gauge theories with an application to chern-simons theory · 2020. 9. 15. · of a gauge theory,...

Gauge Theories with an Application to

Chern-Simons Theory

Anton Quelle

May 31, 2013

Contents

1 Introduction 5

2 Connections in principal bundles 92.1 Koszul connections . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Ehresmann connections . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Connection forms for a frame . . . . . . . . . . . . . . . . 152.2.2 Ehresmann connections . . . . . . . . . . . . . . . . . . . 172.2.3 The meaning of the structure group . . . . . . . . . . . . 20

2.3 Further properties of connections . . . . . . . . . . . . . . . . . . 212.3.1 The horizontal distribution . . . . . . . . . . . . . . . . . 212.3.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . 242.3.3 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Gluing of connections . . . . . . . . . . . . . . . . . . . . 29

3 Classical field theories 313.1 Lagrangian mechanics in symplectic terms . . . . . . . . . . . . . 31

3.1.1 The case of strictly convex Lagrangians . . . . . . . . . . 313.1.2 The general case . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Classical Field theories . . . . . . . . . . . . . . . . . . . . . . . . 413.2.1 scalar field theories . . . . . . . . . . . . . . . . . . . . . . 423.2.2 First order field theories . . . . . . . . . . . . . . . . . . . 46

3.3 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Chern-Simons theory 554.1 The Chern-Simons form and the action . . . . . . . . . . . . . . 554.2 Chern-Simons as a Lagrangian theory . . . . . . . . . . . . . . . 59

4.2.1 The theory without boundary . . . . . . . . . . . . . . . . 594.2.2 The theory with boundary . . . . . . . . . . . . . . . . . . 60

4.3 The E-L equations and the field theory . . . . . . . . . . . . . . . 654.4 The generalisation to toric bundles . . . . . . . . . . . . . . . . . 70

4.4.1 Bundle extensions . . . . . . . . . . . . . . . . . . . . . . 704.4.2 The Lagrangian formulation . . . . . . . . . . . . . . . . . 714.4.3 The Hamiltonian theory . . . . . . . . . . . . . . . . . . . 73

4.5 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3

4 CONTENTS

A Lie groups 79A.1 Lie groups and the exponential map . . . . . . . . . . . . . . . . 79A.2 The Lie algebra of a Lie group . . . . . . . . . . . . . . . . . . . 80A.3 Smooth actions on a manifold . . . . . . . . . . . . . . . . . . . . 81A.4 The Maurer-Cartan form . . . . . . . . . . . . . . . . . . . . . . 82

B Fibre bundles 85B.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . 85B.2 Principal bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

C Symplectic geometry 89C.1 Symplectic linear algebra . . . . . . . . . . . . . . . . . . . . . . 89C.2 symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 90C.3 Hamiltonian geometry . . . . . . . . . . . . . . . . . . . . . . . . 92

D Category theory 95

Chapter 1

Introduction

Classical Chern-Simons theory is an example of a field theory. It is of interest toboth the mathematics and the physics communities. In physics it is of interestboth because of its applications in solid-state physics [1] to the quantum Halleffect and in string theory [2] because it is an example of a topological fieldtheory (its action is not dependent on any metric structure on the space-time).In mathematics the interest mostly comes from the strong interplay betweenmathematics and string theory. In [2] it is shown that quantising Chern-Simonstheory yields knot-invariants for example. This is clearly a purely mathemati-cal consideration. Because of this importance it is desirable to have a rigorousmathematical treatment of Chern-Simons theory. Truly, several papers alreadydo so ( [3] and [4] are notable examples). However, both these papers share adeficiency that this paper seeks to remedy. They treat the Chern-Simons theoryfrom a very mathematical point of view, without giving much physical justifica-tion, or context, for the notions introduced. Himpel ( [4]) is at once very formalbut also very introductory. With the notable exception of the application ofChern-Simons to the Casson invariant for 3-manifolds, most results that Him-pel gives are also found in this paper. In contrast, Freed ( [3]) assumes muchpreliminary knowledge we treat in detail. Apart from a treatment of AbelianChern-Simons, which we give but is skipped there, Freed goes much deeper intothe theory. We have opted to treat the Chern-Simons theory at a more basiclevel to facilitate the inclusion of much content that helps give context and alsogives physical motivation for the constuctions involved in the theory.

The second chapter will be about connections in principal bundles, whichare purely mathematical constructs. The reason for treating them so thoroughlyis that they are a necessary basis for the formulation of Gauge theories. Thenecessary preliminaries from Lie groups and fibre bundles are assumed to beknown. However, for completeness, they are briefly reviewed in the appen-dices. In the treatment of connections we start with the concept of a Koszulconnection, these give the basic notion of covariant differentiation that everyphysicist will be familiar with. Every mathematician that has taken a course

5

6 CHAPTER 1. INTRODUCTION

in Riemannian geometry will know them as well. To give the reader a feelingfor Koszul connections, and the related concept of curvature, some basic resultsfrom Riemannian geometry are proven. We then show how the set of Koszulconnections on a vector bundle is in bijection with a specific set of 1-forms on itsframe-bundle. The characterisation of this set is generalisable to any principalbundle, and objects belonging to it are called connections. We then constructthe curvature of a connection, and other related concepts, in this more generalsetting. We finish by proving some properties that these general connectionsobey. We will need these properties in our discussion of Chern-Simons theory.

In the third chapter we will introduce the modern formulation of classicalfield theories, which is in terms of symplectic geometry. This formulation isgreatly abstract, and any reader not familiar with the reasoning leading to itwill not see the link with the more cavalier definition in use by physicists. Whilethere are some general methods of approach in general field theories every sin-gle field theory needs to be tackled in a slightly different way. As a result littlecan be proven about field theories in general, and our formulation will be in-troduced by example. We will start by treating classical mechanics, which is a1-dimensional field theory. We show how the Hamiltonian formulation in termsof symplectic geometry, which is treated in an appendix, is generally inadequate.As a remedy we show how the Lagrangian formulation can be done in symplecticterms. This will lead us the modern formulation of the theory we are lookingfor. We then define modern field theories by generalising this construction. Todefend this generalisation we show that several well known field theories fromphysics, such as electrostatics and electrodynamics, can be formulated in theseterms. Electrodynamics is also the archetypal example of a gauge theory, thekind of field theory we are ultimately most interested in. The constructionswe use in the formulation of electrodynamics will lead us to the definition of ageneral gauge theory. We will briefly mention Yang-Mills theory as an exampleof a gauge theory, but will not go into any details. Apart from electrodynamicsChern-Simons theory is the only gauge theory we rigorously treat.

The treatment of this Chern-Simons theory is what is done in the fourthand final chapter. It is here that we will need the full machinery that we havedeveloped in the second chapter. Every single result from the fourth chapter canbe understood without reading the third. However, without reading the thirdchapter the motivation behind all the constructions will be missing. Specificallywe will mirror the treatment of electrodynamics in the third chapter as closelyas possible to show that Chern-Simons theory is a gauge theory according to ourdefinition. This is done by defining the set of space-times on which the theorycan be formulated together with the space of fields and the action. We thenshow this data defines a field theory in the sense of chapter three. Obviouslythe formulation of Chern-Simons theory will be more complicated; otherwiseit would not require an entire chapter. All the constructions involved have aparallel in electrodynamics however. A major difference is that the action forthe Chern-Simons theory is not dependent on any choice of metric on the space-

7

time; it only depends on the topologies involved. For this reason Chern-Simonstheory is generally refered to as a topological field theory. From a practicalpoint of view this means that the proofs in Chern-Simons theory often usecohomological arguments, whereas in electrodynamics results from PDEs areused that depend less on topology.

8 CHAPTER 1. INTRODUCTION

Chapter 2

Connections in principalbundles

In the introduction we briefly mentioned the mathematical structure needed todefine gauge theories. The purpose of this chapter is to serve as a mathematicalpreliminary in which this mathematical structure is introduced in detail. Specif-ically we will give an exposition on connections in principal bundles. We willdefine what they are and prove those properties we will need later on, togetherwith a host of periferial results that serve to give the reader a good understand-ing of the concepts,their geometrical interpretation, and their possible use inother disciplines of mathematics. We start with defining the notion of a Koszulconnection, which is a basic construct in differential geometry, and show thereader its importance in geometry. We then slowly work up the ladder of ab-staction to show how this eventually leads to the much more general definitionof an Ehresmann connection on a principal bundle.

2.1 Koszul connections

What we have to say about Koszul connections is essentially a pruned versionof the exposition in [5].

Definition 2.1.1. A (Koszul) connection in a vector bundle π : V → M is abilinear operator

∇ : Γ(TM)× Γ(V )→ Γ(V ) : (X, v) 7→ ∇Xv

such that:

1. ∇fXv = f∇Xv

2. ∇X(fv) = f∇Xv +X(f)v

9

10 CHAPTER 2. CONNECTIONS IN PRINCIPAL BUNDLES

If we take the trivial vector bundle M ×R and identify a section with a realfunction we see that the directional derivative (X, f) 7→ X(f) is a connection(it is in fact the only one [6]); connections are therefore direct generalisationsof directional derivatives to arbitrary vector bundles. That every vector bundleallows a connection at all is at this point not yet clear, but later an easy proofwill become available. Note also that because the Γ(TM) argument is linearover functions a connection defines a map v ∈ Γ(V ) 7→ ∇v ∈ Γ(T ∗M) ⊗ Γ(V ),and vice versa. In what follows these two viewpoints on connections will beused interchangably. For now we will content ourselves with some trickeryconnections allow us. First of all a connection in a vector bundle V gives riseto a connection on the bundles ⊗mV ⊗n V ∗. The construction is given in thefollowing theorem.

