Chevalley-Eilenberg differential for the direct sum L∞ algebra L ⊕M. Note that the Lie bracket on InnerAct(L,M) is of cohomological degree +1.

17.4.0.4 Definition. An inner action of L onM is a Maurer-Cartan element

SL ∈ InnerAct(L,M).

Thus, SL is of cohomological degree 0, and satisfies the master equation

dSL + 12S

L, SL.

17.4.0.5 Lemma. Suppose we have an action of L on a field theoryM. Then there is an obstruc-tion class in H1(Oloc(L[1])) such that the action extends to an inner action if and only if this classvanishes.

PROOF. There is a short exact sequence of dg Lie algebras

0→ Oloc(L[1])→ InnerAct(L,M)→ Act(L,M)→ 0

and Oloc(L[1]) is central. The result follows from general facts about Maurer-Cartan sim-plicial sets.


More explicitly, the obstruction is calculated as follows. Suppose we have an actionfunctional

SL ∈ Act(L,M) ⊂ C∗red(L(X))⊗ C∗red(M(X)).Then, let us view SL as a functional in

SL InnerAct(L,M) ⊂ C∗red(L(X))⊗ C∗(M(X))

using the natural inclusion C∗red(M(X)) → C∗(M(X)). The obstruction is simply thefailure of SL to satisfy the Maurer-Cartan equation in InnerAct(L,M).

Let us now briefly remark on some refinements of this lemma, which give some morecontrol over obstruction class.

Recall that we sometimes use the notation C∗red,loc(L) for the complex Oloc(L[1]). Thus,C∗red,loc(L) is a subcomplex of C∗red(L(X)). We let C≥2

,loc(L) be the kernel of the natural map

C∗red,loc(L)→ L!(X)[−1].

Thus, C≥2red,loc(L) is a subcomplex of C≥2

red(L(X)).

17.4.0.6 Lemma. If a local L∞ algebra L acts on a classical field theoryM, then the obstructionto extending L to an inner action lifts naturally to an element of the subcomplex

C≥2red,loc(L) ⊂ C∗red,loc(L).

PROOF. Suppose that the action of L onM is encoded by an action functional SL, asbefore. The obstruction is(

dLSL + dMSL + 12S

L, SL)|Oloc(L[1])∈ Oloc(L[1]).

Here, dL and dM are the Chevalley-Eilenberg differentials for the two L∞ algebras.

We need to verify that no terms in this expression can be linear in L. Recall thatthe functional SL has no linear terms. Further, the differentials dL and dM respect thefiltration on everything by polynomial degree, so that they can not produce a functionalwith a linear term from a functional which does not have a linear term.

Remark: There is a somewhat more general situation when this lemma is false. Whenone works with families of classical field theories over some dg ring R with a nilpotentideal I, one allows the L∞ algebraM describing the field theory to be curved, as long asthe curving vanishes modulo I. This situation is encountered in the study of σ-models:see [Cos11a]. WhenM is curved, the differential dM does not preserve the filtration bypolynomial degree, so that this argument fails.

Let us briefly discuss a special case when the obstruction vanishes.

17.5. CLASSICAL NOETHER’S THEOREM 371

17.4.0.7 Lemma. Suppose that the action of L onM, when viewed as an action of L on the sheafof formal moduli problems BM, preserves the base point of BM. In the language of L∞ algebras,this means that the L∞ structure on L⊕M defining the action has no terms mapping

L⊗n →M

for some n > 0.

Then, the action of L extends canonically to an inner action.

PROOF. We need to verify that the obstruction(dLSL + dMSL + 1

2SL, SL

)|Oloc(L[1])∈ Oloc(L[1]).

is identically zero. Our assumptions on SL mean that it is at least quadratic as a functiononM[1]. It follows that the obstruction is also at least quadratic as a function ofM[1], sothat it is zero when restricted to being a function of just L[1].

17.5. Classical Noether’s theorem

As we showed in lemma ??, there is a bijection between classes in H1(C∗red,loc(L)) andlocal central extensions of Lc shifted by −1.

17.5.0.8 Theorem. LetM be a classical field theory with an action of a local L∞ algebra L. LetLc be the central extension corresponding to the obstruction class α ∈ H1(C∗red,loc(L)) for lifting

L to an inner action. Let Obscl

be the classical observables of the field theoryM, equipped withits P0 structure. Then, there is an L∞-map of precosheaves of L∞-algebras

Lc → Obscl[−1]

which sends the central element c to the unit 1 ∈ Obscl[−1] (note that, after the shift, the unit 1

is in cohomological degree 1, as is the central element c).

Remark: The linear term in the L∞-morphism is a map of precosheaves of cochain com-plexes from Lc → Obscl [−1]. The fact that we have such a map of precosheaves impliesthat we have a map of commutative dg factorization algebras

Sym∗(Lc[1])→ Obs

cl.

which, as above, sends the central element to 1. This formulation is the one that willquantize: we will find a map from a certain Chevalley-Eilenberg chain complex of Lc[1]to quantum observables.


Remark: Lemma 17.4.0.6 implies that the central extension Lc is split canonically as apresheaf of cochain complexes:

Lc(U) = C[−1]⊕Lc(U).

Thus, we have a map of precosheaves of cochain complexes

Lc → Obscl .

The same proof will show that that this cochain map to a continuous map from the distri-butional completion Lc(U) to Obscl .

PROOF. Let us first consider a finite-dimensional version of this statement, in the casewhen the central extension splits. Let g, h be L∞ algebras, and suppose that h is equippedwith an invariant pairing of degree−3. Then, C∗(h) is a P0 algebra. Suppose we are givenan element

G ∈ C∗red(g)⊗ C∗(h)of cohomological degree 0, satisfying the Maurer-Cartan

dG + 12G, G = 0

where dg, dh are the Chevalley-Eilenberg differentials for g and h respectively, and −,−denotes the Poisson bracket coming from the P0 structure on C∗(h).

Then, G is precisely the data of an L∞ map

g→ C∗(h)[−1].

Indeed, for any L∞ algebra j, to give a Maurer-Cartan element in C∗red(g)⊗ j is the same asto give an L∞ map g→ j. Further, the simplicial set of L∞-maps and homotopies betweenthem is homotopy equivalent to the Maurer-Cartan simplicial set.

Let us now consider the case when we have a central extension. Suppose that we havean element

G ∈ C∗red(g)⊗ C∗(h)of degree 0, and an obstruction element

α ∈ C∗red(g)

of degree 1, such thatdG + 1

2G, G = α⊗ 1.

Let g be the −1-shifted central extension determined by α, so that there is a short exactsequence

0→ C[−1]→ g→ g→ 0.

Then, the data of G and α is the same as a map of L∞ algebras g→ C∗(h)[−1] which sendsthe central element of g to 1 ∈ C∗(h).

17.5. CLASSICAL NOETHER’S THEOREM 373

To see this, let us choose a splitting g = g ⊕ C · c where the central element c is ofdegree 1. Let c∨ be the linear functional on g which is zero on g and sends c to 1.

Then, the image of α under the natural map C∗(g) → C∗(g) is made exact by c∨,viewed as a zero-cochain in C∗(g). It follows that

G + c∨ ⊗ 1 ∈ C∗red(g)⊗ C∗(h)

satisfies the Maurer-Cartan equation, and therefore defines (as above) an L∞-map g →C∗(h)[−1]. This L∞-map sends c → 1: this is because G only depends on c by the termc∨ ⊗ 1.

Let us apply these remarks to the setting of factorization algebras. First, let us remarka little on the notation: we normally use the notation Oloc(L[1]) to refer to the complex oflocal functionals on L[1], with the Chevalley-Eilenberg differential. However, we can alsorefer to this object as C∗red,loc(L), the reduced, local cochains of L. It is the subcomplex ofC∗red(L(X)) of cochains which are local.

Suppose we have an action of a local L∞-algebra L on a classical field theoryM. Let

α ∈ Oloc(L[1]) = C∗red,loc(L)

be a 1-cocycle representing the obstruction to lifting to an inner action onM. Let

Lc = Lc ⊕C[−1]

be the corresponding central extension.

By the definition of α, we have a functional

SL ∈ C∗red,loc(L⊕M)/C∗red,loc(M)

of cohomological degree 0 satisfying the Maurer-Cartan equation

dSL + 12S

L, SL = α.

For every open subset U ⊂ M, we have an injective cochain map

Φ : C∗red,loc(L⊕M)/C∗red,loc(M)→ C∗red(Lc(U))⊗C∗(M(U)),

where ⊗ refers to the completed tensor product and C∗(M(U)) refers to the subcomplexof C∗(M(U)) consisting of functionals with smooth first derivative. The reason we havesuch a map is simply that a local functional on U is defined when at least if its inputs iscompactly supported.

The cochain map Φ is in fact a map of dg Lie algebras, where the Lie bracket arises asusual from the pairing onM. Thus, for every U, we have an element

SL(U) ∈ C∗red(Lc(U))⊗C∗(M(U))


satisfying the Maurer-Cartan equation

dSL(U) + 12S

L(U), SL(U) = α(U).

It follows, as in the finite-dimensional case discussed above, that SL(U) gives rise to amap of L∞ algebras

Lc(U)→ C∗(M(U))[−1] = Obscl(U)[−1]

sending the central element c in Lc(U) to the unit 1 ∈ Obscl(U). The fact that SL is local

implies immediately that this is a map of precosheaves.

17.6. Conserved currents

Traditionally, Noether’s theorem states that there is a conserved current associatedto every symmetry. Let us explain why the version of (classical) Noether’s theorem pre-sented above leads to this more traditional statement. Similar remarks will hold for thequantum version of Noether’s theorem.

In the usual treatment, a current is taken to be an d− 1-form valued in Lagrangians (ifwe’re dealing with a field theory on a manifold X of dimension d). In our formalism, wemake the following definition (which will be valid at the quantum level as well).

17.6.0.9 Definition. A conserved current in a field theory is a map of precosheaves

J : Ω∗c [1]→ Obscl

to the factorization algebra of classical observables.

Dually, J can be viewed as a closed, degree 0 element of

J(U)Ω∗(U)[n− 1]⊗Obscl(U)

defined for every open subset U, and which is compatible with inclusions of open subsetsin the obviious way.

In particular, we can take the component of J(U)n−1,0 which is an element

J(U)n−1,0 ∈ Ωn−1(U)⊗Obscl(U)0.

The superscript in Obscl(U)0 indicates cohomological degree 0. Thus, J(U)n−1,0 is ann− 1-form valued in observables, which is precisely what is traditionally called a current.

Let us now explain why our definition means that this current is conserved (up tohomotopy). We will need to introduce a little notation to explain this point. If N ⊂ X is aclosed subset, we let

Obscl(N) = holimN⊂U Obscl(U)

17.6. CONSERVED CURRENTS 375

be the homotopy limit of observables on open neighbourhoods of N. Thus, an element ofObscl(N) is an observable defined on every open neighbourhood of N, in a way compati-ble (up to homotopy) with inclusions of open sets. The fact that we are taking a homotopylimit instead of an ordinary limit is not so important for this discussion, it’s to ensure thatthe answer doesn’t depend on arbitrary choices.

For example, if p ∈ X is a point, then Obscl(p) should be thought of as the space oflocal observables at p.

Suppose we have a conserved current (in the sense of the definition above). Then, forevery compact codimension 1 oriented submanifold N ⊂ X, the delta-distribution on Nis an element

[N] ∈ Ω1(U)

defined for every open neighbourhood U of N. Applying the map defining the closedcurrent, we get an element

J[N] ∈ Obscl(U)

for every neighbourhood U of N. This element is compatible with inclusions U → U′, sodefines an element of Obscl(N).

Let M ⊂ X be a top-dimensional submanifold with boundary ∂M = N q N′. Then,

dJ[M] = J[N]− J[N′] ∈ Obscl(M).

It follows that the cohomology class [J[N]] of J[N] doesn’t change if N is changed by acobordism.

In particular, let us suppose that our space-time manifold X is a product

X = N ×R.

Then, the observable [J[Nt]] associated to the submanifold N × t is independent of t.

This is precisely the condition (in the traditional formulation) for a current to be con-served.

Now let us explain why our version of Noether’s theorem, as explained above, pro-duces a conserved current from a symmetry.

17.6.0.10 Lemma. Suppose we have a classical field theory on a manifold X which has an in-finitesimal symmetry. To this data, our formulation of Noether’s theorem produces a conservedcurrent.

PROOF. A theory with an infinitesimal symmetry is acted on by the abelian Lie algebraR (or C). Lemma ?? shows us that such an action is equivalent to the action of the Abelianlocal dg Lie algebra Ω∗X. Lemma 17.4.0.6 implies that the central extension Lc is split as a


cochain complex (except in the case that we work in families and the classical field theoryM has curving). We thus get a map

Ω∗X,c[1]→ Obscl .

The remark following theorem 17.5.0.8 tells us that this map extends to a continuouscochain map

Ω∗X,c[1]→ Obscl

which is our defintion of a conserved current.

17.7. Examples of classical Noether’s theorem

Let’s give some simple examples of this construction. All of the examples we willconsider here will satisfy the criterion of lemma ?? which implies that the central extensionof the local L∞ algebra of symmetries is trivial.

Example: Suppose that a field theory on a manifold X of dimension d has an inner actionof the Abelian local L∞ algebra Ωd

X[−1]. Then, we get a map of presheaves of cochaincomplexes

ΩdX → Obscl .

Since, for every point p ∈ X, the delta-function δp is an element of Ωd(X), in this way we

get a local observable in Obscl(p) for every point. This varies smoothly with p.

For example, consider the free scalar field theory on X. We can define an action ofΩd

X[−1] on the free scalar field theory as follows. If φ ∈ C∞(X) and ψ ∈ Ωd(X)[−1] arefields of the free scalar field theory, and γ ∈ Ωd(X) is an element of the Abelian local L∞algebra we want to act, then the action is described by the action functional describinghow L

SL(φ, ψ, γ) =∫

φγ.

The corresponding map L∞ map

(†) Ωdc (U)→ Obscl(U)

is linear, and sends γ ∈ Ωdc (U) to the observable

∫φγ.

Example: Let’s consider the example of a scalar field theory on a Riemannian manifold X,described by the action functional ∫

Xφ4φ + φ3dVol.

This is acted on by the dg Lie algebra LRiem describing deformations of X as a Riemannianmanifold.

17.7. EXAMPLES OF CLASSICAL NOETHER’S THEOREM 377

If U is an open subset of X, then LRiemc (U) consists, in degree 0, of the compactly-

supported first-order deformations of the Riemannian metric g0 on U ⊂ X. If

α ∈ Γc(U, Sym2 TX)

is such a deformation, let us Taylor expand the Laplace-Beltrami operator φ4g0+αφ as asum

4g0+α = 4g0 ∑n≥1

1n! Dn(α, . . . , α)

where Dn are poly-differential operators from Γ(X, TX)⊗n to the space of order ≤ 2 dif-ferential operators on X. Explicit formula for the operators Dn can be derived from theformula for the Laplace-Beltrami operator in terms of the metric.

Note that if α has compact support in U, then∫φDn(α, . . . , α)φ

defines an observable in Obscl(U), and in fact in Obscl(U) (as it has smooth first derivative

in φ.

The L∞ mapLRiem

c (U)→ Obscl(U)[−1]has Taylor terms

Φn : LRiemc (U)⊗n → Obscl(U)

defined by the observables

Φn((α1, V1), . . . , (αn, Vn))(φ, ψ) =

∫φDn(α1, . . . , αn)φ if n > 1∫φD1(α)φ +

∫(Vφ)ψ if n = 1.

One is often just interested in the cochain map

LRiemc (U)→ Obscl(U),

corresponding to φ1 above, and not in the higher terms. This cochain map has two terms:one given by the observable describing the first-order variation of the metric, and onegiven by the observable

∫(Vφ)ψ describing the action of vector fields on the fields of the

theory.

A similar analysis describes the map from LRiemc to the observables of a scalar field

theory with polynomial interaction.

Example: Let us consider the βγ system in one complex dimension, on C. The dg LiealgebraM describing this theory is

M(C) = Ω0,∗(C, V)[−1]⊕Ω1,∗(C, V∗)[−1].

LetL = Ω0,∗(C, TC)


be the Dolbeault resolution of holomorphic vector fields on C. L acts onM by Lie deriv-ative. We can write down the action functional encoding this action by

SL(β, γ, V) =∫(LV β)γ.

Here, β ∈ Ω0,∗(C, V), γ ∈ Ω1,∗(C, V∗) and V ∈ Ω0,∗(C, TC).

Lemma ?? implies that in this case there is no central extension. Therefore, we have amap

Φ : Lc[1]→ Obscl

of precosheaves of cochain complexes. At the cochain level, this map is very easy todescribe: it simply sends a compactly supported vector field V ∈ Ω0,∗

c (U, TU)[1] to theobservable

Φ(V)(β, γ) =∫

U(LV β) γ.

We are interested in what this does at the level of cohomology. Let us work on anopen annulus A ⊂ C. We have seen (section ????) that the cohomology of Obscl(A) canbe expressed in terms of the dual of the space of holomorphic functions on A:

H0(Obscl(A)) = Sym∗ (

Hol(A)∨ ⊗V∨ ⊕Ω1hol(A)∨ ⊗V

).

Higher cohomology of Obscl(A) vanishes.

Here, Hol(A) denotes holomorphic functions on A, Ω1hol(A) denotes holomorphic 1-

forms, and we are taking the continuous linear duals of these spaces. Further, we use, asalways, the completed tensor product when defining the symmetric algebra.

In a similar way, we can identify

H∗(Ω0,∗c (A, TA)) = H∗(Ω0,∗(A, K⊗2

A )∨[−1].

The residue pairing gives a dense embedding

C[t, t−1]dt ⊂ Hol(A)∨.

A concrete mapC[t, t−1][−1]→ Ω0,∗

c (A)

which realizes this map is defined as follows. Choose a smooth function f on the annuluswhich takes value 1 near the outer boundary and value 0 near the inner boundary. Then,∂ f has compact support. The map sends a polynomial P(t) to ∂( f P). One can check,using Stokes’ theorem, that this is compatible with the residue pairing: if Q(t)dt is aholomorphic one-form on the annulus,∮

P(t)Q(t)dt =∫

A∂( f (t, t)P(t))Q(t)dt.

17.7. EXAMPLES OF CLASSICAL NOETHER’S THEOREM 379

In particular, the residue pairing tells us that a dense subspace of H1(Lc(A)) is

C[t, t−1]∂t ⊂ H1(Ω0,∗c (A, TA)).

We therefore need to describe a map

Φ : C[t, t−1]∂t → Sym∗ (

Hol(A)∨ ⊗V∨ ⊕Ω1hol(A)∨ ⊗V

).

In other words, given an element P(t)∂t ∈ C[t, t−1]dt, we need to describe a functionalΦ(P(t)∂t) on the space of pairs

(β, γ) ∈ Hol(A)⊗V ⊕Ω1hol(A)⊗V∨.

From what we have explained so far, it is easy to calculate that this functional is

Φ(P(t)∂t)(β, γ) =∮

(P(t)∂tβ(t)) γ(t).

The reader familiar with the theory of vertex algebras will see that this is the classical limitof a standard formula for the Virasoro current.

CHAPTER 18

Noether’s theorem in quantum field theory

18.1. Quantum Noether’s theorem

So far, we have explained the classical version of Noether’s theorem, which states thatgiven an action of a local L∞ algebra L on a classical field theory, we have a central exten-sion Lc of the precosheaf Lc of L∞-algebras, and a map of precosheaves of L∞ algebras

Lc → Obscl [−1].

Our quantum Noether’s theorem provides a version of this at the quantum level. Beforewe explain this theorem, we need to introduce some algebraic ideas about envelopingalgebras of homotopy Lie algebras.

Given any dg Lie algebra g, one can construct its P0 envelope, which is the universalP0 algebra containing g. This functor is the homotopy left adjoint of the forgetful functorfrom P0 algebras to dg Lie algebras. Explicitly, the P0 envelope is

UP0(g) = Sym∗ g[1]

with the obvious product. The Poisson bracket is the unique bi-derivation which on thegenerators g is the given Lie bracket on g.

Further, if we have a shifted central extension g of g by C[−1], determined by a classα ∈ H1(g), we can define the twisted P0 envelope

UP0α (g) = UP0(g)⊗C[c] Cc=1

obtained from the P0 envelope of g by specializing the central parameter to 1.

We will reformulate the classical Noether theorem using the factorization P0 envelopeof a sheaf of L∞ algebras on a manifold. The quantum Noether theorem will then beformulated in terms of the factorization BD envelope, which is the quantum version ofthe factorization P0 envelope. The factorization BD envelope is a close relative of thefactorization envelope of a sheaf of L∞ algebras that we discussed in Section 3.6 of Chapter3.

For formal reasons, a version of these construction holds in the world of L∞ algebras.One can show that a commutative dg algebra together with a 1-shifted L∞ structure with

381

382 18. NOETHER’S THEOREM IN QUANTUM FIELD THEORY

the property that all higher brackets are multi-derivations for the product structure de-fines a homotopy P0 algebra. (The point is that the operad describing such gadgets isnaturally quasi-isomorphic to the operad P0).

If g is an L∞ algebra, one can construct a homotopy P0 algebra which has underlyingcommutative algebra Sym∗ g[1], and which has the unique shifted L∞ structure whereg[1] is a sub-L∞ algebra and all higher brackets are derivations in each variable. This L∞structure makes Sym∗ g[1] into a homotopy P0 algebra, and one can show that it is thehomotopy P0 envelope of g.

We can rephrase the classical version of Noether’s theorem as follows.

18.1.0.11 Theorem. Suppose that a local L∞ algebra L acts on a classical field theory, and that theobstruction to lifting this to an inner action is a local cochain α. Then there is a map of homotopyP0 factorization algebras

UP0α (Lc)→ Obscl .

Here UP0α (Lc) is the twisted homotopy P0 factorization envelope, which is defined by saying that

on each open subset U ⊂ M it is UP0α (Lc(U)).

The universal property of UP0α (Lc) means that this theorem is a formal consequence

of the version of Noether’s theorem that we have already proved. At the level of com-mutative factorization algebras, this map is obtained just by taking the cochain mapLc(U) → Obscl(U) and extending it in the unique way to a map of commutative dgalgebras

Sym∗(Lc(U))→ Obscl(U),

before specializing by setting the central parameter to be 1. There are higher homotopiesmaking this into a map of homotopy P0 algebras, but we will not write them down ex-plicitly (they come from the higher homotopies making the map Lc(U) → Obscl(U) intoa map of L∞ algebras).

This formulation of classical Noether’s theorem is clearly ripe for quantization. Wemust simply replace classical observables by quantum observables, and the P0 envelopeby the BD envelope.

Recall that the BD operad is an operad over C[[h]] which quantizes the P0 operad.A BD algebra cochain complex with a Poisson bracket of degree 1 and a commutativeproduct, such that the failure of the differential to be a derivation for the commutativeproduct is measured by h times the Poisson bracket.

There is a map of operads over C[[h]] from the Lie operad to the BD operad. At thelevel of algebras, this map takes a BD algebra A to the dg Lie algebra A[−1] over C[[h]],with the Lie bracket given by the Poisson bracket on A. The BD envelope of a dg Lie

18.1. QUANTUM NOETHER’S THEOREM 383

algebra g is defined to be homotopy-universal BD algebra UBD(g) with a map of dg Liealgebras from g[[h]] to UBD(g)[−1].

One can show that for any dg Lie algebra g, the homotopy BD envelope of g is theRees module for the Chevalley chain complex C∗(g) = Sym∗(g[−1]), which is equippedwith the increasing filtration defined by the symmetric powers of g. Concretely,

UBD(g) = C∗(g)[[h]] = Sym∗(g[−1])[[h]]

with differential dg+ hdCE, where dg is the internal differential on g and dCE is the Chevalley-Eilenberg differential. The commutative product and Lie bracket are the h-linear exten-sions of those on the P0 envelope we discussed above. A similar statement holds for L∞algebras.

This discussion holds at the level of factorization algebras too: the BD envelope ofa local L∞ algebra L is defined to be the factorization algebra which assigns to an opensubset U the BD envelope of Lc(U). Thus, it is the Rees factorization algebra associatedto the factorization envelope of U. (We will describe this object in more detail in section18.5 of this chapter).

Now we can state the quantum Noether theorem.

18.1.0.12 Theorem. Suppose we have a quantum field theory on a manifold M acted on by a localL∞ algebra L. Let Obsq be the factorization algebra of quantum observables of this field theory.

In this situation, there is a h-dependent local cocycle

α ∈ H1(Oloc(L[1]))[[h]]

and a homomorphism of factorization algebras from the twisted BD envelope

UBDα (Lc)→ Obsq .

The relationship between this formulation of quantum Noether’s theorem and thetraditional point of view on Noether’s theorem was discussed (in the classical case) insection ??.Let us explain, however, some aspects of this story which are slightly differentin the quantum and classical settings.

Suppose that we have an action of an ordinary Lie algebra g on a quantum field theoryon a manifold M. Then the quantum analogue of the result of lemma ?? (which we willprove below) shows that we have an action of the local dg Lie algebra Ω∗X ⊗ g on thefield theory. It follows that we have a central extention of Ω∗X ⊗ g, given by a class α ∈H1(Oloc(Ω∗X ⊗ g[1])), and a map from the twisted BD envelope of this central extension toobservables of our field theory.


Suppose that N ⊂ X is an oriented codimension 1 submanifold. (We assume forsimplicity that X is also oriented). Let us choose an identification of a tubular neighbour-hood of N with N × R. Let πN : N × R → R denote the projection map to R. Thepush forward of the factorization algebra UBD(Ω∗X ⊗ g) along the projection πR defines alocally-constant facotrization algebra on R, and so an associative algebra.

Let us assume, for the moment, that the central extension vanishes. Then a variant oflemma ?? shows that there is an isomorphism of associative algebras

H∗(

πNUBD(Ω∗X ⊗ g))∼= Rees(U(H∗(N)⊗ g)).

The algebra on the right hand side is the Rees algebra for the universal enveloping algebraof H∗(N)⊗ g. This algebra is a C[[h]]-algebra which specializes at h = 0 to the completedsymmetric algebra of H∗(N)⊗ g, but is generically non-commutative.

In this way, we see that Noether’s theorem gives us a map of factorization algebras onR

Rees(U(g))→ H0(πN Obsq)

where on the right hand side we have quantum observables of our theory, projected to R.

Clearly this is closely related to the traditional formulation of Noether’s theorem: weare saying that every symmetry (i.e. element of g) gives rise to an observable on everycodimension 1 manifold (that is, a current). The operator product between these observ-ables is the product in the universal enveloping algebra.

Now let us consider the case when the central extension is non-zero. A small calcula-tion shows that the group containing possible central extensions can be identified as

H1(Oloc(Ω∗X ⊗ g[1]))[[h]] = Hd+1(X, C∗red(g)) = ⊕i+j=d+1Hi(X)⊗ H jred(g)[[h]],

where d is the real dimension of X, and C∗red(g) is viewed as a constant sheaf of cochaincomplexes on X.

Let us assume that X is of the form N×R, where as above N is compact and oriented.Then the cocycle above can be integrated over N to yield an element in H2

red(g), which canbe viewed as an ordinary, unshifted central extension of the Lie algebra g (which dependson h). We can form the twisted universal enveloping algebra Uα(g), obtained as usual bytaking the universal enveloping algebra of the central extension of g and then setting thecentral parameter to 1. This twisted enveloping algebra admits a filtration, so that we canform the Rees algebra. Our formulation of Noether’s theorem then produces a map offactorization algebras on R

Rees(Uα(g))→ H0(π∗N Obsq).

18.2. ACTIONS OF A LOCAL L∞-ALGEBRA ON A QUANTUM FIELD THEORY 385

18.2. Actions of a local L∞-algebra on a quantum field theory

Let us now turn to the proof of the quantum version of Noether’s theorem. As in thediscussion of the classical theory, the first thing we need to pin down is what it means fora local L∞ algebra to act on a quantum field theory.

As in the setting of classical field theories, there are two variants of the definition weneed to consider: one defining a field theory with an L action, and one a field theory withan inner L-action. Just as in the classical story, the central extension that appears in ourformulation of Noether’s theorem appears as the obstruction to lifting a field theory withan action to a field theory with an inner action.

We have used throughout the definition of quantum field theory given in [Cos11c].The concept of field theory with an action of a local L∞-algebra L relies on a refined defi-nition of field theory, also given in [Cos11c]: the concept of a field theory with backgroundfields. Let us explain this definition.

Let us fix a classical field theory, defined by a local L∞ algebra M on X with an in-variant pairing of cohomological degree −3. Let us choose a gauge fixing operator QGF

onM, as discussed in section ??. Then as before, we have an elliptic differential operator[Q, QGF] (where Q refers to the linear differential onM). As explained in section ??, thisleads to the following data.

(1) A propagator P(Φ) ∈ M[1]⊗2, defined for every parametrix Φ. If Φ, Ψ are para-metrices, then P(Φ)− P(Ψ) is smooth.

(2) A kernel KΦ ∈ M[1]⊗2 for every parametrix Φ, satisfying

Q(P(Φ)− P(Ψ)) = KΨ − KΦ.

These kernels lead, in turn, to the definition of the RG flow operator and of the BV Lapla-cian

W(P(Φ)− P(Ψ),−) : O+P,sm(M[1])[[h]]→ O+

P,sm(M[1])[[h]]

4Φ : O+P,sm(M[1])[[h]]→ O+

P,sm(M[1])[[h]]

associated to parametrices Φ and Ψ. There is also a BV bracket −,−Φ which satisfiesthe usual relation with the BV Laplacian 4Φ. The space O+

P,sm(M[1])[[h]] is the spaceof functionals with proper support and smooth first derivative which are at least cubicmodulo h.

The homological interpretation of these objects are as follows.


(1) For every parametrix Φ, we have the structure of 1-shifted differential graded Liealgebra on O(M[1])[[h]]. The Lie bracket is −,−Φ, and the differential is

Q + I[Φ],−Φ + h4Φ.

The subspace O+sm,P(M[1])[[h]] is a nilpotent sub-dgla. The Maurer-Cartan equa-

tion in this space is called the quantum master equation.(2) The map W(P(Φ)− P(Ψ),−) takes solutions to the QME with parametrix Ψ to

solutions with parametrix Φ. Equivalently, the Taylor terms of this map definean L∞ isomorphism between the dglas associated to the parametrices Ψ and Φ.

If L is a local L∞ algebra, then O(L[1]), with its Chevalley-Eilenberg differential, isa commutative dg algebra. We can identify the space O(L[1]⊕M[1]) of functionals onL[1]⊕M[1] with the completed tensor product

O(L[1]⊕M[1]) = O(L[1])⊗πO(M[1]).

The operations 4Φ, −,−Φ and ∂P(Φ) associated to a parametrix on M extend, byO(L[1])-linearity, to operations on the space O(L[1] ⊕M[1]). For instance, the opera-tor ∂P(Φ) is associated to the kernel

P(Φ) ∈ (M[1])⊗2 ⊂ (M[1]⊕L[1])⊗2.

If dL denotes the Chevalley-Eilenberg differential on O(L[1]), then we can form an op-erator dL ⊗ 1 on O(L[1] ⊕M[1]). Similarly, the linear differential Q on M induces aderivation of O(M[1]) which we also denote by Q; we can for a derivation 1 ⊗ Q ofO(L[1]⊕M[1]).

The operators 4Φ and ∂Φ both commute with dL ⊗ 1 and satisfy the same relationdescribed above with the operator 1⊗Q.

We will letO+

sm,P(L[1]⊕M[1])[[h]] ⊂ O(L[1]⊕M[1])

denote the space of those functionals which satisfy the following conditions.

(1) They are at least cubic modulo h when restricted to be functions just on M[1].That is, we allow functionals which are quadratic as long as they are either qua-dratic in L[1] or linear in both L[1] and inM[1], and we allow linear functionalsas long as they are independent ofM[1]. Further, we work modulo the constantsC[[h]]. (This clause is related to the superscript + in the notation).

(2) We require our functionals to have proper support, in the usual sense (as func-tionals on L[1]⊕M[1]).

(3) We require our functionals to have smooth first derivative, again in the sense wediscussed before. Note that this condition involves differentiation by elements ofboth L[1] andM[1].

18.2. ACTIONS OF A LOCAL L∞-ALGEBRA ON A QUANTUM FIELD THEORY 387

The renormalization group flow operator W(P(Φ)− P(Ψ),−) on the space O+sm,P(M[1])[[h]]

extends to an O(L)-linear operator on the space

O+sm,P(L[1]⊕M[1])[[h]].

It is defined by the equation, as usual,

W(P(Φ)− P(Ψ), I) = h log exp(h∂P(Φ) − h∂P(Ψ)) exp(I/h).

We say that an element

I ∈ O+sm,P(L[1]⊕M[1])[[h]]

satisfies the quantum master equation for the parametrix Φ if it satisfies the equation

dL I + QI + I, IΦ + h4Φ I = 0.

Here dL indicates the Chevalley differential on O(L[1]), extended by tensoring with 1 toan operator on O(L[1]⊕M[1]), and Q is the extension of the linear differential onM[1].

The renormalization group equation takes solutions to the quantum master equationfor the parametrix Φ to those for the parametrix Ψ.

There are two different versions of quantum field theory with an action of a Lie algebrathat we consider: an actionn and an inner action. For theories with just an action, thefunctionals we consider are in the quotient

O+P,sm(L[1]⊕M[1] | L[1])[[h]] = O+

P,sm(L[1]⊕M[1])/OP,sm(L[1])[[h]]

of our space of functionals by those which only depend on L.

Now we can define our notion of a quantum field theory acted on by the local L∞algebra L.

18.2.0.13 Definition. Suppose we have a quantum field theory on M, with space of fieldsM[1].Thus, we have a collection of effective interactions

I[Φ] ∈ O+P,sm(M[1])[[h]]

satisfying the renormalization group equation, BV master equation, and locality axiom, as detailedin subsection 14.2.9.1.

An action of L on this field theory is a collection of functionals

IL[Φ] ∈ OP,sm(L[1]⊕M[1] | L[1])[[h]]satisfying the following properties.

(1) The renormalization group equation

W(P(Φ)− P(Ψ), IL[Ψ]) = IL[Φ].


(2) Each I[Φ] must satisfy the quantum master equation (or Maurer-Cartan equation) forthe dgla structure associated to the parametrix Φ. We can explicitly write out the variousterms in the quantum master equation as follows:

dL IL[Φ] + QIL[Φ] + 12IL[Φ], IL[Φ]Φ + h4Φ IL[Φ] = 0.