Proposition 2.1.2. Given a connection ∇ on V this is uniquely extended to afamily of connections on ⊗mV ⊗n V ∗ satisfying

1. ∇Xf = X(f)

2. ∇X(A⊗B) = (∇XA)⊗B +A⊗ (∇XB)

3. ∇X C = C ∇X where C is any contraction of a covariant with acontravariant index. Here we interpret the case m = n = 0 as the trivialbundle of real valued functions (c.f. item 1).

Proof. First we define the connection on V ∗. To do this notice that for v ∈ Γ(V )and ω ∈ Γ(V ∗) we have

ω(v) = C(v ⊗ ω)

where C is the unique contraction for this tensor. Then applying the axioms weget

X(ω(v)) = ∇X(ω(v)) = ∇X(C(v ⊗ ω)) = C∇X(v ⊗ ω)

= C(∇Xv ⊗ ω + v ⊗∇xω) = ω(∇Xv) + (∇Xω)(v)

⇒ (∇Xω)(v) = X(ω(v))− ω(∇Xv).

This obviously defines a connection. That item 2 correctly extends the connec-tion to all higher tensor powers is an easy induction argument in the ranks ofA and B and is left to the reader.

Another application of connections, and the one they derive their name from,is that they give a notion of vector fields being parallel along a curve.

Definition 2.1.3. Given a (local) section v of V and a smooth curve γ we saythat v is parallel along γ if and only if

∇γ′v = 0

for all points along the curve.

2.1. KOSZUL CONNECTIONS 11

This definition, though simple is unfortunately not sufficient. Consider forexample the following problem: we are given a curve γ : [0, 1]→M and a vectorvγ(0) ∈ π−1(γ(0)) and we want to extend this vector field to a parrallel onealong γ. The problem in this case is that the vector field we seek will only bedefined along γ so we cannot use our connection just yet. Fortunately we canfix this shortcoming.

Lemma 2.1.4. Given a connection ∇ there is a unique operator, which wesomewhat abusively denote ∇γ′ : V

∣∣γ→ V

∣∣γ

, such that

• ∇γ′(v + w) = ∇γ′v +∇γ′w

• ∇γ′(fv) = dfdtv + f∇γ′v

• If v extends v to an open set around γ then ∇γ′v = ∇γ′ v.

In this very last step we use the actual connection.

Proof. Consider first the curve restricted to a coordinate system (x, U) ⊂ Msuch that we also have a trivialisation φU of V . Above U we then have a framee1, ..., en where n is the dimension of V . Write v =

∑ni=1 v

iei for functions vi

along γ. Then we have

∇γ′v =

n∑i=1

(∇γ′(viei)) =∑i=1

(dvi

dtei + vi∇γ′ei) (2.1)

From the above formula we see that we have uniqueness. It remains to checkthat this expression is independent of the chosen trivialisation. It then followsthat the above expressions correctly glue together to give a single well-definedoperator. The required calculation is standard change of basis work and willnot be performed here.

With this theorem we can define the parallel transport of a single vectorvγ(0) along a curve by defining it to be the unique v along γ extending vγ(0)

satisfying ∇γ′v = 0. That there always exists a unique solution follows becausewe have expression (2.1). This is a system of linear first order ODE’s, whichhas a unique solution along all of γ given initial conditions vγ(0). Since theflowlines of an ODE are disjoint the maps τt : vγ(0) 7→ vγ(t) given by the paralleltransport are linear isomorphisms for all t. This is where the name connectionoriginates.

It is an interesting fact that a connection is uniquely characterised by itsparallel translation.

Proposition 2.1.5. Let γ be a curve with γ(0) = p and γ′(0) = Xp and ∇ aconnection on V with parallel transport τ along γ. We have the formula

∇Xpv = lim

h→0

1

h(τ−1h v(γ(h))− v(γ(0))

for any section v of V along γ


Proof. Choose a basis ui(γ(0)) of π−1(γ(0)), and extend it to a parallel frameui along γ. Every vector field v along gamma is then given by

∑ni=1 v

iui for aset of functions vi. Using Einstein summation convention we calculate

limh→0

1

h(τ−1h v(γ(h))− v(p) = lim

h→0

1

h(τ−1h

(vi(γ(h))ui(γ(h))

)− vi(p)ui(p))

= limh→0

1

h(τ−1h

(vi(γ(h))

)ui(γ(h))− vi(p)ui(p))

= limh→0

1

h(τ−1h

(vi(γ(h))

)− vi(p))ui(p)

=dvi

dt(p)ui(p) = ∇Xp

(viui) = ∇Xpv

When a metric is also present on our manifold M then one can ask thequestion whether or not the parallel transport is not only an invertible, but alsoan isometric mapping. It is easy to determine whether or not this is the case.Recall that a metric g on V is a smooth section of V ∗ ⊗ V ∗ that gives an innerproduct on each fibre.

Definition 2.1.6. A connection∇ on V is compatible with a metric g if∇g = 0.

A connection being compatible with a metric is equivalent to its paralleltransport being an isometry. To see this let γ be a curve, and v, w two vectorfields parallel along it. Then

d

dtg(v, w) = γ′(g(v, w)) = (∇γ′g)(v, w)+g(∇γ′v, b)+g(v,∇γ′w) = (∇γ′g)(v, w),

proving the statement.

So far we have only seen a single example of a connection, the action ofvector fields on functions. After one more definition we are ready to give anotherbroad class of examples that has been the subject of extensive study. For thiswe restrict ourselves to connections on the tangent bundle. In this case boththe arguments of ∇ lie in TM and one can wonder if it is symmetric.

Definition 2.1.7. For a connection ∇ on TM we define the torsion by theformula

T (X,Y ) := ∇XY −∇YX − [X,Y ]

It is easily checked that both arguments of the torsion are linear over func-tions and therefore the torsion defines a tensor. The Lie bracket appears in thedefinition to ensure linearity over the functions, and in coordinate expressionsit disappears. Indeed

T (∂

∂xi,∂

∂xj) = ∇ ∂

∂xi

∂

∂xj−∇ ∂

∂xj

∂

∂xi,

2.1. KOSZUL CONNECTIONS 13

so that the torsion measures commutativity of the connection with respect tocoordinate vectorfields. We are now ready for the fundamental theorem ofRiemannian geometry:

Lemma 2.1.8 (Fundamental lemma of Riemannian geometry). Given a Rie-mannian manifold (M, g) there is a unique connection on TM that is compatiblewith the metric and torsion-free (its torsion is 0). This connection is called theLevi-Civita connection for the metric.

Proof. The statement is proven by finding an explicit formula for the connection.The compatibility of the connection is expressed by

X(g(Y, Z)) = g(∇XY,Z) + g(Y,∇XZ)

Y (g(Z,X)) = g(∇Y Z,X) + g(Z,∇YX)

Z(g(X,Y )) = g(∇ZX,Y ) + g(X,∇ZY ).

Now add the first 2 equalities and substract the last. Using that the torsiondisappears then gives

g(∇XY, Z) = g(∇ZX −∇XZ, Y ) + g(X,∇ZY −∇Y Z)− g(Z,∇YX)

+X(g(Y,Z)) + Y (g(Z,X))− Z(g(X,Y ))

= g([Z,X], Y ) + g(X, [Z, Y ])− g(Z,∇XY )− g(Z, [Y,X])

+X(g(Y,Z)) + Y (g(Z,X))− Z(g(X,Y ))

2g(∇XY, Z) = g(Y, [Z,X]) + g(X, [Z, Y ]) + g(Z, [X,Y ])

+X(g(Y,Z)) + Y (g(Z,X))− Z(g(X,Y ))

This final expression is linear over functions in X and Z and satisfies the Leibnizrule in Y so it defines a connection.

Remark. In contrast to the rest of this section this proof is adapted from [7].

It is certainly very nice that we can have such a well behaved connection,and that it is unique. However, one can wonder what disappearance of the tor-sion means and why we should single out the Levi-Civita connection for study.The precise meaning of the torsion is unfortunately a bit of a delicate subject,but there is an easy way out. On Rn in the standard coordinate system theLevi-Civita connection is given by ∇ ∂

∂xi

∂∂xj = 0. This is of course the standard

formula for deriving vector fields that one sees in analysis. The Levi-Civitaconnection is therefore the proper generalisation of this connection to arbitraryRiemann manifolds.

The astute reader might have noticed that the above lemma has a rather loftyname. This is the case because the Levi-Cevita connection is the primary objectof study in Riemannian geometry. One of the standard tools of this study is thecurvature of the connection (which in the case of the Levi-Civita connection isoften called the curvature of the manifold itself). The goal of this thesis is not


an exposition of the many beautiful results from Riemannian geometry, but thecurvature of a connection turns out to be an integral concept of gauge theory.With this in mind we shall define the curvature of an arbitrary connection, andmotivate this definition by showing its behavior for the Levi-Civita connection.

Definition 2.1.9. The curvature of a connection ∇ on a vector bundle π : V →M is an element R ∈ Ω2(M,End(V )), the space of End(V )-valued 2-forms. Itis defined by the following formula:

R(X,Y )v = ∇X∇Y v −∇Y∇Xv −∇[X,Y ]v.

That R is End(V )-valued is the statement that it is linear in all its argu-ments and hence defines an object in Ω2⊗V ∗⊗V. This means it defines a tensorof rank 4, the curvature tensor.

Let us finally turn to the geometrical interpretation of the curvature. Thecrucial fact here is that on Rn in the standard coordinates the Levi-Civitaconnection obeys ∇ ∂

∂xi

∂∂xj = 0. This has two consequences, the curvature is

0 and the coordinate vectorfields are always parallel. It turns out that thecurvature measures the extent to which this last fact can hold. What thismeans becomes clear from the proof of the following theorem.

Theorem 2.1.10. Let (M,g) be a Riemann manifold, ∇ the Levi-Civita con-nection and R its curvature. M is locally isometric to Rn if and only if R = 0.

Proof. Take any point p ∈M . We must show that there is a coordinate system(x, U) 3 p such that the coordinate vectors are orthonormal. As a first step weshow that any Xp can be extended to a vectorfield with ∇X = 0; to do this usean induction argument. Assume that X is defined on the (y1, ..., yk, 0, ..., 0)-plane, so that ∇ ∂

∂yiX = 0;here and in the rest of this proof i ≤ k. Extend

X by parallel translation along the curves t 7→ (y1, ..., yk, t, 0, ..., 0), so that∇ ∂

∂yk+1X = 0. From the vanishing of the curvature we have:

0 = R(∂

∂yk+1,∂

∂yi)X = ∇ ∂

∂yk+1∇ ∂

∂yiX −∇ ∂

∂yi∇ ∂

∂yk+1X = ∇ ∂

∂yk+1∇ ∂

∂yiX

Now in the (y1, ..., yk, 0, ..., 0)-plane ∇ ∂

∂yiX = 0. Because parallel transport is

an isometry for ∇ it thus holds everywhere, which concludes the inductive step.The case k = 0 is true by assumption, yielding our X.

If we have an orthonormal basis of TpM we can thus extend it to a frameX1, ..., Xn such that ∇Xj = 0. This implies that the Xj are everywhere or-thonormal. We finish the proof by showing that the Xj are coordinate vector-fields. This is the case precisely if they all commute. Since ∇ is torsion-free wealso have

0 = T (Xi, Xj) = ∇XiXj −∇XjXi − [Xi, Xj ] = [Xj , Xi].

2.2. EHRESMANN CONNECTIONS 15

Before ending the discussion of Koszul connections there is one final remark.A Pseudo-Riemannian manifold is a manifold that merely has a smoothly vary-ing non-degenerate bilinear form on each tangent space. The non-degeneracymeans that the proof of uniqueness of the Levi-Civita connection still holds.Also redefining an orthonormal basis as an orthogonal basis of vectors withlength ±1 the above proof carries through unchanged as well. In this case oneconsiders Rn to be equipped with the diagonal form with entries ±1. We will en-counter Pseudo-Riemannian manifolds in the sequel because of their importancein relativity and the description of wave-like behaviour.

2.2 Ehresmann connections

It turns out that a Koszul connection on V can be described in terms of a specific1-form on the corresponding bundle of frames F (V ). It can be classified preciselywhich 1-forms correspond to a Koszul connection and these 1-forms will be theEhresmann connections. These 1-forms exist on any principal bundle, makingthis definition more general. Ehresmann connections are also more flexible towork with and are thus useful in a practical sense as well.

2.2.1 Connection forms for a frame

To find our definition for Ehresmann connections first we want to describeKoszul connections in terms of 1-forms assigned to (local) frames. In orderto do so let ∇ be a connection on V and choose a (local) frame s; note that sis a local section of F (V ). The frame s consists of n vector fields si. There isnow a unique set of 1-forms ωjs,i such that

∇Xsj =

n∑j=1

siωis,j(X).

Since gln ' End(Rn) we can also view the ωis,j as the components of a gln-valued1-form ωs. Using matrix notation the above equation then reads

∇Xs = s · ωs(X).

Definition 2.2.1. The form ωs described above is called the connection formfor the frame s.

Any other frame s′ is of the form s · a for some GLn-valued function a. Thequestion we now want to answer is, how do ωs and ωs·a relate? We calculate

s′ · ωs′ = ∇s′ = ∇(s · a) = ∇s · a+ s · da = s · ωs · a+ s′ · a−1 · da= s′ · (a−1 · ωs · a+ a−1 · da).


It follows that the connection forms satisfy the following transformationproperty:

ωs·a = a−1 · ωs · a+ a−1 · da.

By reversing the above calculation one sees that the inverse is also true: a setof 1-forms as above consistently defines a connection on V . Furthermore, ifone gives connection forms satisfying the above transformation rule for a set oftrivialisations covering M then these uniquely extend to a set of curvature formsfor every trivialisation. This follows from the cocycle condition ωs·(ab) = ω(s·a)·b,which shows that the extension does not depend on the order in which oneextends.

Before we move on to Ehresmann connections we want to see what form thecurvature has in the current formulation. Since the ωs are gln-valued 1-formsand the curvature is a set of gln-valued 2-forms one might hope the latter isneatly expressible in the former; this turns out to be the case as we will seebelow. First however we adress the statement that the curvature is a set ofgln-valued 2-forms. This is because it is an End(V ) valued 2-form and everyframe s induces an isomorphism End(V ) ' gln.

We now give a theorem which states the form of the curvature tensor interms of local frames.

Theorem 2.2.2. Consider a (local) frame s of V and a connection ∇ with con-nection form ωs, and curvature R. If Ωs is the 2-form such that s ·Ωs(X,Y ) =R(X,Y )s then

Ωs = dωs +1

2[ωs ∧ ωs].

Proof. Using Einstein summation convention we perform a series of calculations.First we have

∇X∇Y sj = ∇X(siωis,j(Y )) = siX

(ωis,j(Y )

)+∇Xskωks,j(Y )

= siX(ωis,j(Y )

)+ siω

is,k(X)ωks,j(Y )

= si(X(ωis,j(Y )

)+ ωis,k(X)ωks,j(Y )).

Using this we get

si(dωis,j(X,Y ) + ωis,k ∧ ωks,j(X,Y ))

= si(X(ωis,j(Y )

)+ ωis,k(X)ωks,j(Y )

− Y(ωis,j(X)

)− ωis,k(Y )ωks,j(X)− ωis,j([X,Y ]))

= ∇X∇Y sj −∇Y∇Xsj −∇[X,Y ]sj = R(X,Y )sj .

This completes the frame dependent treatment of connections. All the proofsabout Koszul connections have their analogues in this language. The majordifference is that the appearance of PDE’s is more explicit in this case. In the


case of Koszul connections their appearance is mostly hidden by the geometricallanguage in use. These proofs will not be given in this case and can be foundin [5]. Instead we will now reformulate the above again in the language ofprincipal bundles.

2.2.2 Ehresmann connections

Consider a connection to be an assignment of a form ωs to a section s of F (V ).We can then wonder if there is a gln-valued 1-form ω on F (V ) such that ωs =s∗(ω). The answer is that such a form exists; we can also specify precisely whichforms pull back to a set of connection forms. The way in which this is done,and the way in which this can be generalised, is shown below.

As a first step we would like to characterise which ω give us a connectionat all. That is we want to characterise the gln-valued 1-forms such that for aGLn-valued function a

(s · a)∗ω = a−1 · s∗ω · a+ a−1 · da.

With the same ammount of difficulty we can tackle a more general question.We will classify which ω on a general principal bundle P satisfy

(s · a)∗ω∣∣p

= Ada−1(p)s∗ω∣∣p

+ La−1(p)∗ a∗∣∣p

= Ada−1(p)s∗ω∣∣p

+ a∗θ∣∣p. (2.2)

Here L is left multiplication in G, s is a section over U , a is a G-valued functonover U and θ is the Maurer-Cartan form on G. That (2.2) gives the formulapreceding it for G = GLn is left as an exercise to the reader.

Lemma 2.2.3. Let s be a (local) section of P over an open U and a : U → Ga function. For a tangent vector field X we have

(s · a)∗Xp = Ra(p)∗ s∗Xp + σ(La−1(p)∗ a∗Xp)∣∣s(p)·a(p)

= Ra(p)∗ s∗Xp + σ(a∗θ(Xp)∣∣s(p)·a(p)

Here σ is defined in (A.1) and θ is the Maurer-Cartan form on G.

Proof. Let m : P ×G→ G : (p, g) 7→ p · g and let c be a curve with c′(0) = Xp.We have s · a = m(s, a) so we get

(s · a)∗Xp = m∗(s∗Xp, a∗Xp) = m∗(s∗Xp, 0) +m∗(0, a∗Xp)

=d

dts c(t) · a(p) +

d

dts(p) · a c(t)

For the first term we can write

d

dts c(t) · a(p) =

d

dt(Ra(p) s c(t)) = Ra(p)∗ s∗Xp.


For the second term we write

d

dts(p) · a (t) =

d

dts(p) · a(p) · a−1(p) · a c(t) = σ

(d

dta−1(p)a c(t)

) ∣∣∣∣s(p)·a(p)

= σ(La−1(p)∗ a∗Xp)∣∣s(p)·a(p)

This lemma allows us our our characterisation, which is the content of thefollowing theorem.