Here dL refers to the Chevalley-Eilenberg differential on O(L[1]), and Q to the linear dif-ferential onM[1]. As above, −,−Φ is the Lie bracket on O(M[1]) which is extendedin the natural way to a Lie bracket on O(L[1]⊕M[1]).

(3) The locality axiom, as explained in subsection 14.2.9.1, holds: saying that the support ofIL[Φ] converges to the diagonal as the support of Φ tends to zero, with the same boundsexplained in section 14.2.9.1.

(4) The image of IL[Φ] under the natural map

O+sm,P(L[1]⊕M[1] | L[1])[[h]]→ O+

sm,P(M[1])[[h]]

(given by restricting to functions just of M[1]) must be the original action functionalI[Φ] defining the original theory.

An inner action is defined in exactly the same way, except that the functionals IL[Φ] areelements

IL[Φ] ∈ OP,sm(L[1]⊕M[1])[[h]].

That is, we don’t quotient our space of functionals by functionals just of L[1]. We require thataxioms 1− 5 hold in this context as well.

Remark: One should interpret this definition as a variant of the definition of a family oftheories over a pro-nilpotent base ring A. Indeed, if we have an L-action on a theory onM, then the functionals IL[Φ] define a family of theories over the dg base ring C∗(L(M))of cochains on the L∞ algebra L(M) of global sections of L. In the case that M is compact,the L∞ algebra L(M) often has finite-dimensional cohomology, so that we have a familyof theories over a finitely-generated pro-nilpotent dg algebra.

Standard yoga from homotopy theory tells us that a g-action on any mathematicalobject (if g is a homotopy Lie algebra) is the same as a family of such objects over the basering C∗(g) which restrict to the given object at the central fibre. Thus, our definition of anaction of the sheaf L of L∞ algebras on a field theory on M gives rise to an action (in thishomotopical sense) of the L∞ algebra L(M) on the field theory.

However, our definition of action is stronger than this. The locality axiom we imposeon the action functionals IL[Φ] involves both fields in L and inM. As we will see later,this means that we have a homotopy action of L(U) on observables of our theory on U,for every open subset U ⊂ M, in a compatible way. ♦

18.3. OBSTRUCTION THEORY FOR QUANTIZING EQUIVARIANT THEORIES 389

18.3. Obstruction theory for quantizing equivariant theories

The main result of [Cos11c] as explained in section ?? states that we can constructquantum field theories from classical ones by obstruction theory. If we start with a classi-cal field theory described by an elliptic L∞ algebraM, the obstruction-deformation com-plex is the reduced local Chevalley-Eilenberg cochain complex C∗red,loc(M), which by defi-nition is the complex of local functionals onM[1] equipped with the Chevalley-Eilenbergdifferential.

A similar result holds in the equivariant context. Suppose we have a classical fieldtheory with an action of a local L∞ algebra L. In particular, the elliptic L∞ algebraM isacted on by L, so we can form the semi-direct product L nM. Thus, we can form thelocal Chevalley-Eilenberg cochain complex

Oloc((LnM)[1]) = C∗red,loc(LnM).

The obstruction-deformation complex for quantizing a classical field theory with anaction of L into a quantum field theory with an action of L is the same as the deformationcomplex of the original classical field theory with an action of L. This is the complexC∗red,loc(LnM | L), the quotient of C∗red,loc(LnM) by C∗red,loc(L).

One can also study the complex controlling deformations of the action of L on M,while fixing the classical theory. This is the complex we denoted by Act(L,M) earlier: itfits into an exact sequence of cochain complexes

0→ Act(L,M)→ C∗red,loc(LnM | L)→ C∗red,loc(M)→ 0.

There is a similar remark at the quantum level. Suppose we fix a non-equivariant quan-tization of our original L-equivariant classical theory M. Then, one can ask to lift thisquantization to an L-equivariant quantization. The obstruction/deformation complex forthis problem is the group Act(L,M).

We can analyze, in a similar way, the problem of quantizing a classical field theorywith an inner L-action into a quantum field theory with an inner L-acion. The relevantobstruction/deformation complex for this problem is C∗red,loc(LnM). If, instead, we fixa non-equivariant quantization of the original classical theoryM, we can ask for the ob-struction/deformation complex for lifting this to a quantization with an inner L-action.The relevant obstruction-deformation complex is the complex denoted InnerAct(L,M)in section 17.4. Recall that InnerAct(L,M) fits into a short exact sequence of cochaincomplexes (of sheaves on X)

0→ InnerAct(L,M)→ C∗red,loc(LnM)→ C∗red,loc(M)→ 0.

A more formal statement of these results about the obstruction-deformation com-plexes is the following.


Fix a classical field theoryM with an action of a local L∞ algebra L. Let T (n)L denote

the simplicial set of L-equivariant quantizations of this field theory defined modulo hn+1.The simplicial structure is defined exactly as in chapter 14.2: an n-simplex is a familyof theories over the base ring Ω∗(4n) of forms on the n-simplex. Let T (n) denote thesimplicial set of quantizations without any L-equivariance condition.

Theorem. The simplicial sets T (n)L are Kan complexes. Further, the main results of obstruction

theory hold. That is, there is an obstruction map of simplicial sets

T (n)L → DK

(C∗red,loc(LnM | L)[1]

).

(Here DK denotes the Dold-Kan functor). Further, there is a homotopy fibre diagram

T(n+1)L

// 0

T(n)L

// DK(

C∗red,loc(LnM | L)[1])

.

Further, the natural map

T(n)L → T (n)

(obtained by forgetting the L-equivariance data in the quantization) is a fibration of simplicial sets.

Finally, there is a homotopy fibre diagram

T(n+1)L

// T (n+1) ×T (n) T(n)L

T

(n)L

// DK (Act(L,M)[1]) .

We should interpret the second fibre diagram as follows. The simplicial set T (n+1)×T (n)

T(n)L describes pairs consisting of an L-equivariant quantization modulo hn+1 and a non-

equivariant quantization modulo hn+2, which agree as non-equivariant quantizations mod-ulo hn+1. The deformation-obstruction group to lifting such a pair to an equivariant quan-tization modulo hn+2 is the group Act(L,M). That is, a lift exists if the obstruction classin H1(Act(L,M)) is zero, and the simplicial set of such lifts is a torsor for the simpli-cial Abelian group associated to the cochain complex Act(L,M)). At the level of zero-simplices, the set of lifts is a torsor for H0(Act(L,M)).

18.3. OBSTRUCTION THEORY FOR QUANTIZING EQUIVARIANT THEORIES 391

This implies, for instance, that if we fix a non-equivariant quantization to all orders,then the obstruction-deformation complex for making this into an equivariant quantiza-tion is Act(L,M)).

Further elaborations, as detailed in chapter 14.2, continue to hold in this context. Forexample, we can work with families of theories over a dg base ring, and everything isfibred over the (typically contractible) simplicial set of gauge fixing conditions. In addi-tion, all of these results hold when we work with translation-invariant objects on Rn andimpose “renormalizability” conditions, as discussed in section ??.

The proof of this theorem in this generality is contained in [Cos11c], and is essentiallythe same as the proof of the corresponding non-equivariant theorem. In [Cos11c], theterm “field theory with background fields” is used instead of talking about a field theorywith an action of a local L∞ algebra.

For theories with an inner action, the same result continues to hold, except that theobstruction-deformation complex for the first statement is C∗red,loc(LnM), and in the sec-ond case is InnerAct(L,M).

18.3.1. Lifting actions to inner actions. Given a field theory with an action of L, wecan try to lift it to one with an inner action. For classical field theories, we have seen thatthe obstruction to doing this is a class in H1(Oloc(L[1])) (with, of course, the Chevalley-Eilenberg differential).

A similar result holds in the quantum setting.

18.3.1.1 Proposition. Suppose we have a quantum field theory with an action of L. Then there isa cochain

α ∈ Oloc(L[1])[[h]] = C∗red,loc(L)of cohomological degree 1 which is closed under the Chevalley-Eilenberg differential, such thattrivializing α is the same as lifting L to an inner action.

PROOF. This follows immediately from the obstruction-deformation complexes forconstructing the two kinds of L-equivariant field theories. However, let us explain explic-itly how to calculate this obstruction class (because this will be useful later). Indeed, letus fix a theory with an action of L, defined by functionals

IL[Φ] ∈ O+P,sm(L[1]⊕M[1] | L[1])[[h]].

It is always possible to lift I[Φ] to a collection of functionals

IL[Φ] ∈ O+P,sm(L[1]⊕M[1])[[h]]

which satisfy the RG flow and locality axioms, but may not satisfy the quantum masterequation. The space of ways of lifting is a torsor for the graded abelian group Oloc(L[1])[[h]]


of local functionals on L. The failure of the lift IL[Φ] to satisfy the quantum master equa-tion is, as explained in [Cos11c], independent of Φ, and therefore is a local functionalα ∈ Oloc(L[1]). That is, we have

α = dL IL[Φ] + QIL[Φ] + 12 IL[Φ], IL[Φ]Φ + h4Φ IL[Φ].

Note that functionals just of L are in the centre of the Poisson bracket −,−Φ, and arealso acted on trivially by the BV operator4Φ.

We automatically have dLα = 0. It is clear that to lift IL[Φ] to a functional IL[Φ] whichsatisfies the quantum master equation is equivalent to making α exact in C∗red,loc(L)[[h]].

18.4. The factorization algebra associated to an equivariant quantum field theory

In this section, we will explain what structure one has on observables of an equivari-ant quantum field theory. As above, letM denote the elliptic L∞ algebra on a manifoldM describing a classical field theory, which is acted on by a local L∞-algebra L. Let usdefine a factorization algebra C∗f act(L) by saying that to an open subset U ⊂ M it assignsC∗(L(U)). (As usual, we use the appropriate completion of cochains). Note that C∗f act(L)is a factorization algebra valued in (complete filtered differentiable) commutative dg al-gebras onM.

In this section we will give a brief sketch of the following result.

18.4.0.2 Proposition. Suppose we have a quantum field theory equipped with an action of a localLie algebra L; letM denote the elliptic L∞ algebra associated to the corresponding classical fieldtheory. Then there is a factorization algebra of equivariant quantum observables which is a factor-ization algebra in modules for the factorization algebra C∗(L) of cochains on L. This quantizesthe classical factorization algebra of equivariant observables constructed in proposition 17.3.0.3.

PROOF. The construction is exactly parallel to the non-equivariant version which wasexplained in chapter 14.2, so we will only sketch the details. We define an element ofObsq

L(U) of cohomological degree k to be a family of functionals O[Φ], of cohomologicaldegree k one for every parametrix, on the space L(M)[1]⊕M(M)[1] of fields of the the-ory. We require that, if ε is a parameter of cohomological degree −k and square zero, thatIL[Φ] + εO[Φ] satisfies the renormalization group equation

W(P(Φ)− P(Ψ), IL[Ψ] + εO[Ψ]) = IL[Φ] + εO[Φ].

Further, we require the same locality axiom that was detailed in section 15.4, sayingroughly that O[Φ] is supported on U for sufficiently small parametrices U.


The differential on the complex ObsqL(U) is defined by

(dO)[Φ] = dLO[Φ] + QO[Φ] + IL[Φ], O[Φ]Φ + h4ΦO[Φ],

where Q is the linear differential onM[1], and dL corresponds to the Chevalley-Eilenbergdifferential on C∗(L).

We can make ObsqL(U) into a module over C∗(L(U)) as follows. If O ∈ Obsq

L(U) andα ∈ C∗(L(U)), we can define a new observable α ·O defined by

(α ·O)[Φ] = α · (O[Φ]).

This makes sense, because α is a functional on L(U)[1] and so can be made a functional onM(U)[1]⊕L(U)[1]. The multiplication on the right hand side is simply multiplication offunctionals onM(U)[1]⊕L(U)[1].

It is easy to verify that α ·O satisfies the renormalization group equation; indeed, theinfinitesimal renormalization group operator is given by differentiating with respect to akernel inM[1]⊗2, and so commutes with multiplication by functionals of L[1]. Similarly,we have

d(α ·O) = (dα) ·O + α · dOwhere dO is the differential discussed above, and dα is the Chevalley-Eilenberg differen-tial applied to α ∈ C∗(L(U)[1]).

As is usual, at the classical level we can discuss observables at scale 0. The differentialat the classical level is dL + Q + IL,− where IL ∈ Oloc(L[1] ⊕M[1]) is the classicalequivariant Lagrangian. This differential is the same as the differential on the Chevalley-Eilenberg differential on the cochains of the semi-direct product L∞ algebra LnM. Thus,it is quasi-isomorphic, at the classical level, to the one discussed in proposition 17.3.0.3.

18.5. Quantum Noether’s theorem

Finally, we can explain Noether’s theorem at the quantum level. As above, supposewe have a quantum field theory on a manifold M with space of fieldsM[1]. Let L be alocal L∞ algebra which acts on this field theory. Let α ∈ H1(C∗red,loc(L))[[h]] denote theobstruction to lifting this action to an inner action.

Recall that the factorization envelope of the local L∞ algebra L is the factorizationalgebra whose value on an open subset U ⊂ M is the Chevalley chain complex C∗(Lc(U)).Given a cocycle β ∈ H1(C∗red,loc(L)), we can form a shifted central extension

0→ C[−1]→ Lc → Lc → 0

of the precosheaf Lc of L∞ algebras on M. Central extensions of this form have alreadybeen discussed in section ??.


We can then form the twisted factorization envelope Uβ(L), which is a factorizationalgebra on M. The twisted factorization envelope is defined by saying that it’s value onan open subset V ⊂ M is

Uβ(L)(V) = Cc=1 ⊗C[c] C∗(Lc(V)).

Here the complex C∗(Lc(V)) is made into a C[c]-module by multiplying by the centralelement.

We have already seen ?? that the Kac-Moody vertex algebra arises as an example of atwisted factorization envelope.

There is a C[[h]]-linear version of the twisted factorization envelope construction too:if our cocycle α is in H1(C∗red,loc(L))[[h]], then we can form a central extension of the form

0→ C[[h]][−1]→ Lc[[h]]→ Lc[[h]]→ 0.

This is an exact sequence of precosheaves of L∞ algebras on M in the category of C[[h]]-modules. By performing the C[[h]]-linear version of the construction above, one finds thetwisted factorization envelope Uα(L). This is a factorization algebra on M in the categoryof C[[h]]-modules, whose value on an open subset V ⊂ M is

Uα(L)(V) = C[[h]]c=1 ⊗C[[h]][c] C∗(Lc[[h]]).

Here Chevalley chains are taken in the C[[h]]-linear sense.

Our version of Noether’s theorem will relate the factorization envelope of Lc, twistedby the cocycle α, to the factorization algebra of quantum observables of the field theoryon M. The main theorem is the following.

18.5.0.3 Theorem. Suppose that the local L∞-algebra L acts on a field theory on M, and that theobstruction to lifting this to an inner action is a cocycle α ∈ H1(C∗red,loc(L))[[h]]. Then, there is aC((h))-linear homomorphism of factorization algebras

Uα(L)[h−1]→ Obsq[h−1].

(Note that on both sides we have inverted h).

One can ask how this relates to Noether’s theorem for classical field theories. In orderto provide such a relationship, we need to state a version of quantum Noether’s theoremwhich holds without inverting h. For every open subset V ⊂ M, we define the Reesmodule

Rees Uα(L)(V) ⊂ Uα(L)(V)

to be the submodule spanned by elements of the form hkγ where γ ∈ Sym≤k(Lc(V)). Thisis a sub-C[[h]]-module, and also forms a sub-factorization algebra. The reason for the ter-minology is that in the case α = 0, or more generally α is independent of h, Rees Uα(L)(V)is the Rees module for the filtered chain complex Cα

∗(Lc(V)).


One can check that Rees Uα(V) is a free C[[h]]-module and that, upon inverting h, wefind

(Rees Uα(V)) [h−1] = Uα(V).

18.5.0.4 Theorem. The Noether map of factorization algebras

Uα(L)[h−1]→ Obsq[h−1]

over C((h)) refines to a mapRees Uα(L)→ Obsq

of factorization algebras over C[[h]].

We would like to compare this statement to the classical version of Noether’s theorem.Let α0 denote the reduction of α modulo h. Let Lc denote the central extension ofLc arisingfrom α0. We have seen that the classical Noether’s theorem states that there is a map ofprecosheaves L∞ algebras

Lc → Obscl[−1]

where on the right hand side, Obscl[−1] is endowed with the structure of dg Lie algebra

coming from the shifted Poisson bracket on Obscl

. Further, this map sends the central

element in Lc to the unit element in Obscl[−1].

In particular, the classical Noether map gives rise to a map of precosheaves of cochaincomplexes

Lc[1]→ Obscl .

We will not use the fact that this arises from an L∞ map in what follows. Because Obscl isa commutative factorization algebra, we automatically get a map of commutative prefac-torization algebras

Sym∗ Lc[1]→ Obscl .Further, because the Noether map sends the central element to the unit observable, we geta map of commutative factorization algebras

(†) Cc=1 ⊗C[c] Sym∗ Lc[1]→ Obscl .

Now we have set up the classical Noether map in a way which is similar to the quantumNoether map. Recall that the quantum Noether map with h not inverted is expressed interms of the Rees module Rees Uα(L). When we set h = 0, we can identify

Rees Uα(L)(V)⊗C[[h]] Ch=0 = Sym∗(Lc(V))⊗C[c] Cc=1.

18.5.0.5 Lemma. The quantum Noether map

Rees Uα(L)→ Obsq

of factorization algebras becomes, upon setting h = 0, the map in equation (†).


18.5.1. Proof of the quantum Noether theorem. Before we being our proof of quan-tum Noether’s theorem, it will be helpful to discuss the meaning (in geometric terms) ofthe chains and cochains of an L∞ algebra twisted by a cocycle.

If g is an L∞ algebra, then C∗(g) should be thought of as functions on the formalmoduli problem Bg associated to g. Similarly, C∗(g) is the space of distributions on Bg. Ifα ∈ H1(C∗(g)), then α defines a line bundle on Bg, or equivalently, a rank 1 homotopyrepresentation of g. Sections of this line bundle are C∗(g) with the differential dg − α, i.e.we change the differential by adding a term given by multiplication by −α. Since α isclosed and of odd degree, it is automatic that this differential squares to zero. We willsometimes refer to this complex as C∗α(g).

Similarly, we can define C∗,α(g) to be C∗(g) with a differential given by adding theoperator of contracting with −α to the usual differential. We should think of C∗,α(g) asthe distributions on Bg twisted by the line bundle associated to α; i.e. distributions whichpair with sections of this line bundle.

Let g be the shifted central extension of g associated to α. Then C∗(g) is a module overC[c], where c is the central parameter. Then we can identify

C∗(g)⊗C[c] Cc=1 = C∗,α(g).

A similar remark holds for cochains.

In particular, if L is a local L∞ algebra on a manifold M and α ∈ H1(C∗red,loc(L)) is alocal cochain, then

Uα(L)(V) = C∗,α(Lc(V))

for an open subset V ⊂ M.

Now we will turn to the proof of theorems 18.5.0.3 and 18.5.0.4 and lemma 18.5.0.5, allstated in the previous section.

The first thing we need to do is to produce, for every open subset V ⊂ M, a chain map

Cα∗(Lc(V))→ Obsq(V)[h−1].

A linear map

f : Sym∗(Lc(V)[1])→ Obsq(V)[h−1]

is the same as a collection of linear maps

f [Φ] : Sym∗(Lc(V)[1])→ O(M(M)[1])((h))

one for every parametrix Φ, which satisfy the renormalization group equation and thelocality axiom. This, in turn, is the same as a collection of functionals

O[Φ] ∈ O(Lc(V)[1]⊕M(M)[1])((h))


satisfying the renormalization group equation and the locality axiom. We are using thenatural pairing between the symmetric algebra of Lc(V)[1] and the space of functionalson Lc(V)[1] to identify a linear map f [Φ] with a functional O[Φ].

We will write down such a collection of functionals. Recall that, because we have anaction of the local L∞ algebra L on our theory, we have a collection of functionals

IL[Φ] ∈ O(Lc(M)[1]⊕Mc(M)[1])[[h]]

which satisfy the renormalization group equation and the following quantum masterequation:

(dL + Q)IL[Φ] + 12IL[Φ], IL[Φ]Φ + h4Φ

(IL[Φ]

)= α.

We will use the projections from Lc(M)[1]⊕Mc(M)[1] to Lc(M)[1] andMc(M)[1] to liftfunctionals on these smaller spaces to functionals on Lc(M)[1]⊕Mc(M)[1]. In particular,if as usual I[Φ] denotes the effective action of our quantum field theory, which is a func-tion of the fields inMc(M)[1], we will use the same notation to denote the lift of I[Φ] to afunction of the fields in Lc(M)[1]⊕Mc(M)[1].

LetIL[Φ] = IL[Φ]− I[Φ] ∈ O(Lc(M)[1]⊕Mc(M)[1])[[h]].

This functional satisfies the following master equation:

(dL + Q) IL[Φ] + 12 IL[Φ], IL[Φ]Φ + I[Φ], IL[Φ]Φ + h4Φ

(IL[Φ]

)= α.

The renormalization group equation for the functionals IL[Φ] states that

exp(

h∂P(Φ) − h∂P(Ψ)

)exp (I[Ψ]/h) exp

(IL[Ψ]/h

)= exp (I[Φ]/h) exp

(IL[Φ]/h

).

This should be compared with the renormalization group equation that an observableO[Φ] in Obsq(M) satisfies:

exp(

h∂P(Φ) − h∂P(Ψ)

)exp (I[Ψ]/h)O[Ψ] = exp (I[Φ]/h)O[Φ].

Note also thatIL[Φ] ∈ O(Lc(V)[1]⊕M(M))[[h]].

The point is the following. Let

ILi,k,m[Φ] : Lc(M)⊗k ×Mc(M)⊗m → C

denote the coefficients of hi in the Taylor terms of this functional. This Taylor term is zerounless k > 0, and further it has proper support (which can be made as close as we like tothe diagonal by making Φ small). The proper support condition implies that this Taylorterm extends to a functional

Lc(M)⊗k ×M(M)⊗m → C,


that is, only one of the inputs has to have compact support and we can choose this to bean L-input.

From this, it follows that

exp(

IL[Φ]/h)∈ O(Lc(V)[1]⊕M(M)[1])((h)).

Although there is a h−1 in the exponent on the left hand side, each Taylor term of thisfunctional only involves finitely many negative powers of h, which is what is required tobe in the space on the right hand side of this equation.

Further, the renormalization group equation satisfied by exp(

IL[Φ]/h)

is preciselythe one necessary to define (as Φ varies) an element which we denote

exp(

IL/h)∈ C∗α(Lc(V), Obsq(M))[h−1].

The locality property for the functionals IL[Φ] tells us that for Φ small these functionalsare supported arbitrarily close to the diagonal. This locality axiom immediately impliesthat

exp(

IL/h)∈ C∗α(Lc(V), Obsq(V))[h−1].

Thus, we have produced the desired linear map

F : Cα∗(Lc(V))→ Obsq(V)[h−1].

Explicitly, this linear map is given by the formula

F(l)[Φ] =⟨

l, exp(

IL[Φ]/h)⟩

where 〈−,−〉 indicates the duality pairing between Cα∗(Lc(V)) and C∗α(Lc(V)).

Next, we need to verify that F is a cochain map. Since the duality pairing betweenchanges and cochains of Lc(V) (twisted by α) is a cochain map, it suffices to check thatthe element exp

(IL/h

)is closed. This is equivalent to saying that, for each parametrix

Φ, the following equation holds:

(dL − α + h4Φ + I[Φ],−Φ) exp(

IL/h)= 0.

Here, dL indicates the Chevalley-Eilenberg differential on C∗(Lc(V)) and α indicates theoperation of multiplying by the cochain α in C1(Lc(V)).

This equation is equivalent to the statement that

(dL − α + h4Φ) exp(

I[Φ]/h + IL[Φ]/h)= 0

which is equivalent to the quantum master equation satisfied by IL[Φ].


Thus, we have produced a cochain map from Cα∗(Lc(V)) to Obsq(V)[h−1]. It remains

to show that this cochain map defines a map of factorization algebras.

It is clear from the construction that the map we have constructed is a map of pre-cosheaves, that is, it is compatible with the maps coming from inclusions of open setsV ⊂W. It remains to check that it is compatible with products.

Let V1, V2 be two disjoint subsets of M, but contained in W. We need to verify that thefollowing diagram commutes:

Cα∗(Lc(V1))× Cα

∗(Lc(V2)) //

Obsq(V1)[h−1]×Obsq(V2)[h−1]

Cα∗(Lc(W)) // Obsq(W)[h−1].

Let li ∈ Cα∗(Lc(Vi)) for i = 1, 2. Let · denote the factorization product on the factorization

algebra Cα∗(Lc). This is simply the product in the symmetric algebra on each open set,

coupled with the maps coming from the inclusions of open sets.

Recall that if Oi are observables in the open sets Vi, then the factorization productO1O2 ∈Obsq(W) of these observables is defined by

(O1O2)[Φ] = O1[Φ] ·O2[Φ]

for Φ sufficiently small, where · indicates the obvious product on the space of functionsonM(M)[1]. (Strictly speaking, we need to check that for each Taylor term this identityholds for sufficiently small parametrices, but we have discussed this technicality manytimes before and will not belabour it now).

We need to verify that, for Φ sufficiently small,

F(l1)[Φ] · F(l2)[Φ] = F(l1 · l2)[Φ] ∈ O(M(M)[1])((h)).

By choosing a sufficiently small parametrix, we can assume that IL[Φ] is supported asclose to the diagonal as we like. We can further assume, without loss of generality, thateach li is a product of elements in Lc(Vi). Let us write li = m1i . . . mkii for i = 1, 2 and eachmji ∈ Lc(Vi). (To extend from this special case to the case of general li requires a smallfunctional analysis argument using the fact that F is a smooth map, which it is. Since werestrict attention to this special case only for notation convenience, we won’t give moredetails on this point).

Then, we can explicitly write the map F applied to the elements li by the formula

F(li)[Φ] =

∂

∂m1i. . .

∂

∂mkiiexp

(IL[Φ]/h

)|0×M(M)[1] .


In other words, we apply the product of all partial derivatives by the elements mji ∈Lc(Vi) to the function exp

(IL[Φ]/h

)(which is a function on Lc(M)[1]⊕M(M)[1]) and

then restrict all the Lc(Vi) variables to zero.

To show thatF(l1 · l2)[Φ] = F(l1)[Φ] · F(l2)[Φ]

for sufficiently small Φ, it suffices to verify that∂

∂m11. . .

∂

∂mk11exp

(IL[Φ]/h

) ∂

∂m12. . .

∂

∂mk22exp

(IL[Φ]/h

)=

∂

∂m11. . .

∂

∂mk11

∂

∂m12. . .

∂

∂mk22exp

(IL[Φ]/h

).

Each side can be expanded, in an obvious way, as a sum of terms each of which is aproduct of factors of the form

(†)∂

∂mj1i1. . .

∂

∂mjr irIL[Φ]

together with an overall factor of exp(

IL[Φ]/h)

. In the difference between the two sides,all terms cancel except for those which contain a factor of the form expressed in equation(†) where i1 = 1 and i2 = 2. Now, for sufficiently small parametrices,

∂

∂mj11

∂

∂mj22IL[Φ] = 0

because IL[Φ] is supported as close as we like to the diagonal and mj11 ∈ Lc(V1) andmj22 ∈ Lc(V2) have disjoint support.

Thus, we have constructed a map of factorization algebras

F : Uα(L)→ Obsq[h−1].

It remains to check the content of theorem 18.5.0.4 and of lemma 18.5.0.5. For theorem18.5.0.4, we need to verify that if

l ∈ Symk(Lc(V))

(for some open subset V ⊂ M) then

F(l) ∈ h−k Obsq(V).

That is, we need to check that for each Φ, we have

F(l)[Φ] ∈ h−kO(M(M)[1])[[h]].

Let us assume, for simplicty, that l = m1 . . . mk where mi ∈ Lc(V). Then the explicitformula

F(l)[Φ] =

∂m1 . . . ∂mk exp(

IL[Φ]/h)|M(M)[1]

18.6. THE QUANTUM NOETHER THEOREM AND EQUIVARIANT OBSERVABLES 401

makes it clear that the largest negative power of h that appears is h−k. (Note that IL[Φ] iszero when restricted to a function of justM(M)[1].)

Finally, we need to check lemma 18.5.0.5. This states that the classical limit of ourquantum Noether map is the classical Noether map we constructed earlier. Let l ∈ Lc(V).Then the classical limit of our quantum Noether map sends l to the classical observable

limh→0

hF(l) = limΦ→0

limh→0

h∂l exp

(IL[Φ]/h

)|M(M)[1]

= limΦ→0

∂l ILclassical [Φ]

|M(M)[1]

=

∂l ILclassical

|M(M)[1] .

Note that by ILclassical [Φ] we mean the scale Φ version of the functional on L[1] ⊕M[1]which defines the inner action (at the classical level) of L on our classical theory, and byILclassical we mean the scale zero version.

Now, our classical Noether map is the map appearing in the last line of the abovedisplayed equation.

This completes the proof of theorems 18.5.0.3 and 18.5.0.4 and lemma 18.5.0.5.

18.6. The quantum Noether theorem and equivariant observables

So far in this chapter, we have explained that if we have a quantum theory with anaction of the local L∞ algebra L, then one finds a homotopical action of L on the quantumobservables of the theory. We have also stated and proved our quantum Noether theorem:in the same situation, there is a homomorphism from the twisted factorization envelopeof L to the quantum observables. It is natural to expect that these two constructions areclosely related. In this section, we will explain the precise relationship. Along the way, wewill prove a somewhat stronger version of the quantum Noether theorem. The theoremswe prove in this section will allow us to formulate later a definition of the local index of anelliptic complex in the language of factorization algebras.

Let us now give an informal statement of the main theorem in this section. Quantumobservables on U have a homotopy action of the sheaf L(U) of L∞ algebras. By restrictingto compactly-supported sections, we find that Obsq(U) has a homotopy action of Lc(U).This action is compatible with the factorization structure, in the sense that the productmap

Obsq(U)⊗Obsq(V)→ Obsq(W)

(defined when UqV ⊂W) is a map of Lc(U)⊕Lc(V)-modules, where Obsq(W) is madeinto an Lc(U) ⊕ Lc(V) module via the natural inclusion map from this L∞ algebra toLc(W).


We can say that the factorization algebra Obsq is an Lc-equivariant factorization alge-bra.

It can be a little tricky formulating correctly all the homotopy coherences that go intosuch an action. When we state our theorem precisely, we will formulate this kind of actionslightly differently in a way which captures all the coherences we need.

In general, for an L∞ algebra g, an element α ∈ H1(C∗(g)) defines an L∞-homomorphismg → C (where C is given the trivial L∞ structure). In other words, we can view such acohomology class as a character of g. (More precisely, the choice of a cochain representa-tive of α leads to such an L∞ homomorphism, and different cochain representatives giveL∞-equivalent L∞-homomorphisms).

Suppose thatL is a local L∞ algebra on M and that we have an element α ∈ H1(C∗red,loc(L)).This means, in particular, that for every open subset V ⊂ M we have a character of Lc(V),and so an action of Lc(V) on C. Let C denote the trivial factorization algebra, which as-signs the vector space C to each open set. The fact that α is local guarantees that the actionof each Lc(V) on C makes C into an Lc-equivariant factorization algebra. Let us denotethis Lc-equivariant factorization algebra by Cα.

More generally, given any Lc-equivariant factorization algebra F on M, we can forma new Lc-equivariant factorization algebra Fα, defined to be the tensor product of F andCα in the category of factorization algebras with multiplicative Lc-actions.

Suppose that we have a field theory on M with an action of a local L∞ algebra L, andwith factorization algebra of quantum observables Obsq. Let α ∈ H1(C∗red,loc(L))[[h]] bethe obstruction to lifting this to an inner action. Let Cα[[h]] denote the trivial factorizationalgebra C[[h]] viewed as an Lc-equivariant factorization algebra using the character α. Aswe have seen above, the factorization algebra Obsq is an Lc-equivariant factorization al-gebra. We can tensor Obsq with Cα[[h]] to form a new Lc-equivariant factorization algebraObsq

α (the tensor product is of course taken over the base ring C[[h]]). As a factorizationalgebra, Obsq

α is the same as Obsq. Only the Lc-action has changed.

The main theorem is the following.

18.6.0.1 Theorem. The Lc-action on Obsqα[h−1] is homotopically trivial.

In other words, after twisting the action of Lc by the character α, and inverting h,the action of Lc on observables is homotopically trivial. The trivialization of the actionrespects the fact that its multiplicative.


An alternative way to state the theorem is that the action of Lc on Obsq[h−1] is homo-topically equivalent to the α-twist of the trivial action. That is, on each open set V ⊂ Mthe action of Lc(V) on Obsq(V)[h−1] is by the identity times the character α.

Let us now explain how this relates to Noether’s theorem. If we have anLc-equivariantfactorization algebra F (valued in convenient or pro-convenient vector spaces) then onevery open subset V ⊂ M we can form the Chevalley chain complex C∗(Lc(V),F (V))of Lc(V) with coefficients in F (V). This is defined by taking the (bornological) tensorproduct of C∗(Lc(V)) with F (V) on every open set, with a differential which incorpo-rates the usual Chevalley differential as well as the action of Lc(V) on F (V). The cochaincomplexes C∗(Lc(V),F (V)) form a new factorization algebra which we call C∗(Lc,F ).(As we will see shortly in our more technical statement of the theorem, the factorizationalgebra C∗(Lc,F ), with a certain structure of C∗(Lc)-comodule, encodes the F as an Lc-equivariant factorization algebra).

In particular, when we have an action of L on a quantum field theory on a manifoldM, we can form the factorization algebra C∗(Lc, Obsq), and also the version of this twistedby α, namely C∗(Lc, Obsq

α). We can also consider the chains of Lc with coefficients on Obsq

with the trivial action: this is simply U(Lc)⊗Obsq (where we complete the tensor productto the bornological tensor product).

Then, the theorem above implies that we have an isomorphism of factorization alge-bras

Φ : C∗(Lc, Obsqα)[h−1] ∼= U(Lc)⊗Obsq[h−1].

(Recall that U(Lc) is another name for C∗(Lc) with trivial coefficients, and that the tensorproduct on the right hand side is the completed bornological one).

Now, the action of Lc on Obsq preserves the unit observable. This means that the unitmap

1 : C[[h]]→ Obsq

of factorization algebras is Lc-equivariant. Taking Chevalley chains and twisting by α, weget a map of fatorization algebras

1 : C∗(Lc, Cα[[h]]) = Uα(Lc)→ C∗(Lc, Obsqα).

Further, there is a natural map of factorization algebras

ε : U(Lc)→ C,

which on every open subset is the map C∗(Lc(V))→ C which projects onto Sym0 Lc(V).(This map is the counit for a natural cocommutative coalgebra structure on C∗(Lc(V)).Tensoring this with the identity map gives a map

ε⊗ Id : U(Lc)⊗Obsq → Obsq .


The compatibility of the theorem stated in this section with the Noether map is the fol-lowing.

Theorem. The composed map of factorization algebras

Uα(Lc)1−→ C∗(Lc, Obsq

α)[h−1]∼=−→ U(Lc)⊗Obsq[h−1]

ε⊗Id−−→ Obsq[h−1]

is the Noether map from theorem 18.5.0.3.

This theorem therefore gives a compatibility between the action of Lc on observablesand the Noether map.

18.6.1. Let us now turn to a more precise statement, and proof, of theorem 18.6.0.1.We first need to give a more careful statement of what it means to have a factorizationalgebra with a multiplicative action of Lc, where, as usual, L indicates a local L∞ algebraon a manifold M.

The first fact we need is that the factorization algebra U(L), which assigns to everyopen subset V ⊂ M the complex C∗(Lc(V)), is a factorization algebra valued in commu-tative coalgebras (in the symmetric monoidal category of convenient cochain complexes).

To see this, first observe that for any L∞ algebra g, the cochain complex C∗(g) is acocommutative dg coalgebra. The coproduct

C∗(g)→ C∗(g)⊗ C∗(g)

is the map induced from the diagonal map of Lie algebras g → g ⊕ g, combined withthe Chevalley-Eilenberg chain complex functor. Here we are using the fact that there is anatural isomorphism

C∗(g⊕ h) ∼= C∗(g)⊗ C∗(h).

We want, in the same way, to show that C∗(Lc(V)) is a cocommutative coalgebra, forevery open subset V ⊂ M. The diagonal map Lc(V)→ Lc(V)⊕Lc(V) gives us, as above,a putative coproduct map

C∗(Lc(V))→ C∗(Lc(V)⊕Lc(V)).

The only point which is non-trivial is to verify that the natural map

C∗(Lc(V))⊗βC∗(Lc(V))→ C∗(Lc(V)⊕Lc(V))

is an isomorphism, where on the right hand side we use the completed bornological tensorproduct of convenient vector spaces.

The fact that this map is an isomorphism follows from the fact that for any two mani-folds X and Y, we have a natural isomorphism

C∞c (X)⊗βC∞

c (Y) ∼= C∞c (X×Y).


Thus, for every open subset V ⊂ M, U(L)(V) is a cocommutative coalgebra. The factthat the assignment of the cocommutative coalgebra C∗(g) to an L∞ algebra g is functorialimmediately implies that U(L)(V) is a prefactorization algebra in the category of cocom-mutative coalgebras. (It is a factorization algebra, and not just a pre-algebra, because theforgetful functor from cocommutative coalgebras to cochain complexes preserves colim-its).

Now we can give a formal definition of a factorization algebra with a multiplicativeLc-action.

18.6.1.1 Definition. LetF be a factorization algebra on a manifold M valued in convenient vectorspaces, and let L be a local L∞ algebra. Then, a multiplicative Lc-action on F is is the following:

(1) A factorization algebra FL in the category of (convenient) comodules for U(L).(2) Let us give F the trivial coaction of U(L). Then, we have a map of dg U(L)-comodule

factorization algebras F → FL.(3) For every open subset V ⊂ M, FL is quasi-cofree: this means that there is an isomor-

phismFL ∼= U(Lc)(V)⊗βF (V).

of graded, but not dg, U(Lc)(V) comodules, such that the given map from F (V) is ob-tained by tensoring the identity onF (V) with the coaugmentation map C→ U(Lc)(V).(The coaugmentation map is simply the natural inclusion of C into C∗(Lc(V)) = Sym∗ Lc(V)[1]).

More generally, suppose that F is a convenient factorization algebra with a complete decreas-ing filtration. We give U(L) a complete decreasing filtration by saying that Fi(U(L))(V) = 0for i > 0. In this situation, a multiplicative Lc-action on F is a complete filtered convenient U(L)comodule FLc with the same extra data and properties as above, except that the tensor product isthe one in the category of complete filtered convenient vector spaces.

One reason that this is a good definition is the following.

18.6.1.2 Lemma. Suppose that F is a (complete filtered) convenient factorization algebra with amultiplicative Lc action in the sense above. Then, for every open subset V ⊂ M, there is an L∞action of Lc(V) on F (V) and an isomorphism of dg C∗(Lc(V))-comodules

C∗(Lc(V),F (V)) ∼= FLc(V),

where on the left hand side we take chains with coefficients in the L∞-module F (V).

PROOF. Let g be any L∞ algebra. There is a standard way to translate between L∞g-modules and C∗(g)-comodules: if W is an L∞ g-module, then C∗(g, W) is a C∗(g)-comodule. Conversely, to give a differential on C∗(g)⊗W making it into a C∗(g)-comodulewith the property that the map W → C∗(g)⊗W is a cochain map, is the same as to give


an L∞ action of g on W. Isomorphisms of comodules (which are the identity on the copyof W contained in C∗(g, W)) are the same as L∞ equivalences.

This lemma just applies these standard statements to the symmetric monoidal cate-gory of convenient cochain complexes.

To justify the usefulness of our definition to the situation of field theories, we need toshow that if we have a field theory with an action of L then we get a factorization algebrawith a multiplicative Lc action in the sense we have explained.

18.6.1.3 Proposition. Suppose we have a field theory on a manifold M with an action of a localL∞ algebra L. Then quantum observables Obsq of the field theory have a multiplicative Lc-actionin the sense we described above.

PROOF. We need to define the space of elements of Obsq,Lc .

Now we can give the precise statement, and proof, of the main theorem of this section.Suppose that we have a quantum field theory on a manifold M, with an action of a localL∞ algebra L. Let α ∈ H1(C∗red,loc(L))[[h]] be the obstruction to lifting this to an inneraction.

18.7. Noether’s theorem and the local index

In this section we will explain how Noether’s theorem – in the stronger form formu-lated in the previous section – gives rise to a definition of the local index of an ellipticcomplex with an action of a local L∞ algebra.

Let us explain what we mean by the local index. Suppose we have an elliptic complexon a compact manifold M. We will let E (U) denote the cochain complex of sections ofthis elliptic complex on an open subset U ⊂ M.

Then, the cohomology of E (M) is finite dimensional, and the index of our ellipticcomplex is defined to be the Euler characteristic of the cohomology. We can write this as

Ind(E (M)) = STrH∗(E (M)) Id .

That is, the index is the super-trace (or graded trace) of the identity operator on cohomol-ogy.

More generally, if g is a Lie algebra acting on global sections of our elliptic complexE (M), then we can consider the character of g on H∗(E (M)). If X ∈ g is any element, thecharacter can be written as

Ind(X, E (M)) = STrH∗(E (M))X.

18.7. NOETHER’S THEOREM AND THE LOCAL INDEX 407

Obviously, the usual index is the special case when g is the one-dimensional Lie algebraacting on E (M) by scaling.

We can rewrite the index as follows. For any endomorphism X of H ∗ (E (M)), thetrace of X is the same as the trace of X acting on the determinant of H∗(E (M)). Notethat for this to work, we need H∗(E (M)) to be treated as a super-line: it is even or odddepending on whether the Euler characteristic of H∗(E (M)) is even or odd.

It follows that the character of the action of a Lie algebra g on E (M)) can be encodedentirely in the natural action of g on the determinant of H∗(E (M)). In other words,the character of the g action is the same data as the one-dimensional g-representationdet H∗(E (M)).

Now suppose that g is global sections of a sheaf L of dg Lie algebras (or L∞ algebras)on M. We will further assume that L is a local L∞ algebra. Let us also assume that theaction of g = L(M) on E (M) arises from a local action of the sheaf L of L∞ algebras onthe sheaf E of cochain complexes.

Then, one can ask the following question: is there some way in which the character ofthe L(M) action on E (M) can be expressed in a local way on the manifold? Since, as wehave seen, the character of the L(M) action is entirely expressed in the homotopy L(M)action on the determinant of the cohomology of E (M), this question is equivalent to thefollowing one: is it possible to express the determinant of the cohomology of E (M) in away local on the manifold M, in an L-equivariant way?

Now, E (M) is a sheaf, so that we can certainly describe E (M) in a way local on M.Informally, we can imagine E (M) as being a direct sum of its fibres at various points inM. More formally if we choose a cover U of M, then the Cech double complex for U withcoefficients in the sheaf E produces for us a complex quasi-isomorphic to E (M). Thisdouble complex is an additive expression describing E (M) in terms of sections of E in theopen cover U of M.

Heuristically, the Cech double complex gives a formula of the form

E (M) ∼∑i

E (Ui)−∑i,j

E (Ui ∩Uj) + ∑i,j,k

E (Ui ∩Uj ∩Uk)− . . .

which we should imagine as the analog of the inclusion-exclusion formula from combi-natorics. If U is a finite cover and each E (U) has finite-dimensional cohomology, thisformula becomes an identity upon taking Euler characteristics.

Since M is compact, one can also view E (M) as the global sections of the cosheaf ofcompactly supported sections of E , and then Cech homology gives us a similar expres-sion.


The determinant functor from vector spaces to itself takes sums to tensor products.We thus could imagine that the determinant of the cohomology of E (M) can be expressedin a local way on the manifold M, but where the direct sums that appear in sheaf theoryare replaced by tensor products.

Factorization algebras have the feature that the value on a disjoint union is a tensorproduct (rather than a direct sum as appears in sheaf theory). That is, factorization alge-bras are multiplicative versions of cosheaves.

It is therefore natural to express that the determinant of the cohomology of E (M) canbe realized as global sections of a factorization algebra, just as E (M) is global sections ofa cosheaf.

It turns out that this is the case.

18.7.0.4 Lemma. Let E be any elliptic complex on a compact manifold M. Let us form the freecotangent theory to the Abelian elliptic Lie algebra E [−1]. This cotangent theory has ellipticcomplex of fields E ⊕ E ![−1].

Let ObsqE denote the factorization algebra of observables of this theory. Then, there is a quasi-

isomorphism

H∗(ObsqE (M)) = det H∗(E (M))[d]

where d is equal to the Euler characteristic of H∗(E (M)) modulo 2.

Recall that by det H∗(E (M)) we mean

det H∗(E (M)) = ⊗i

det Hi(E (M))

(−1)i

.

This lemma therefore states that the cohomology of global observables of the theory is thedeterminant of the cohomology of E (M), with its natural Z/2 grading. The proof of thislemma, although easy, will be given at the end of this section.

This lemma shows that the factorization algebra ObsqE is a local version of the deter-

minant of the cohomology of E (M). One can then ask for a local version of the index.Suppose that L is a local L∞ algebra on M which acts linearly on E . Then, as we haveseen, the precosheaf of L∞-algebras given by compactly supported sections Lc of L actson the factorization algebra Obsq. We have also seen that, up to coherent homotopieswhich respect the factorization algebra structure, the action of Lc(U) on Obsq(U) is by acharacter α times the identity matrix.

18.7.0.5 Definition. In this situation, the local index is the multiplicative Lc-equivariant fac-torization algebra Obsq

E .

18.7. NOETHER’S THEOREM AND THE LOCAL INDEX 409

This makes sense, because as we have seen, the action of L(M) on ObsqE (M) is the

same data as the character of the L(M) action on E (M), that is, the index.

Theorem 18.6.0.1 tells us that the multiplicative action of Lc on ObsqE is through the

character α of Lc, which is also the obstruction to lifting the action to an inner action.

18.7.1. Proof of lemma 18.7.0.4. Before we give the (simple) proof, we should clar-ify some small points. Recall that for a free theory, there are two different versions ofquantum observables we can consider. We can take our observables to be polynomialfunctions on the space of fields, and not introduce the formal parameter h; or we can takeour observables to be formal power series on the space of fields, in which case one needsto introduce the parameter h. These two objects encode the same information: the secondconstruction is obtained by applying the Rees construction to the first construction. Wewill give the proof for the first (polynomial) version of quantum observables. A similarstatement holds for the second (power series) version, but one needs to invert h and tensorthe determinant of cohomology by C((h)).

Globally, polynomial quantum observables can be viewed as the space P(E (M)) ofpolynomial functions on E (M), with a differential which is a sum of the linear differentialQ on E (M) with the BV operator. Let us compute the cohomology by a spectral sequenceassociated to a filtration of Obsq

E (M). The filtration is the obvious increasing filtrationobtained by declaring that

Fi ObsqE (M) = Sym≤i(E (M)⊕ E !(M)[−1])∨.

The first page of this spectral sequence is cohomology of the associated graded. The asso-ciated graded is simply the symmmetric algebra

H∗Gr ObsqE (M) = Sym∗

(H∗(E (M))∨ ⊕ H∗(E !(M)[−1])∨

).

The differential on this page of the spectral sequence comes from the BV operator asso-ciated to the non-degenerate pairing between H∗(E (M)) and H∗(E !(M))[−1]. Note thatH∗(E !(M)) is the dual to H∗(E (M)).

It remains to show that the cohomology of this secondary differential yields the deter-minant of H∗(E (M)), with a shift.

We can examine a more general problem. Given any finite-dimensional graded vectorspace V, we can give the algebra P(V⊕V∗[−1]) of polynomial functions on V⊕V∗[−1] aBV operator4 arising from the pairing between V and V∗[−1]. Then, we need to producean isomorphism

H∗(P(V ⊕V∗[−1]),4) ∼= det(V)[d]

where the shift d is equal modulo 2 to the Euler characteristic of V.


Sending V to H∗(P(V⊕V∗[−1]),4) is a functor from the groupoid of finite-dimensionalgraded vector spaces and isomorphisms between them, to the category of graded vectorspaces. It sends direct sums to tensor products. It follows that to check whether or not itreturns the determinant, one needs to check that it does in the case that V is a graded line.

Thus, let us assume that V = C[k] for some k ∈ Z. We will check that our functorreturns V[1] if k is even and V∗ if k is odd. Thus, viewed as a Z/2 graded line, our functorreturns det V with a shift by the Euler characteristic of V.

To check this, note thatP(V ⊕V∗[−1]) = C[x, y]

where x is of cohomological degree k and y is of degree −1− k. The BV operator is

4 =∂

∂x∂

∂y.

A simple calculation shows that the cohomology of this complex is 1 dimensional, spannedby x if k is odd and by y if k is even. Since x is a basis of V∗ and y is a basis of V, thiscompletes the proof.

18.8. The partition function and the quantum Noether theorem

Our formulation of the quantum Noether theorem goes beyond a statement just aboutsymmetries (in the classical sense of the word). It also involves deformations, which aresymmetries of cohomological degree 1, as well as symmetries of other cohomological de-gree. Thus, it has important applications when we consider families of field theories.

The first application we will explain is that the quantum Noether theorem leads to adefinition of the partition function of a perturbative field theory.

Suppose we have a family of field theories which depends on a formal parameter c,the coupling constant. (Everything we will say will work when the family depends on anumber of formal parameters, or indeed on a pro-nilpotent dg algebra). For example, wecould start with a free theory and deform it to an interacting theory. An example of sucha family of scalar field theories is given by the action functional

S(φ) =∫

φ(4+ m2)φ + cφ4.

We can view such a family of theories as being a single theory – in this case the free scalarfield theory – with an action of the Abelian L∞ algebra C[1]. Indeed, by definition, anaction of an L∞ algebra g on a theory is a family of theories over the dg ring C∗(g) whichspecializes to the original theory upon reduction by the maximal ideal C>0(g).

We have seen (lemma ??) that actions of g on a theory are the same thing as actions ofthe local Lie algebra Ω∗X ⊗ g. In this way, we see that a family of theories over the base

18.8. THE PARTITION FUNCTION AND THE QUANTUM NOETHER THEOREM 411

ring C[[c]] is the same thing as a single field theory with an action of the local abelian L∞algebra Ω∗X[−1].

Here is our definition of the partition function. We will give the definition in a generalcontext, for a field theory acted on by a local L∞ algebra; afterwards, we will analyze whatit means for a family of field theories depending on a formal parameter c.

The partition function is only defined for field theories with some special properties.Suppose we have a theory on a compact manifold M, described classically by a localL∞ algebra M with an invariant pairing of degree −3. Suppose that H∗(M(M)) = 0;geometrically, this means we are perturbing around an solated solution to the equationsof motion on the compact manifold M. This happens, for instance, with a massive scalarfield theory.

This assumption implies that H∗(Obsq(M)) = C[[h]]. There is a preferred C[[h]]-linearisomorphism which sends the observable 1 ∈ H0(Obsq(M)) to the basis vector of C.

Suppose that this field theory is equipped with an inner action of a local L∞ algebraL. Then, proposition ?? tells us that for any open subset U ⊂ M, the action of Lc(U)on Obsq(U) is homotopically trivialized. In particular, since M is compact, the action ofL(M) on Obsq(M) is homotopically trivialized.

A theory with an L-action is the same as a family of theories over BL. The complexObsq(M)L(M) of L(M)-equivariant observables should be interpreted as the C∗(L(M))-module of sections of the family of observables over BL(M).

Since the action of L(M) is trivialized, we have a quasi-isomorphism of C∗(L)[[h]]-modules

Obsq(M)L(M) ' Obsq(M)⊗ C∗(L).

Since Obsq(M) is canonically quasi-isomorphic to C[[h]], we get a quasi-isomorphism

Obsq(M)L(M) ' C∗(L)[[h]].

18.8.0.1 Definition. The partition function is the element in C∗(L)[[h]] which is the image of theobservable 1 ∈ Obsq(M)L(M).

Another way to interpret this is as follows. The fact that L acts linearly on our the-ory implies that the family of global observables over BL(M) is equipped with a flatconnection. Since we have also trivialized the central fibre, via the quasi-isomorphismObsq(M) ' C[[h]], this whole family is trivialized. The observable 1 then becomes a sec-tion of the trivial line bundle on BL(M) with fibre C[[h]], that is, an element of C∗(L(M))[[h]].

Let us now analyze some examples of this definition.


Example: Let us first see what this definition amounts to when we work with one-dimensionaltopological field theories, which (in our formalism) are encoded by associative algebras.

Thus, suppose that we have a one-dimensional topological field theory, whose factor-ization algebra is described by an associative algebra A. We will view A as a field theoryon S1. For any interval I ⊂ S1, observables of theory are A, but observables on S1 are theHochschild homology HH∗(A).

Suppose that g is a Lie algebra with an inner action on A, given by a Lie algebrahomomorphism g → A. The analog of proposition ?? in this situation is the statementthat that the action of g on the Hochschild homology of A is trivialized. At the level ofHH0, this is clear, as HH0(A) = A/[A, A] and bracketing with any element of A clearlyacts by zero on A/[A, A].

The obstruction to extending an action of Ω∗X[−1] to an inner action is an element inH1(C∗red,loc(Ω

∗X[−1])). The D-module formulation ?? of the complex of local cochains of a

local Lie algebra gives us a quasi-isomorphism

C∗red,loc(Ω∗X[−1]) ∼= cΩ∗(X)[[c]][d]

where d = dim X and c is a formal parameter. Therefore, there are no obstructions tolifting an action of Ω∗X[−1] to an inner action. However, there is an ambiguity in givingsuch a lift, coming from the space cHd(X)[[c]].

Let us suppose that we have a field theory with an inner action of Ω∗X[−1]. Let Obsq

denote observables of this theory. Then, Noether’s theorem gives us a map of factorizationalgebras

UBD(Ω∗X,c)→ Obsq

where Obsq denotes the factorization algebra of observables of our theory, and UBD(Ω∗X[−1])is the completed BD envelope factorization algebra of Ω∗X. Since Ω∗X[−1] is abelian, thenthe BD envelope is simply a completed symmetric algebra:

UBD(Ω∗c (U)[−1]) = Sym∗(Ω∗c (U))[[h]].

In particular, if M is a compact manifold, then we have a natural isomorphism

Sym∗(H∗(M))[[h]] = H∗

(UBD(Ω∗c (M)[−1])

).

Applying Noether’s theorem, we get a map

Sym∗(H0(M))[[h]]→H0(Obsq(M)).

Part 6

Appendices

APPENDIX A

Background

We use techniques from disparate areas of mathematics throughout this book and notall of these techniques appear in the standard graduate curriculum, so here we provide aterse introduction to

• simplicial sets and simplicial techniques;• operads, colored operads (or multicategories), and algebras over colored oper-

ads;• differential graded (dg) Lie algebras, L∞ algebras, their (co)homology, and their

relation to deformation theory;• sheaves, cosheaves, and their homotopical generalizations;• elliptic complexes, formal Hodge theory, and parametrices,

along with pointers to more thorough treatments. By no means does the reader need tobe expert in all these areas to use our results or follow our arguments. She just needs aworking knowledge of this background machinery, and this appendix aims to provide thebasic definitions, to state the relevant results for us, and to explain the essential intuition.

We do assume that the reader is familiar with basic homological algebra and basic cate-gory theory. For homological algebra, there are numerous excellent sources, in books andonline, among which we recommend the complementary texts by Weibel [Wei94] andGelfand-Manin [GM03]. For category theory, the standard reference [ML98] is more thanadequate to our needs; we also recommend the series by Borceux [Bor94].

Remark: Our references are not meant to be complete, and we apologize in advance for theomission of many important works. We simply point out sources that we found pedagog-ically oriented or particularly accessible. ♦

A.1. Simplicial techniques

Simplicial sets are a combinatorial substitute for topological spaces, so it should be nosurprise that they can be quite useful. On the one hand, we can borrow intuition for themfrom algebraic topology; on the other, simplicial sets are extremely concrete to work withbecause of their combinatorial nature. In fact, many constructions in homological algebra

415

416 A. BACKGROUND

are best understood via their simplicial origins. Analogs of simplicial sets (i.e., simplicialobjects in other categories) are useful as well.

In this book, we use simplicial sets in two ways:

• when we want to talk about a family or space of QFTs or parametrices, we willusually construct a simplicial set of such objects instead; and• we accomplish some homological constructions by passing through simplicial

sets (e.g., in constructing the extension of a factorization algebra from a basis).

After giving the essential definitions, we state the main theorems we use.

A.1.0.2 Definition. Let ∆ denote the category whose objects are totally ordered finite sets andwhose morphisms are non-decreasing maps between ordered sets. We usually work with the skeletalsubcategory whose objects are

[n] = 0 < 1 < · · · < n.

A morphism f : [m]→ [n] then satisfies f (i) ≤ f (j) if i < j.

We will relate these objects to geometry below, but it helps to bear in mind the follow-ing picture. The set [n] corresponds to the n-simplex4n equipped with an ordering of itsvertices, as follows. View the n-simplex4n as living in Rn+1 as the solution to

(x0, x1, . . . , xn)

∣∣∣∣ ∑j

xj = 1 and 0 ≤ xj ≤ 1 ∀j

.

Identify the element 0 ∈ [n] with the zeroth basis vector e0 = (1, 0, . . . , 0), 1 ∈ [n] withe1 = (0, 1, 0, . . . , 0), and k ∈ [n] with the kth basis vector ek. The ordering on [n] thenprescribes a path along the edges of 4n, starting at e0, then going to e1, and on till thepath ends at en.

Every map f : [m] → [n] induces a linear map f∗ : Rm+1 → Rn+1 by setting f∗(ek) =e f (k). This linear map induces a map of simplices f∗ : 4m → 4n.

There are particularly simple maps that play an important role throughout the subject.Note that every map f factors into a surjection followed by an injection. We can then factorevery injection into a sequence of coface maps, namely maps of the form

fk : [n] → [n + 1]

i 7→

i, i ≤ ki + 1, i > k

.

A.1. SIMPLICIAL TECHNIQUES 417

Similarly, we can factor every surjection into a sequence of codegeneracy maps, namelymaps of the form

dk : [n] → [n− 1]

i 7→

i, i ≤ ki− 1, i > k

.

The names face and degeneracy fit nicely with the picture from above: a coface map corre-sponds to a choice of n-simplex in the boundary of the n + 1-simplex, and a codegeneracymap corresponds to “collapsing” an edge of the n-simplex to project the n-simplex ontoan n− 1-simplex.

We now introduce the main character.

A.1.0.3 Definition. A simplicial set is a functor X : ∆op → Sets, often denoted X•. Theset X([n]) (often denoted Xn) is called the “set of n-simplices of X.” A map of simplicial setsF : X → Y is a natural transformation of functors.

Let’s quickly examine what the coface maps tell us about a simplicial set X. For exam-ple, by definition, a map fk : [n] → [n + 1] in ∆ goes to X( f ) : Xn+1 → Xn. We interpretthis map X( f ) as describing the kth n-simplex sitting as a “face on the boundary” of ann + 1-simplex of X. A similar interpretation applies to the dk.

A.1.1. Simplicial sets and topological spaces. When working in a homotopical set-ting, simplicial sets often provide a more tractable approach than topological spaces them-selves. In this book, for instance, we describe “spaces of field theories” as simplicial sets.Below, we sketch how to relate these two kinds of objects.

One can use a simplicial set X• as the “construction data” for a topological space: eachelement of Xn labels a distinct n-simplex 4n, and the structure maps of X• indicate howto glue the simplices together. In detail, the geometric realization is the quotient topologicalspace

|X•| =(

än

Xn ×4n

)/ ∼

under the equivalence relation∼where (x, s) ∈ Xm×4m is equivalent to (y, t) ∈ Xn×4n

if there is a map f : [m]→ [n] such that X( f ) : Xn → Xm sends y to x and f∗(s) = t.

A.1.1.1 Lemma. Under the Yoneda embedding, [n] defines a simplicial set

∆[n] : [m] ∈ ∆op 7→ ∆([m], [n]) ∈ Sets.

The geometric realization of ∆[n] is the n-simplex4n (more accurately, it is homeomorphic to then-simplex).

418 A. BACKGROUND

In general, every (geometric) simplicial complex can be obtained by the geometricrealization of some simplicial set. Thus, simplicial sets provide an efficient way to studycombinatorial topology.

One can go the other way, from topological spaces to simplicial sets: given a topolog-ical space X, there is a simplicial set Sing X, known as the “singular simplicial set of S.”The set of n-simplices (Sing X)n is simply Top(4n, X), the set of continuous maps fromthe n-simplex 4n into X. The structure maps arise from pulling back along the naturalmaps of simplices.

A.1.1.2 Theorem. Geometric realization and the singular functor form an adjunction

| − | : sSets Top : Sing

between the category of simplicial sets and the category of topological spaces.

This relationship suggests, for instance, how to transport notions of homotopy to sim-plicial sets: the homotopy groups of a simplicial set X• are the homotopy groups of itsrealization |X•|, and maps of simplicial sets are homotopic if their realizations are. Wewould then like to say that these functors | − | and Sing make the categories of simplicialsets and topological spaces equivalent, in some sense, particularly when it comes to ques-tions about homotopy type. One sees immediately that some care must be taken, sincethere are topological spaces very different in nature from simplicial or cell complexes andfor which no simplicial set could provide an accurate description. The key is only to thinkabout topological spaces and simplicial sets up to the appropriate notion of equivalence.

Remark: It is more satisfactory to define these homotopy notions directly in terms of sim-plicial sets and then to verify that these match up with the topological notions. We directthe reader to the references for this story, as the details are not relevant to our work in thebook. ♦

We say a continuous map f : X → Y of topological spaces is a weak equivalence ifit induces a bijection between connected components and an isomorphism πn( f , x) :πn(X, x) → πn(Y, f (x)) for every n > 0 and every x ∈ X. Let Ho(Top), the homotopycategory of Top, denote the category of topological spaces where we localize at the weakequivalences. (“Localizing” means that we formally invert the weak equivalences.) Thereis a concrete way to think about this homotopy category. For every topological space,there is some CW complex weakly equivalent to it, under a zig zag of weak equivalences;and by the Whitehead theorem, a weak equivalence between CW complexes is in fact ahomotopy equivalence. Thus, Ho(Top) is equivalent to the category of CW complexeswith morphisms given by continuous maps modulo homotopy equivalence.

Likewise, let Ho(sSets) denote the homotopy category of simplicial sets, where welocalize at the appropriate notion of simplicial homotopy. Then Quillen proved the fol-lowing wonderful theorem.

A.1. SIMPLICIAL TECHNIQUES 419

A.1.1.3 Theorem. The adjunction induces an equivalence

| − | : Ho(sSets) Ho(Top) : Sing

between the homotopy categories. (In particular, they provide a Quillen equivalence between thestandard model category structures on these categories.)

This theorem justifies the assertion that, from the perspective of homotopy type, weare free to work with simplicial sets in place of topological spaces. In addition, it help-ful to know that algebraic topologists typically work with a better behaved categories oftopological spaces, such as compactly generated spaces.

Among simplicial sets, those that behave like topological spaces are known as Kancomplexes or fibrant simplicial sets. Their defining property is a simplicial analogue of thehomotopy lifting property, which we now describe explicitly. The horn for the kth face ofthe n-simplex, denoted Λk[n], is the subsimplicial set of ∆[n] given by the union of the allthe faces ∆[n − 1] → ∆[n] except the kth. (As a functor on ∆, the horn takes the [m] tomonotonic maps [m] → [n] that do not have k in the image.) A simplicial set X is a Kancomplex if for every map of a horn Λk[n] into X, we can extend the map to the n-simplex∆[n]. Diagrammatically, we can fill in the dotted arrow

Λk[n] //

X

∆[n]

==

to get a commuting diagram. In general, one can always find a “fibrant replacement”of a simplicial set X• (e.g., by taking Sing |X•| or via Kan’s Ex∞ functor) that is weaklyequivalent and a Kan complex.

A.1.2. Simplicial sets and homological algebra. Our other use for simplicial sets re-lates to homological algebra. We always work with cochain complexes, so our conventionswill differ from those who prefer chain complexes. For instance, the chain complex com-puting the homology of a topological space is concentrated in non-negative degrees. Wework instead with the cochain complex concentrated in non-positive degrees.

A.1.2.1 Definition. A simplicial abelian group is a simplicial object A• in the category ofabelian groups, i.e., a functor A : ∆op → AbGps.

By composing with the forgetful functor U : AbGps → Sets that sends a group to itsunderlying set, we see that a simplicial abelian group has an underlying simplicial set.To define simplicial vector spaces or simplicial R-modules, one simply replaces abeliangroup by vector space or R-module everywhere. All the work below will carry over to thesesettings in a natural way.

420 A. BACKGROUND

There are two natural ways to obtain a cochain complex of abelian groups (respec-tively, vector spaces) from a simplicial abelian group. The unnormalized chains CA is thecochain complex

(CA)m =

A|m|, m ≤ 0

0, m > 0with differential

d : (CA)m → (CA)m+1

a 7→ ∑|m|k=0(−1)k A( fk)(a),where the fk run over the coface maps from [|m| − 1] to [|m|]. The normalized chains NA isthe cochain complex

(NA)m =|m|−1⋂k=0

ker A( fk),

where m ≤ 0 and where the fk run over the coface maps from [|m| − 1] to [|m|]. Thedifferential is A( f|m|), the remaining coface map. One can check that the inclusion NA →CA is a quasi-isomorphism (in fact, a chain homotopy equivalence).

Example: Given a topological space X, its singular chain complex C∗(X) arises as a com-position of three functors in this simplicial world. (Note that because we prefer cochaincomplexes, the singular chain complex is, in fact, a cochain complex concentrated innonpositive degrees.) First, we make the simplicial set Sing X, which knows about allthe ways of mapping a simplex into X. Then we apply the free abelian group functorZ− : Sets→ AbGps levelwise to obtain the simplicial abelian group

Z Sing X : [n] 7→ Z(Top(4n, X)).

Then we apply the unnormalized chains functor to obtain the singular chain complex

C∗(X) = CZ Sing X.

In other words, the simplicial language lets us decompose the usual construction into itsatomic components. ♦

It is clear from the constructions that we only ever obtain cochain complexes concen-trated in nonpositive degrees from simplicial abelian groups. In fact, the Dold-Kan corre-spondence tells us that we are free to work with either kind of object — simplicial abeliangroup or such a cochain complex — as we prefer.

Let Ch≤0(AbGps) denote the category of cochain complexes concentrated in nonposi-tive degrees, and let sAbGps denot the category of simplicial abelian groups.

A.1.2.2 Theorem (Dold-Kan correspondence). The normalized chains functor

N : sAbGps→ Ch≤(AbGps)

is an equivalence of categories. Under this correspondence,

πn(A) ∼= H−n(NA)

A.2. OPERADS AND ALGEBRAS 421

for all n ≥ 0, and simplicial homotopies go to chain homotopies.

Throughout the book, we will often work with cochain complexes equipped with al-gebraic structures (e.g., commutative dg algebras or dg Lie algebras). Thankfully, it iswell-understood how the Dold-Kan correspondence intertwines with the tensor struc-tures on sAbGps and Ch≤0(AbGps).

Let A and B be simplicial abelian groups. Then their tensor product A⊗ B is the sim-plicial abelian group with n-simplices

(A⊗ B)n = An ⊗Z Bn.

There is a natural transformation

∇A,B : CA⊗ CB→ C(A⊗ B),

known as the Eilenberg-Zilber map or shuffle map, which relates the usual tensor product ofcomplexes with the tensor product of simplicial abelian groups. It is a quasi-isomorphism.

A.1.2.3 Theorem. The unnormalized chains functor and the normalized chains functor are bothlax monoidal functors via the Eilenberg-Zilber map.

Thus, with a little care, we can relate algebra in the setting of simplicial modules withalgebra in the setting of cochain complexes.

A.1.3. References. Friedman’s paper [Fri12] is a very accessible and concrete intro-duction to simplicial sets, with lots of intuition and pictures. Weibel [Wei94] explainsclearly how simplicial sets appear in homological algebra, notably for us, the Dold-Kancorrespondence and the Eilenberg-Zilber map. As usual, Gelfand-Manin [GM03] pro-vides a nice complement to Weibel. The expository article [GS07] provides a lucid andquick discussion of how simplicial methods relate to model categories and related issues.For the standard, modern reference on simplicial sets and homotopy theory, see the thor-ough and clear Goerss-Jardine [GJ09].