Theorem 2.2.4. A g-valued 1-form ω on a principal bundle π : P → M withfibre G obeys (2.2) for any section s and any a ∈ G if and only if

ω(σ(Y1)) = Y1 ∀Y1 ∈ g

ω(Ra∗Y2) = Ad(a−1)ω(Y2) ∀Y2 ∈ TP.

Proof. By the above lemma we need to prove

ω(Ra(p)∗ s∗Xp

)+ ω

(σ(La−1(p)∗ a∗Xp)

)= Ada−1(p)ω(s∗Xp) + La−1(p)∗ a∗(Xp).

(?)

We can split this into 2 separate equations. First choose a(p) = e in (?) toobtain

ω(σ(a∗Xp)s(p)

)= a∗Xp. (I)

Then we choose a to have the constant value and obtain

ω (Ra∗ s∗Xp) = Ada−1ω(s∗Xp). (II)

It is left as an exercise for the reader that (I) and (II) together also give (?).We thus continue to work with (I) and (II). Since in (I) we only required thata(p) = e it follows that a∗ is a surjection onto g. Therefore (I) is equivalent toω(σ(Y1)) = Y1 ∀Y1 ∈ g. As for (II), the image of s∗ is any vector that is notvertical. The theorem is thus proven if (I) gives (II) for vertical vectors. Thisis true precisely if

Ra∗σ(Y1) = σ(Ada−1Y1) ∀Y1 ∈ g,

which is a standard fact from the theory of Lie groups, and has A.3.2 as animmediate consequence by taking the derivative to a.

This theorem motivates a definition.

Definition 2.2.5. An Ehresmann connection in a principal bundle π : P →Mwith structure group G is a smooth g-valued 1-form such that

ω(σ(X)) = X ∀X ∈ g

ω(Ra∗Y ) = Ad(a−1)ω(Y ) ∀a ∈ G, ∀Y ∈ TP.


By the above treatment an Ehresmann connection on a frame bundle F (V )defines a Koszul connection in V , which motivates the name connection. Theseare the connections whose definition we have been working to motivate. Beforewe continue on with the definition of the curvature of an Ehresmann connectionwe show that every set of connection forms ωs comes from a connection ω onthe corresponding frame bundle. It is clear that if such an ω exists it shouldobey

ω(σ(X)) = X ∀X ∈ g

ω(s∗Y ) = ωs(Y ) ∀Y ∈ TM.

What is left is showing that this actually defines a connection. First weshow that the above is actually well-defined. This ammounts to showing thatif s1∗Yp = s2∗Yp for all then ωs1(Yp) = ωs2(Yp). We can write s2 = s1 · a witha(p) = e. Then we have

s2∗Yp = (s1 · a)∗ = s1∗Yp + σ(a∗Yp)∣∣s(p)·a(p)

.

From this it follows that a∗(Yp) = 0. Then

ωs2(Yp) = ωs1·a(Yp) = ωs1(Yp) + a∗(Yp) = ωs1(Yp).

Finally we have to show that ω(Ra∗s∗Yp) = Ada−1ω(s∗Yp). From the defi-nition this is equivalent to

ωs·a(Yp) = Ada−1ωs(Yp).

This is true by the transformation law for the connection forms and the factthat a is the constant function. We thus see that every Koszul connection isinduced by an Ehresmann connection, making the later a true generalisation.

With this out of the way it is time to associate a curvature to every Ehres-mann connection.

Definition 2.2.6. To a connection ω on a principal bundle P we associate itscurvature Ω.

Ω := dω +1

2[ω ∧ ω]

A connection for which Ω = 0 is called flat.

If P = F (V ) then s∗Ω = Ωs by virtue of theorem 2.2.2. Which validatescalling this 2-form the curvature. The curvature will play an important rolein gauge theory. We are nearing the end of the application of connectionsto Riemannian geometry. There is however a claim from the previous sectionthat remains unproven. The claim in question being that any vector bundle Vallows a Koszul connection. This is implied by the more general fact that every


principal bundle allows a connection. First we show that every trivial bundleM ×G allows a connection. We can choose

ω(σ(X)) = X ∀X ∈ g

ω(Y ) = 0 ∀Y ∈ TM.

Now we use these trivial connections to build one on a general principal bundle.To do this choose a set of trivialisations φU of π : P →M , such that any x ∈Mis contained in one of the opens U . Choose a partition of unity subordinateto the open cover of U ’s and use the trivialisations over U to obtain locallydefined 1-forms ωU . Then we glue these together into a connection defined byω =

∑U φUωU . That this is a connection is easily checked.

2.2.3 The meaning of the structure group

As a final topic in Riemannian geometry it is worthwhile to say something aboutthe meaning of the structure group of the principal bundle. Depending on theammount of structure we put on a vector bundle V there are several differentframe bundles we can associate with it: F (V ), SF (V ), O(V ), SO(V ). What wehave shown above is that every Koszul connection comes from an Ehresmannconnection on F (V ) and vice versa. One can wonder what the effect of chang-ing the structure group is. We know from above that a connection is uniquelydetermined if we know the corresponding connection forms on a set of frameswhose domain of definition cover M . This has the immediate consequence thata connection on O(V ) is uniquely extended to one on F (V ) and so forth. Al-ternatively a connection ω on F (V ) restricts to one on O(V ) if it takes valuesin on ⊂ gln on every orthonormal frame.

This last statement automatically holds for F (V ), S(V ) and O(V ), SO(V )whose structure groups have the same Lie algebra. This shows that connectionsare insensitive to a choice of orientation on a manifold. The reason behind thisis that parallel transport is a local problem. On the other hand orientabilityis a global one; the obtructions to gluing local orientations together are of noconsequence to the connection.

As suggested above in the presence of a metric the restriction question is anon-trivial one. The question is which connections actually have values in onon orthonormal frames. The answer is related to the presence of a metric in aprofound way.

Theorem 2.2.7. Let (M,g) be a Riemannian manifold. A connection ω onπ : F (V ) → M restricts to a connection on π : O(V ) → M if and only if it iscompatible with the metric.

Proof. We need to check that on an orthonormal frame si the connection formsωij are anti-symmetric if and only if the connection is compatible. Here the

2.3. FURTHER PROPERTIES OF CONNECTIONS 21

subsript s on the connection forms is supressed for simplicity. Let gij be thematrix corresponding to the metric:

g(si, sj) = gij .

Because the frame is orthonormal this is the identity matrix. The connection isthen compatible with the metric if and only if for all tangent vectors X

0 = X (gij) = X (g(si, sj)) = g(∇Xsi, sj) + g(si∇Xsj)= g(ωki (X)sk, sj) + g(si, ω

ljX, sl) = gkjω

ki + gilω

lj

= ωij + ωji

It follows that connections are compatible with the metric if and only if theycome from connections on the bundle of orthonormal frames O(V ).

2.3 Further properties of connections

2.3.1 The horizontal distribution

There is another, equivalent, characterisation of connection that will prove use-ful in what follows. Let m = Dim(M) and ω a connection on π : P →M . Theconnection ω defines a smooth m-dimensional distribution H on TP by the for-mula Hp = ker(ωp) called the horizontal distribution. That H is m-dimensionalis true because ω gives a linear isomorphism surjection from VpP to g. Thisimplies that

TP ' Hp ⊕ VpP,

which explains the name horizontal distribution. If Dim(G) = n then Dim(P ) =n + m so Dim(H) = m. The distribution H satisfies an additional importantproperty

Hp·a = Ra∗Hp.

To see this let Xp ∈ Hp; then ω(Ra∗X) = Ada−1ω(Xp) = 0. Since Ra∗ is aninjective linear map it must be onto Hp·a. The above horizontal distributionsturn out to be in bijection with the connections.

Proposition 2.3.1. The map ω 7→ ker(ω) is a bijection between connectionsand smooth distributions satisfying

• TpP = Hp ⊕ VpP

• Hp·a = Ra∗Hp

Proof. If the inverse of the map exists it must couple to a distribution H theform defined by ω(H) = 0 and ωσ = id : g→ g. This clearly defines a g-valued1-form. It is actually a connection if ω(Ra∗Xp) = Ada−1ω(Xp), since the otherproperty is true by definition. From a previous proof we know that this is true


on vertical vectors. Therefore it is true everywhere if it is true on horizontalvectors. But then we use that Ra∗Hp = Hp·a to obtain for all Xp ∈ Hp

Ada−1ω(Xp) = 0 = ω(Ra∗Xp).

Because of the above theorem from now on we will use the characterisationof a connection as a form ω or a distribution H interchangably. A horizontaldistribution in a principal bundle π : P →M defines the concept of a horizontallift of a curve.

Definition 2.3.2. Let π : P →M be a principal bundle with connection H. Ifγ is a curve in M , then a horizontal lift γ of γ is a curve in P such that π γ = γand γ′ is horizontal.

The important thing in this case is the following:

Proposition 2.3.3. Assume a principal bundle π : P →M with connection ω,a curve γ in M satisfying γ(0) = x and a point p ∈ π−1(x). There is a uniquehorizontal lift γ of γ such that γ(0) = p.

Proof. We first prove local existence and uniqueness. Using trivialisations wecan then assume the bundle is trivial. In this case it is clear that we can finda lift γ of γ if we drop the horizontality condition. We want to show there isa unique a with a(0) = e such that γ · a is a horizontal lift. Using the knownformula

(γ · a)′ = Ra(t)∗γ′ + σ(La−1(t)∗ a′)

We want

0 = ω ((γ · a)′) = Ada−1(t)ω (γ′) + La−1(t)∗a′.

Thus

La−1(t)∗a′ = −La−1(t)∗Ra(t)∗ω (γ′)

or

a′ = −Ra(t)∗ω (γ′) .

This is an ODE and has a unique solution on its domain of definition. We nowneed to prove that we can extend any definition to a larger open subset. Assumethat γ is defined on (tb, te), we show that γ can be extended past te, the othercase is analogous and left to the reader. Pick any point p ∈ π−1(γ(te) and finda horizontal lift γ of γ through p. By uniqueness of horizontal lifts and thetranslational properties of H there is, for t < te, a g ∈ G such that γ = γ · g.Clearly we can extend γ past te by defining γ = γ · g there as well.


The use of viewing a connection as a horizontal distribution lies in the factthat it provides a very geometric notion of parallel transport. To see how thisworks consider a curve γ in a manifold M and a frame s of a vector bundleV . The vectors in the frame are all parallel along γ if and only if ωs(γ

′) = 0.This is precisely the case if s

∣∣γ

is a horizontal lift of γ for the connection ω.

Horizontal lifts of curves are thus choices of parallel frames along these curves.The uniqueness of a lift through a specific point is merely the statement that aparallel frame is determined by its value at a single point. Since parallel trans-lation along a curve is a linear isomorphism this should come as no surprise.

There is one other statement worth making about horizontal distributions.In the proof of theorem 2.1.10 the first major step was finding a frame s forwhich ωs = 0. To do so we used that the curvature of the connection we usedwas flat. It turns that this condition is both necessary and sufficient. Before wegive the proof a lemma is needed.

Lemma 2.3.4. The curvature is a horizontal 2-form. This means that Ω(Xp, Yp) =0 if either Xp or Yp is vertical

Proof. Without loss of generality we assume that Xp = σ(X)p and Yp = Yp +

σ(Y )p with Yp horizontal. We extend to vectorfields X = σ(X) and Y =

Y +σ(Y ). We demand that Y is everywhere horizontal and satisfies Rg∗Y = Yfor all g ∈ G. Now use dω(X,Y ) = Xω(Y )− Y ω(X)− ω([X,Y ]) and A.3.2 toobtain

Ω(X,Y ) = Ω(σ(X), σ(Y )) + Ω(σ(X), Y ) = dω(σ(X), σ(Y ))

+ [ω(σ(X)

), ω(σ(Y )

)] + dω(σ(X), Y ) + [ω

(σ(X)

), ω(Y ))

]

= σ(X)(Y )− σ(Y )(X)

− [X, Y ]) + [X, Y ] + σ(X)ω(Y )− Y (X)− ω([σ(X), Y ])

+ [X, ω(Y )] = ω([σ(X), Y ])

We now show that [σ(X), Y ] = 0 to finish the proof. The flow of σ(X) is righttranslation by exp(tX). We thus get

[σ(X), Y ] =d

dt

∣∣∣∣t=0

Rexp(−tX)∗Y − Yt

=d

dt

∣∣∣∣t=0

Y − Yt

= 0

Proposition 2.3.5. The horizontal distribution H of a connection is integrableif and only if the connection is flat.

Proof. We use the frobenius theorem. H is integrable if and only if [X,Y ] liesin H for any X,Y in H. Take X,Y lying in H, then ω(X) = ω(Y ) = 0. It


follows that

Ω(X,Y ) = dω(X,Y ) + [ω(X), ω(Y )] = dω(X,Y )

= Xω(Y )− Y ω(X)− ω([X,Y ]) = −ω([X,Y ])

The distribution H is thus integrable if and only if Ω vanishes on the horizontaldistribution. The lemma now gives that this is equivalent to Ω = 0.

In the case of the bundle of frames of the tangent bundle F (TM) we couldalso define the torsion of a connection in terms of ω. We could then use theabove result to give another proof of theorem 2.1.10. Instead we leave the realmof Riemannian gemoetry behind us and move on to other topics.

2.3.2 Gauge transformations

In the appendix the concept of a principal bundle homomorphism was defined.Of special interest to us are principal bundle homomorphisms covering the iden-tity map id : M →M .

Definition 2.3.6. A smooth bundle homomorphism φ from a principal bundleπ : P →M to itself covering the identity is called a gauge transformation.

In the case of a trivialisable bundle the gauge transformations correspondto changes of trivialisations. The name gauge transformation comes from theintended application of principal bundles from gauge theory. There is a cor-respondence between gauge transformations in P and Ad-invariant G-valuedfunctions on P . To find this correspondence choose a point p ∈ P . There isthen a unique element gp ∈ G such that φ(p) = p · gp. Clearly p 7→ gp is asmooth function. From the condition that φ intertwines the action we deduce

p · h · gp·h = φ(p · h) = φ(p) · h = p · gp · h.

This is equivalent to gp·h = h−1 · gp · h. That any function function p 7→ gpsatisfying this last requirement actually defines a gauge transformation is easyto check. We can thus identify gauge transformations with G-equivariant mapsa : P → G where G has a right action on itself by conjugation. We want tofind how connections transform under these gauge transformations; a formulafor this behaviour can be obtained in terms of the functions a. The reason wewant to do this lies in our intended application to gauge theory. Here a gaugetransformation corresponds to a ”change of basis” for our theory, and shouldnot have any physical significance. We should therefore only look at structuresdefined on equivalence classes of connections under gauge transformations. Itshould come as no surprise that the transformation behaviour of the connectionis closely related to that of the connection forms under changes of frame.

Theorem 2.3.7. Let φ be a gauge transformation with corresponding G-equivariantmap a : P → G. If ω is a connection on P with curvature Ω then

φ∗ω = Ada−1 ω + a∗θ


with θ being the Maurer-Cartan form. This is again a connection. Furthermorewe have

φ∗Ω = Ada−1 Ω

and this is the curvature of φ∗ω.

Remark. The transformation property here is essentially that of (2.2) which aconnection satisfies by construction. Since the formulation of the behaviour isslightly different we give a full proof.

Proof. We first check the claims about φ∗ω, starting with the transformationequation. We check this on vertical and non-vertical vectors separately. Let Xp

be non vertical, then there is a section s and a vector Y such that Xp = s∗Y .In this case by 2.2.3 we have

φ∗ω(Xp) = φ∗ω(s∗Y ) = ω((s · a)∗Y ) = ω(Ra(p)∗s∗Y ) + (a s)∗θ(Y )

= Ada1 ω(Xp) + a∗θ(Xp).

On vertical vectors the left hand side is calculated to be

φ∗ω(σ(X)p) = ω(φ∗σ(X)p) = ω

(d

dtφ(p · exp(tX))

)= ω

(d

dtφ(p) exp(tX)

)= ω

(σ(X)φ(p)

)= ω(σ(X)p) = X.

On the other hand the right hand side is

Ada−1 X + θ(a∗(σ(X))) = Ada−1 X + θ(d

dt

∣∣∣∣t=0

a(p · exp(tX)))

= Ada−1 X + θ(d

dt

∣∣∣∣t=0

exp(−tX)a(p) · exp(tX))

= Ada−1 X + θ(La(p)∗X)− θ(Ra(p)∗X)

= Ada−1 X +X −Ada−1 X = X.

We have thus proven the formula for φ∗ω and that it takes the correct values onvertical vectors. For it to be a connection it must behave correctly under righttranslation. But here we have for g ∈ G

R∗gφ∗ω = φ∗R∗gω = φ∗Adg−1 ω = Adg−1 φ∗ω.

The first equality follows because gauge transformations commute with the rightaction. Finally we check the transformation of the curvature. By 2.3.4 we onlyhave to check it on horizontal vectors, since clearly both sides are 0 if one ofthe vectors is vertical. Let Xp, Yp be two horizontal vectors and write them ass∗X, s∗Y respectively. Once more using 2.2.3 we get

φ∗Ω(Xp, Yp) = Ω ((s · a)∗X, (s · a)∗Y ) = Ω(Ra(p)∗s∗X,Ra(p)∗s∗Y )

= R∗a(p)Ω(Xp, Yp) = (dR∗a(p)ω +1

2[R∗a(p)ω ∧R

∗a(p)ω])(Xp, Yp)

= Ada(p)−1(dω +1

2[ω ∧ ω])(Xp, Yp) = Ada(p)−1 Ω(Xp, Yp).


In going from the second to the third line it was used that the Lie bracketcommutes with Ad. That φ∗Ω is the curvature of φ∗ω follows because takingpull-backs commutes with exterior differentiation and wedge products.

It is clear that the pullback as described above gives a right action of gaugetransformations on connections and their curvature. Let φ be a gauge transfor-mation with G-equivariant map a, then we denote ω ·a := φ∗ω and Ω ·a := φ∗Ωaccordingly. Finally note that gauge transformations take flat connections toflat connections.

This concludes the part on gauge transformations. We now move on to atopic that is needed specifically for our treatment of Chern-Simons theory later.