A.2. Operads and algebras

An operad is a way of describing the essential structure underlying some class of alge-braic objects. For instance, there is an operad Ass that captures the essence of associativealgebras, and there is an operad Lie that captures the essence of Lie algebras. An algebraover an operad is an algebraic object with that kind of structure: a Lie algebra is an alge-bra over the operad Lie. Although their definition can seem abstract and unwieldy at first,operads provide an efficient language for thinking about algebra and proving theoremsabout large classes of algebraic objects. As a result, they have become nearly ubiquitousin mathematics.

422 A. BACKGROUND

A colored operad is a way of describing a collection of objects that interact alge-braically. For example, there is a colored operad that describes a pair consisting of analgebra and a module over that algebra. Another name for a colored operad is a sym-metric multicategory, which emphasizes a different intuition: it is a generalization of thenotion of a category in which we allow maps with n inputs and one output.

In the book, we use these notions in several ways:

• we capture the algebraic essence of the Batalin-Vilkovisky notion of a quantiza-tion via the Beilinson-Drinfeld operad;• the notion of a prefactorization algebra — perhaps the central notion in the book

— is an algebra over a colored operad made from the open sets of a topologicalspace; and• we define colored operads that describe how observables vary under translation

and that generalize the notion of a vertex algebra.

The first use has a different flavor than the others, so we begin here by focusing on operadswith a linear flavor before we introduce the general formalism of colored operads. Wehope that by being concrete in the first part, the abstractions of the second part will notseem arid.

A.2.1. Operads. In the loosest sense — encompassing Lie, associative, commutative,and more — an algebra is a vector space A with some way of combining elements mul-tilinearly. Typically, we learn first about examples determined by a binary operationµ : A⊗ A→ A such that

• µ has some symmetry under permutation of the inputs (e.g., µ(a, b) = −µ(b, a)for a Lie algebra), and• the induced 3-ary operations µ (µ⊗ 1) and µ (1⊗ µ) satisfy a linear relation

(e.g., associativity or the Jacobi identity).

But we recognize that there should be more elaborate notions involving many differentn-ary operations required to satisfy complicated relations. As a basic example, a Poissonalgebra has two binary operations.

Before we give the general definition of an operad in (dg) vector spaces, we explainhow to visualize such algebraic structures. An n-ary operation τ : A⊗n → A is picturedas a rooted tree with n labeled leaves and one root.


τ

a1 a2 . . . an

µ(a1, . . . , an)

For us, operations move down the page. To compose operations, we need to specifywhere to insert the output of each operation. We picture this as stacking rooted trees. Forexample, given a binary operation µ, the composition µ (µ⊗ 1) corresponds to the tree

µ

µ

1

a1 a2 a3

µ(µ(a1, a2), a3)

whereas µ (1⊗ µ) is the tree

µ

µ

1

a1 a2 a3

µ(a1, µ(a2, a3))

with the first µ on the other side. We allow permutations of the inputs, which rearrangesthe inputs. Thus the vector space of n-ary operations τ : A⊗n → A has an action of thepermutation group Sn. We also want the permutations to interact in the natural way withcomposition.

We now give the formal definition. We will describe an operad in vector spaces overa field K of characteristic zero (e.g., C or R) and use ⊗ to denote ⊗K. It is straightforwardto give a general definition of an operad in an arbitrary symmetric monoidal category. Itshould be clear, for instance, how to replace vector spaces with cochain complexes.

A.2.1.1 Definition. An operad O in vector spaces consists of

424 A. BACKGROUND

(i) a sequence of vector spaces O(n) | n ∈N, called the operations,(ii) a collection of multilinear maps

n;m1,...,mn : O(n)⊗ (O(m1)⊗ · · · ⊗ O(mn))→ O(

n

∑j=1

mj

),

called the composition maps,(iii) a unit element η : K→ O(1).

This data is equivariant, associative, and unital in the following way.

(1) The n-ary operations O(n) have a right action of Sn.(2) The composition maps are equivariant in the sense that the diagram below commutes,

O(n)⊗ (O(m1)⊗ · · · ⊗ O(mn))

σ⊗σ−1// O(n)⊗ (O(mσ(1))⊗ · · · ⊗ O(mσ(n)))

O(

∑nj=1 mj

) σ(mσ(1),...,mσ(n)) // O(

∑nj=1 mj

)where σ ∈ Sn acts as a block permutation on the ∑n

j=1 mj inputs, and the diagram belowalso commutes,

O(n)⊗ (O(m1)⊗ · · · ⊗ O(mn))

1⊗(τ1⊗···⊗τn))

))O(n)⊗ (O(mσ(1))⊗ · · · ⊗ O(mσ(n)))

O(

∑nj=1 mj

)τ1⊕···⊕τn // O

(∑n

j=1 mj

)where each τj is in Smj and τ1 ⊕ · · · ⊕ τn denotes the blockwise permutation in S∑n

j=1 mj.

(3) The composition maps are associative in the following sense. Let n, m1, . . . , mn, `1,1, . . . ,`1,m1 , `2,1, . . . , `n,mn be positive integers, and set M = ∑n

j=1 mj, Lj = ∑mji=1 `j,i, and

N =n

∑i=1

Lj = ∑(j,k)∈M

`j,k.


Then the diagram

O(n)⊗(⊗n

j=1

(O(mj)⊗

⊗mjk=1O(`j,k)

))shuffle

,,

1⊗(⊗j)

(O(n)⊗⊗n

j=1O(mj))⊗(⊗n

j=1⊗mj

k=1O(`j,k))

⊗1

O(n)⊗(⊗n

j=1O(Lj))

O(M)⊗(⊗n

j=1⊗mj

k=1O(`j,k))

rrO(N)

commutes.(4) The unit diagrams commute:

O(n)⊗K⊗n1⊗η⊗n

//

∼=

((O(n)⊗O(1)⊗n

// O(n)

and

K⊗O(n)η⊗1//

∼=

''O(1)⊗O(n)

// O(n) .

A map of operads f : O → P is a sequence of linear maps f (n) : O(n) → P(n)interwining in the natural way with all the structure of the operads.

Example: The operad Com that describes commutative algebras has Com(n) ∼= K, with thetrivial Sn action, for all n. This is because there is only one way to multiply n elements:even if we permute the inputs, we have the same output. ♦

Example: The operad Ass that describes associative algebras has Ass(n) = K[Sn], the reg-ular representation of Sn, for all n. This is because the product of n elements only dependson their left-to-right ordering, not on a choice of parantheses. We should have a distinctn-ary product for each ordering of n elements. ♦

426 A. BACKGROUND

Remark: One can describe operads via generators and relations. The two examples aboveare generated by a single binary operation. In Ass, there is a relation between the 3-aryoperations generated by that binary operation — the associativity relation — as alreadydiscussed. For a careful treatment of this style of description, we direct the reader to thereferences. ♦

We now explain the notion of an algebra over an operad. Our approach is modeled ondefining a representation of a group G on a vector space V as a group homomorphism ρ :G → GL(V). Given a vector space V, there is an operad EndV that contains all imaginablemultilinear operations on V and how they compose, just like GL(V) contains all linearautomorphisms of V.

A.2.1.2 Definition. The endomorphism operad EndV of a vector space V has n-ary operationsEndV(n) = Hom(V⊗n, V) and compositions

n;m1,...,mn : O(n)⊗ (O(m1)⊗ · · · ⊗ O(mn)) → O(

∑nj=1 mj

)µn ⊗ (µm1 ⊗ · · · ⊗ µmn) 7→ µn (µm1 ⊗ · · · ⊗ µmn)

are simply composition.

A.2.1.3 Definition. For O an operad and V a vector space, we make V an algebra over O bychoosing a map of operads ρ : O → EndV .

For example, an associative algebra A is given by specifying a vector space A and alinear map µ : A⊗2 → A satisfying the associativity relation. This data is equivalent tospecifying an operad map Ass→ EndA.

There is one final notion from the theory of linear operads that we use: a Hopf op-erad. Algebras over a Hopf operad are closed under tensor product, just as commutativealgebras can be tensored to form a new commutative algebra.

A.2.1.4 Definition. A Hopf operad is a reduced operad P (i.e., P(0) = 0) equipped with acounit, a map of operads

εP : P → Com,

and coproduct, a map of operads

∆P : P → P ⊗H P ,

that is counital and coassociative.

Here P ⊗H Q denotes the Hadamard product of operads. The n-ary operations are

P ⊗H Q(n) = P(n)⊗Q(n),


and the composition maps are obtained by tensoring the compositions in P and Q “sideby side.” If A is a P-algebra and B is a Q-algebra, then A ⊗ B is naturally a P ⊗H Q-algebra. Building on this observation, we obtain the following.

A.2.1.5 Proposition. For P a Hopf operad, the tensor product A⊗ B of P- algebra A and B isagain a P-algebra. Moreover, there is a natural isomorphism of P-algebras

(A⊗ B)⊗ C ∼= A⊗ (B⊗ C)

for C another P-algebra.

A.2.1.1. References. For a very brief introduction to operads, see Stasheff’s “What is”column [Sta04]. The recent article by Vallette [Val] provides a nice motivation and overviewfor linear operads and their relation to homotopical algebra. For a systematic treatmentwith an emphasis on Koszul duality, see Loday-Vallette [LV12]. In [Cos04], there is a de-scription of the basics emphasizing a diagrammatic approach: an operad is a functor on acategory of rooted trees. Finally, Fresse’s book-in-progress is wonderful [Fre].

A.2.2. Colored operads aka multicategories. We now introduce a natural generaliza-tion of the notions of a category and of an operad. The essential idea is to have a collectionof objects that we can “combine” or “multiply.” If there is only one object, we recover thenotion of an operad. If there are no ways to combine multiple objects, then we recover thenotion of a category.

We will give the symmetric version of this concept, just as we did with operads.

Let (C,) be a symmetric monoidal category, such as (Sets,×) or (VectK,⊗K). Werequire C to have all reasonable colimits.

A.2.2.1 Definition. A multicategory (or colored operad)M over C consists of

(i) a collection of objects (or colors) ObM,(ii) for every n + 1-tuple of objects (x1, . . . , xn | y), an object

M(x1, . . . , xn | y)

in C called the maps from the xj to y,(iii) a unit element ηx : 1C →M(x | x) for every object x.(iv) a collection of composition maps in C

M(x1, . . . , xn | y)(M(x1

1, . . . , xm11 | x1) · · ·M(x1

n, . . . , xmnn | xn)

)→M(x1

1, . . . , xmnn | y),

(v) for every n + 1-tuple (x1, . . . , xn | y) and every permutation σ ∈ Sn a morphism

σ∗ :M(x1, . . . , xn | y)→M(xσ(1), . . . , xσ(n) | y)

in C.

428 A. BACKGROUND

This data satisfies conditions of associativity, units, and equivariance directly analogous to that ofoperads. For instance, given σ, τ ∈ Sn, we require

σ∗τ∗ = (τσ)∗,

so we have an analog of a right Sn action on maps out of n objects. Each unit ηx is a two-sided unitfor composition inM(x | x).

A.2.2.2 Definition. A map of multicategories (or functor between multicategories) F :M→N consists of

(i) an object F(x) in N for each object x inM, and(ii) a morphism

F(x1, . . . , xn | y) :M(x1, . . . , xn | y)→ N (F(x1), . . . , F(xn) | F(y))

in the category (C,) for every tuple (x1, . . . , xn | y) of objects inM

such that the structure of a multicategory is preserved (namely, the units, associativity, and equiv-ariance).

There are many familiar examples.

Example: Let B be an ordinary category, so that each collection of morphisms B(x, y) isa set. Then B is a multicategory over the symmetric monoidal category (Sets,×) whereB(x | y) = B(x, y) and B(x1, . . . , xn | y) = ∅ for n > 1. ♦

Similarly, a K-linear category is a multicategory over the symmetric monoidal cate-gory (VectK,⊗K) with no compositions between two or more elements.

Example: An operad O, in the sense of the preceding subsection, is a multicategory overthe symmetric monoidal category (VectK,⊗K) with a single object ∗ and

O(∗, . . . , ∗︸︷︷︸n

| ∗) = O(n).

Replacing VectK with another symmetric monoidal category C, we obtain a definition foroperad in C. ♦

Example: Every symmetric monoidal category (C,) has an underlying multicategory Cwith the same objects and with maps

C(x1, . . . , xn | y) = C(x1 · · · xn, y).

When C = VectK, these are precisely all the multilinear maps. ♦

For every multicategory M, one can construct a symmetric monoidal envelope SM byforming the left adjoint to the forgetful functor from symmetric monoidal categories tomulticategories. An object of SM is a formal finite sequence [x1, . . . , xm] of colors xi,

A.3. LIE ALGEBRAS AND L∞ ALGEBRAS 429

and a morphism f : [x1, . . . , xm] → [y1, . . . , yn] consists of a surjection φ : 1, . . . , m →1, . . . , n and a morphism f j ∈ M(xii∈φ−1(j) | yj) for every 1 ≤ j ≤ n. The symmetricmonoidal product in SM is simply concatenation of formal sequences.

Finally, an algebra over a colored operad M with values in N is simply a functor ofmulticategories F :M→ N . When we viewO as a multicategory and use the underlyingmulticategory VectK , then F : O → VectK reduces to an algebra over the operad O as inthe preceding subsection.

A.2.2.1. References. For a readable discussion of operads, multicategories, and differ-ent approaches to them, see Leinster [Lei04]. Beilinson and Drinfeld [?] develop pseudo-tensor categories — yet another name for this concept — for exactly the same reasons aswe do in this book. In [Lurb], Lurie provides a thorough treatment of colored operadscompatible with ∞-categories. Another approach to higher operads is provided by thedendroidal sets of Moerdijk, Weiss, and collaborators [MW07].

A.3. Lie algebras and L∞ algebras

Lie algebras, and their homotopical generalization L∞ algebras, appear throughoutthis book in a variety of contexts. It might surprise the reader that we never use their rep-resentation theory or almost any aspects emphasized in textbooks on Lie theory. Instead,we primarily use dg Lie algebras as a convenient language for formal derived geometry.In this section, we overview the language, and in the following section, we overview therelationship with derived geometry.

We use these ideas in the following settings.

• We use the Chevalley-Eilenberg complex to construct a large class of factorizationalgebras, via the factorization envelope of a sheaf of dg Lie algebras. This classincludes the observables of free field theories and the Kac-Moody vertex algebras.• We use the Lie-theoretic approach to deformation functors to motivate our ap-

proach to classical field theory.• We introduce the notion of a local Lie algebra to capture the symmetries of a field

theory and prove generalizations of Noether’s theorem.

We also use Lie algebras in the construction of gauge theories in the usual way.

A.3.1. A quick review of homological algebra with ordinary Lie algebras. Let K bea field of characteristic zero. (We always have in mind K = R or C.) A Lie algebra over K

is a vector space g with a bilinear map [−,−] : g⊗K g→ g such that

• [x, y] = −[y, x] (antisymmetry) and

430 A. BACKGROUND

• [x, [y, z]] = [[x, y], z] + [y, [x, z]] (Jacobi rule).

The Jacobi rule is the statement that [x,−] acts as a derivation. A simple example is thespace of n× n matrices Mn(K), usually written as gln in this context, where the bracket is

[A, B] = AB− BA,

using the commutator with the usual matrix multiplication on the right hand side. An-other classic example is given by Vect(M), the vector fields on a smooth manifold M, viathe commutator bracket.

A module over g, or representation of g, is a vector space M with a bilinear map ρ :g⊗K M→ M such that

ρ(x⊗ ρ(y⊗m))− ρ(y⊗ ρ(x⊗m)) = ρ([x, y]⊗m).

Usually, we will suppress the notation ρ and simply write x · m or [x, m]. Continuingwith the examples from above, the matrices gln acts on Kn by left multiplication, so Kn

is naturally a gln-module. Analogously, vector fields Vect(M) act on smooth functionsC∞(M) as derivations, and so C∞(M) is a Vect(M)-module.

There is a category g− mod whose objects are g-modules and whose morphisms arethe natural structure-preserving maps. To be explicit, a map f ∈ g − mod(M, N) of g-modules is a linear map f : M → N such that [x, f (m)] = f ([x, m]) for every x ∈ g andevery m ∈ M.

Lie algebra homology and cohomology arise as the derived functors of two naturalfunctors on the category of g-modules. We define the invariants as the functor

g−mod → VectK

M 7→ Mg

where Mg = m | [x, m] = 0 ∀x ∈ g. A nonlinear analog is taking the fixed points of agroup action on a set. The coinvariants is the functor

g−mod → VectK

M 7→ Mg

where Mg = M/gM = M/[x, m] | x ∈ g, m ∈ M. A nonlinear analog is taking thequotient, or orbit space, of a group action on a set (i.e., the collection of orbits).

To define the derived functors, we rework our constructions into the setting of mod-ules over associative algebras so that we can borrow the Tor and Ext functors. The universalenveloping algebra of a Lie algebra g is

Ug = Tens(g)/(x⊗ y− y⊗ x− [x, y])

where Tens(g) = ∑n≥0 g⊗n denotes the tensor algebra of g. Note that the ideal by which

we quotient ensures that the commutator in Ug agrees with the bracket in g: for all x, y ∈


g,x · y− y · x = [x, y],

where · denotes multiplication in Ug. It is straightforward to verify that there is an ad-junction

U : LieAlgK AssAlgK : Forget,

where Forget(A) views an associative algebra A over K as a vector space with bracketgiven by the commutator of its product. As a consequence, we can view g− mod as thecategory of left Ug-modules, which we’ll denote Ug−mod, without harm.

Now observe that K is a trivial g-module for any Lie algebra g: x · k = 0 for all x ∈ gand k ∈ K. Moreover, K is the quotient of Ug by the ideal (g) generated by g itself, so thatK is a bimodule over Ug. It is then straightforward to verify that

Mg = K⊗Ug M and Mg = HomUg(K, M)

for every module M.

A.3.1.1 Definition. For M a g-module, the Lie algebra homology of M is

H∗(g, M) = TorUg∗ (K, M),

and the Lie algebra cohomology of M is

H∗(g, M) = Ext∗Ug(K, M).

As usual, there are concrete interpretations of the lower (co)homology groups, interms of derivations and extensions. Notice that H0 and H0 recover invariants and coin-variants, respectively.

There are standard cochain complexes for computing Lie algebra (co)homology, andtheir generalizations will play a large role throughout the book. The key is to find anefficient, tractable resolution of K as a Ug module. We use again that K is a quotient ofUg:

(· · · → ∧ng⊗K Ug→ · · · → g⊗K Ug→ Ug)'−→ K,

where the final map is given by the quotient and the remaining maps are of the form

(y1 ∧ · · · ∧ yn)⊗ (x1 · · · xm) 7→n

∑k=1

(−1)n−k(y1 ∧ · · · yk · · · ∧ yn)⊗ (yk · x1 · · · xm)

− ∑1≤j<k≤n

(−1)j+k−1([yj, yk] ∧ y1 ∧ · · · yj · · · yk · · · ∧ yn)⊗ (x1 · · · xm),

where the hat yk indicates removal. As this is a free resolution of K, we use it to computethe relevant Tor and Ext groups: for coinvariants we have

K⊗LUg M ' (· · · → ∧ng⊗K M→ · · · → g⊗K M→ M)

432 A. BACKGROUND

and for invariants we have

R HomUg(K, M) ' (M→ g∨ ⊗K M→ · · · → ∧ng∨ ⊗K M→ · · · ),

where g∨ = HomK(g, K) is the linear dual. These resolutions were introduced by Cheval-ley and Eilenberg and so their names are attached.

A.3.1.2 Definition. The Chevalley-Eilenberg complex for Lie algebra homology of the g-module M is

C∗(g, M) = (SymK(g[1])⊗K M, d)

where the differential d encodes the bracket of g on itself and on M. Explicitly, we have

d(x1 ∧ · · · ∧ xn ⊗m) = ∑1≤j<k≤n

(−1)j+k[xj, xk] ∧ x1 ∧ · · · xj · · · xk · · · ∧ xn ⊗m

+n

∑j=1

(−1)n−jx1 ∧ · · · xj · · · ∧ xn ⊗ [xj, m].

We often call this complex the Chevalley-Eilenberg chains.

The Chevalley-Eilenberg complex for Lie algebra cohomology of the g-module M is

C∗(g, M) = (SymK(g∨[−1])⊗K M, d)

where the differential d encodes the linear dual to the bracket of g on itself and on M. Fixing alinear basis ek for g and hence a dual basis ek for g∨, we have

d(ek ⊗m) = −∑i<j

ek([ei, ej])ei ∧ ej ⊗m + ∑l

ek ∧ el ⊗ [el , m]

and we extend d to the rest of the complex as a derivation of cohomological degree 1 (i.e., use theLeibniz rule repeatedly to reduce to the case above). We often call this complex the Chevalley-Eilenberg cochains.

When M is the trivial module K, we simply write C∗(g) and C∗(g). It is important forus that C∗(g) is a commutative dg algebra and that C∗(g) is a cocommutative dg coalgebra,as a little work shows. This property suggests a geometric interpretation of the Chevalley-Eilenberg complexes: under the philosophy that every commutative algebra should beinterpreted as the functions on some “space,” we view C∗(g) as “functions on a space Bg”and C∗(g) as ”distributions on Bg.” Here we interpret the natural pairing between the twocomplexes as providing the pairing between functions and distributions. This geometricperspective on the Chevalley-Eilenberg complexes motivates the role of Lie algebras indeformation theory, as we explain in the following section.

A.3.1.1. References. Weibel [Wei94] contains a chapter on the homological algebra ofordinary Lie algebras, of which we have given a gloss. In [Lura], Lurie gives an efficienttreatment of this homological algebra in the language of model and infinity categories.


A.3.2. Dg Lie algebras and L∞ algebras. We now quickly extend and generalize ho-mologically the notion of a Lie algebra. Our base ring will now be a commutative algebraR over a characteristic zero field K, and we encourage the reader to keep in mind thesimplest case: where R = R or C. Of course, one can generalize the setting considerably,with a little care, by working in a symmetric monoidal category (with a linear flavor); thecleanest approach is to use operads.

Before introducing L∞ algebras, we treat the simplest homological generalization.

A.3.2.1 Definition. A dg Lie algebra over R is a Z-graded R-module g such that

(1) there is a differential

· · · d→ g−1 d→ g0 d→ g1 → · · ·

making (g, d) into a dg R-module;(2) there is a bilinear bracket [−,−] : g⊗R g→ g such that

• [x, y] = −(−1)|x||y|[y, x] (graded antisymmetry),• d[x, y] = [dx, y] + (−1)|x|[x, dy] (graded Leibniz rule),• [x, [y, z]] = [[x, y], z] + (−1)|x||y|[y, [x, z]] (graded Jacobi rule),

where |x| denotes the cohomological degree of x ∈ g.

In other words, a dg Lie algebra is an algebra over the operad Lie in the category of dgR-modules. In practice — and for the rest of the section — we require the graded piecesgk to be projective R-modules so that we do not need to worry about the tensor productor taking duals.

Here are several examples.

(a) Let (V, dV) be a cochain complex over K. Then End(V) = ⊕n Homn(V, V), thegraded vector space where Homn denotes the linear maps that shift degree by n,becomes a cochain complex via the differential

dEnd V = [dV ,−] : f 7→ dV f − (−1)| f | f dV .

The commutator bracket makes End(V) a dg Lie algebra over K. When V = Kn,this example simply reduces to gln.

(b) For M a smooth manifold and g an ordinary Lie algebra (such as su(2)), the ten-sor product Ω∗(M)⊗R g is a dg Lie algebra where the differential is simply theexterior derivative and the bracket is

[α⊗ x, β⊗ y] = α ∧ β⊗ [x, y].

We can view this dg Lie algebra as living over K or over the commutative dgalgebra Ω∗(M). This example appears naturally in the context of gauge theory.

434 A. BACKGROUND

(c) For X a simply-connected topological space, let g−nX = π1+n(X)⊗Z Q and use the

Whitehead product (or bracket) to provide the bracket. Then gX is a dg Lie alge-bra with zero differential. This example appears naturally in rational homotopytheory.

We now introduce a generalization where we weaken the Jacobi rule on the bracketsin a systematic way. After providing the (rather convoluted) definition, we sketch somemotivations.

A.3.2.2 Definition. An L∞ algebra over R is a Z-graded, projective R-module g equipped witha sequence of multilinear maps of cohomological degree 2− n

`n : g⊗R · · · ⊗R g︸︷︷︸n times

→ g,

with n = 1, 2, . . ., satisfying the following properties.

(1) Each bracket `n is graded-antisymmetric, so that

`n(x1, . . . , xi, xi+1, . . . , xn) = −(−1)|xi ||xi+1|`n(x1, . . . , xi+1, xi, . . . , xn)

for every n-tuple of elements and for every i between 1 and n− 1.(2) Each bracket `n satisfies the n-Jacobi rule, so that

0 =n

∑k=1

(−1)k ∑i1<···<ik

jk+1<···<jni1,...,jn=1,...,n

(−1)ε`n−k+1(`k(xi1 , . . . , xik), xjk+1 , . . . , xjn).

Here (−1)ε denotes the following sign for the permutation(1 · · · k k + 1 · · · ni1 · · · ik jk+1 · · · jn

)acting on the element x1 ⊗ · · · ⊗ xn: the sign arises from the alternating-Koszul signrule, where the transposition ab 7→ ba acquires sign −(−1)|a||b|.

For small values of n, we recover familiar relations. For example, the 1-Jacobi rulesays that `1 `1 = 0. In other words, `1 is a differential! Momentarily, let’s denote `1 by dand `2 by the bracket [−,−]. The 2-Jacobi rule then says that

−[dx1, x2] + [dx2, x1] + d[x1, x2] = 0,

which encodes the graded Leibniz rule. Finally, the 3-Jacobi rule rearranges to

[[x1, x2], x3] + [[x2, x3], x1] + [[x3, x1], x2]

= d`3(x1, x2, x3) + `3(dx1, x2, x3) + `3(dx2, x3, x1) + `3(dx3, x1, x2).


In other words, g does not satisfy the usual Jacobi rule on the nose but the failure is de-scribed by the other brackets. In particular, at the level of cohomology, the usual Jacobirule is satisfied.

Example: There are numerous examples of L∞ algebras throughout the book, but many aresimply dg Lie algebras spiced with analysis. We describe here a small, algebraic exampleof interest in topology and elsewhere (see, for instance, [?], [?], [?]). The String Lie 2-algebrastring(n) is the graded vector space so(n)⊕Rβ, where β has degree 1 — thus string(n) isconcentrated in degrees 0 and 1 — equipped with two nontrivial brackets:

`2(x, y) =

[x, y], x, y ∈ so(n)0, x = β

`3(x, y, z) = µ(x, y, z)β x, y, z ∈ so(n),

where µ denotes 〈−, [−,−]〉, the canonical (up to scale) 3-cocycle on so(n) arising fromthe Killing form. This L∞ algebra arises as a model for the “Lie algebra” of String(n),which itself appears in various guises (as a topological group, as a smooth 2-group, or asa more sophisticated object in derived geometry). ♦

There are two important cochain complexes associated to an L∞ algebra, which gen-eralize the two Chevalley-Eilenberg complexes we defined earlier.

A.3.2.3 Definition. For g an L∞ algebra, the Chevalley-Eilenberg complex for homologyC∗g is the dg cocommutative coalgebra

SymR(g[1]) =∞⊕

n=0

((g[1])⊗n)Sn

equipped with the coderivation d whose restriction to cogenerators dn : Symn(g[1]) → g[1] areprecisely the higher brackets `n.

Remark: The coproduct ∆ : C∗g → C∗g⊗R C∗g is just the natural way one can “break amonomial into two smaller monomials.” Namely,

∆(x1 · · · xn) = ∑σ∈Sn

∑1≤k≤n−1

(xσ(1) · · · xσ(k)⊗ (xσ(k+1) · · · xσ(n)).

A coderivation respects the coalgebra analog of the Leibniz property, and so it is deter-mined by its behavior on cogenerators. ♦

This coalgebra C∗g conveniently encodes all the data of the L∞ algebra g. The coderiva-tion d puts all the brackets together into one operator, and the fact that d2 = 0 encodesall the higher Jacobi relations. It also allows for a concise definition of a map between L∞algebras.

A.3.2.4 Definition. A map of L∞ algebras F : g → h is given by a map of dg cocommutativecoalgebras F : C∗g→ C∗h.

436 A. BACKGROUND

Note that a map of L∞ algebras is not determined just by its behavior on g. Un-winding the definition above, one discovers that it is necessary to specify a linear mapSymn(g[1])→ h for each n, satisfying some compatibility conditions.

To define the other Chevalley-Eilenberg complex C∗g, we use the graded linear dualof g,

g∨ =⊕n∈Z

HomR(gn, R)[n],

which is the natural notion of dual in this context.

A.3.2.5 Definition. For g an L∞ algebra, the Chevalley-Eilenberg complex for cohomologyC∗g is the dg commutative algebra

SymR(g[1]∨) =

∞

∏n=0

((g[1]∨)⊗n)

Sn

equipped with the derivation d whose Taylor coefficients dn : g[1]∨ → Symn(g[1]∨) are dual tothe higher brackets `n.

We emphasize that this dg algebra is completed with respect to the filtration by powersof the ideal generated by g[1]∨. This filtration will play a crucial role in the setting ofdeformation theory.

A.3.3. References. We highly recommend [?] for an elegant and efficient treatmentof L∞ algebras (over K) (and simplicial sets and also how these constructions fit togetherwith deformation theory). The book of Kontsevich and Soibelman [KS] provides a wealthof examples, motivation, and context.

A.4. Derived deformation theory

In physics, one often studies very small perturbations of a well-understood system,wiggling an input infinitesimally or deforming an operator by a small amount. Askingquestions about how a system behaves under small changes is ubiquitous in mathemat-ics, too, and there is an elegant formalism for such problems in the setting of algebraicgeometry, known as deformation theory. Here we will give a very brief sketch of derived de-formation theory, where homological ideas are mixed with classical deformation theory.

A major theme of this book is that perturbative aspects of field theory — both classicaland quantum — are expressed cleanly and naturally in the language of derived deforma-tion theory. In particular, many constructions from physics, like the the Batalin-Vilkoviskyformalism, obtain straightforward interpretations. Moreover, derived deformation theorysuggests how to rephrase standard results in concise, algebraic terms and also suggests

A.4. DERIVED DEFORMATION THEORY 437

how to generalize these results substantially (see, for instance, the discussion on Noether’stheorem).

In this section, we begin with a quick overview of formal deformation theory in alge-braic geometry. We then discuss its generalization in derived algebraic geometry. Finally,we explain the powerful relationship between deformation theory and L∞ algebras, whichwe exploit throughout the book.

A.4.1. The formal neighborhood of a point. Let S denote some category of spaces,such as smooth manifolds or complex manifolds or schemes. The Yoneda lemma implieswe can understand any particular space X ∈ S by understanding how other spaces Y ∈ Smap into X. That is, the functor represented by X,

hX : S op → SetsY 7→ S(Y, X)

,

knows everything about X as an S type of space. We call hX the functor of points of X, andthis functorial perspective on geometry will guide our work below. Although abstract atfirst acquaintance, this perspective is especially useful for thinking about general featuresof geometry.

Suppose we want to describe what X looks like near some point p ∈ X. Motivatedby the perspective of functor of points, we might imagine describing “X near p” by somekind of functor. The input category ought to capture all possible “small neighborhoods ofa point” permitted in S , so that we can see how such models map into X near p. We nowmake this idea precise in the setting of algebraic geometry.

Let S = SchC denote the category of schemes over C. Every such scheme X consists ofa topological space Xtop equipped with a sheaf of commutative C-algebras OX (satisfyingvarious conditions we will not specify). We interpret the algebra OX(U) on the open set Uas the “algebra of functions on U.” Every commutative C-algebra R determines a schemeSpec R where the prime ideals of R provide the set for underlying topological space andwhere the stalk of O at a prime ideal P is precisely the localization of R with respect toR−P. We call such a scheme Spec R an affine scheme. By definition, every scheme admitsan open cover by affine schemes.

It is a useful fact that the functor of points hX of a scheme X is determined by its be-havior on the subcategory A f fC of affine schemes. By construction, A f fC is the oppositecategory to CAlgC, the category of commutative C-algebras. Putting these facts together,we know that every scheme X provides a functor from CAlgC to Sets.

Example: Consider the polynomial q(x, y) = x2 + y2 − 1. The functor

hX : CAlgC → SetsR 7→ (a, b) ∈ R2 | 0 = q(a, b) = a2 + b2 − 1

438 A. BACKGROUND

corresponds to the affine scheme Spec S for the algebra S = C[x, y]/(q). This functorsimply picks out solutions to q in the algebra R, which we might call the “unit circle” inR2. It should be clear how any system of polynomials (or ideal in an algebra) defines asimilar functor of “solutions to the system of equations.” ♦

Example: Consider the scheme SL2, viewed as the functor

SL2 : CAlgC → Sets

R 7→

M =

(a bc d

) ∣∣∣∣∣ a, b, c, d ∈ R such that 1 = ad− bc

where CAlgC denotes the category of the commutative C-algebras. Note that SL2(C) isprecisely the set that we usually mean. One can check as well that this functor factorsthrough the category of groups. ♦

The notion of “point” in this category is given by Spec C, which is the locally ringedspace given by a one-point space ∗ equipped with C as its algebra of functions. A pointin the scheme X is then a map p : Spec C → X.1 Every point is contained in some affinepatch U ∼= Spec R ⊂ X, so it suffices to understand points in affine schemes. It is nowpossible to provide an answer to the question, “What are the affine schemes that look likesmall thickenings of a point?”

A.4.1.1 Definition. A commutative C-algebra A is artinian if A is finite-dimensional as a C-vector space. A local algebra A with unique maximal ideal m is artinian if and only if there issome integer n such that mn = 0.

Any local artinian algebra (A,m) provides a scheme Spec A whose underlying topo-logical space is a point but whose scheme structure has “infinitesimal directions” in thesense that every function f ∈ m is “small” because f n = 0. Let ArtC denote the categoryof local artinian algebras, which we will view as the category encoding “small neighbor-hoods of a point.”2

A point p : Spec C→ Spec R corresponds to a map of algebras P : R→ C. Every localartinian algebra (A,m) has a distinguished map Q : A → A/m ∼= C. Given a point p inSpec R, we obtain a functor

hp : ArtC → Sets(A,m) 7→ F : R→ A | P = Q F .

1As practice in translation, try phrasing this notion using the functor of points perspective.2Hopefully it seems reasonable to choose ArtC as a model for “small neighborhoods of a point.” There

are other approaches imaginable but this choice is quite useful. In particular, the most obvious topology forschemes — the Zariski topology — is quite coarse, so that open sets are large and hence do not reflect theidea of “zooming in near the point.” Instead, we use schemes whose space is just a point but have interestingbut tractable algebra.


Geometrically, this condition on φ means p is the composition Spec C → Spec ASpec F→

Spec R. The map F thus describes some way to “extend infinitesimally” away from thepoint p in X.

A concrete example is in order. Our favorite point in SL2 is given by the identityelement 1. Let h1 denote the associated functor of artinian algebras. We can describe thetangent space T1SL2 using it, as follows. Consider the artinian algebra D = C[ε]/(ε2),often called the dual numbers. Then

h1(D) =

M =

(1 + sε tε

uε 1 + vε

) ∣∣∣∣∣ s, t, u, v ∈ C and1 = (1 + sε)(1 + vε)− tuε2 = 1 + (s + v)ε

∼= N ∈ M2(C) | Tr N = 0= sl2(C),

where the isomorphism is given by M = 1+ εN. Thus, we have recovered the underlyingset of the Lie algebra.

For any point p in a scheme X, the set hp(D) is the tangent space to p in X. By consid-ering more complicated artinian algebras, one can study the higher order jets at p. We saythat hp describes the formal neighborhood of p in X. The following proposition motivatesthis terminology.

A.4.1.2 Proposition. Let P : R → C be a map of algebras (i.e., we have a point p : Spec C →Spec R). Then the functor hp : ArtC → Sets is given by

A 7→ CAlgC(Rp, A),

whereRp = lim←− R/mn

p

is the completed local ring given by the inverse limit over powers of mp = ker P, the maximal idealgiven by the functions vanishing at p.

In other words, the functor hp is not represented by an artinian algebra (unless R isartinian), but it is represented inside the larger category CAlgC. When R is noetherian, thering Rp is given by an inverse system of artinian algebras, so we say hp is pro-represented.When R is a regular ring (such as a polynomial ring over C), Rp is isomorphic to formalpower series. This important example motivates the terminology of formal neighborhood.

There are several properties of such a functor hp we would like to emphasize, as theyguide our generalization in the next section. First, by definition, hp(C) is simply a point,namely the point p. Second, we can study hp in stages, by a process we call artinianinduction. Observe that every artinian algebra (A,m) is equipped with a natural filtration

A ⊃ m ⊃ m2 ⊃ · · · ⊃ mn = 0.

440 A. BACKGROUND

Thus, every artinian algebra can be constructed iteratively by a sequence of small exten-sions, namely a short exact sequence of vector spaces

I → Bf→ A

where f : B → A is a map of algebras and I is an ideal in B such that mB I = 0. Wecan thus focus on understanding the maps hp( f ) : hp(B) → hp(A), which are simplerto analyze. In summary, hp is completely determined by how it behaves with respect tosmall extensions.

A third property is categorical in nature. Consider a pullback of artinian algebras

B×A C

// B

C // A

and note that B×A C is artinian as well. Then the natural map

hp(B×A C)→ hp(B)×hp(A) hp(C)

is surjective — in fact, it is an isomorphism. (This property will guide us in the nextsubsection.)

As an example, we describe how to study small extensions for the model case. Let(R,mR) be a complete local ring with residue field R/mR ∼= C and with finite-dimensionaltangent space TR = (mR/m2

R)∨. Consider the functor hR : A 7→ CAlg(R, A), which de-

scribes the formal neighborhood of the closed point in Spec R. The following propositionprovides a tool for understanding the behavior of hR on small extensions.

A.4.1.3 Proposition. For every small extension

I → Bf→ A,

there is a natural exact sequence of sets

0→ TR ⊗C I → hR(B)f −→ hR(A)

ob→ OR ⊗ I,

where exact means that a map φ ∈ hR(A) lifts to a map φ ∈ hR(B) if and only if ob(φ) = 0 andthe space of liftings is an affine space for the vector space TR ⊗C I.

Here ob denotes the obstruction to lifting maps, and OR is a set where an obstructionlives. An obstruction space OR only depends on the algebra R, not on the small extension.One can construct an obstruction space as follows. If d = dimC TR, there is a surjection ofalgebras

r : S = C[[x1, . . . , xd]]→ Rsuch that J = ker r satisfies J ⊂ m2

S, where mS = (x1, . . . , xd) is the maximal ideal of S. Inother words, Spec R can be embedded into the formal neighborhood of the origin in Ad,


and minimally, in some sense. Then OR is (J/mS J)∨. For a proof of the proposition, seechapter 6 of [FGI+05].

This proposition hints that something homotopical lurks behind the scenes, and thatthe exact sequence of sets is the truncation of a longer sequence. For a discussion of theseideas and the modern approach to deformation theory, we highly recommend the 2010ICM talk of Lurie [Lur10].

A.4.1.1. References. The textbook of Eisenbud and Harris [EH00] is a lovely introduc-tion to the theory of schemes, full of examples and motivation. There is an extensivediscussion of the functor of points approach to geometry, carefully compared to the lo-cally ringed space approach. For an introduction to deformation theory, we recommendthe article of Fantechi and Gottsche in [FGI+05]. Both texts provide extensive referencesto the literature.

A.4.2. Formal moduli spaces. The functorial perspective on algebraic geometry sug-gests natural generalizations of the notion of a scheme by changing the source and targetcategories. For instance, stacks arise as functors from CAlgC to the category of groupoids,allowing one to capture the idea of a space “with internal symmetries.” It is fruitful togeneralize even further, by enhancing the source category from commutative algebras todg commutative algebras (or simplicial commutative algebras) and by enhancing the tar-get category from sets to simplicial sets. (Of course, one needs to simultaneously adopta more sophisticated version of category theory, namely ∞-category theory.) This gener-alization is the subject of derived algebraic geometry, and much of its power arises fromthe fact that it conceptually integrates geometry, commutative algebra, and homotopicalalgebra. As we try to show in this book, the viewpoint of derived geometry providesconceptual interpretations of constructions like Batalin-Vilkovisky quantization.

We now outline the derived geometry version of studying the formal neighborhood ofa point. Our aim to pick out a class of functors that capture our notion of a formal derivedneighborhood.

A.4.2.1 Definition. An artinian dg algebra A is a dg commutative algebra over C such that

(1) each component Ak is finite-dimensional, dimC Ak = 0 for k << 0 and for k > 0, and(2) A has a unique maximal ideal m, closed under the differential, and A/m = C.

Let dgArtC denote the category of artinian algebras, where morphisms are simply maps of dgcommutative algebras.

442 A. BACKGROUND

Note that, as we only want to work with local rings, we simply included it as partof the definition. Note as well that we require A to be concentrated in nonpositive de-grees. (This second condition is related to the Dold-Kan correspondence: we want A tocorrespond to a simplicial commutative algebra.)

We now provide an abstract characterization of a functor that behaves like the formalneighborhood of a point, motivated by our earlier discussion of functors hp.

A.4.2.2 Definition. A formal moduli problem is a functor

F : dgArtC → sSets

such that

(1) F(C) is contractible,(2) F sends a surjection of dg artinian algebras to a fibration of simplicial sets, and(3) for every pullback diagram in dgArt

B×A C //

B

C // A

the map F(B×A C)→ F(B)×F(A) F(C) is a weak homotopy equivalence.

Note that since surjections go to fibrations, the strict pullback F(B)×F(A) F(C) agreeswith the homotopy pullback F(B)×h

F(A) F(C).

We now describe a large class of examples. Let R be a commutative dg algebra overC whose underlying graded algebra is SymV, where V is a Z-graded vector space, andwhose differential dR is a degree 1 derivation. It has a unique maximal ideal generated byV. Let hR denote the functor into simplicial sets whose n-simplices are

hR(A)n = f : R→ A⊗Ω∗(4n) | f a map of unital dg commutative algebras

and whose structure maps arise from those between the de Rham complexes of simplices.

A.4.2.1. References. We are modeling our approach on Lurie’s, as explained in his ICMtalk [Lur10] and his paper on deformation theory [Lura]. For a discussion of these ideasin our context of field theory, see [Cos].

A.4.3. The role of L∞ algebras in deformation theory. There is another algebraicsource of formal moduli functors — L∞ algebras — and, perhaps surprisingly, formalmoduli functors arising in geometry often manifest themselves in this form. We begin byintroducing the Maurer-Cartan equation for an L∞ algebra g and how it provides a formal


moduli functor. This construction is at the heart of our approach to classical field theory. Wethen describe several examples from geometry and algebra.

A.4.3.1 Definition. Let g be an L∞ algebra. The Maurer-Cartan equation (or MC equation) is∞

∑n=1

1n!`n(α

⊗n) = 0,

where α denotes a degree 1 element of g.

Note that when we consider the dg Lie algebra Ω∗(M)⊗ g, with M a smooth manifoldand g an ordinary Lie algebra, the MC equation becomes the equation

dα +12[α, α] = 0

for a flat g-connection α ∈ Ω1 ⊗ g on the trivial principal G-bundle on M. (This is thesource of the name Maurer-Cartan.)

There are two other perspectives on the MC equation. First, observe that a map ofcommutative dg algebras α : C∗g → C is determined by its behavior on the generatorsg∨[−1] of the algebra C∗g. Hence α is a linear functional of degree 0 on g∨[−1] — or,equivalently, a degree 1 element α of g — that commutes with differentials. This conditionα d = 0 is precisely the MC equation for α. The second perspective uses the coalgebraC∗g, rather than the algebra C∗g. A solution to the MC equation α is equivalent to givinga map of cocommutative dg coalgebras α : C→ C∗g.

Now observe that L∞ algebras behave nicely under base change: if g is an L∞ algebraover C and A is a commutative dg algebra over C, then g⊗ A is an L∞ algebra (over Aand, of course, C). Solutions to the MC equation go along for the ride as well. For instance,a solution α to the MC equation of g⊗ A is equivalent to both a map of commutative dgalgebras α : C∗g → A and a map of cocommutative dg coalgebras α : A∨ → C∗g. Again,we simply unravel the conditions of such a map restricted to (co)generators. As mapsof algebras compose, for instance, solutions play nicely with base change. Thus, we canconstruct a functor out of the MC solutions.

A.4.3.2 Definition. The Maurer-Cartan functor of an L∞ algebra g

MCg : dgArtC → sSets

sends (A,m) to the simplicial set whose n-simplices are solutions to the MC equation in g⊗m⊗Ω∗(4n).

We remark that tensoring with the nilpotent ideal m makes g⊗ m is nilpotent. Thiscondition then ensures that the simplicial set MCg(A) is a Kan complex [?] [?]. In fact,their work shows the following.

444 A. BACKGROUND

A.4.3.3 Theorem. The Maurer-Cartan functor MCg is a formal moduli problem.

In fact, every formal moduli problem is represented — up to a natural notion of weakequivalence — by the MC functor of an L∞ algebra [?].

A.4.3.1. References. For a clear, systematic introduction with an expository emphasis,we highly recommend Manetti’s lecture [Man09], which carefully explains how dg Liealgebras relate to deformation theory and how to use them in algebraic geometry. Theunpublished book [KS] contains a wealth of ideas and examples; it also connects theseideas to many other facets of mathematics. The article of Hinich [?] is the original pub-lished treatment of derived deformation theory, and it provides one approach to necessaryhigher category theory. For the relation with L∞ algebras, we recommend [?], which con-tains elegant arguments for many of the ingredients too. Finally, see Lurie’s [Lura] for aproof that every formal moduli functor is described by a dg Lie algebra.

A.5. Sheaves, cosheaves, and their homotopical generalizations

Sheaves appear throughout geometry and topology because they capture the idea ofgluing together local data to obtain something global. Cosheaves are an equally naturalconstruction that are nonetheless used less frequently. We will give a very brief discussionof these ideas. As we always work with sheaves and cosheaves of a linear nature, we givedefinitions in that setting.

A.5.1. Sheaves.

A.5.1.1 Definition. A presheaf of vector spaces on a space X is a functorF : OpensopX → VectK

where OpensX is the category encoding the partially ordered set of open sets in X (i.e., the objectsare open sets in X and there is a map from U to V exactly when U ⊂ V).

In other words, a presheaf F assigns a vector space F (U) to each open U and a re-striction map resV⊃U : F (V) → F (U) whenever U ⊂ V. Bear in mind the following twostandard examples. The constant presheaf F = K has K(U) = K (hence it assigns the samevector space to every open) and its restriction map is always the obvious isomorphism.The presheaf of continuous functions C0

X assigns the vector space C0X(U) of continuous func-

tions from U to R (or C, as one prefers) and the restriction map resV⊃U consists preciselyof restricting a continuous function from V to a smaller open U.

A sheaf is a presheaf whose value on a big open is determined by its behavior on smallopens.

A.5. SHEAVES, COSHEAVES, AND THEIR HOMOTOPICAL GENERALIZATIONS 445

A.5.1.2 Definition. A sheaf of vector spaces on a space X is a presheaf F such that for everyopen U and every cover U = Vii∈I of U, we have

F (U)∼=→ lim

(∏i∈IF (Vi) ⇒ ∏

i,j∈IF (Vi ∩Vj)

),

where the map out of F (U) is the product of the restriction maps from U to the Vi and where, inthe limit diagram, the top arrow is restriction from Vi to Vi ∩Vj and the bottom arrow is restrictionfrom Vj to Vi ∩Vj.

This gluing condition says precisely that an element s ∈ F (U), called a section of F onU, is given by sections on the cover, (si ∈ F (Vi))i∈I , that agree on the overlapping opensVi ∩Vj. It captures in a precise way how to reconstruct F on big opens in terms of data ona cover.

It is a good exercise to verify that C0X is always a sheaf and that K is not a sheaf on a

disconnected space.

In this book, our spaces are nearly always smooth manifolds, and most of our sheavesarise in the following way. Let E → X be a vector bundle on a smooth manifold. Let Edenote the presheaf where E (U) is the vector space of smooth sections E|U → U on theopen set U. It is quick to show that E is a sheaf.

A.5.2. Cosheaves. We now discuss the dual notion of a cosheaf.

A.5.2.1 Definition. A precosheaf of vector spaces on a space X is a functor G : OpensX → Vect.A cosheaf is a precosheaf such that for every open U and every cover U = Vii∈I of U, we have

colim

(äi,j∈IG(Vi ∩Vj) ⇒ ä

i∈IG(Vi)

)∼=→ G(U),

where the map into G(U) is the coproduct of the extension maps from the Vi to U and where, inthe colimit diagram, the top arrow is extension from Vi ∩Vj to Vi and the bottom arrow is extensionfrom Vi ∩Vj to Vj.

The crucial example of a cosheaf (for us) is the functor Ec that assigns to the open U,the vector space Ec(U) of compactly-supported smooth sections of E on U. If U ⊂ V, we canextend a section s ∈ Ec(U) to a section extU⊂V(s) ∈ Ec(V) on V by setting it equal to zeroon V \U.

A closely related example is the cosheaf of compactly-supported distributions dualto the sheaf E of smooth sections of a bundle. When E denotes the sheaf of fields for afield theory, this cosheaf describes the linear observables on the fields and, importantly,organizes them by their support.

446 A. BACKGROUND

A.5.3. Homotopical versions. Often, we want our sheaves or cosheaves to take val-ues in categories of a homotopical nature. For instance, we might assign a cochain com-plex to each open set, rather than a mere vector space. In this homotopical setting, onetypically works with a modified version of the gluing axioms above. The modificationsare twofold:

(1) we work with all finite intersections of the opens in the cover (i.e., not just over-laps Vi ∩Vj but also Vi1 ∩ · · · ∩Vin ), and

(2) we use the homotopy limit (or colimit).

The first modification is straightforward — and familiar to anyone who has seen a Cechcomplex — but the second is more subtle. (We highly recommend [Dug] for an introduc-tion and development of these notions.)

In our main examples, we give explicit complexes that encode the relevant informa-tion.

For completeness’ sake, a homotopy cosheaf with values in dg vector spaces is a functorG : OpensX → dgVect such that for every open U and every cover U = Vii∈I of U, wehave a quasi-isomorphism

Totoplus

∞⊕n=0

⊕~i∈In+1

G(∩nj=0Vij)[n]

'→ G(U),

where the total differential on the left hand side is the sum of the internal differentials ofG(V~i) and a differential that takes the alternating sum of the structure maps. We say thatG satisfies Cech (co)descent.

A.5.3.1. References. Nearly any modern book on differential or algebraic geometrycontains an introduction to sheaves. See, for instance, [GH94] or [Ram05]. For a clearexposition (and more) of cosheaves, see [Cur].

A.6. Elliptic complexes

Classical field theory involves the study of systems of partial differential equations (orgeneralizations), which is an enormously rich and sophisticated subject. We focus in thisbook on a tractable and well-understood class of PDE that appear throughout differen-tial geometry: elliptic complexes. Here we will spell out the basic definitions and someexamples, as these suffice for our work in this book. (In the development of substantialexamples, deeper aspects of the theory of elliptic complexes will undoubtedly appear.)

We start with a local description of differential operators before homing in on ellipticoperators. Let U be an open set in Rn. A linear differential operator is an R-linear map

A.6. ELLIPTIC COMPLEXES 447

L : C∞(U)→ C∞(U) of the form

L( f ) = ∑α∈Nn

aα(x)∂α f (x),

where we use the multi-index α = (α1, . . . , αn) to efficiently denote

∂α =∂α1

∂xα11· · · ∂αn

∂xαnn

,

where the coefficients aα are smooth functions, and where only finitely many of the aα

are nonzero functions. We call |α| = ∑j αj the order of the index, and thus the order ofL is the maximum order |α| among the nonzero coefficients aα. (For example, the orderof the Laplacian ∑j ∂2/∂x2

j is two.) The principal symbol (or leading symbol) of a kth orderdifferential operator L is the “fiberwise polynomial”

σL(ξ) = ∑|α|=k

aα(x)ikξα

obtained by summing over the indices of order k and replacing the partial derivative ∂/∂xjby variables iξ j, where i is the usual square root of one. (It is the standard convention toinclude the factor of i, due to the role of Fourier transforms in motivating many of theseconstructions and definitions.) It is natural to view the principal symbol as a functionon the cotangent bundle T∗U that is a homogeneous polynomial of degree k along thecotangent fibers, where ξ j is the linear functional dual to dxj. The principal symbol con-trols the qualitative behavior of L. It also behaves nicely under changes of coordinates,transforming as a section of the bundle Symk(TU)→ U.

Remark: Elsewhere in the text, we talk about differential operators in a more abstract, ho-mological setting. For instance, we say the BV Laplacian is a second-order differentialoperator. Our use of the differential operators in this homological setting is inspired bytheir use in analysis, but the definitions are modified, of course. For instance, we view theodd directions as geometric in the BV formalism, so the BV Laplacian is second-order.

It is straightforward to extend these notions to a smooth manifold and to smooth sec-tions of vector bundles on that manifold. Let E → X be a rank m vector bundle and letF → X be a rank n vector bundle. Let E denote the smooth global sections of E and letF denote the smooth global sections of F. A differential operator from E to F is an R-linearmap L : E → F such that for a local choice of coordinates on X and trivializations of Eand F,

L(s)i = ∑α∈Nn

aijα (x)∂αsj(x),

where s = (s1, . . . , sm) is a section of E (U) ∼= C∞(U)m on a small neighborhood U inX, Ls = ((Ls)1, . . . , (Ls)n) is a section of F (U) ∼= C∞(U)n on that small neighborhood,and where the aij

α (x) are smooth functions. In other words, we have a matrix-valueddifferential operator locally. The principal symbol σL of a kth order differential operator L

448 A. BACKGROUND

defines a section of the bundle Symk(TX)⊗ E∨ ⊗ F → X. Thus, by pulling back along thecanonical projection π : T∗X → X, we can view the principal symbol as a map of vectorbundles σL : π∗E→ π∗F over the cotangent bundle T∗X.

A.6.0.1 Definition. An elliptic operator L : E → F is a differential operator whose principalsymbol is σL : π∗E→ π∗F is an isomorphism of vector bundles on T∗X \X, the cotangent bundlewith the zero section removed.

This definition says that the principal symbol is an invertible linear operator after eval-uating at every nonzero covector. As an example, consider the Laplacian on Rn: its prin-cipal symbol is ∑ ξ2

j , which only vanishes when all the ξ j are zero.

Ellipticity is purely local and is an easy property to check in practice. Globally, it haspowerful consequences, of which the following is the most famous.

A.6.0.2 Theorem. For X a closed manifold (i.e., compact and boundaryless), an elliptic operatorQ : E → F is Fredholm, so that its kernel and cokernel are finite-dimensional vector spaces.

Thus, an elliptic operator on a closed manifold is invertible up to a finite-dimensional“error.” Moreover, there is a rich body of techniques for constructing these partial in-verses, especially for classical operators such as the Laplacian.

The notions from above extend naturally to cochain complexes. A differential complexon a manifold X is a Z-graded vector bundle ⊕nEn → X (with finite total rank) and adifferential operator Qn : E n → E n+1 for each integer n such that Qn+1 Qn = 0. Wetypically denote this by (E , Q). There is an associated principal symbol complex (π∗E, σQ)on T∗X by taking the principal symbol of each operator Qn.

A.6.0.3 Definition. An elliptic complex is a differential complex whose principal symbol com-plex is exact on T∗X \ X (i.e., the cohomology of the symbol complex vanishes away from the zerosection of the cotangent bundle).

Every elliptic operator Q : E → F defines a two-term elliptic complex

· · · → 0→ EQ→ F → 0→ · · · .

The other standard examples of elliptic complexes are the de Rham complex (Ω∗(X), d)of a smooth manifold and the Dolbeault complex (Ω0,∗(X), ∂) of a complex manifold.

The analog of the Fredholm result above is the following, sometimes known as theformal Hodge theorem (by analogy to the Hodge theorem, which is usually focused onthe de Rham complex).


A.6.0.4 Theorem. Let (E , Q) be an elliptic complex on a closed manifold X. Then the cohomologygroups Hk(E , Q) are finite-dimensional vector spaces.

In proving these theorems, one constructs a partial inverse with nice, geometric prop-erties. We will not give a full definition here (because we do not want to delve into pseu-dodifferential operators) but will state the important properties.

Recall that every continuous linear operator F : E → F between smooth sections of avector bundles possesses a Schwartz kernel, KF, a section of the bundle (E∨ ⊗Dens) Fon X×X that is distributional along the first copy of X (i.e., in the E direction) and smoothalong the second copy of X (i.e., in the F direction). Then

F(s)(x) =∫

y∈Xs(y)KF(y, x).

If the kernel KF is a smooth over all of X × X, then F is called a smoothing operator. ForX a closed manifold, a smoothing operator F is compact (viewed as an operator betweenSobolev space completions of the smooth sections).

The Fredholm operators are stable under compact perturbations: if T is Fredholm andC is compact, then T + C is also Fredholm. In particular, 1 + C is always Fredholm.

For Q : E → F an elliptic operator, a parametrix is an operator P : F → E satisfying

(1) 1E − PQ = S for some smoothing operator S,(2) 1F −QP = T for some smoothing operator T, and(3) the Schwartz kernel of P is smooth away from the diagonal X ⊂ X× X.

Thus, a parametrix P for Q is a partial or approximate inverse to Q. Armed with aparametrix P, we know that Q is Fredholm. The theory of pseudodifferential operatorsprovides a construction of such a parametrix for any elliptic operator.

We want to explain how to generalize the parametrix approach to elliptic complexes.The essential idea to make a contraction of the elliptic complex E onto its cohomologyH∗E (viewed as a complex with zero differential),

Eπ //η 88 H∗Eι

oo ,

with some special properties. Recall that a retraction means that ι includes the cohomol-ogy as a subcomplex, π projects away everything but cohomology, and η is a degree −1map on E , such that these maps satisfy

1H∗E = π ι and 1E − ι π = [Q, η] = Qη + ηQ.

Thus, a contraction is a homological generalization of a partial inverse.

450 A. BACKGROUND

We want the contractions for elliptic complexes to have two important properties.First, we require all the maps above to be continuous with respect to the usual Frechettopologies. Second, the operator C = ι π should be a cochain map that is smoothing.This property ensures that 1E − C is Fredholm.

The existence of such a contraction implies the theorem. The operator C : E → Esends any cocycle s to a distinguished representative C(s) of its cohomology class. Infact, C2 = C, so it is a projection operator whose image is isomorphic to the cohomologyof E . Moreover, C annihilates exact cocycles (since π does) and elements that are notcocycles. As 1E − C is a projection operator, we know the kernel of this operator is finite-dimensional, and so the image of C is finite-dimensional.

Finding the necessary contraction η exploits the existence of parametrices for ellipticoperators. In the situations relevant to this book, the general idea is simple. First, onefinds a differential operator Q∗ of degree −1 such that the commutator D = [Q, Q∗] isa generalized Laplacian (i.e., its principal symbol looks like that of a Laplacian).3 In oursetting, the existence of such a Q∗ is a hypothesis. This operator D is Fredholm by theearlier theorem, so we know it has finite-dimensional kernel.

Second, let G denote the parametrix for D (where G is for “Green’s function”). Setη = Q∗G. We need to show it a contracting homotopy with the desired properties. Let

1− DG = S and 1− GD = T,

where S and T are smoothing endomorphisms of cohomological degree 0. Note that

QD = QQ∗Q = DQ,

so

G(QD)G = G(DQ)G =⇒ (GQ)(1− S) = (1− T)(QG).

Hence we find GQ = QG − U, where U = QS − TQG is smoothing because Q is adifferential operator. In consequence,

[Q, η] = QQ∗G + Q∗GQ= QQ∗G + Q∗QG−Q∗U= DG−Q∗U

= 1− (S + Q∗U).

The final term in parantheses is smoothing. We have verified that η has the desired prop-erties.

3To do this, one can pick inner products on the bundles Ej (hermitian, if complex bundles) and a Rie-mannian metric on X. This provides an adjoint to Q, i.e., a differential operator Q∗ of degree −1.


A.6.1. References. As usual, Atiyah and Bott [?] explain beautifully the essential ideasof elliptic complexes and pseudodifferential techniques and show how to use them effi-ciently. For an accessible development of the analytic methods in the geometric setting,we recommend [Wel08]. The full story and much more is available in the classic works ofHormander [Hor03].

APPENDIX B

Homological algebra with differentiable vector spaces

The factorization algebras we consider take values in vector spaces of an analyticalnature, like the space of smooth functions on a manifold. We would thus like to performhomological algebra in this setting. The standard approach to working with objects of thisnature is to treat them as topological vector spaces. However, it is not completely obvioushow one should set up homological algebra when using topological vector spaces. It isalso not straightforward to construct the topology on vector spaces which appear in ourmost important examples of factorization algebras: the observables of a quantum fieldtheory.

Thus we will work with a weaker and more flexible concept, that of differentaible vec-tor space. This Appendix develops homological algebra in the category of differentiablevector spaces. Related approaches to functional analysis are developed in [Pau10] and in[KM97].

B.1. Diffeological vector spaces

Let us remind the reader of the concept of diffeological space [Sta11].

B.1.0.1 Definition. The site of smooth manifolds is the site whose objects are smooth manifolds,morphisms are smooth maps, and where a map M → N is an open covering if it is a surjectivelocal diffeomorphism.

A diffeological space X is a sheaf of sets on the site of smooth manifolds with the propertythat, for all smooth manifolds M, the map

X(M)→ HomSets(M, X(∗))

is injective. (On the right hand side, X(∗) is the value of X on a point.)

A map of diffeological spaces is a map of sheaves of sets on the smooth site.

We will sometimes refer to maps of diffeological spaces as smooth maps, to distin-guish them from maps of the underlying sets.

453

454 B. HOMOLOGICAL ALGEBRA WITH DIFFERENTIABLE VECTOR SPACES

We can rewrite the axioms of a diffeological space in more explicit terms, as follows.A diffeological space is determined by a set X = X(∗), and for each smooth manifold M,a subset X(M) ⊂ Maps(M, X) of smooth maps from M to X. These subsets must satisfy thefollowing conditions. If f : M → X is a smooth map and if g : N → M is a smooth mapof ordinary manifolds, then f g : N → X is a smooth map. Further, a map f : M → X issmooth if and only if it is smooth locally on M. Finally, all constant maps to X are smooth.We call this collection of smooth maps the diffeology of X.

Note that if X, Y are diffeological spaces, then so is X × Y: a map M → X × Y issmooth if the composition with both projection maps is smooth.

B.1.0.2 Definition. A diffeological vector space is a vector space V together with a diffeologycompatible with the vector space structure. Thus, the sum map V × V → V and the scalarmultiplication map R× V → V are maps of diffeological spaces (where R is given the standarddiffeology).

A diffeological vector space V has enough structure to talk about smooth maps froma manifold M to V. We also want to be able to differentiate such maps. This requires extrastructure.

We use the notation C∞(M, V) to denote the C∞(M)-module of smooth maps from Mto V. Thus, C∞(M, V) is, as a vector space, just V(M), equipped with the natural structureof module over the (discrete) algebra C∞(M).

Similarly, we let

Ωk(M, V) = Ωk(M)⊗C∞(M) C∞(M, V)

denote the space of k-forms with values in V. This is just the algebraic tensor product.This is a reasonable thing to do because Ωk(M) is a finitely-generated projective C∞(M)module: it is a direct summand of a free finite rank C∞(M)-module. This implies thatΩk(M, V) is a direct summand of C∞(M, V)⊕l for some l.

B.1.0.3 Definition. A differentiable vector space is a diffeological vector space together with,for each smooth manifold M, a flat connection

∇M,V : C∞(M, V)→ Ω1(M, V)

such that, for all smooth maps f : N → M,

f ∗∇M,V = ∇N,V .

To say that∇M,V is a flat connection means, of course, that it satisfies the Leibniz rule,

∇M,V( f · s) = (d f )s + f∇M,Vs,

B.1. DIFFEOLOGICAL VECTOR SPACES 455

and that the curvature

F(∇M,V) = (∇M,V)2 : C∞(M, V)→ Ω2(M, V)

vanishes.

The flat connection ∇M,V allows us to differentiate smooth maps M → V. If f : M →V is a smooth map and if X ∈ Vect(M) is a vector field on M, we define

X( f ) = 〈X,∇M,V f 〉 ∈ C∞(M, V),

where 〈−,−〉 indicates the C∞(M)-linear pairing

Vect(M)×Ω1(M, V)→ C∞(M, V).

Differentiable vector spaces form a category that we denote DVS. An object is a dif-ferentiable vector space V. A morphism φ : V → W is a linear map such that for everysmooth map f : M → V, the map φ f is smooth, and which is compatible with connec-tions in the sense that, for all smooth manifolds M,

φ ∇M,V = ∇M,V φ.

We will often refer to morphisms of differentiable vector spaces as smooth linear maps.

Differentiable vector spaces appear naturally in geometry. In section B.2 below, weshow that for M a manifold and E is a vector bundle on M, the space C∞(N, E) of smoothsections of E has a natural structure of differentiable vector space. The same holds forthe space of compactly supported or distributional sections of E. Most of our examples ofdifferentiable vector spaces arise in this way.

Below, we will examine various properties and examples of differentiable vector spaces.Many of these constructions work equally well for diffeological vector spaces.

Remark: One of the main reasons we consider differentiable vector spaces instead of topo-logical vector spaces is that homological algebra for sheaves of vector spaces on a site isrelatively standard, whereas homological algebra for topological vector spaces is trickier.

Another reason is that, when we consider our construction of factorization algebrasin families, we will only use families where the base is a smooth manifold. Differentiablevector spaces have just enough structure to talk about such families.

B.1.1. Limits and colimits. Let V be a differentiable vector space, and let i : W ⊂ V bea sub-vector space. The subspace diffeology on W is defined by saying that a map f : M →W is smooth if the composed map i f : M → V is smooth. We say that the diffeologicalsubspace W is a differentiable subspace if, for all smooth manifolds M, the connection∇M,Vmaps C∞(M, W) ⊂ C∞(M, V) to Ω1(M, W) ⊂ Ω1(M, V).


Differentiable subspaces have the usual universal property: if A is another differen-tiable vector space, a linear map A → W is smooth if and only if the composed mapA→ V is.

If W ⊂ V is a differentiable subspace, then we can form the quotient V/W. A mapfrom M to V/W is smooth if, locally on M, it lifts to a smooth map to V. The connection onV/W is uniquely determined by the requirement that the map V → V/W is compatiblewith connections (and so a map of differentiable spaces). We call V/W a differentiablequotient of V.

Again, this has the usual universal property: if A is another differentiable space, alinear map V/W → A is smooth if and only if the composed map V → A is smooth.

B.1.1.1 Lemma. The category of differentiable spaces admits all products and coproducts.

These can be described explicitly as follows. Let Vi | i ∈ I be some family of differentiablespaces indexed by a set I. The product ∏i Vi differentiable space has, as underlying vector space,the product vector space ∏ Vi. A map M → ∏i Vi is smooth if and only if the composed mapsM→ Vi are smooth for all i. The connection map

∇M,∏ Vi : C∞(M, ∏ Vi) = ∏ C∞(M, Vi)→ Ω1(M, ∏ Vi) = ∏ Ω1(M, Vi)

is the product of the connections ∇M,Vi .

Similarly, the differentiable coproduct of the Vi has, as underlying vector space, the ordinarydirect sum ⊕Vi. A map f : M → ⊕Vi is smooth if, locally on M, f can be written as a finite sumof smooth maps to some Vi1 , . . . , Vik . The connection

∇M,⊕Vi : C∞(M,⊕Vi)→ Ω1(M,⊕Vi)

is the unique connection which restricts to ∇M,Vi on the subspace C∞(M, Vi).

PROOF. We need to verify that the product and coproduct as described above havethe desired universal properties. For the product, this is immediate. Let’s verify it for thecoproduct. Let A be another differentiable vector space. Let f : ⊕Vi → A be a linear map.Suppose that the maps fi : Vi → A are all smooth. We need to show that f is smooth.Let φ : M → ⊕Vi be a smooth map. To show that f φ is smooth, it suffices to do solocally on M. Thus, we can assume that φ can be written as a finite sum of smooth mapsφi : M → Vi. Then, f φ is a finite sum of f φi, and by assumption, f φi : M → A aresmooth. It is straightforward to verify that the fact that the maps fi are compatible withthe connections on Vi and A imply that f is compatible with connections.

B.1.1.2 Corollary. The category of differentiable vector spaces admits all limits (and so is com-plete).


PROOF. Arbitrary limits are obtained from products and kernels.Thus, we need toverify that the category of differentiable vector spaces admits kernels.

Let f : V → W be a map of differentiable vector spaces. Let us consider the kernelKer f ⊂ V, just as an ordinary vector space. We say that a map M → Ker f is smooth ifand only if the composed map to V is smooth: this gives Ker f the subspace diffeology.Then, the sequence

0→ C∞(M, Ker f )→ C∞(M, V)→ C∞(M, W)

is exact.