2.3.3 Holonomy

This subject is that of holonomy. The idea of holonomy is to measure the extentto which a horizontal lift of a loop fails to close. More precisely take a pointp in a principal bundle π : P → M and a loop γ : [0, 1] → M with start- andendpoint π(p). There is a unique horizontal lift γ of γ starting at p. There isalso a unique g ∈ G such that γ(1) = p · g.

Definition 2.3.8. We define the holonomy of γ through p by

holω(γ, p) = g.

Holonomy behaves especially nicely in case of a flat connection. As we willsee below Chern-Simons theory has as its solutions gauge equivalency classes offlat connections. It is the nice behaviour of holonomy in this case that makes itan important concept in Chern-Simons theory, and hence in this thesis.

Proposition 2.3.9. Let π : P →M be a principal bundle with flat connection ωthen the holonomy map holω(·, p) is constant on homotopy equivalence classes.It thus descends to a map holω(·, p) : π1(M,π(p)) → G on the fundamentalgroup of M .

Proof. We need to show that if two loops are homotopic then they have the sameholonomy. Two loops α and β are homotopic if and only if α∗β−1 is contractible.It is therefore enough to show that the holonomy on a contactible loop γ is e. Letγ be a contractible loop with horizontal lift γ and H the homotopy contractingit: H(1, ·) = γ, H(0, ·) = γ(0). We use that the horizontal distribution isintegrable. Since the image of H is a compact subset of M it can thus becovered with finitely many opens Ui above which horizontal sections exist. Theimage of H is connected so these sections can be glued together (after possibletranslations by gi ∈ G) to obtain a unique horizontal section s extending γ.The homotopy H lifts to a homotopy H := s H contracting γ. Specificallythis means that γ is a loop and therefore has holonomy e.

Since any small enough loop on a manifold is contractible the above theoremcan also be used to create a horizontal section, inverting the argument in the


proof. From previous results we had already seen that the curvature measuresthe way in which parallel transport is path dependent. The above propositioncan be seen as another formulation of this fact.

The good behaviour of holonomy in the case of flat connections goes furtherthan just the above however. The holonomy also behaves well under changes ofbase point.

Proposition 2.3.10. Let ω be a flat connection in π : P →M , p ∈ P and γ aloop based at x = π(p). Then

• For any g ∈ G we have

holω(γ, p · g) = g−1 · holω(γ, p) · g.

• For any gauge transformation with G-equivariant map a

holω·a(γ, p) = a−1p · holω(γ, p) · ap.

• For p′ a point in a different fibre with π(p′) = x′ and α a path from x tox′ there exists a g ∈ G such that

holω(α ∗ γ ∗ α−1, p′) = g · holω(γ, p) · g−1

Proof. Let γ be the horizontal lift of γ through p. Then for the first point wesee that γ · g is the horizontal lift through p · g. We then see

p · g · holω(γ, p · g) = γ · g(1) = p · holω(γ, p) · g= p · g · g−1 · holω(γ, p) · g.

This proves the first statement. As for the second statement: using lemma 2.2.3and proposition 2.3.7 it is easy to check that γ · a−1 is a horizontal lift for ω · a.This lift starts at p · a−1

p and ends at

p · holω(γ, p) · a−1p·holω(γ,p) = p · a−1

p · holω(γ, p).

Using this we get

p · holω·a(γ, p) = γ · a−1(1) · ap = p · a−1p · holω(γ, p) · ap

Lastly we check the change of base-point. To this end let α−1 be the horizontallift of α−1 starting at p′. It ends at p·g for some g ∈ G. Therefore the horizontallift of γ ∗α−1 is γ · g ∗ α−1 which ends at p · holω(γ, p) · g. The horizontal lift ofα ∗ γ ∗ α−1 is then

α · g−1 · holω(γ, p) · g ∗ γ · g ∗ α−1.

This loop ends at p′ · g−1 · holω(γ, p) · g, giving the stated holonomy.


The above two propositions show that the holonomy of a flat connectiongives an element of

∏α Hom(π1(Mα), G)/G. Here G has a right action on

Hom(π1(Mα), G) via conjugation and Mα are the connected components of M .Clearly if (P, ω) ∼ (P ′ω′) in the sense that there is a bundle isomorphismφ : P → P ′ with ω = φ∗ω′ then they induce the same element. It turns outthat up to this equivalence of bundles the holonomy is a bijection. Since flatconnections will play a major role in Chern-Simons theory we will prove this fact.

Theorem 2.3.11. Let P be a principal bundle with connection ω. The mapping

[P, ω] 7→ holω

is a bijection from the equivalence classes of principal bundles with flat connec-tions to ∏

α

Hom(π1(Mα), G)/G.

Here Mα are the connected components of M .

Proof. We will start by considering M connected, since we can carry throughthe construction below for every component of M indivually if it is not. We willbegin by constructing a right inverse of the holonomy. Choose a point x0 ∈M .As a preliminary we will construct a bundle Pρ with connection ωρ that has

hol(·, x0) = ρ. Let M be the universal cover of M , regarded as the space ofhomotopy classes of paths starting at a point x0 ∈ M [8]. Recall that there isa right action of π1(M,x0) on M . For an element ρ ∈ Hom(π1(M,x0), G) wedefine the principal bundle

Pρ := M ×ρ G := M ×G/ ∼

where (x, g) ∼ (x · α, ρ(α)−1) · g for all α ∈ π1(M). Note that Pρ is a bundle

over M , not M . Let π : M ×G→ G be projection. We define ω := π∗θ whereθ is the Maurer-Cartan form on G. This defines a connection on the bundleM × G → M by the properties of the Maurer-Cartan form. Furthermore forany α ∈ π1(M) we have

ω(Rα∗X ⊕ Lρ(α)−1∗Y ) = θ(Lρ(α)−1∗Y ) = θ(Y ) = ω(X ⊕ Y ),

so ω is invariant under the equivalence relation generating M ×ρ G and hencedescends to a connection ωρ on Pρ. We now show that ωρ has ρ as its holonomy

map. For our calculation we work in M ×G and with the connection ω. Let γbe a loop through x0 with [γ] = α, and γ a lift of this curve in M . The curve(γ, e) is clearly horizontal in M × G. The end point of this lift is (γ(0) · α, e).In Pρ this lies in the class [γ(0), ρ(α)] = [γ(0), e]ρ(α). The curve (γ, e) is clearlya horizontal lift of γ and we just calculated that it has holonomy ρ(α). It thusfollows that ωρ has holonomy map ρ through x0. This gives us a correspondence

ρ ∈ Hom(π1(M,x0), G) 7→ (Pρ, ωρ).


If ρ = g−1ρg then the map M ×G→ M ×G : (x, h) 7→ (x, g · h) descends to abundle isomorphism Φ : Pρ → Pρ. It also obeys

ωρ(X ⊕ Y ) = θ(Y ) = θ(Lg∗Y ) = ωρ(Φ∗X ⊕ Y ).

The above correspondence therefore descends to a map

[ρ] ∈ Hom(π1(M), G)/G 7→ [Pρ, ωρ].

By construction this map is a right inverse to the holonomy map. It is theright inverse we were looking for. We finish the proof by showing that it is aleft-inverse, i.e. that (P, ω) ' (Pρ, ωρ) if holω = [ρ]. To this end π : P →M be

the projection in P and choose p0 ∈ π−1(x0). Let βx be a representative of thehomotopy class of paths in M representing x in M . Let βx be the horizontallift through p0 of βx. By the properties of holonomy βx(1) is independent of thechoice of βx. It follows that

Ψ : M ×G→ P : (x, g) 7→ (βx(1) · g)

is a well defined function. By the properties of holonomy it descends to a bundleisomorphism Ψ : Pρ → P . Furthermore Ψ(M × e) is precisely the horizontalintegral manifold through p0, so ωρ = Ψ∗ω and the claim is proved.

If M contains multiple components clearly we can use the resulting isomor-phism for each connected component to obtain the claim for M .

2.3.4 Gluing of connections

The final subject that we will need some knowledge on for our applications isthe gluing of connections, which will be the final subject in this chapter. We willneed to know in what sense two principal bundles, with a common boundarycomponent and connections agreeing thereon can be glued together into one. Itturns out that this can always be done if one only looks at gauge equivalenceclasses. To prove this fact we first need a lemma.

Lemma 2.3.12. Let π1 : P →M be a principal bundle with structure group Gand π2 : [0,∞) × P → [0,∞) ×M the pullback to the cylinder over M . Anyconnection ω on [0,∞)× P can be written as

ω = ωt + ψtdt,

where ωt is a connection on t × P . In this notation there is a unique gaugetransformation φ with G-equivariant map a such that

a∣∣0×P = e

andω · a = ωt;

i.e. ω · a has no dt component.


Remark. Since a∣∣0×P = e we have that ω0 = ω0.

Proof. Denote by at the G-equivariant map corresponding to φ∣∣t×P , and by

θ the Maurer-Cartan form on G. We then get that the dt component of φ∗ω isequal to

Ada−1tψt +

d

dt(a∗t θ) .

By putting this equal to zero we obtain a first order PDE which can be solveduniquely via the method of characteristics.

Using this lemma we obtain the gluability of connections.

Proposition 2.3.13. Let π : P → M be a principal G-bundle and N a codi-mension 1 submanifold of M such that M \N has two components. Denote byMcut the manifold obtained by cutting M along N , by g : Mcut →M the gluingmap and Pcut := g∗P . Use N1 and N2 for the 2 copies of N in g−1(N). Nowsuppose we have a connection ωcut on Pcut and a connection ω on P

∣∣N

suchthat

ωcut

∣∣N1∪N2

= g∗ω.