We need to give Ker f a connection. Since the map C∞(M, V) → C∞(M, W) is com-patible with connections, the connection ∇M,V on C∞(M, V) must map C∞(M, Ker f ) toΩ1(M, Ker f ).

It is easy to verify that Ker f satisfies the universal property of a kernel.

Note that the forgetful functor DVS→ Vect preserves all limits.

B.1.2. Cokernels and exact sequences. The category of differentiable spaces does notadmit all cokernels. Here is the prime example. Let W be a differentiable space, and letV ⊂ W be an arbitrary linear subspace. We equip V with the initial diffeology, by sayingthat the space of smooth maps M → V is the algebraic tensor product C∞(M) ⊗alg V,rather than the subspace diffeology. Suppose these two diffeologies differ. The quotientW/V has a natural diffeology, by saying that a map M→W/V is smooth if locally it liftsto a smooth map to W. The fact that the sequence

C∞(M)⊗alg V = C∞(M, V)→ C∞(M, W)→ C∞(M, W/V)→ 0

is not exact means that the connection on C∞(M, W) need not descend to one on C∞(M, V/W).

Happily, this example is the only way that things can go wrong.

B.1.2.1 Definition. A map f : V →W of differentiable spaces is admissible if, for all manifoldsM and all maps φ : M→ Im f , the composed map M→W is smooth if and only if φ lifts locallyto a smooth map to V.

In other words, f is admissible if the two natural pre-diffeologies on Im f (where we view it asa quotient of V or a subspace of W) coincide.

B.1.2.2 Lemma. Cokernels of admissible maps exist.

PROOF. If f : V → W is an admissible map, we give Coker f the quotient diffeology:a map M → Coker f is smooth if locally it lifts to a smooth map to W. Let C∞

M(V) denote


the sheaf on M which sends U ⊂ M to C∞(U, V). Then, the sequence of sheaves

C∞M(V)→ C∞

M(W)→ C∞M(Coker f )→ 0

is exact. This implies that the connection on C∞(M, W) descends uniquely to one onC∞(M, Coker f ).

Another simple class of colimits that exist is the following.

B.1.2.3 Lemma. The category of differentiable vector spaces is closed under taking sequentialcolimits of injective maps.

By injective, we just mean that the map on the underlying vector space is injective.

PROOF. Let Vi for i ∈ Z≥0 be a sequence of differentiable vector spaces, and let fij :Vi → Vj be injective maps with f jk fij = fik. Let V denote the ordinary vector space

V = colim Vi = ∪Vi.

We say a map from a smooth manifold M to V is smooth if, locally on M, it comes froma smooth map to one of the Vi. Let C∞

M(V) denote the sheaf on M of smooth maps to V;then

C∞M(V) = colim C∞

M(Vi).This identification uses the fact that the maps in our directed system are injective. Recallalso that the colimit in the category of sheaves is defined to be the sheafification of thecolimit in the category of presheaves.

Now, we define the flat connection ∇M,V to be the map of sheaves

∇M,V : C∞M(V)→ Ω1

M(V)

which arises as the colimit of the maps of sheaves

C∞M(Vi)→ Ω1

M(Vi).

B.1.2.4 Definition. A sequence 0 → A → B → C → 0 of differentiable spaces is exact if itis exact as a sequence of ordinary vector spaces, A ⊂ B is a differentiable subspace, and C is adifferentiable quotient.

Equivalently, the sequence is exact if A is the kernel of the map B → C and C is the cokernelof the map A→ B.

Let V be a differentiable vector space. By evaluating V on open subsets of Rn, Vbecomes a sheaf on Rn. We can thus define the stalk

Stalkn(V) = colim0∈U⊂Rn V(U)


of V at the origin in Rn. The colimit above is taken over open subsets of Rn containingthe origin.

Note that the stalk of V at a point in any manifold can be defined in the same way, butthe stalk at a point in a n-dimensional manifold is the same as the stalk at the origin in Rn.

B.1.2.5 Lemma. A sequence of differentiable vector spaces 0→ A→ B→ C → 0 is exact if andonly if, for all n, the sequence

0→ Stalkn(A)→ Stalkn(B)→ Stalkn(C)→ 0

of vector spaces is exact.

PROOF. Suppose 0 → A → B → C → 0 is an exact sequence of differentiable vec-tor spaces. Then, for any manifold M, the sequence 0 → C∞(M, A) → C∞(M, B) →C∞(M, C) is exact. Further, a map M → C is smooth if locally on M it lifts to a smoothmap to B. Thus, if C∞

M(B) denotes the sheaf on M of smooth maps to B, the sequence

(†) 0→ C∞M(A)→ C∞

M(B)→ C∞M(C)→ 0

of sheaves on M is exact. This implies that the corresponding sequence on stalks is exact.

Conversely, if the sequence of stalks is exact for all n, then the sequence (†) of sheavesis exact for all manifolds M. This implies that the sequence

0→ Ω1M(A)→ Ω1

M(B)→ Ω1M(C)→ 0

is also exact, where we define

Ω1M(A) = Ω1

M ⊗C∞M

C∞M(A).

It follows that the connection ∇M,C on C is the unique connection which descends fromthat on B, and that the connection on A is the restriction of the connection on B to A. Theseare the conditions we imposed for A to be a differentiable subspace of B and for C to be adifferentiable quotient.

B.1.3. The multicategory structure. We will consider the category of differentiablevector spaces as a multicategory, instead of a symmetric monoidal category.

B.1.3.1 Definition. If V1, . . . , Vk, W are differentiable vector spaces, then a smooth multilinearmap

φ : V1 × · · · ×Vk →Wis a multilinear map with the following properties.

(1) It is smooth: if fi : M→ Vi are smooth maps from a manifold M, then

φ( f1, . . . , fk) : M→W

is a smooth map.


(2) It is compatible with flat connections: if fi : M→ V are smooth and if X is a vector fieldon M, then

∇Xφ( f1, . . . , fk) = ∑i

φ( f1, . . . ,∇X fi, . . . , fk).

Differentiable vector spaces form a multicategory where the multi-morphisms Hom(V1, . . . Vk; W)are smooth multilinear maps.

B.1.4. In general, we can not tensor differentiable vector spaces together. Nonethe-less, certain tensor products arise naturally and we will use them repeatedly.

First, we can tensor a differentiable vector space V with the algebra of smooth func-tions on a manifold: we use the notation

C∞(M)⊗V = C∞(M, V)

when V is a differentiable vector space.

Similarly, if E is a vector bundle on M, then C∞(M, E) is a projective module overC∞(M). Thus, it’s reasonable to form the algebraic tensor product

C∞(M, E)⊗C∞(M) C∞(M, V).

We interpret the output as a kind of completed tensor product of V with C∞(M, E). Wewill use the notation C∞(M, E⊗V) to denote this tensor product. This notation is natural:we can tensor a differentiable vector space with a finite-dimensional vector space, so thatE⊗V can be thought of as a bundle of differentiable vector spaces on M.

B.1.4.1 Lemma. Let E, F be vector bundles on M. Let D : C∞(M, E) → C∞(M, F) be a differ-ential operator. Let V be a differentiable vector space. Then, the map

D⊗ 1 : C∞(M, E)⊗alg V → C∞(M, F)⊗alg V

extends canonically to a map

C∞(M, E⊗V)→ C∞(M, F⊗V).

PROOF. Let’s start with the case when E and F are trivial of rank 1. Then we areasserting that differential operators on M act naturally on C∞(M, V). This action arisesin the standard way from the connection ∇M,V : C∞(M, V) → Ω1(M, V) which we aregiven as part of the structure of a differentiable vector space.

In the case when E and F are non-trivial, one constructs the desired map in localtrivializations and then verifies that it is independent of the choice of local trivializations.This is standard.

B.2. DIFFERENTIABLE VECTOR SPACES FROM SECTIONS OF A VECTOR BUNDLE 461

Here is a formal way to describe these structures. Let C denote the following sym-metric monoidal category: the objects are smooth manifolds M equipped with a vectorbundle E; a morphism (M, E) → (N, F) is a smooth map f : M → N together with adifferential operator C∞(M, f ∗F)→ C∞(M, E); and the tensor product is defined by

(M, E)⊗ (N, F) = (M× N, E F).

Then, the category of differentiable vector spaces DVS is tensored over Cop.

B.2. Differentiable vector spaces from sections of a vector bundle

In this section, we will describe various classes of sections of a vector bundle on amanifold M and show that each is equipped with a natural diffeology and flat connec-tion. There are various ways to express this construction; we begin in a more geometriclanguage and then rephrase in the language of topological vector spaces.

The most basic example is as follows. Let M be a manifold, and let E be a vectorbundle on M. Then the space

E = Γ(M, E)

of smooth sections of E is a differentiable vector space. We let

Ec = Γc(M, E)

be the space of compactly supported smooth sections of E on M.

B.2.0.2 Definition. Equip E with the diffeology where, for N a smooth manifold, a smooth mapf : N → E is a section of the pullback bundle π∗ME on N ×M arising from the projection mapπM : N ×M→ M.

Similarly, give Ec a diffeology by saying that a smooth map N → Ec is a section s of π∗ME onN ×M with the property that the map Supp(s) → N is proper (where Supp(s) is the closure ofthe locus on which s is non-vanishing).

Note that the spaces E and Ec are complete locally-convex topological vector spaces,using the standard topologies for these spaces. As is explained in [KM97], for example,one has a notion of smooth map N → V for any manifold N and for any such topologicalvector space V. Thus, V defines a diffeological space. The diffeologies described aboveon E and Ec arise from the standard topologies on these spaces.

Notice as well that C∞(N, E ) is the vector space given by the completed projectivetensor product C∞(N)⊗ E , which is their natural tensor product as nuclear spaces.

Next, we explain the flat connections on the spaces E and Ec.


B.2.0.3 Definition. Let N be a smooth manifold. Equip the pullback bundle π∗ME on N × Mwith the natural flat connection along the fibers of the projection map πM : N × M → M. Wethus obtain a map

∇N,E : Γ(N ×M, π∗ME)→ Γ(N ×M, T∗N E)or, equivalently, a map

∇N,E : C∞(N, E )→ Ω1(N, E ).This defines the flat connection on C∞(N, E ) and so gives E the structure of a differentiable vectorspace.

This flat connection preserves the subspace C∞(N, Ec), giving Ec the structure of a differen-tiable vector space.

B.2.1. We are also interested in distributional sections of a vector bundle E. LetD(M)denote the space of distributions on M, that is, the continuous dual of the space C∞

c (M).Let Dc(M) denote the space of compactly supported distributions on M, which is thecontinuous dual of C∞(M).

We letE (M) = E (M)⊗C∞(M) D(M)

be the space of distributional sections of E. (These are sections whose coefficients aredistributions rather than functions.) We let

E c(M) = Ec(M)⊗C∞c (M) Dc(M)

be the space of compactly supported distributional sections of E.

B.2.1.1 Definition. Equip the space D(M) of distributions on M with the diffeology where asmooth map N → D(M) is a continuous linear map C∞

c (M)→ C∞(N). Similarly, give Dc(M)a diffeology by saying that a smooth map N → Dc(M) is a continuous linear map C∞(M) →C∞(N).

Equip E (M) with the diffeology in which the vector space of smooth maps N → E (M) is

C∞(N, E (M)) = C∞(N, E (M))⊗C∞(M×N) C∞(N,D(M))

(where we use the notation C∞(N, V) to indicate the space of smooth maps from N to a diffeologicalvector space V).

Similarly, give E c(M) a diffeology by saying that

C∞(N, E c(M)) = C∞(N, Ec(M))⊗C∞(N,C∞c (M)) C∞(N,Dc(M)).

Remark: The diffeologies we have defined on these spaces of distributions again arise fromthe standard topology on these spaces. In particular, note that

C∞(N,D(M)) = C∞(N)⊗ D(M)

B.3. DIFFERENTIABLE VECTOR SPACES FROM HOLOMORPHIC SECTIONS OF A HOLOMORPHIC VECTOR BUNDLE463

as vector spaces, where⊗ denotes the completed projective tensor product. Compare thisto the analogous definition of C∞(N, C∞(M)) from earlier.

We need to equip these diffeological vector spaces with flat connections to make theminto differentiable vector spaces. There is a natural choice.

B.2.1.2 Definition. We extend the diffeological vector spaceD(M) to a differentiable vector spaceas follows. There is a natural inclusion

C∞(N,D(M)) → D(N ×M).

The Lie algebra Vect(N) of vector fields on N acts naturally on D(N ×M); this action preservesthe subspace C∞(N,D(M)), giving this subspace the desired flat connection.

The action of Vect(N) also preserves the smaller subspace

C∞(N,Dc(M)) ⊂ D(N ×M)

and so gives C∞(N,Dc(M)) the structure of differentiable vector space.

Similarly, the spaceD(N×M, π∗ME) of distributional sections on N×M of the vector bundleπ∗ME has a natural action of vector fields on M. This preserves the subspaces C∞(N, E(M)) andC∞(N, Ec(M)), and gives those the structure of differentiable vector spaces.

Let E! denote the vector bundle E⊗DensM, where DensM denotes the bundle of den-sities on M. Then, as above, we can define vector spaces

E ! = Γ(M, E!)

E !c = Γc(M, E!).

The vector spaces have natural diffeologies and topologies. There are natural identifica-tions

E (N) = Homcont(E!

c , C∞(N))

E c(N) = Homcont(E!, C∞(N))

where Homcont denotes the vector space of continuous linear maps.

B.3. Differentiable vector spaces from holomorphic sections of a holomorphic vectorbundle

For a complex manifold X and a holomorphic vector bundle E on X, the space ofglobal holomorphic sections OX(E) provides another important and natural example of atopological vector space arising from geometry. We now show how such spaces and theircontinuous duals yield differentiable vector spaces.


We explain the simplest example, as the general case is completely parallel. Let X bea complex manifold. Let O(X) denote the vector space of holomorphic functions on X.Recall that it has a natural topology inherited from smooth functions on X (this topologyis also given by uniform convergence on compact sets).

B.3.0.3 Definition. Equip O(X) with the diffeology where, for N a smooth manifold, a smoothmap f : N → O(X) is a smooth function on N × X such that fn(x) := f (n, x) is a holomorphicfunction on X for every n ∈ N.

Alternatively, we could simply use the construction from [KM97], turning a completelocally-convex vector space into a diffeological vector space.

Note that O(X) is a diffeological subspace of the differentiable vector space C∞(X).Moreover, it is preserved by the flat connection on C∞(X) and hence inherits a differen-tiable structure.

Let O(X)∨ denote the vector space of continuous linear functionals on O(X).

B.3.0.4 Definition. Equip O(X)∨ with the diffeology where, for N a smooth manifold, a smoothmap f : N → O(X) is a continuous linear map from O(X) to C∞(N).

This definition states that C∞(N, O(X)) is the vector space C∞(N)⊗ O(X), where ⊗here denotes the completed projective tensor product. The flat connection is then simplydN ⊗ 1O(X), where dN is the exterior derivative on N.

B.4. Differentiable cochain complexes

B.4.0.5 Definition. A differentiable cochain complex is a cochain complex V where each Vi isa differentiable vector space and each differential d : Vi → Vi+1 is a smooth map.

A map of differentiable cochain complexes is simply a cochain map V → W whose con-stituent maps Vi →W i are all smooth.

A cochain homotopy of such maps is a cochain homotopy whose constituent maps Vi →W i−1 are all smooth.

We want to perform standard constructions from homological algebra with differ-entiable cochain complexes. Of course, we need to make sure the definitions take intoaccount the diffeological structure.

B.4.0.6 Definition. (1) A sequence 0 → A → B → C → 0 of differentiable cochaincomplexes is exact if the component sequences 0→ Ai → Bi → Ci → 0 are exact.

B.4. DIFFERENTIABLE COCHAIN COMPLEXES 465

(2) A map f : A → B of differentiable cochain complexes is a cofibration if it fits into an

exact sequences 0→ Af→ B→ C → 0.

(3) Similarly, a map f : A → B is a fibration if it fits into an exact sequence 0 → C →A

f→ B→ 0.

Note that A → B is a cofibration (respectively, fibration) if and only if, for all n, themap Stalkn(A) → Stalkn(B) is an injective (respectively, surjective) map of cochain com-plexes. Equivalently, the map A → B is a cofibration if in each cohomological degree, Ai

is a differentiable subspace of Bi. This means that Ai → Bi is injective and that a mapM → Ai is smooth if and only if the composed map to Bi is smooth. Similarly, A → B isa fibration if and only if every Ai → Bi is surjective and a map M → Bi is smooth if andonly if it locally lifts to a map to Ai.

We have seen above that one can take kernels and cokernels in the category of differ-entiable vector spaces. Thus, we can define the cohomology groups Hi(A) of any differ-entiable cochain complex. These are differentiable vector spaces.

B.4.0.7 Definition. A map A → B is a weak equivalence if and only if, for all n, the map ofcochain complexes Stalkn A→ Stalkn B is a quasi-isomorphism.

Standard constructions and lemmas from ordinary homological algebra hold in thissetting. For example, if f : V → W is a map of differentiable cochain complexes, we canform the cone Cone( f ), whose underlying graded differentiable space is V[1] ⊕W, butequipped with differential (

dV[1] f0 dW

).

If f is a fibration, then the map

Ker f [1]→ Cone( f )

is an equivalence. If f is a cofibration, then the map

Cone( f )→ Coker( f )

is an equivalence.

B.4.1. We will often use versions of spectral sequence arguments in the category ofdifferentiable complexes.

Suppose that Vi is a directed system of differentiable cochain complexes indexed byi ∈ Z≥0. Thus, we have maps fi : Vi → Vi+1.


Let us suppose that the maps fi are cofibrations. Then, since the category of differen-tiable vector spaces is closed under colimits of cofibrant maps, we can form the differen-tiable cochain complex colimi V, which in cohomological degree k is colimi Vk

i .

B.4.1.1 Lemma. Let V∗, W∗ be sequential directed systems where the maps Vi → Vi+1, Wi →Wi+1 are cofibrations. Let V∗ →W∗ be a map of directed systems.

Suppose that the maps Vi/Vi−1 →Wi/Wi−1 are all weak equivalences.

Then the mapcolim Vi → colim Wj


PROOF. We need to verify that the maps are equivalences at the level of stalks. Theforgetful functors

Stalkn : DVS→ Vect

commute with all colimits of cofibrations. It follows that, in the situation above,

colim Stalkn Vi = Stalkn colim Vi.

Stalkn(Vi/Vi−1) = Stalkn Vi/ Stalkn Vi−1.

Now, Stalkn Vi is a directed system of cochain complexes where the maps are injective, andlikewise for Stalkn Wi. Therefore, by the usual spectral sequence argument, if the map

Stalkn Vi/ Stalkn Vi−1 → Stalkn Wi Stalkn Wi−1

is a quasi-isomorphism for all n, the map

colim Stalkn Vi → colim Stalkn Wi

is also a weak equivalence, giving the desired result.

Similarly, we have the following.

B.4.1.2 Lemma. Let V∗, W∗ be sequential directed systems of differentiable cochain complexes,where the maps Vi → Vi+1, Wi →Wi+1 are cofibrations. Let V∗ →W∗ be a map of systems, suchthat the constituent maps Vi → Wi are quasi-isomorphisms. Then the map colim Vi → colim Wiis a quasi-isomorphism.

PROOF. The proof is almost identical to that of the previous lemma.

We have a similar statement for inverse systems, but only under some stronger hy-pothesis.

B.4. DIFFERENTIABLE COCHAIN COMPLEXES 467

B.4.1.3 Lemma. Let V∗, W∗ be sequential inverse systems of differentiable cochain complexes.Thus, there are maps fi : Vi → Vi−1 and gi : Wi → Wi−1. Suppose that these maps are fi-brations and that the systems Vi, Wi are eventually constant. In other words, the maps fi, gi areisomorphisms for i sufficiently large.

Let V∗ →W∗ be a map of inverse systems, which induces a quasi-isomorphism

Ker fi → Ker gi

for each i.

Then the map lim←−V → lim←−W is a quasi-isomorphism.

PROOF. The functor of stalks commutes with finite limits of fibrations. The fact thatthe maps Vi → Vi−1 are isomorphisms for sufficiently large i ≥ N thus implies that

Stalkn lim←−Vi = lim←− Stalkn Vi.

Because the maps Vi → Vi−1 are fibrations, the maps Stalkn Vi → Stalkn Vj are surjectivemaps of cochain complexes. The spectral sequence argument implies that the map

lim←− Stalkn Vi → lim←− Stalkn Wi

is a quasi-isomorphism as desired.

B.4.2. With these definitions, we can define a factorization algebra valued in the mul-ticategory of differentiable cochain complexes. Indeed, we have already given a generaldefinition of factorization algebra valued in a multicategory. The definition presentedhere is just an exegesis of the general definition.

B.4.2.1 Definition. A prefactorization algebra on a manifold M valued in the multicategoryof differentiable cochain complexes is the assignment of a differentiable cochain complex F (U) toevery open subset U ⊂ M, together with smooth multilinear cochain maps

F (U1)× · · · × F (Un)→ F (V)

if U1, . . . , Un are disjoint open subsets of V, and satisfying the coherence axioms explained earlier.

Given any such prefactorization algebra F and any Weiss cover

U = Ui | i ∈ Iof an open set V ⊂ M, we can form the Cech complex

C(U,F ).As usual, this is the direct sum

C(U,F ) = ⊕i1,...,ik⊂IF (Ui1 ∩ · · · ∩Uin)[k− 1]

of all finite intersections of elements of the open cover.


Since differentiable cochain complexes admit all coproducts, this Cech complex isagain a differentiable cochain complex.

B.4.2.2 Definition. A differentiable factorization algebra on M is a differentiable prefactor-ization algebra F on M with the property that, for every factorizing cover U of an open subsetV ⊂ M, the map

C(U,F )→ F (V)

is a weak equivalence of differentiable cochain complex (as defined above).

B.5. Pro-cochain complexes

This section involves ordinary vector spaces, not differentiable vector spaces, but itprepares us for an important notion we use throughout the book.

Most of the examples of factorization algebras we construct will take values not in thecategory of ordinary cochain complexes but in the category of pro-cochain complexes or,equivalently, of complete filtered cochain complexes.

B.5.0.3 Definition. A complete filtered cochain complex is a cochain complex V equippedwith a decreasing filtration FiV ⊂ V by sub-cochain complexes indexed by i ∈ Z≥0, such thatF0V = V and

V = lim←−V/FiV.

A map of complete filtered cochain complexes is a map V →W which preserves the filtration.

Such a map is a weak equivalence if the map Gri V → Gri W is a quasi-isomorphism for alli. (Note that this implies that the map V →W is a quasi-isomorphism.)

Some care is needed when defining colimits of complete filtered cochain complexes.

B.5.0.4 Definition. Let Vα | α ∈ A be a collection of complete filtered cochain complexesindexed by some set A. Then the direct sum

⊕α∈A Vα is defined by

⊕α∈A

Vα = lim←−i∈Z≥0

(⊕α

Vα/FiVα

).

(On the right hand side of this equation,⊕

Vα/FiVα indicates the ordinary direct sum of cochaincomplexes.)

The reason for making this definition is that the filtration on the naive direct sum ofthe Vα is not complete. It is easy to verify that the direct sum defined above is a coproductin the category of complete filtered cochain complexes.

B.6. DIFFERENTIABLE PRO-COCHAIN COMPLEXES 469

Similarly, the tensor product of complete filtered cochain complexes needs to be com-pleted.

B.5.0.5 Definition. Let V, W be complete filtered cochain complexes. The tensor product V⊗Wis defined as the limit

V ⊗W = lim←−i,j

(V/FiV)⊗ (V/FjW).

The filtration on V ⊗W is defined by

Fk(V ⊗W) = colimi+j≥k

FiV ⊗ FjW,

where the tensor product FiV ⊗ FjW is defined as the limit of FiV/FrV ⊗ FjW/FsV.

Again, the reason for this definition is that the filtration on the naive tensor productof V ⊗W is not complete.

With this definition of completed direct sum and tensor product, it is straightforwardto modify our definition of factorization algebra to take values in the category of completefiltered cochain complexes.

B.5.0.6 Definition. A complete filtered factorization algebra F on X is a prefactorizationalgebraF taking values in the symmetric monoidal category of complete filtered cochain complexes,using the tensor product described above, such that, for every factorizing open cover U of an opensubset U of X, the map

C(U,F )→ F (U)

is an equivalence. The direct sums and tensor products appearing in the definition of the Cechcomplex are completed, as above.

B.6. Differentiable pro-cochain complexes

As a final elaboration on the concept of cochain complex, we will put together the twoideas described above.

B.6.0.7 Definition. A differentiable pro-cochain complex is a differentiable cochain complexV equipped with a decreasing filtration by differentiable subcomplexes FiV with the followingproperties.

(1) F0V = V.(2) The maps FiV → FjV if i > j are cofibrations. This means that they are injective and

that in each cohomological degree, the map FiVk → FjVk has the property that a mapM→ FjVk is smooth if it lifts to a smooth map to FiVk.


This implies that we can form the quotient differentiable vector space V/FiV, andthat the maps

V/FiV → V/FjV

are fibrations (again for i > j).(3) We require that

V = lim←−V/FiV.

A map of differentiable pro-cochain complexes is a filtration-preserving map V → W.Such a map is a weak equivalence if the maps Gri V → Gri W are weak equivalences of differ-entiable cochain complexes. Note that this implies that the maps V/FiV → W/FiW are weakequivalences of differentiable cochain complexes.

A map V → W of differentiable pro-cochain complexes is a fibration (respectively, a cofibra-tion) if the map V/FiV →W/FiW are fibrations (cofibrations) for all i.

As before, we need to define the completed direct sum and multilinear maps of differ-entiable cochain complexes.

B.6.0.8 Definition. If Vα | α ∈ A is a collection of complete filtered differentiable cochaincomplexes, indexed by some set A, then the completed direct sum

⊕α∈AVα is defined to be the

inverse limit ⊕α∈A

Vα = lim←−i∈Z≥0

⊕α∈A

(Vα/FiVα

),

where on the right hand side we use the ordinary direct sum of differentiable spaces.

We can define the stalks of a differentiable pro-cochain complex Stalkn(V) as the col-imit

Stalkn(V) = colim0∈U⊂Rn

V(U),

where the colimit of the pro-cochain complexes V(U) is completed as above. Thus, Stalkn(V)is a pro-cochain complex, and

Stalkn(V)/Fi Stalkn(V) = Stalkn(V/FiV).

B.6.0.9 Lemma. A map V → W of differentiable pro-cochain complexes is an equivalence if andonly if the maps Stalkn(V)→ Stalkn(W) are weak equivalences of pro-cochain complexes.

PROOF. Immediate.

B.6.0.10 Lemma. Let V∗, W∗ be sequential directed systems of differentiable pro-cochain com-plexes, and let V∗ →W∗ be a map of directed systems.

B.6. DIFFERENTIABLE PRO-COCHAIN COMPLEXES 471

Suppose that the maps Vi → Vj and Wi → Wj are all cofibrations and suppose that the mapsVi →Wi are all equivalences.

Then the mapcolim Vi → colim Wj

is an equivalence.

PROOF. The proof is almost identical to the proof of lemma B.4.1.1.

Similarly, we can have spectral sequences for inverse systems, but only under somemore restrictive hypotheses.

B.6.0.11 Lemma. Let V∗, W∗ be sequential inverse systems of differentiable pro-cochain com-plexes, and let V∗ →W∗ be a map of inverse systems. Let V = lim V∗ and W = lim W∗.

Suppose that

(1) The maps fi : Vi → Vi−1, gi : Wi → Wi−1 are fibrations of differentiable cochaincomplexes.

(2) For each k, the inverse systems V∗/FkV∗ and W∗/FkW∗ are eventually constant, as inlemma B.4.1.3.

(3) The maps Ker fi → Ker gi are quasi-isomorphisms of differentiable cochain complexes.

Then, the map V →W is a quasi-isomorphism of differentiable pro-cochain complexes.

PROOF. This follows immediately from lemma B.4.1.3.

B.6.1. Differentiable pro-cochain complexes form a multicategory, just like differen-tiable cochain complexes.

B.6.1.1 Definition. Let V1, . . . , Vk, W be differentiable pro-cochain complexes. In the multicate-gory of differentiable pro-cochain complexes, an element of Hom(V1, . . . , Vk; W) is a smooth mul-tilinear cochain map

Φ : V1 × · · · ×Vk →W = lim W/FiW

which preserves filtrations: if vi ∈ Fri(Vi), then

Φ(v1, . . . , vk) ∈ Fr1+···+rkW.

Factorization algebras valued in differentiable pro-cochain complexes are defined asbefore.


B.7. Differentiable cochain complexes over a differentiable dg ring

The category of differentiable cochain complexes is a differential graded multi-category.Thus, we can talk about commutative differentiable dg algebras R. This is just a commu-tative dg algebra R, with the structure of a differentiable vector space, such that all thestructure maps are smooth. Similarly, a commutative differentiable pro-algebra is a com-mutative dg algebra in the multi-category of differentiable pro-cochain complexes.

In either context, we can define an R-module M to be a differentiable (pro-)cochaincomplex equipped with an action of the commutative differentiable (pro-)algebra R, inthe obvious way. We say a map M→ M′ is a weak equivalence if it is a weak equivalence(as defined above) in the category of differentiable (pro-)cochain complex.

In either context, we say a sequence of R-modules 0→ M1 → M2 → M3 → 0 is exactif it is exact in the category of differentiable (pro-)cochain complexes. A map M1 → M2 isa cofibration if it can be extended to an exact sequence 0 → M1 → M2 → M3 → 0; it is afibration if it can be extended to an exact sequence 0→ M3 → M1 → M2 → 0.

The category of modules over a differentiable (pro-)dg algebra R is, as above, multi-category. In either case, the multi-maps

HomR(M1, . . . , Mn; N)

are the multi-maps in the category of differentiable (pro-)cochain complexes whose un-derlying multilinear map M1 × · · · ×Mn → N are R-multilinear.

B.8. Classes of functions on the space of sections of a vector bundle

Let M be a manifold and E a graded vector bundle on M. Let U ⊂ M be an opensubset. In this section we will introduce some notation for various classes of functionalson sections E (U) of E on U. These spaces of functionals will all be graded differentiablepro-vector spaces.

B.8.1. We are interested in symmetric algebras on vector spaces of the form E!c(U),

E !c (U), etc. These symmetric algebras can be defined in two ways: either using the com-

pleted projective tensor product of topological vector spaces, or in terms of sections ofbundles on Un. We will explain both points of view.

Thus, let us first define (E (U))⊗n to be the tensor power defined using the completedprojective tensor product on the topological vector space E (U). Then a more concretedescription of this space is as follows. Let En denotes the vector bundle on Mn obtainedas the external tensor product, so

(E (U))⊗n = Γ(Un, En)

B.8. CLASSES OF FUNCTIONS ON THE SPACE OF SECTIONS OF A VECTOR BUNDLE 473

is the space of smooth sections of En on Un. Similarly, we can identify

(Ec(U))⊗n = Γc(Un, En)

(E c(U))⊗n = Γc(Un, En)

(E (U))⊗n = Γ(Un, En)

where Γ indicates the space of distributional sections and the subscript c indicates com-pactly supported distributional sections.

We have already seen how to equip the various kinds of spaces of sections of a vectorbundle with the structure of a differentiable vector space. Since the spaces listed above areexpressed as sections of various kinds of a vector bundle on Un, they all have the structureof differentiable vector spaces.

Symmetric (or exterior) powers of the spaces Ec(U), E c(U), E (U), E (U) are definedby taking coinvariants of the tensor powers defined above with respect to the action of thesymmetric group. These symmetric powers inherit the structure of differentiable vectorspace.

Thus, we can define, for example, the completed symmetric algebra

Sym∗E !

c (U) = ∏n

Symn E !c (U)

Sym∗E

!c(U) = ∏

nSymn E

!c(U)

Note that since E!c(U) is dual to E (U), we can view Sym E

!c(U) as the algebra of formal

power series on E (U). Thus, we often write

Sym E!c(U) = O(E (U)).

Similarly, Sym E !c (U) is the algebra of formal power series on E (U).

In a similar way, we can construct

O(E (U)) = ∏n

Symn(E !c (U))

O(E c(U)) = ∏n

Symn(E !(U)).

These spaces of functionals are all products of the differentiable vector spaces of sym-metric powers, and so they are themselves differentiable vector spaces. We will equipall of these spaces of functionals with the structure of a differentiable pro-vector space,induced by the filtration

FiO(E (U)) = ∏n≥i

Symi E!c(U)


(and similarly for O(Ec(U)), O(E (U)) and O(E c(U))).

The natural product O(E (U)) is compatible with the differentiable structure, mak-ing O(E (U)) into a commutative algebra in the multicategory of differentiable gradedpro-vector spaces. The same holds for the spaces of functionals O(Ec(U)), O(E (U)) andO(E c(U)).

B.8.2. One-forms. Recall that if V is a vector space, we can define the space of one-forms on V (treated as formal scheme) as

Ω1(V) = O(V)⊗V∨.

Similarly, we can define

Ω1(E (U)) = O(E (U))⊗ E!c(U),

where ⊗ denotes the completed projective tensor product.

In concrete terms,

Ω1(E (U)) = ∏n

Symn(E!c(U))⊗ E

!c(U)

and we can identify the space

Symn(E!c(U))⊗ E

!c(U)) ⊂ E

!c(U)⊗n+1 = Γc(Un+1, (E!)n+1)

as the space of compactly supported distributional sections of (E!)n+1 that are symmetricin the first n variables.

In this way, Ω1(E (U)) becomes a differentiable pro-cochain complex, where the filtra-tion is defined by

FiΩ1(E (U)) = ∏n≥i−1

Symn(E!c(U))⊗ E

!c(U).

Further, Ω1(E (U)) is a module for the commutative algebra O(E (U)), where the modulestructure is defined in the multicategory of differentiable pro-vector spaces.

If V is a finite-dimensional vector space, the exterior derivative map

d : O(V)→ O(V)⊗V∨

is, in components, just the composition

Symn+1 V∨ → (V∨)⊗n+1 → Symn(V∨)⊗V∨

where the maps are the inclusion followed by the natural projection (up to an overallcombinatorial constant).

We can, in a similar way, define the exterior derivative

d : O(E (U))→ Ω1(E (U))


by saying that on components it is given by the same formula as in the finite-dimensionalcase.

B.8.3. Other classes of sections of a vector bundle. Before we introduce our nextclass of functionals — those with proper support — we need to introduce some furthernotation concerning classes of sections of a vector bundle.

Let M be a manifold, and let f : M→ N be a fibration. Let E be a vector bundle on M.We say a section s ∈ Γ(M, E) has relative compact support if the map

f : Supp(s)→ N

is proper. We let Γc/ f (M, E) denote the space of sections with relative compact support.This is a differentiable vector space: if X is an auxiliary manifold, a smooth map X →Γc/ f (M, E) is a section of the bundle π∗ME on X ×M which has relative compact supportrelative to the map

M× X → N × X.(It is straightforward to write down a flat connection on C∞(X, Γc/ f (M, E)), using argu-ments of the type described in section B.2.)

Next, we need to consider spaces of the form E (M) ⊗ F (N), where M and N aremanifolds and E, F are vector bundles on M and N respectively. Of course, we can givean abstract definition using the projective tensor product, but we want a more geometricinterpretation.

There are several ways to identify this space geometrically. We will view E (M) ⊗F (N) as a subspace

E (M)⊗F (N) ⊂ E (M)⊗F (N).It consists of those elements D with the property that, if φ ∈ E !

c (M), then map

D(φ) : F !c(N) → R

ψ 7→ D(φ⊗ ψ)

comes from an element of F (N).

Alternatively, E (M) ⊗ F (N) is the space of continuous linear maps from E !c (M) to

F (N).

We can similarly define E c(M)⊗F (N) as the subspace of those elements of E (M)⊗F (N) that have compact support relative to the projection M× N → N.

These spaces form differentiable vector spaces in a natural way: a smooth map from anauxiliary manifold X to E (M)⊗F (N) is an element of E (N)⊗F (N)⊗C∞(X). Similarly,a smooth map to E c(M)⊗F (N) is an element of E (M)⊗F (N)⊗ C∞(X) whose supportis compact relative to the map M× N × X → N × X.


B.8.4. Functions with proper support. Recall that

Ω1(Ec(U)) = O(Ec(U))⊗ E!(U).

We can thus define a subspace

O(E (U))⊗ E!(U) ⊂ Ω1(Ec(U)).

The Taylor components of elements of this subspace are in the space

Symn(E!c(U))⊗ E

!(U),

which in concrete terms is the Sn invariants of

E!c(U)⊗n ⊗ E

!(U).

B.8.4.1 Definition. A function Φ ∈ O(Ec(U)) has proper support if

dΦ ∈ O(E (U))⊗ E!(U) ⊂ O(Ec(U))⊗ E

!(U)∨.

The reason for the terminology is as follows. Let Φ ∈ O(Ec(U)) and let

Φn ∈ Hom(Ec(U)⊗n, R)

be the nth term in the Taylor expansion of Φ.

Then, Φ has proper support if and only if, for all n, the composition with a projectionmap

Supp(Φn) ⊂ Un → Un−1

is proper.

We will let

OP(Ec(U)) ⊂ O(Ec(U))

be the subspace of functions with proper support. Note that functions with proper sup-port are not a subalgebra.

Because OP(Ec(U)) fits into a fiber square

OP(Ec(U)) → O(E (U))⊗ Ec(U)∨

↓ ↓O(Ec(U)) → O(Ec(U))⊗ Ec(U)∨

it has a natural structure of a differentiable pro-vector space.


B.8.5. Functions with smooth first derivative.

B.8.5.1 Definition. A function Φ ∈ O(Ec(U)) has smooth first derivative if dΦ, which is apriori an element of

Ω1(Ec(U)) = O(Ec(U))⊗ E!(U)

is an element of the subspaceO(Ec(U))⊗ E !(U).

Note that we can identify, concretely, O(Ec(U))⊗ E !(U) with the space

∏n

Symn E!(U)⊗ E !(U)

andSymn E

!(U)⊗ E !(U) ⊂ E

!(U)⊗n ⊗ E !(U).

(Spaces of the form E (U)⊗ E (U) were described concretely above.)

Thus O(Ec(U))⊗ E !(U) is a differentiable pro-vector space. It follows that the spaceof functionals with smooth first derivative is a differentiable pro-vector space, since it isdefined by a fiber diagram of such objects.

An even more concrete description of the space O sm(Ec(U)) of functionals with smoothfirst derivative is as follows.

B.8.5.2 Lemma. A functional Φ ∈ O(Ec(U)) has smooth first derivative if each of its Taylorcomponents

DnΦ ∈ Symn E!(U) ⊂ E

!(U)⊗n

lies in the intersection of all the subspaces

E!(U)⊗k ⊗ E !(U)⊗ E

!(U)⊗n−k−1

for 0 ≤ k ≤ n− 1.

PROOF. The proof is a simple calculation.

Note that the space of functions with smooth first derivative is a subalgebra of O(Ec(U)).We will denote this subalgebra by O sm(Ec(U)). Again, the space of functions with smoothfirst derivative is a differentiable pro-vector space, as it is defined as a fiber product.

We can also define the space of functions on E (U) with smooth first derivative, byrequiring that the exterior derivative lies in

O(E (U))⊗ E !c (U) ⊂ Ω1(E (U)).


B.8.6. Functions with smooth first derivative and proper support. We are particu-larly interested in those functions which have both smooth first derivative and propersupport. We will refer to this subspace as OP,sm(Ec(U)). The differentiable structure onOP,sm(Ec(U)) is, again, given by viewing it as defined by the fiber diagram

OP,sm(Ec(U)) → O(E (U))⊗ E !(U)↓ ↓

O(Ec(U)) → O(Ec(U))⊗ E!(U)

We have inclusions

O sm(E (U)) ⊂ OP,sm(Ec(U)) ⊂ O sm(Ec(U)),

where each inclusion has dense image.

B.9. Derivations

As before, let M be a manifold, E a graded vector bundle on M, and U an open subsetof M. In this section we will define derivations of algebras of functions on E (U).

To start with, recall that, for V a finite dimensional vector space (which we treat asa formal scheme) and O(V) = ∏ Symn V∨ the algebra of formal power series on V, weidentify the space of continuous derivations of O(V) with O(V) ⊗ V. We view thesederivations as the space of vector fields on V and use the notation Vect(V).

In a similar way, we can define the space of vector fields Vect(E (U)) of vector fieldson E (U) as

Vect(E (U)) = O(E (U))⊗ E (U) = ∏n

(Symn(E

!c(U))⊗ E (U)

),

using the completed projective tensor product. We have already seen (section B.8) how todefine the structure of diffeological pro-vector space on spaces of this nature.

In concrete terms, the Taylor expansion of an element of X ∈ Vect(E (U)) is given bya sequence of continuous symmetric multilinear maps

DnX : E (U)× · · · × E (U)→ E (U).

More generally, if M is a smooth manifold and if X : M → Vect(E (U)) is a smooth map,then the Taylor expansion of X is a sequence of continuous symmetric multilinear maps

E (U)× · · · × E (U)→ E (U)⊗ C∞(M) = Γ(U ×M, E∣∣U).

In this section we will show the following.

B.9. DERIVATIONS 479

B.9.0.1 Proposition. Vect(E (U)) has a natural structure of Lie algebra in the multicategory ofdiffeological pro-vector spaces. Further, O(E (U)) has an action of the Lie algebra Vect(E (U)) byderivations, where the structure map Vect(E (U))×O(E (U))→ O(E (U)) is smooth.

PROOF. To start with, let’s look at the case of a finite-dimensional vector space V, toget an explicit formula for the Lie bracket on Vect(V), and the action of Vect(V) on O(V).Then, we will see that these formulae make sense when V = E (U).

Let X ∈ Vect(X), and let us consider the Taylor components DnX, which are multilin-ear maps

V × · · · ×V → V.Our conventions are such that

Dn(X)(v1, . . . , vn) =

(∂

∂v1. . .

∂

∂vnX)(0) ∈ V

Here, we are differentiating vector fields on V using the trivialization of the tangent bun-dle to this formal scheme arising from the linear structure.

Thus, we can view DnX as in the endomorphism operad of the vector space V.

If A : V×n → V and B : V×m → V, let us define

A i B(v1, . . . , vn+m−1) = A(v1, . . . , vi−1, B(vi, . . . , vi+m−1), vi+m, . . . , vn+m−1).

If A, B are symmetric (under Sn and Sm, respectively), then define

A B =n

∑i=1

A i B.

Then, if X, Y are vector fields, the Taylor components of [X, Y] satisfy

Dn([X, Y]) = ∑k+l=n+1

ck,l (DkX DlY− DlY DkX)

where ck,l are combinatorial constants which are irrelevant for our purposes.

Similarly, if f ∈ O(V), the Taylor components of f are multilinear maps

Dn f : V×n → C.

In a similar way, if X is a vector field, we have

Dn(X f ) = ∑k+l=n+1

c′k,l Dk(X) Dk( f ).

Thus, we see that in order to define the Lie bracket on Vect(E (U)), we need to givemaps of diffeological vector spaces

i : Hom(E (U)⊗n, E (U))×Hom(E (U)⊗m, E (U))→ Hom(E (U)⊗(n+m−1), E (U))


where here Hom indicates the space of continuous linear maps, treated as a diffeologicalvector space. Similarly, to define the action of Vect(E (U)) on O(E (U)), we need to definea composition map

i : Hom(E (U)⊗n, E (U))×Hom(E (U)⊗m)→ Hom(E (U)⊗n+m−1).

We will treat the first case; the second is similar.

Now, if X is an auxiliary manifold, a smooth map

X → Hom(E (U)⊗m, E (U))

is the same as a continuous multilinear map

E (U)×m → E (U)⊗ C∞(X).

Here, “continuous” means for the product topology.

This is the same thing as a continuous C∞(X)-multilinear map

Φ : (E (U)⊗ C∞(X))×m → E (U)⊗ C∞(X).

IfΨ : (E (U)⊗ C∞(X))×n → E (U)⊗ C∞(X).

is another such map, then it is easy to define Φ i Ψ by the usual formula:

Φ i Ψ(v1, . . . , vn+m−1) = Φ(v1, . . . vi−1, Ψi(vi, . . . , vm+i−1), . . . , vn+m−1)

if vi ∈ E (U)⊗ C∞(X). This map is C∞(X) linear.

Remark: It is not hard to show that Vect(E (U)), as defined above, is the space of all con-tinuous derivations of the topological algebra O(E (U)); we will not need this fact.

B.10. The Atiyah-Bott lemma

In [AB67], Atiyah and Bott showed that for an elliptic complex (E , d) on a compactclosed manifold M, with E the smooth sections of a Z-graded vector bundle, there is ahomotopy equivalence (E , d) → (E , d) into the elliptic complex of distributional sections.The argument follows from the existence of parametrices for elliptic operators. This resultwas generalized by N.N. Tarkhanov to the non-compact case [Tar87].

Let M be a smooth manifold (which, in general, will not be compact).

B.10.0.2 Definition. An elliptic complex on M is a graded vector bundle E, whose space ofsmooth sections we denote by E , together with a square-zero differential operator Q : E → E ofcohomological degree 1 possessing the following property, known as ellipticity. Let π∗E denote thepullback bundle along the projection map for the cotangent bundle π : T∗M → M. The symbolσ(Q) of Q is a cohomological degree 1 endomorphism of the vector bundle π∗E. We require thatthe complex of vector bundles (π∗E, σ(Q)) on T∗M is exact away from the zero section.

B.10. THE ATIYAH-BOTT LEMMA 481

Let E denote the complex of distributional sections of E . We will endow both E andE with their natural topologies.

B.10.0.3 Lemma (Tarkhanov, [Tar87]). There is a continuous homotopy inverse to the naturalinclusion

(E , Q) → (E , Q).

This is Lemma 1.7 of [Tar87]. The continuous homotopy inverse

Φ : E → E

is given by a kernel KΦ ∈ E ! ⊗ E with proper support. The homotopy S : E → E is acontinuous linear map with

[d, S] = Φ− Id .

The kernel KS for S is a distribution, that is, an element of E! ⊗ E , with proper support.

PROOF. We will reproduce the proof in [Tar87]. Choose a metric on E (Hermitian, ifE is a complex vector bundle) and a volume form on M. Let Q∗ be the adjoint to Q. Weform the graded commutator D = [Q, Q∗]. This is an elliptic operator on each space E i,the cohomological degree i part of E . Thus, by standard results in the theory of pseudo-differential operators, there is a parametrix P for D. The kernel KP is an element of E

!⊗ E ,and the corresponding operator P : Ec → E is an inverse for D up to smoothing operators.

By multiplying KP by a smooth function on M×M which is 1 in a neighborhood ofthe diagonal, we can assume that KP has proper support. This means that P will extendto a map E → E and will still be a parametrix: thus P D and D P both differ from theidentity by smoothing operators.

The homotopy S is now defined by

S = Q∗P.

Note that the homotopy inverse Φ : E → E and the homotopy S : E → E we haveconstructed are maps of differentiable vector spaces (as well as being continuous).

APPENDIX C

Homological algebra with topological vector spaces

For the bulk of the book, we use the language of differentiable vector spaces as a wayto keep track of the analytic structure on the vector spaces that form our factorizationalgebras. The advantage of working with differentiable vector spaces is that homologicalalgebra with sheaves on a site is very well developed, so we don’t need to develop anynew techniques. The disadvantage is that we need to treat the category of differentiablevector spaces as a multi-category, and not as a symmetric monoidal category. Becausedifferentiable vector spaces form a multi-category, it doesn’t make sense to ask that afactorization algebra with values in this category takes disjoint unions to tensor products.Thus, factorization algebras with values in a multicategory are not as local as one wouldlike: for instance, they do not form a sheaf of categories.

If we treat the observables of our field theory as topological vector spaces, however,then it is possible to restore the axiom that the factorization algebra assigns a tensor prod-uct to a disjoint union. For instance, we will see that for every field theory on a manifoldM, the quantum observables Obsq(U) on any open subset have the structure of a topolog-ical vector space over C[[h]], and that for a suitable completed tensor product ⊗ we havea quasi-isomorphism

Obsq(U)⊗C[[h]] Obsq(V) ∼= Obsq(U qV).

(We will explain precisely which tensor product we will use shortly).

Other aspects of homological algebra, however, are much more difficult with topolog-ical vector spaces than with differentiable vector spaces. For instance, it is not obviouswhat one should mean by a quasi-isomorphism of topological vector spaces. If we usethe weakest notion – just a quasi-isomorphism when we forget the topology – then thecompleted tensor product will almost never respect quasi-isomorphisms. With strongernotions – for example, asking for a continuous homotopy equivalence – it is much harderto check that a map is a quasi-isomorphism.

The main result of this appendix is that there is a symmetric monoidal dg categorywhich has the best of both worlds: it is a full subcategory of both differentiable cochaincomplexes and of topological cochain complexes, quasi-isomorphisms are the same asthose in differentiable cochain complexes, and tensor product and Hom’s in this category

483

484 C. HOMOLOGICAL ALGEBRA WITH TOPOLOGICAL VECTOR SPACES

respect quasi-isomorphisms. Further, complexes of observables of a quantum field theorylive in this category.

More precisely, we have the following result.

Theorem. There exists a full subcategory GeoVSast of the category DVS∗ of differentiable vectorspaces, whose objects are called geometric cochain complexes. The category GeoVSast has thefollowing properties.

(1) GeoVSast is also a full subcategory of the category of locally-convex topological vectorspaces.

(2) GeoVSast has the structure of a symmetric monoidal category, such that the functorGeoVSast → DVS∗ is a full embedding of multi-categories.

(3) A map in GeoVSast is a quasi-isomorphism (in DVS∗) if and only if it is a homotopyequivalence. In particular, the symmetric monoidal structure on GeoVSast preservesquasi-isomorphisms, as does the functor

Hom :(GeoVSast)op ×GeoVSast → DVS∗ .

(4) GeoVSast, viewed as a full subcategory of DVS∗, is closed under the formation of allcones and summands. This implies that the category GeoVSast is a pre-triangulated dgcategory, and so a stable ∞-category, in the terminology of [].

(5) If E is an elliptic complex on a manifold M, then the complexes Ec and E c are objects ofGeoVSast.

This theorem asserts the existence of a nice category of geometric cochain complexes.The reason this category is useful for us is that complexes of observables of a quan-tum field theory, which a priori are objects of the category ProDVS∗ of pro-differentiablecochain complexes, actually are pro-objects in GeoVSast.

Let Pro GeoVSast denote the full subcategory of the category of ProDVS∗ consisting ofinverse sequences · · · → V1 → V0 where the objects Vi are in GeoVSast and where the mapsVi → Vi−1 are smoothly split (that is, they are split degreewise in the category DVS butthe splitting might not be a cochain map). For formal reasons, the category Pro GeoVSast

has the same properties we discussed above for GeoVSast. Namely,

(1) Pro GeoVSast is a symmetric-monoidal dg category,(2) the functor Pro GeoVSast → DVS∗ is a full embedding of multicategories,(3) a map in Pro GeoVSast is a quasi-isomorphism if and only if it is a homotopy

equivalence,(4) tensor product and Hom respect quasi-isomorphisms,(5) Pro GeoVSast is a stable infinity-category.

C. HOMOLOGICAL ALGEBRA WITH TOPOLOGICAL VECTOR SPACES 485

Further, the category of quasi-free C[[h]]-modules in Pro GeoVSast inherits all these niceproperties. Here, the tensor product is taken over the ring C[[h]]. We let Pro GeoVSast

hdenote this category (and, similarly, we let ProDVS∗h denote the category of quasi-freeC[[h]]-modules in ProDVS∗.

The tensor product on GeoVSast and on Pro GeoVSast will be denoted by ⊗β (for rea-sons that will be clear later).

Theorem. For any quantum field theory on a manifold M, the factorization algebra Obsq of ob-servables, which a priori is a factorization algebra in the multi-category ProDVS∗h, is actually afactorization algebra in the symmetric monoidal category Pro GeoVSast

h .

More precisely, for every open subset U ⊂ M, the pro-diffeological complex Obsq(U) is anobject in the full subcategory Pro GeoVSast

h ⊂ ProDVS∗h. Further, because Pro GeoVSasth is a full

sub multi-category of ProDVS∗, the C[[h]]-multilinear maps

Obsq(U1)× · · · ×Obsq(Un)→ Obsq(V)

(for disjoint opens Ui in V) extend to maps

Obsq(U1)⊗β,C[[h]] . . . ⊗β,C[[h]] Obsq(Un)→ Obsq(V).

The map

Obsq(U1)⊗β,C[[h]] . . . ⊗β,C[[h]] Obsq(Un)→ Obsq(U1 q · · · qUn)

is a quasi-isomorphism in Pro GeoVSasth (and therefore a homotopy equivalence).

There is a simplicial set of quantizations of a given classical field theory, where dif-ferent quantizations connected by a homotopy should be regarded as equivalent. Thissimplicial set allows us to track the dependence of the field theory on various auxiliarychoices, like that of a gauge-fixing condition.

If we have an n-simplex in this simplicial set, the factorization algebra of observablesbecomes a factorization algebra over Ω∗(4n). We will see that this family of factorizationalgebras also lives in the world of geometric pro-cochain complexes.

Theorem. Suppose we have an n-simplex in the simplicial set of quantizations of a fixed classicalfield theory. Then, the factorization algebra Obsq

4n of observables of this family of theories is afactorization algebra in the category ProGeoVS∗h over the dg ring Ω∗(4n) of cochains on then-simplex.

Further, [n] → [m] is a face or degeneracy map, then the corresponding restriction map offactorization algebras

Obsq4n → Obsq

4m

is a quasi-isomorphism in ProGeoVS∗h, and therefore a homotopy equivalence.


Because Obsq is a factorization algebra valued in ProDVS∗h, we know that there isa quasi-isomorphism in ProDVS∗h between Obsq(U) and the Cech complex constructedfrom any Weiss cover of U.

If U is a Weiss cover of an open subset U, the Cech complex is denoted C(U, Obsq).This Cech complex arises from a simplicial Cech complex C4(U, Obsq). We defined theCech complex just by taking the usual realization of the simplicial Cech complex, andobserving that this makes sense in the category ProDVS∗h. However, in homotopy theory,this is not really sufficient: we would like the Cech complex to be the homotopy colimitof the simplicial Cech complex.

It turns out that this is true in the dg category ProGeoVS∗. More precisely, we havethe following theorem.

Theorem. Fix a quantum field theory on a manifold M. For any Weiss cover U of an open subsetU of M, the Cech complex C(U, Obsq) is an object of ProGeoVS∗h.

It follows that the quasi-isomorphism in ProDVS∗h

C(U, Obsq)→ Obsq(U)

is a homotopy equivalence.

Further, the Cech complex is the homotopy colimit in the dg category ProGeoVS∗ of the sim-plicial Cech object C4(U, Obsq).

In general, a homotopy colimit in a dg category (or (∞, 1)-category) has a homotopyuniversal property similar to the universal property of a colimit in an ordinary category.We will show that the Cech complex has this homotopy universal property.

C.1. Bornological vector spaces

Our construction of the category GeoVS∗ of geometric cochain complexes is a little in-tricate, and requires a detour into the theory of bornological and convenient vector spaces.Our main reference for this theory is [KM97].

C.1.0.4 Definition. If E is a locally-convex topological vector space, the bornologification of E isthe finest locally convex topology on E which has the same bounded sets as E. We say that E isbornological if it is the same as its bornologification.

Remark: All locally-convex topological vector spaces will be Hausdorff, but not necessarilycomplete.

C.1. BORNOLOGICAL VECTOR SPACES 487

We let BVS denote the category of complete bornological vector spaces, which is a fullsubcategory of the category LCTVS of complete locally-convex topological vector spaces.The bornologification functor LCTVS → BVS is a right adjoint to the inclusion functorBVS → LCTVS. An alternative definition of BVS is that it is equivalent to the categoryof locally convex topological vector spaces with bounded (rather than continuous) linearmaps. Bounded is, of course, a weaker condition.

We let DVS be the category of differentiable vector spaces. There is a notion of smoothmap from a manifold M to a locally-convex topological vector space (see [KM97], Chapter1). We can also differentiate smooth maps. In this way, we get a functor LCTVS → DVS.This functor factors through the bornologification functor LCTVS→ BVS.

C.1.0.5 Proposition. When restricted to BVS, this functor embeds BVS as a full subcategory ofDVS.

It follows that BVS is equivalent to the essential image of LCTVS in TVS.

This proposition is Corollary 4.6 in [KM97]. The point is that smooth maps betweenbornological vector spaces are the same as continuous maps, and that any topologicalvector space and its bornologification have the same smooth maps from any manifold.

The multicategory structure on DVS restricts to one on BVS.

C.1.0.6 Lemma. The multicategory structure on BVS arises from a symmetric monoidal struc-ture.

The tensor product in this symmetric monoidal structure preserves all colimits.

PROOF. We need to show that, given any objects E1, . . . , En of BVS, there is an objectE1⊗β . . .⊗βEn such that

Hom(E1 × · · · × En, G) = Hom(E1⊗β . . .⊗βEn, G)

for all objects G of BVS. On the left hand side of this equation we have the space of smoothmultilinear maps. According to Lemma 5.5 of [KM97], a map is smooth if and only if it isbounded, and in section 5.7 of [KM97] it is shown that the tensor product E1⊗β . . .⊗βEnexists. It is called the bornological tensor product.

The fact that the tensor product preserves colimits is the theorem in section 5.7 of[KM97].

Thus, we have constructed a symmetric monoidal category BVS with a full embeddingBVS → DVS of multicategories.

The functor BVS→ LCTVS preserves all colimits, as it is a left adjoint.


The functor BVS→ DVS does not preserve all colimits, but it does preserve a class ofcolimits.

C.1.0.7 Proposition. The functor BVS → DVS preserves countable coproducts and sequentialcolimits of closed embeddings.

PROOF. Since the case of countable coproducts is a special case of that of sequentialcolimits of closed embeddings, we will prove the latter. Suppose we have a sequenceV1 → V2 . . . of closed embeddings. We let V = colim Vi.

We need to show that for all manifolds M, C∞(M, V) is the colimit, in the categoryof sheaves on M, of the sheaf of smooth maps from M to Vi. By definition, a section ofthe colimit of these sheaves is a map from M to V which locally on M factors through asmooth map to some Vi.

It is clear that this colimit is a subspace of C∞(M, V). Since Vi is a closed subspace ofV, lemma 3.8 of [KM97] tells us that a map to Vi is smooth if and only if the composd mapto V is smooth. We thus need to show that every smooth map from M to V locally factorsthrough some Vi.

By [KM97], 52.8, a subset of V is bounded if and only if it is a bounded subset of someVi. It thus suffices to show that a smooth map to V locally lies in a bounded subset. In factthis is true for continuous maps to any locally convex topological vector space. Indeed,given any point p ∈ M, choose a neighbourhood U of p whose closure U in M is compact.If f : M→ V is continuous, then f (U) is also compact and so bounded.

Note that the functor LCTVS → BVS preserves limits, because it is a right adjoint. Itfollows that BVS admits all limits (since LCTVS does). The functor BVS → LCTVS doesnot, however, preserve all limits.

C.1.0.8 Proposition. The functor BVS→ DVS preserves all limits.

PROOF. Lemma 3.8 of [KM97] show that the functor LCTVS→ DVS preserves limits.We need to show that BVS is closed inside DVS under formation of limits. Since BVS isthe essential image of LCTVS, this is immediate.

C.1.0.9 Corollary. The functor BVS → DVS has a left adjoint, as does the functor LCTVS →DVS.

PROOF. It suffices to construct the adjoint functor DVS → BVS, as the one to LCTVSis obtained by composing with the full embedding BVS → LCTVS. This follows fromthe adjoint functor theorem, since this functor preserves limits and since BVS admits all


limits. One can check easily that the set-theoretic criteria of the adjoint functor theoremholds.

We won’t really use this left adjoint much, but it’s useful to know that it exists. It isimmediate that the composition

BVS→ DVS→ BVS

is the identity, because BVS is a full subcategory.

C.1.0.10 Corollary. BVS admits all colimits.

PROOF. Indeed, the category DVS admits all colimits and the functor DVS → BVS iscolimit preserving.

Alternatively, see page 53 of [KM97].

C.1.1. Completeness. We are interested in topological vector spaces which have somenotion of completeness. A good definition of completeness for our purposes was devel-oped in [KM97].

C.1.1.1 Definition. An object V in BVS or LCTVS is c∞-complete if every smooth map f :R→ V admits an antiderivative.

A c∞-complete bornological vector space will be called a convenient vector space. The cate-gory of convenient vector spaces (which is a full subcategory of that of bornological vector spaces)will be called ConVS. This is equivalent to the category of c∞-complete topological vector spaceswith bounded linear maps.

Equivalent formulations of this definition are presented in [KM97], theorem 2.14.

C.1.1.2 Proposition. The full subcategory ConVS ⊂ DVS is closed under formation of all limitsand under sequential colimits of sequences of closed embeddings.

PROOF. This is theorem 2.15 of [KM97].

In particular, by the adjoint functor theorem, there are left adjoints BVS→ ConVS andDVS → Convs. The functor BVS → ConVS sends a bornological vector space to its c∞

completion, see theorem 4.29 of [KM97].

C.1.1.3 Lemma. The multi-category structure on ConVS is represented by the c∞ completion ofthe bornological tensor product. We will denote this completed tensor product by V⊗βW.


PROOF. The bornological tensor product V ⊗β W has the universal property that amap from it is the same as a smooth bilinear map from V ×W. The completion F of abornological vector space F has the universal property that a map from F to a convenientvector space is the same as a map from F. It follows that, for any convenient vector spaceU, smooth bilinear maps from V ×W to U are the same as maps from the completedbornological tensor product V⊗βW to U.

C.1.1.4 Lemma. The category ConVS admits all colimits, and the tensor product ⊗β commuteswith the tensor product.

PROOF. The functor DVS → ConVS is a left adjoint and hence colimit preserving.The category DVS admits all colimits. It follows that ConVS admits all colimits and thatthese colimits can be computed by first computing them in DVS and then applying theleft adjoint functor DVS→ ConVS.

The fact that the tensor product commutes with colimits follows from the correspond-ing fact for BVS and the fact that the completion functor BVS → ConVS is symmetricmonoidal and colimit preserving.

The following fact will be useful sometimes.

C.1.1.5 Lemma. The antiderivative map∫: C∞(R, F)→ C∞(R, F)

f 7→ f (t) =∫ t

0f (x)dx.

is smooth.

(Note that the antiderivative of a smooth curve in F is unique if we require that it vanishes atthe origin in R.)

PROOF. To check that the antiderivative map is smooth, it suffices to show that it takessmooth curves in C∞(R, F) to smooth curves. Thus, let A : Rx×Ry → F be a smooth map,so that A gives a smooth map Rx → C∞(Ry, F). We need to show that∫

yA : Rx ×Rx → F

is also smooth.


Lemma 3.7 of [KM97] shows that C∞(R, F) is convenient whenever F is convenient.Thus, the curve Ry → C∞(Rx, F) arising from A has an anti-derivative which is alsosmooth. By uniqueness of antiderivatives, the result follows.

C.1.1.6 Theorem. The category ConVS has internal Hom’s. That is, there is a convenient vectorspace structure on Hom(E, F) for any two convenient vector spaces E, F, characterized by the factthat a smooth map from a manifold M to Hom(E, F) is a continuous map E → C∞(M, F), andthe composition bilinear maps are smooth.

Further, there is a Hom-tensor adjunction:

Hom(E⊗βF, G) = Hom(E, Hom(F, G)).

PROOF. This is essentially the Theorem in section 5.7 of chapter 1 of [KM97]. Moreprecisely, this theorem shows that the space of bounded linear maps between bornologicalvector spaces has a bornological topology such that there is a Hom-tensor adjunction

Hom(E, Hom(F, G)) = Hom(E⊗β F, G).

It remains to check that a smooth map from a manifold M to Hom(E, F) is the same thingas a bounded linear map E→ C∞(M, F).

In [KM97], section 3.1.1, a topology is given to the space of all smooth (possiblynon-linear) maps E → F between bornological vector spaces. Call this space C∞(E, F).Theorem 3.12 shows that the category of bornological vector spaces, with smooth (possi-bly non-linear) maps is Cartesian closed. This implies, that for any manifold M, a mapM→ C∞(E, F) is smooth if and only if the map E×M→ F is smooth.

To complete the proof that bornological vector spaces have a Hom-tensor adjunction,we need to show that a map from a manifold to Hom(E, F) is smooth if and only if it issmooth as a map to C∞(E, F), so that a map to Hom(E, F) is smooth if and only if it arisesfrom a map E → C∞(M, F). Note that this will also show that the composition maps inBVS are smooth multi-linear maps.

In general, a map from a manifold to a closed subspace of a topological vector space issmooth if and only if it is smooth to the ambient vector space. Thus, it suffices to show thatHom(E, F) (with its topology for which Hom-tensor adjunction holds) is bornologicallyisomorphic to a closed subspace of C∞(E, F).

Note that the condition that a smooth map is linear is obviously a closed condition,so that there is a topology on the space of smooth maps so that it is a closed subspace ofC∞(E, F). Let us call the space of smooth linear maps, with this topology, Hom′(E, F).

We need to show that the bornologifications of Hom′(E, F) and of Hom(E, F) agree. Toshow this, it suffices to show that they have the same bounded sets. Because Hom′(E, F)is a closed subspace of C∞(E, F), a subset of Hom′(E, F) is bounded if and only if it is


bounded as a subspace of C∞(E, F). Proposition 5.6 of [KM97] shows that a subset ofHom(E, F) is bounded if and only if it is bounded when viewed as a subset of C∞(E, F).

So far, we have shown that the category BVS has internal Hom’s and a Hom-tensoradjunction. Next, we need to show that the full subcategory ConVS also has internalHom’s. That is, if E, F are convenient vector spaces, we need to show that Hom(E, F) isalso convenient.

It suffices to show that C∞(E, F) is convenient, since a closed subspace of a convenientspace is convenient. Now, according to the lemma in section 3.11 of [KM97], the spaceC∞(E, F) is a limit of the spaces C∞(R, F) over smooth maps R→ E. Since, by lemma 3.7,C∞(R, F) is convenient whenever F is, and by theorem 2.15, convenient vector spaces areclosed under the formation of limits, the result follows.

Alternatively, to show that Hom(E, F) is convenient we need to show that any smoothmap f : R→ Hom(E, F) admits an antiderivative.

To see this, note that such a smooth map is the same as a smooth map E→ C∞(R, F).Sincethe antiderivative map from C∞(R, F) to itself is smooth, the result follows.

So, we have shown that ConVS has internal Hom’s. We finally need to show thatConVS has a Hom-tensor adjuntion. But this follows immediately from the fact that thefunctor ConvS→ BVS is a full embedding of multicategories, and the fact that BVS has aHom-tensor adjunction.

Let us summarize the results we have so far.

(1) The categories LCTVS, BVS, ConVS, and DVS admit all (small) limits and colim-its.

(2) ConVS and BVS are symmetric monoidal categories, whereas we treat DVS justas a multicategory. The tensor product on both ConVS and BVS are colimit-preserving.

(3) The categories ConVS, BVS, DVS all admit internal Hom’s, and the functorsConVS → BVS → DVS are all functors of categories enriched in the multicat-egory DVS.

(4) Both ConVS and BVS admit a Hom-tensor adjunction.(5) The colimit-preserving functor

BVS→ ConVS

is symmetric monoidal.(6) The functors

ConVS→ BVS→ DVSare all right-adjoints (so limit preserving), and are full embeddings of multicate-gories.

C.2. EXAMPLES OF BORNOLOGICAL VECTOR SPACES 493

(7) The functorsDVS→ BVS→ ConVS

are left adjoints, and so colimit-preserving.(8) The functors

LCTVS→ BVS→ DVS

are all limit preserving, so that their adjoints are colimit preserving.(9) The functor

BVS → LCTVS

is a full embedding of categories and commutes with colimits. If we equip LCTVSwith the multicategory structure coming from continuous multilinear maps, thisfunctor is not a full embedding of multicategories.

C.2. Examples of bornological vector spaces

Any topological vector space can be regarded as a bornological vector space by ap-plying the bornologification functor. In what follows we will give all topological vectorspaces their bornological topology. In general this might differ from the usual topology,but it will have the same spaces of smooth maps. We will explain some examples wherethe bornological and the usual topologies coincide.

For any smooth connected manifold M, C∞(M) is a Frechet bornological vector space.The bornological structure is determined either by taking the bornologification of theusual topology, or by declaring that a smooth map N → C∞(M) is an element of C∞(N×M).