There is an extension of ω to P such that ωcut is gauge equivalent to g∗(ω).

Remark. The condition in the equation means that if we identify N1 and N2

using the gluing map ωcut is the same along N1 and N2, but not necessarilytransverse to it. By performing a gauge transformation this transverse compo-nent can be annihilated, allowing for gluing of the connections.

Proof. Choose tubular neighbourhoods Ti ' [0,∞) × Ni of the Ni. By usinglocal trivialisations and possibly shrinking our Ti we see that Pcut

∣∣Ti' [0,∞)×

P∣∣N

. Let φi denote the gauge transformations from the previous lemma and let

ρ : [0,∞) → [0, 1] be a smooth map such that t 7→ t on [0, 12 ] and ρ(t) = 0 for

t > 1. Using these maps define φi := φi ρ, which is the identity for large t.Since the manifold N cuts nicely these gauge transformations can be smoothlyexended to the identity outside of Ti. Applying these gauge transformations weget a connection ωcut, which clearly is the pullback of a form ω on P . Since theφi are the identity on the Ni we get that this ω is the extension we’re lookingfor.

Since in our applications we will only be considering connections up to equiv-alence under gauge transformations the above proposition allows us to glue to-gether connections in a very wide range of circumstances. Why we would liketo do so will become clearer in the next chapter, which is about the modernformulation of fields theories.

Chapter 3

Classical field theories

The modern formulation of classical field theories is a good deal more abstractthan the one used when field theories were first studied. From a mathematicalpoint of view this abstaction has been a great benefit. The Chern-Simons theoryfor example is subject to a number of subtleties that cannot be tackled in theolder, and more naive, formulations of field theories. It has reached the pointhowever that the current formulation of field theories is not immediately recon-cilable with the older formulations. The purpose of this chapter is to introducethe constructions necessary to give a detailed and rigorous treatment of Chern-Simons, which will be then be given in the next chapter. The approach will beto introduce the new concepts in the case of Lagrangian mechanics, which willbe seen as a 1-dimensional field theory. The language in which we eventuallyformulate this is easily generalised to arbitrary field theories. We will then giveexamples of some well-known field theories, and show how they fit into thisformulation. Due to the inherent mathematical difficulties in field theories amore general approach is unfortunately not possible. Specific proofs have to betailored to the example at hand. Through our examples we will develop a senseof how other examples might be tackled.

3.1 Lagrangian mechanics in symplectic terms

In the appendix on symplectic geometry we briefly reviewed how Hamiltonianmechanics can be formulated in terms of symplectic geometry on the cotan-gent bundle of the configuration manifold. Unfortunately this procedure is notgeneral enough for our purposes since the Hamiltonian formulation for a givenLagrangian does not always exist.

3.1.1 The case of strictly convex Lagrangians

One situation where the Hamiltonian formulation does exist is if the Lagrangianis strictly convex in the velocities v, a concept we will define below. We will show

31

32 CHAPTER 3. CLASSICAL FIELD THEORIES

how in this case the hamiltonian formulation can be used to give a symplecticstructure on TM , and use this structure to give the Lagrangian mechanics. Thisaccount closely follows that of [9]. Before we delve into the definitions we recallthe construction from Lagrangian mechanics. In Lagrangian mechanics we workwith

• A manifold M called the configuration space.

• A smooth function L on the trivial fibre bundle π : R × TM → R calledthe Lagrangian. The standard coordinate system on the base R is denotedby t.

One reason why in the above definition we work with the fibre bundle R×TMis because we can have Lagrangians that are dependent on time: the base man-ifold is the time-axis. In the present formulation the use of time-dependent co-ordinate systems is also more natural (consider for example rotating coordinatesystems). The significance of the above setup is that it gives us a functionalon parametrised smooth paths γ : I → M where I is a closed interval. Theparametrised paths are in bijection with local sections γ : I → R ×M of thetrivial bundle and these sections induce sections t 7→ (t, γ(t), γ′(t)) of R× TM .If we, somewhat sloppily, identify the curves γ with their induced sections ofR× TM , then we get the action functional

S(γ) :=

∫I

L γ(t)dt.

We call a section γ admissable if it is a critical point of the action undervariations keeping end-points fixed. Let (q, U) be a coordinate system on M ,and ((q, v), U×Rm) the induced coordinate system on the tangent bundle. Beingan admissable section is then equivalent to the Euler-Lagrange equations

γ′ = v

d

dt

(∂L

∂v(γ, γ′, t)

)=∂L

∂q(γ, γ′, t).

Note that these equations are first-order on the tangent bundle. We first wantto formulate the above in terms of symplectic geometry. We find clues as tothe correct procedure by looking at the Hamiltonian formulation of classicalmechanics. To do so we assume that the Hamiltonian formulation is well posed.To state the specifics of what we assume we introduce some terminology.

Definition 3.1.1. A Lagrangian is

• regular if the hessian matrix ∂2L∂vi∂vj is invertible everywhere.

• strictly convex if the hessian matrix has strictly positive eigenvalues ev-erywhere.

3.1. LAGRANGIAN MECHANICS IN SYMPLECTIC TERMS 33

Remark. A convex Lagrangian would be one where the hessian matrix has non-negative eigenvalues everywhere. This is not compatible with the notion ofregularity in the first item and therefore not considered.

Example 3.1.2. In most applications in physics the Lagrangian will be of theform L(q, v) = 1

2m‖v‖2 + V (q, v) where V is at most first order in v. In this

case clearly the Lagrangian is strictly convex since the hessian is equal to themetric. For many applications in physics the restriction of strict convexity isnot overly stringent.

Assuming now the strict convexity of the Lagrangian we have the Hamilto-nian dynamics on the cotangent bundle coming from its canonical symplecticstructure as follows.

The strict convexity of L implies that the Legendre mapping LT : TM →T ∗M : (q, v) 7→ (q, p) =

(q, ∂L∂v

)is injective with invertible derivative, and hence

a diffeomorphism onto its image. The map LT induces the Legendre transformH of L on its image by the definition H(q, p) = pv−L(q, v), here v is seen as afunction of p via the inverse of LT .

Using the Legendre mapping we can pull back the canonical structure onthe cotangent bundle to the tangent bundle. Recall that the canonical 1-formα on T ∗M is in canonical coordinates expressed as

∑i pidq

i. Since pi = ∂L∂vi we

clearly have

αL := LT ∗α =∑i

∂L

∂vidqi.

From this it follows that

ωL := LT ∗ω = −dTM∗α =∑i,j

(∂2L

∂qj∂vidqi ∧ dqj +

∂2L

∂vi∂vjdqi ∧ dvj

).

That these forms are well defined follows immediately from the fact that they arepull-backs of forms; it can, however, also be checked directly that the coordinateexpressions transform correctly. That ωL is non-degenerate can also be checkeddirectly; writing it in matrix form we see that it is equivalent to invertibility

of ∂2L∂vi∂vj , which we assumed. Since the Hamiltonian dynamics is equivalent to

the Lagrangian one we have that the Euler-Lagrange equations have LT−1∗ XH

as their vector field, where H is the legendre transform of L. We then calculatefor X a tangent vector field on TM that

ωL(LT−1∗ XH , X) = LT−1,∗ωL(XH , LT∗X) = ω(XH , LT∗X)

= dH(LT∗X) = d(H LT )(X).

This implies that the vector field of the Euler-Lagrange equations is Hamiltonianwith Hamiltonian function H LT .

Definition 3.1.3. The function E := H LT is called the energy function; itclearly has the expression

E(q, v) = v(L)(q, v)− L(q, v) =∑i

vi∂L

∂vi(q, v)− L(q, v).


Summarising we have found a symplectic structure on TM such that theflow corresponding to the Euler-Lagrange equations has vector field XE . Fromthe appendix on symplectic geometry we know that the flow φt of a Hamiltonianvector field is a symplectomorphism for all t where it is defined. This meansthat if we choose a t we get a submanifold

(x, φt(x)) ∈ TM × TM : x ∈ TM .

This submanifold is actually Lagrangian with respect to the symplectic struc-ture ωL ⊕ (−ωL) on TM × TM .

There is one more generalisation we must make before we can make ourreformulation. In the above we used that the Euler-Lagrange equations arecritical points of the action if one only takes variations in the fields with fixedend-points. We would rather work in terms of variations of the action than interms of the Euler-Lagrange equations, since the former is more geometrical innature. There is however considerable generalisation to be made here. To thisend note that for a general variation of the action we have

δS(δγ) =

∫I

(∂L

∂q− d

dt

∂L

∂v) · δγdt+ αL(δγ)|∂I

In the case of fixed end-points the boundary term clearly disappears, butthis is not necessarily the only case in which this happens. To see when thishappens in general observe that a variation of paths induces a path in TM×TMgiven by

π : γt 7→ (γt(t0), γ′t(t0), γt(t1), γ′t(t1)) . (3.1)

The boundary term corresponding to the variation γt disappears precisely if theabove curve disappears on αL ⊕ (−αL). If we want to allow a greater set ofvariations we can choose a submanifold of TM × TM such that αL ⊕ (−αL)disappears on it. If we then restrict our variations to those having boundaryvalues in the chosen submanifold the boundary term in the variation of theaction will disappear. In general we would like to allow for as many variations aspossible; the submanifold we choose should therefore be of maximal dimension.This leads us to the following

Definition 3.1.4. A boundary condition for a Lagrangian system is a familyof Lagrangian submanifolds of (TM × TM,ωL ⊕ (−ωL)).