For any bornological vector space V, C∞(M, V) is again a bornological vector space,where the bornology is determined either from the natural topology, or by saying that asmooth map N → C∞(M, V) is an element of C∞(N ×M, V). (The fact that these definethe same bornology follows from theorem 3.12 of [KM97], which that the category ofbornological vector spaces and smooth (possibly non-linear) maps is cartesian closed).

As we have seen, the category of bornological vector spaces, viewed as a subcategoryof DVS, is closed under sequential colimits of closed embeddings and under limits. Fromthis we can construct bornological vector spaces modelling compactly-supported maps toa bornological vector space.

C.2.0.7 Definition. Let V be a bornological vector space. If K ⊂ M is a compact set, letC∞

K (M, V) denote the space of smooth maps from M → V with support in K. This is the ker-nel of the map

C∞(M, V)→ C∞(M \ K, V)

and hence is a bornological vector space.


We defineC∞

c (M, V) = colimi

C∞Ki(M, V)

where the Ki are an increasing sequence of compact sets whose union is M.

The maps in this colimit are closed embeddings, so that we get the same answer if the colimiscomputed in DVS or BVS. It follows that, for all other manifolds N,

C∞(N, C∞c (M, V)) = C∞

P (M× N, V)

where the P means we consider maps whose support is a subset which maps properly to M.

If V is c∞-complete then so is C∞(M, V) and C∞c (M, V).

Let us explain some further examples. Let E be a vector bundle on a manifold M,and, as before, let E , Ec, E , E c refer to sections of E which are smooth, compactly sup-ported, distributional, or compactly-supported and distributional. All of these spaceshave natural topologies and so can be viewed as bornological vector spaces, and they areall c∞-complete.

We explained in section ?? how to view these vector spaces as being equipped with adiffeological structure. It is easy to check that the diffeological structure discussed there isthe same as the one that arises from the topology. Indeed, the Serre-Swan theorem tells usthat any vector bundle is a direct summand of a trivial vector bundle, so we can reduce tothe case that E is trivial. Then results of Grothendieck summarized in [Gro52] allow oneto describe smooth maps to these various vector spaces using the theory of nuclear vectorspaces; in this way we arrive at the description given earlier.

The following lemma will be useful later.

C.2.0.8 Lemma. For any vector bundle E on a manifold M, the standard (nuclear) topologies onE c, E and Ec are bornological. It follows that these spaces are convenient (because completeness inthe locally-convex sense is stronger than c∞-completeness).

PROOF. Because every vector bundle is a summand of a trivial one, it suffices to provethe statement for the trivial vector bundle. According to [KM97], 52.29, the strong dualof a Frechet Montel space is bornological. The space C∞(M) of smooth functions on amanifold is Frechet Montel, because every nuclear space is Schwartz (see pages 579-581of [KM97]) and every Frechet Schwarz space is Montel. Thus the strong dual of C∞(M)is bornological, as desired.

Next, we will see that any Frechet space is bornological. This follows immediatelyfrom proposition 14.8 of [Tre67] (see also the corollary on the following page). It followsthat C∞(M) is bornological, and that the same holds for C∞

K (M) for any compact subsetK ⊂ M.

C.3. CONVENIENT COCHAIN COMPLEXES 495

Since bornological spaces are closed under formation of colimits, the same holds forC∞

c (M).

C.2.0.9 Lemma. The space of smooth functions on an n-simplex is a complete nuclear Frechetspace and hence bornological (and convenient).

PROOF. The n-simplex 4n is a closed subspace of Rn. We can identify C∞(4n) witthe quotient of C∞(Rn) by the closure of the ideal consisting of those functions whichvanish on4n. (By a theorem of Whitney, this closure consists of those functions f whose∞-jet vanishes at each point in 4n). The quotient of a complete nuclear Frechet space bya closed subspace is again a nuclear Frechet space.

It follows immediately that C∞(4n, E ) and C∞(4n, Ec) (for E , Ec as before) are bornolog-ical when equipped with their nuclear topology.

C.3. Convenient cochain complexes

Let us define the category ConVS∗ of convenient cochain complexes to be the fullsubcategory of the category DVS∗ consisting of those complexes of differentiable vec-tor spaces which are degreewise convenient. The category ConVS∗ is a dg symmetricmonoidal category, and the embedding ConVS∗ → DVS∗ is a full embedding of multicat-egories.

C.3.0.10 Definition. We say that a map of convenient cochain complexes is a quasi-isomorphism(respectively, fibration or cofibration) if it is when viewed as a map of differentiable cochain com-plexes.

The dg symmetric monoidal category ConVS∗ has a notion of quasi-isomorphism, fi-bration, and cofibration. The general yoga of homological algebra tells us that we shouldonly consider tensor products with a flat object. (An object is defined to be flat if tensorproduct with it preserves quasi-isomorphisms). Tensor products between non-flat objectsshould be replaced by derived tensor products, which are computed by replacing one ofthe objects by a flat one. We now see that there is a potential problem: it is not obvi-ous which (if any!) objects of ConVS∗ are flat, so that we don’t know how to define (orcompute) the derived tensor product.

A similar problem occurs when we want to consider the Hom-complexes between twoobjects of ConVS∗. The general philosophy tells us that Hom(V, W) is only a good objectwhen V is projective, that is, when Hom(V,−) takes quasi-isomorphisms in ConVS∗ toquasi-isomorphisms of cochain complexes. If V is not projective, then we should consider


the derived Hom-complex, which is defined by finding a projective replacement of V.Again, it is not obvious which, if any, objects of ConVS are projective.

We solve this problem by restricting to some very small subcategories which containall the objects of interest.

C.3.0.11 Definition. We define the category of geometric objects of ConVS∗ to be the smallestfull subcategory of ConVS∗ which contains objects of the form C∞

c (M, E) where E is an ellipticcomplex on a manifold M; and which is closed under the following operations.

(1) Formation of cones.(2) Homotopy equivalences.(3) Formation of direct summands.

By “cone” we just mean the usual formula for a cone of a map of cochain complexes, where theresulting object is viewed as an element of ConvS∗.

We let GeoVS∗ denote the category of geometric objects in ConVS∗.

C.3.0.12 Theorem. The category of geometric objects in ConVS∗ is closed under the completedbornological tensor product. Thus, the multicategory structure on GeoVS∗ inherited from that onDVS∗ is representable by a symmetric monoidal structure.

Further, if V, W are geometric and W →W ′ is a quasi-isomorphism, then

V⊗βW → V⊗βW ′

is also a quasi-isomorphism. (That is, in the category GeoVS∗, every object is flat).

If V, W, W ′ are as before, then the map

Hom(V, W)→ Hom(V, W ′)

is a quasi-isomorphism. That is, in GeoVS∗, every object is projective.

This result is the main technical result of this appendix. The proof will be presentedover the next several sections.

C.4. Elliptic objects are closed under tensor product

Before we embark on the proof of the statement that GeoVS∗ is closed under tensorproducts, we need a definition.

C.4.0.13 Definition. We say an geometric object V has complexity 0 if it is of the form C∞c (M, E)

for some elliptic complex on a manifold M. We say, inductively, that an object has complexity ≤ nif it can be formed from objects of complexity ≤ n − 1 by applying the operations appearing in

C.4. ELLIPTIC OBJECTS ARE CLOSED UNDER TENSOR PRODUCT 497

the definition of geometric objects: namely, cones, homotopy equivalences, and formation of directsummands. We say that an object has complexity n if it has complexity ≤ n but not ≤ n− 1.

Basically all of our proofs proceed by induction on the complexity.

Let us now prove the first statement.

C.4.0.14 Lemma. Let V, W be geometric objects. Then so is V⊗β.

PROOF. We will do this by induction on the sum of the complexities of V and W.Similar arguments will be used for all the proofs of properties of geometric objects; wewill give full details for this argument and be a little more sketchy later.

The initial case of the induction is when V and W both have complexity zero. In thiscase, V = C∞

c (M, E) and W = C∞c (N, F) for elliptic complexes E, F on manifolds M, N.

Thus, V and W, with their bornological topology, are sequential colimits of the nuclearFrechet complexes C∞

K (M, E) and C∞L (N, F) as K and L range over an exhausting family

of compact subsets of M and N.

Because the completed bornological and completed projective tensor products coin-cide for Frechet spaces, and the completed bornological tensor product commutes withcolimits, we have

V⊗β = colimK,L

C∞K×L(M× N, E F).

This is the same as C∞c (M× N, E F), which is again a compactly supported sections of

an elliptic complex.

Now suppose that V has complexity n and W has complexity m. We can assume (byswitching V and W if necessary) that n > 0. Thus, V is obtained from objects of com-plexity ≤ n − 1 by operations of homotopy equivalence, cones, and sequential colimitsof smoothly-split maps. By induction, we will assume that the tensor product of W withany object of complexity ≤ n− 1 is geometric. Thus, we need to show the following threestatements.

(1) If A, B, W are geometric, and A, B have complexity ≤ n− 1, and if f : A→ B is amap, then Cone( f )⊗βW is geometric.

(2) If A is geometric of complexity ≤ n− 1, and A → B is a homotopy equivalence,then B⊗βW is geometric.

(3) If A has complexity n− 1, and if A splits as a direct sum A = B⊕ B′, then B⊗βWis geometric.

For the first statement, note that Cone( f )⊗βW is the same as Cone( f ⊗β IdW), wheref ⊗β IdW is a map A⊗βW → B⊗βW. The result follows by the induction assumption


that A⊗βW and B⊗βW are geometric. For the second, note that A⊗βW → B⊗βW is ahomotopy equivalence; and for the third, note that B⊗βW is a direct summand of A⊗βW.

Thus, we have proved by induction that GeoVS∗ is closed under tensor product.

C.4.1. Elliptic objects are flat. In this subsection, we will prove the following.

C.4.1.1 Proposition. If V, W, W ′ are geometric objects, and if W → W ′ is a quasi-isomorphism,then so is

V⊗βW → V⊗βW ′.

Before we prove this result, we need some subsidiary lemmas.

C.4.1.2 Lemma. For all geometric objects W and manifolds M, the map

C∞c (M)⊗βW → C∞

c (M, W)


PROOF. We do this by induction on the complexity of W. In the case that W hascomplexity zero, this map is actually an isomorphism, as we have seen above ??. Assumethat the statement is true for all objects of complexity ≤ n − 1. We need to prove it forobjects of complexity n.

Note that the functors C∞c (M,−) and C∞

c (M)⊗β− both take cones to cones, homotopyequivalences to homotopy equivalences, and summands to summands. The inductionstep follows immediately from these facts. We will spell out the case of summands.

Suppose that A is an geometric object of complexity n− 1 and that A can be writtenas a direct sum A = B⊕ B′. Then, both B and B′ are geometric objects of complexity n.

The mapC∞

c (M)⊗β A→ C∞c (M, A)

is a direct sum of the corresponding maps from B and B′. A direct sum of two maps is aquasi-isomorphism if and only if the two factors are quasi-isomorphisms.

C.4.1.3 Lemma. The functor

C∞c (M,−) : DVS∗ → DVS∗

preserves quasi-isomorphisms.

PROOF. By replacing a quasi-isomorphism by its cone, it suffices to show that if W isquasi-isomorphic to 0 then so is C∞

c (M, W).


Saying that W is quasi-isomorphic to 0 is equivalent to saying that the stalks of thesheaves C∞(N, W) on all manifolds N have no cohomology. In particular, the sheaf WMon M of smooth maps to W on M has stalks with no cohomology.

Note also that WM is a sheaf of modules for the sheaf C∞M of smooth functions on

M. Thus, it admits partitions of unity. A partition of unity argument then implies thatC∞

c (M, W) has no cohomology.

More generally, for any manifold N, the space C∞P (N ×M, W) of smooth maps to W

whose support maps properly to N has no cohomology (again by a partition of unityargument). Since this is the space of smooth maps to C∞

c (M, W), it follows that C∞c (M, W)

is quasi-isomorphic to 0 not just as a cochain complex but as an object of DVS∗.

C.4.1.4 Corollary. If W →W ′ is a quasi-isomorphism of geometric objects, then the map C∞c (M)⊗βW →

C∞c (M)⊗βW ′ is a quasi-isomorphism in ConVS∗.

Finally, we will show the following.

C.4.1.5 Proposition. If W → W ′ is a quasi-isomorphism in GeoVS∗, then for all geometricobjects V, V⊗βW → V⊗βW ′ is a quasi-isomorphism.

PROOF. We do this by induction on the complexity of V. Because the functor −⊗βWtakes homotopy equivalences, cones to cones, and summands to summands, the inductiveargument we have used so far reduces us to the case that V is of complexity zero. For thiscase, we need to show that the map

C∞c (M, E)⊗βW → C∞

c (M, E)⊗W ′

is a quasi-isomorphism. If suffices to show this when E is just a single vector bundle, andnot an elliptic complex. Using the fact that every vector bundle is a summand of a trivialbundle, it suffices to show this when E is just the trivial vector bundle. But this is theCorollary above.

C.4.2. Ind-geometric objects are projective.

C.4.2.1 Definition. An object of ConVS∗ is an ind-geometric object if it is a sequential colimit ofa sequence of geometric objects A0 → A1 . . . where the connecting maps Ai → Ai+1 are smoothlysplit.

Smoothly split means split in the category of graded differentiable vector spaces, i.e.the splitting might not be a cochain map.


C.4.2.2 Lemma. The category of ind-geometric objects is closed under the completed bornolog-ical tensor product. Further, if V, W, W ′ are ind-geometric objects and if W → W ′ is a quasi-isomorphism, so is V⊗βW → V⊗βW ′.

PROOF. This follows immediately from the corresponding facts for geometric objectsand the fact that the tensor product ⊗β commuts with colimits.

The main result of this subsection is the following.

C.4.2.3 Proposition. Let V be an ind-geometric object, and let W →W ′ be a quasi-isomorphismbetween two geometric objects. Then, the map


is a quasi-isomorphism of complexes of objects of ConVS.

We have seen that ConVS has internal Hom’s.

The proof relies on the following technical lemma.

C.4.2.4 Lemma. Let Dc(M) denote the space of compactly supported distributions on a manifoldM. For all complete locally convex spaces F, there is a natural isomorphism

HomConVS(Dc(M), Fborn) = C∞(M, F).

PROOF. Theorem 50.4 in [Tre67] states that for all complete nuclear spaces E and allcomplete locally-convex spaces F,

E⊗π F = Homcts(E∨, F)

where ⊗π is the completed projective tensor product and E∨ is the strong dual of E. HereHomcts denotes the space of continous linear maps, and we view this just as an isomor-phism of vector spaces with no topoloy.

In particular, we have

C∞(M)⊗π F = Homcts(Dc(M), F).

A result of Grothendieck [Gro52] tells us that, for all complete locally convex spaces F,

C∞(M)⊗π F = C∞(M, F).

Now, lemma ?? tells us that Dc(M), equipped with its nuclear topology, is bornological.It follows that, for all complete locally convex spaces F, we have

HomBVS(Dc(M), Fborn) = C∞(M, F).


Our ultimate goal is to show that every object in the category of geometric objects isprojective, i.e. that Hom’s between geometric objects preserve quasi-isomorphisms. Itwill be useful, however, to have a stronger result: that the Hom-complex from an geo-metric object preserves quasi-isomorphisms between objects in a larger category. Let usintroduce this (rather technical) larger category.

C.4.2.5 Definition. We say a convenient cochain complex is LC-complete if it is the bornologi-fication of a complex which is complete in the locally convex sense.

We say a convenient cochain complex is weakly LC-complete if it is in the smallest subcat-egory which contains all LC-complete objects and is closed under formation of cones, homotopyequivalences, and direct summands.

Note that all geometric objects are weakly LC-complete.

C.4.2.6 Lemma. Let W be an object of ConVS∗ which is weakly LC-complete.

Then there is a natural quasi-isomorphism in DVS∗

HomConVS(Dc(M), W)→ C∞(M, W).

PROOF. Note that the map δ : M→ Dc(M) sending a point p to δp is smooth. Indeed,a smooth map from N toDc(M) is the same as a continuous linear map C∞(M)→ C∞(N).If f : N → M is smooth, then the map

δ f : N → Dc(M)

arises from the mapf ∗ : C∞(M)→ C∞(N)

and so is obviously smooth.

The natural mapHom(Dc(M), W)→ C∞(M, W)

is obtained by composition with δ : M→ Dc(M).

We need to show that this map is a quasi-isomorphism if W is a finite geometric object.Since we want to show that it’s a quasi-isomorphism in DVS∗, we need to show that forall manifolds N, the map

Hom(Dc(M), C∞(N, W))→ C∞(M× N, W)


Both functors in this equation commute with cone and formation of summands, andtake homotopy equivalences to homotopy equivalences. By induction on the complexityof W, we reduce to the case when W is LC-complete, in which case the result followsimmediately.


Now we arrive at the proof of the main proposition of this section.

C.4.2.7 Proposition. Let V, W, W ′ be convenient cochain complexes where V is ind-geometricand W, W ′ are weakly LC-complete. Let W →W ′ be a quasi-isomorphism.

(In particular, both W and W ′ could be geometric).

Then, the mapHom(V, W)→ Hom(V, W ′)

is a quasi-isomorphism of objects of DVS∗.

PROOF. The functor Hom(−, W) commutes with cones and formation of summands.It takes sequential colimits of smoothly split maps to sequential limits of surjective mapsof cochain complexes. It also takes homotopy equivalenes to homotopy equivalences.Thus, the usual inductive argument reduces us to the case when V = C∞

c (M, E) for Ean elliptic complex on M. But this is homotopy equivalent to E c, the space of compactlysupported distributional sections of E, by the Atiyah-Bott lemma. A further simple in-duction allows us to replace V by Dc(M), the space of compactly supported distributionson M. We have seen that for all weakly LC-complete complexes W, we have a quasi-isomorphism

Hom(Dc(M), W) ∼= C∞(M, W).Since the functor C∞(M,−) preserves quasi-isomorphisms, the result follows.

C.4.2.8 Corollary. Let W, W ′ be both weakly LC-complete and ind-geometric. (For example, theycould both be geometric).

Then, a map f : W →W ′ is a quasi-isomorphism if and only if it’s a homotopy equivalence.

PROOF. Suppose the map is a quasi-isomorphism. Then, the map

Hom(W ′, W)f −−−→ Hom(W ′, W ′)

is a quasi-isomorphism as well, by proposition C.4.2.7. A right homotopy inverse g isprovided by an element in Hom(W ′, W) such that f g is cohomologous to the identityof W ′. In a similar way we can consturct a right homotopy inverse h : W → W ′ to g. Astandard argument shows that h and f are homotopic.

C.5. Convenient pro-cochain complexes

Recall from section ?? that our factorization algebras take values in the category ProDVS∗

of differentiable pro-cochain complexes. Recall that a differentiable pro-cochain complex

C.5. CONVENIENT PRO-COCHAIN COMPLEXES 503

is a differentiable cochain complex V∗ equipped with a complete decreasing filtration, sothat V = lim←−V/FiV. We also require that the maps V/FiV → V/FjV for i > j are fibra-tions. Maps between differentiable pro-cochain complexes must preserve the filtrations.

C.5.0.9 Definition. We say that a differentiable pro-cochain complex is a convenient pro-cochaincomplex if

(1) The maps V/FiV → V/FjV are smoothly split.(2) Each of the quotients V/FiV are convenient cochain complexes.

We say a convenient pro-cochain complex is geometric (or ind-geometric, or weakly LC-complete)if each Gri V is geometric (or ind-geometric, or weakly LC-complete).

We let ProConVS∗ and ProGeoVS∗ denote the category of convenient and geometric pro-cochain complexes. Maps in these categories must preserve filtrations, and a map of convenient orgeometric pro-cochain complexes is a quasi-isomorphism if and only if it is when viewed as a mapof differentiable pro-cochain complexes.

Because we assume that the maps are smoothly split, we can write any convenientprocochain complex

V = ∏i≥0

Vi

where each of the Vi are convenient, and the differential maps Vi to the product ∏j≥i Vj.

C.5.0.10 Definition. If V, W are convenient pro-cochain complexes, we define the tensor product

V⊗βW = lim←−(

V/FiV)⊗β

(W/FjW

).

This tensor product has a natural filtration, where we define Fr(V⊗βW) to be the natural map

V⊗βW → ∏i+j=r

(V/FiV)⊗β(W/FjW).

This tensor product represents the multicategory structure on ProConVS∗ which is inherited fromthat on ProDVS∗.

Let Z>0 denote the category whose objects are the positive integers and where there’sa map n→ m whenever n > m. There’s a full embedding

ProConVS ⊂ Fun(Z>0, ConVS)

by sending V to the sequence V/FiV. The difference between ProConVS and Fun(Z>0, ConVS)is that in the latter we do not require that the maps are smoothly split.

C.5.0.11 Lemma. The tensor product on ProConVS commutes with all colimits which are alsocolimits in Fun(Z>0, ConVS).


PROOF. This is immediate from the fact that the tensor product on ConVS commuteswith colimits.

In particular, the tensor product on ProConVS commutes with coproducts.

C.5.0.12 Definition. We lift the Hom-complexes in ProConVS∗ to objects of ProDVS∗ as fol-lows. We let Fr Hom(V, W) be the space of those maps V → W which, for all i, map FiV →Fi+rW. We make Hom(V, W) into a sheaf on the site of smooth manifolds by declaring that

C∞(M, Hom(V, W)) = Hom(V, C∞(M, W)).

Then, one can check that

(1) The filtraton on Hom(V, W) is smoothly split.(2) Hom(V, W) = lim←−Hom(V, W)/Fr Hom(V, W) in the category DVS∗.(3) Grr Hom(V, W) = ∏i Hom(Gri V, Gri+r W), again in the category DVS∗.

All results stated earlier concerning the flatness and projectivity of geometric objectscarry over to geometric pro-cochain complexes.

C.5.0.13 Proposition. (1) If V, W are geometric pro-cochain complexes then so is V⊗βW.(2) If V, W, W ′ are geometric pro-cochain complexes and if W →W ′ is a quasi-isomorphism,

then so is V⊗βW → V⊗βW ′.(3) Under the same hypothesis, Hom(V, W) → Hom(V, W ′) is a quasi-isomorphism of

objects of ProDVS∗. Therefore, a quasi-isomorphism of geometric pro-cochain complexesis a homotopy equivalence.

(4) More generally, if V is an ind-geometric pro-cochain complex (meaning each Gri V),and if W, W ′ are weakly LC-complete convenient pro-cochain complexes (meaning eachGri W and Gri W ′ are weakly LC-complete), then the map


is a quasi-isomorphism in ProDVS∗.

PROOF. All statements follow essentially immediately by applying spectral sequencearguments together with the corresponding results in the category of ordinary, not pro,convenient cochain complexes.

C.6. Pro-cochain complexes over C[[h]]

Let us view C[[h]] (or R[[h]] if we work over R) as a pro-cochain complex by sayingthat h has degree 2, and that the filtration is induced from this grading.

C.6. PRO-COCHAIN COMPLEXES OVER C[[h]] 505

We can then talk about geometric or convenient pro-cochain complexes which areC[[h]]-modules, meaning that they are C[[h]]-modules in the symmetric monoidal cat-egory of convenient pro-cochain complexes. Any C[[h]]-linear convenient pro-cochaincomplexV has a canonical h-adic filtration defined by saying that Fi

hV = hiV. This fil-tration is automatically complete, because, if FnV denotes the filtration defining the pro-cochain complex structure, we have

FihV ⊂ F2iV

so that modulo FnV, the h-adic filtration is finite. the image of multiplication by hi.

C.6.0.14 Definition. A convenient pro-cochain complex V with an action of C[[h]] is quasi-freeif the h-adic filtration is smoothly split, and if the maps

h : Grih V → Gri+1

h V

given by multiplication by h on the associated graded of the h-adic filtration, are isomorphisms.

Note that this implies thatGrh V = Gr0

h V[[h]]so that the associated graded for the h-adic filtration is actually free.

Since we assumed that the h-adic filtration is smoothly split, there is an isomorphismof graded convenient pro-cochain complexes (without differentials)

V ∼= Gr0h V[[h]].

In what follows, all C[[h]]-linear convenient pro-cochain complexes will be quasi-freealthough we may not always specify this.

C.6.0.15 Definition. Let V, W be quasi-free C[[h]]-modules in the category ProConVS∗. Wedefine the tensor product

V⊗β,C[[h]]W

to be the coequalizer in the category ProConVS∗ of the diagram

V⊗βC[[h]]⊗βW ⇒ V⊗βW.

It is not obvious that this colimit exists in ProConVS∗. However, the fact that V andW are quasi-free means that it does. To see this, note that we may as well assume that Vand W are free, meaning that V = V0[[h]] and W = W0[[h]]. Note also that

V0[[h]] = lim←−V0⊗βC[h]/hn = V0⊗βC[[h]]

by the way we defined the tensor product in the category ProConVS∗ and by the fact thatthe filtration on C[[h]] is defined by giving h weight 2.


Recall also that the bornological tensor product – and it’s completed version – com-mute with colimits. The tensor product on ProConVS also commutes with those colimitswhich are colimits in the category Fun(Z>0, ConVS of sequences . . . Vi → Vi−1 · · · → V1of objects of ConVS. Therefore, in order to show that the desired colimit exists, it sufficesto show that the coequalizer of the diagram

C[[h]]⊗C[[h]]⊗C[[h]] ⇒ C[[h]]⊗C[[h]]

taken in the category Fun(Z>0, ConVS) actually lives in the category ProConVS. This iseasy to see.

C.6.0.16 Lemma. Let us give the category ProConVS∗C[[h]] of quasi-free C[[h]]modules in ProConVS∗

the structure of multicategory induced from the symmetric monoidal structure we have just dis-cussed.

Then, the functorProConVS∗C[[h]] → ProDVS∗C[[h]]

is a full embedding of multicategories, where the right hand side is endowed with the structure ofC[[h]]-multilinear, smooth, filtration-preserving maps.

PROOF. It suffices to show that, given quasi-free C[[h]]-modules V1, . . . , Vn in ProConVS∗,the map

V1 × · · · ×Vn → V1⊗β,C[[h]] . . . ⊗β,C[[h]]Vn

is universal among smooth, filtration-preserving, C[[h]]-multilinear maps. This is straight-forward.

C.7. Observables as geometric pro-cochain complexes

So far, we have proved most of the statements given in the introduction. It remains toshow that, for any quantum field theory, observables form a factorization algebra in thecategory of geometric pro-cochain complex.

C.7.0.17 Theorem. Suppose we have a quantum field theory on a manifold M. Let Obsq denotethe factorization algebra of observables. This is a factorization algebra valued in the multi-categoryProDVS∗C[[h]] of C[[h]]-linear pro diffeological cochain complexes.

Then,

(1) Each Obsq(U) for open subsets U ⊂ M is an geometric pro-cochain complex, which isquasi-free as a C[[h]]-module.

(2) The smooth, filtraton-preserving, C[[h]]-linear maps

Obsq(U)×Obsq(V)→ Obsq(U qV)

C.7. OBSERVABLES AS GEOMETRIC PRO-COCHAIN COMPLEXES 507

for disjoint open subsets U, V ⊂ M, induce a quasi-isomorphism

Obsq(U)⊗β Obsq(V)→ Obsq(U qV).

Thus, Obsq is a factorization algebra valued in the symmetric monoidal category of geometricpro-cochain complexes.

We have seen in ?? that there is a simplicial set of quantizations of a given classicalfield theory. If we have an n-simplex in this simplicial set, we find a family of theoriesover Ω∗(4n), and we have seen in ?? that there is a corresponding family of factorizationalgebras over Ω∗(4n). We would like this factorization algebra to also live in the categoryof geometric pro-cochain complexes, so that everything we do is in this category.

C.7.0.18 Theorem. Further, if we have an n-simplex in the space of quantizations of a givenclassical theory, then observables for this form a factorization algebra in geometric pro-cochaincomplexes over the base ring Ω∗(4n).

In particular, if we have two different quantizations of a given classical theory, which arehomotopic, and whose corresponding factorization algebras are denoted Obsq

0 and Obsq1, then there

is a factorization algebra valued in geometric pro-cochain complexes Obsq[0,1] together with quasi-

isomorphismsObsq

0 ← Obsq[0,1] → Obsq

1 .

The nice thing about geometric pro-cochain complexes is that every quasi-isomorphismis a homotopy equivalence. This, together with standard techniques from homotopicalalgebra, implies that, after replacing Obsq

0 by a free resolution Obsq0, there is a quasi-

isomorphism from Obsq0 to Obsq

1. At some stage, to make such a theorem more clean,we should introduce the infinity-category of factorization algebras (modelling it as a sim-plicially enriched category, for example) by taking as objects, quasi-free factorization al-gebras, with the natural simplicial enrichment

PROOF. The proof of these theorems will take the rest of this section. Note that thestatements are entirely functional-analytic: we have already showed, in chapter ??, thata field theory gives us a factorization algebra valued in differentiable pro-cochain com-plexes, and that an n-simplex in the simplicial set gives rise to a family of factorizationalgebras over Ω∗(4n). We need to verify that these factorization algebras actually live inthe full subcategory of geometric pro-cochain complexes.

However, since this is precisely the purpose for which geometric pro-cochain com-plexes were designed, the result is not so difficult. Suppose we have a family of quantumfield theories on a manifold M over Ω∗(4n). Let Obsq

4n denote the Ω∗(4n)-linear factor-ization algebra associated to this family. Let E denote the sheaf of fields on M.


Recall that there is an isomorphism

Obscl(U) ∼= Sym∗π(E

!c(U))

On the right hand side, Sym∗π refers to the completed tensor product taken using the projec-

tive tensor product ⊗π. We emphasize this point because the tensor product on convenientcochain complexes is defined using the completed bornological tensor product, which isin general different. We will denote the completed projective tensor product by ⊗π andcorresponding symmetric powers by Symk

π. A priori, the completed projective tensorproduct of two convenient vector spaces is a topological vector space which may not bebornological. So we will replace it by its bornologification, which will be convenient (as itis the bornologification of something which is complete in the locally convex sense).

Theorem ?? shows that, for any open subset U ⊂ M, we have an isomorphism ofdifferentiable pro-graded vector spaces complexes

Obsq4n(U) ∼= Obscl

4n(U)[[h]] = Sym∗π(E

!c(U))[[h]]

If we choose a parametrix Φ on U, then we can make this into an isomorphism of differ-ential pro-cochain complexes where we endow the right hand side with the differential

dE + d4n + h4Φ + I[Φ],−Φ

where I[Φ] is the effective action functional for the restriction of the family of field theorieson M to U, dE is the linear differential on the complex of fields, and d4n is the de Rhamdifferential on Ω∗(4n).

We need to show that this is an geometric pro-cochain complex. The first thing to noteis that the defining filtration on Obsq(U) is smoothly split. The associated graded of thisfiltration is

Gr Obsq(U) ∼= Ω∗(4n, Sym∗π(E

!c(U))[[h]]

with the differential dE + dtrn + h4Φ.

To show that this is an geometric pro-cochain complex, we need to show that each Grn

is. The filtration on Obsq(U) is defined in such a way that

Grn Obsq(U) = Ω∗(4n,⊕[n/2]

i=0 hi Symn−2iπ E!

c(U))

.

The differential as before is dE + d4n + h4Φ. Evidently, Grn is obtained from the com-

plexes Ω∗(4n, Symkπ E

!c(U)) with differential dE + d4n by the formation of a finite num-

ber of cones. Since the category of geometric cochain complexes is closed under the for-mation of cones, it suffices to show that Ω∗(4n, Symk

π E!c(U)) is geometric.

Now, geometric cochain complexes are also closed under formation of direct sum-mands, and this complex is a summand of

Ω∗(4n, E!c(U)⊗πn).

C.7. OBSERVABLES AS GEOMETRIC PRO-COCHAIN COMPLEXES 509

Now, Ω∗(4n) is continuously homotopy equivalent to the base field R (or C), so that thiscomplex is homotopy equivalent to E

!c(U)⊗πn.

Since E!c(U) is homotopy equivalent to E !

c (U), this complex is homotopy equivalentE !

c (U)⊗πn, which is, in turn, geometric.

Now we have proved most of the statements we need. It remains to verify that thefactorization structure map

Obsq(U)×Obsq(V)→ Obsq(U qV)

induces a quasi-isomorphism

Obsq(U)⊗β,C[[h]] Obsq(V)→ Obsq(U qV).

Since the C[[h]]-modules Obsq(U) are quasi-free, it suffices to prove the correspondingstatement for classical observables. Thus, we need to show that the map

Sym∗π(E

!c(U))⊗βSym

∗π(E

!c(V))→ Sym

∗π(E

!c(U)⊕ E

!c(V))

is a quasi-isomorphism, where both sides are endowed with the differential arising fromthe linear differential on E

!c. Since E

!c(U) is continuously homotopy equivalent to E !

c (U),it suffices to prove the same statement if E

!c is replaced everywhere by E !

c .

Now, the completed bornological and completed projective tensor products are bornolog-ically isomorphic for spaces of compactly-supported smooth sections of a vector bundleon a manifold (this is lemma ?? need to write a proof of this fact our avoid using it...). Itfollows that we need to show that the map

Sym∗β(E

!c (U))⊗βSym

∗β(E

!c (V))→ Sym

∗β(E

!c (U)⊕ E !

c (V))

is an isomorphism. The way we have completed the bornological tensor product whenwe deal with pro-objects means that it suffices to show, for any convenient vector spacesA and B, that

Symnβ(A⊕ B) = ⊕i+j=n Symi

β(A)⊗β Symjβ(A).

But this statement is true in any symmetric monoidal category where the tensor productcommutes with finite colimits.

Next, we will show the following.

C.7.0.19 Proposition. If we have a countable Weiss cover U of an open subset U of X, and if wehave a prefactorization algebra F on X associated to a quantum field theory, then the Cech complexC(U,F ) is an object of the category of geometric pro-cochain complexes over C[[h]].


PROOF. To prove this, it suffices (since F (U)) is a geometric pro-cochain complex) toshow that the natural map

C(U,F )→ F (U)

is a homotopy equivalence. We already know from ?? that it is a quasi-isomorphism.Further, corollary C.4.2.8 tells us that any quasi-isomorphism between ind-geometric andweakly LC-complete convenient cochain complexes is a homotopy equivalence. The sameresult holds (with the same argument) for convenient pro-cochain complexes. Since F (U)is geometric, it is automatically weakly LC-complete. Further, since U is countable, it isclear that C(U,F ) is ind-geometric. Finally, since the explicit model we write down forF (U) is LC-complete, and since C(U,F ) is (as a graded vector space) a countable directsum of spaces F (V) for various subsets V of U, C(U,F ) is also LC-complete. The resultfollows.

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