Since Lagrangian submanifolds are maximal isotropic submanifolds demand-ing a boundary condition to be Lagrangian ties into allowing as many variationsas possible. The demand of being isotropic is however somewhat weaker thanthat of αL ⊕ (−αL) disappearing. The demand in this case is merely that it isclosed, and hence locally the differential of a function. To see why this is enoughconsider the construction below.

Assume a fibre F of a boundary condition and a point within it, which isπ(γ) for a curve γ. Let us now consider variations of γ lying in F . Since the


form αL ⊕ (−αL) is closed on the boundary condition it can locally be writtenas df for a function f on the boundary condition. Letting π be the projectionmap (3.1) so we can write

δS(δγ) =

∫I

(∂L

∂q− d

dt

∂L

∂v) · δγdt+ π∗df(δγ).

If we now defineS′ = S − π∗f

then we have

δS′(δγ) =

∫I

(∂L

∂q− d

dt

∂L

∂v) · δγdt.

We have thus modified the action so that the boundary term still disappears.Now if the boundary condition is a simply connected manifold we can find a sin-gle function f that works on the entire boundary condition, making the processrelatively easy. In the general case we only have αL ⊕ (−αL) = df locally andseveral different functions will have to be chosen, depending on the path whichwe vary. Though this general case is certainly more laborous it is sufficient fornumerical applications, where critical points of functions are relatively easy tofind.

The reason why we define a boundary condition to be a fibration instead ofa single Lagrangian submanifold is that we want to know what variations areallowed for every single curve, not for just those whose boundary values happento lie on a single submanifold.

Example 3.1.5. Consider as an example the fibration by the manifolds π−1(x)×π−1(y) for x, y ∈ M . It is easily checked that αL ⊕ (−αL) disappears on thesesubmanifolds. Since their dimension is equal to that of TM it is a Lagrangianfibration and hence a boundary condition. Clearly it is the boundary conditioncorresponding to variations with end-points fixed.

We can now reformulate the Lagrangian mechanics in terms of Lagrangiansubmanifolds of a symplectic manifold. This will be the general formulation wehave been aiming for. From the configuration space M and Lagrangian functionL we derive

• A space-time category: objects are subsets of R, morphisms are disjointunions of intervals as in the cobordism category.

• A symplectic category: its objects are symplectic manifolds (M,ω), amorphism M → M ′ is a submanifold of M ×M ′ which is the graph ofa (possible only locally defined) symplectomorphism. The identity mor-phism is the diagonal. If one has two morphisms L1 ⊂ M1 × M2 andL2 ⊂M2 ×M3 then

L2 L1 =

=

(x, z) ∈M1 ×M3

∣∣∃y ∈M2 : (x, y) ∈ L1 ∧ (y, z) ∈ L2

.

(3.2)


This manifold is clearly the graph of the composition of the symplecto-morphisms yielding L1 and L2.

• A functor from the space-time to the symplectic category. It assigns to apoint the symplectic manifold (TM,±ωL), where the sign is given by theorientation of the point, and takes the union of points to the product. Aninterval [t0, t1] is then taken to the graph of the flow along XE for timet1 − t0 and the union of intervals to the product.

Remark. In the case of this last functor there is a technicality in the following:the flow belonging to XE may not be defined for all time. Since the set U onwhich the flow is defined for a certain time is open in TM we can still take thegraph of the flow over U . This will still net a Lagrangian submanifold unless Uis empty.

The functor given in the third item clearly defines the dynamics of the systemcompletely since it gives us the mappings φt. It is possible to reconstruct theLagrangian function from this formulation as well [10], though this process issomewhat involved and won’t be detailed here. As a result the above formulationis actually equivalent to the original Lagrangian formulation.

3.1.2 The general case

The above considerations were made for strictly convex Lagrangians. Althoughthey cover a large portion of physical applications of Lagrangian mechanicsthere are important exeptions to this class. Much of the above work can stillbe carried through in this case, though several relaxations in the symplecticstructures involved must be made. We will first develope the general theory, anoutline of which can be found in [11]. We will then apply this to two systemswhere this degeneracy is clearly not important to demonstrate the reasoning.

First note that the forms αL and ωL can still be defined on the tangentbundle. In this case we define them by their coordinate expressions:

αL :=∑i

∂L

∂vidqi

ωL := −dαL =∑i,j

(∂2L

∂qj∂vidqi ∧ dqj +

∂2L

∂vi∂vjdqi ∧ dvj

).

That these are well defined forms is again easily checked. It now not necessarilytrue that the form ωL is a symplectic form; it will be symplectic if and only ifthe Lagrangian is regular. Since we are expressedly working with non-regularLagrangians also we must work with a weaker concept than being symplectic.

Definition 3.1.6. A pre-symplectic manifold is a pair (M,ω) where M is amanifold and ω a closed 2 form whose kernel is a smooth sub-vectorbundle.

Remark. The demand that the kernel is a sub-vectorbundle is equivalent to themap v 7→ ω(v, ·) having constant rank in finite dimensions. In infinite dimensions


this is clearly not a proper definition so we work with the more general definitiongiven here.

We define isotropic, co-isotropic and Lagrangian submanifolds for of a pre-symplectic manifold in exactly the same way as for a symplectic one. It is toaccomodate these definitions that we require a pre-symplectic form to have con-stant rank. If we did not the maximal dimension of an isotropic manifold couldvary, and the existence of Lagrangian submanifolds would become a tricky busi-ness at best.

Keeping this in mind we now look at Lagrangian systems such that theinduced structure (TM,ωL) is pre-symplectic. We now wish to define the ana-logues of the Lagrangian submanifolds L[t0,t1] ⊂ TM × TM . In this case theywill not be the graph of a symplectomorphism anymore.

To see what happens define the energy functional by the coordinate expres-sion we got previously

E(q, v) = v(L)(q, v)− L(q, v) =∑i

vi∂L

∂vi(q, v)− L(q, v).

Look now at tangent vector fields XE satisfying

ωL(XE , ·) = dE.

Note that for a pre-symplectic form a function f may not always induce a vectorfield Xf since df may not be in the image of ω, and if it does this vector fieldis generally not unique. In the case of the energy function we can make thefollowing calculation in coordinates; note that Einstein summation conventionis used.

dE =

(vi

∂2L

∂qj∂vi− ∂L

∂qj

)dqj + vi

∂2L

∂vj∂vidvj .

Using these coordinate expressions let∑i

(vi ∂∂qi + ai ∂

∂vi

)be a vector field such

that its flow generates solutions of the Euler-Lagrange equation (these do notnecessarily exist); it can be shown that this vector field is a valid XE and hencethe graph of its flow is an isotropic submanifold of (TM ×TM,ωL⊕ (−ωL)) (itwill in general not be Lagrangian since isotropic submanifolds can have greaterdimension for pre-symplectic forms). Clearly by adding to this XE a tangentvector field from the kernel of ωL we get another valid XE with a different flow.Are we then to take the union over the graphs of all such flows, assuming thatany exist at all?

Clearly the above situation is unsatisfactory. We will have to find anotherdefinition of the L[t0,t1]. To do so we go back to the variation of the action. Letπ be the projection from formula 3.1 then we have

δS(δγ) = EL(δγ) + π∗αL(δγ).


Here EL is the bulk term of the variation giving the Euler-Lagrange equations.Note that the δ on the left hand side of this equation is the exterior differential.Take it another time on both sides to get

dEL = π∗ωL.

This equation prompts the following

Definition 3.1.7. L[t0,t1] := π(ker(EL)).

Remark. In the case of strictly convex lagrangians discussed previously thisdefinition is equivalent to that in terms of graphs.

The above consideration gives that the L[t0,t1] are isotropic. In general theywill not be Lagrangian, though it turns out that in our applications they usuallywill be.

Example 3.1.8. As is well known from the theory of Riemannian manifoldsgeodesics are the admissable sections for the free particle on a Riemannianmanifold: L = 1

2‖v‖2. The Euler-Lagrange equations are

v = γ′

∇vv = 0.

Here ∇ is the Levi-Civita connection, from which it is immediately clear that‖γ′‖ = constant. Geodesics are also critical points for the length of a section:L′ = ‖v‖, which has as Euler-Lagrange equations

v = γ′

∇vv

‖v‖= 0.

It is easy to show that the action S′ =∫L′dt is reparametrisation invariant, so

any reparametrisation is also a critical point for length. Conversely any criticalpoint for length can be parametrised with constant velocity and this yields ageodesic. By the above treatment this degeneracy is manifested in terms ofdegeneracy of ωL. To see exactly what happens split every tangent space usingthe canonical diffeomorphism TxM ' Sn−1 × R≥0 : v 7→ (ρ, v) = (‖v‖, v

‖v‖ ).

Using Einstein summation convention we then have

αL =gijv

j√gklvkvl

dqi = v · dq

which has the immediate consequence

ωL = dv · dq.

The kernel of this is a two dimensional distribution spanned by the vector fields∂∂ρ , v ·

∂∂q , making it a pre-symplectic form. This last vector field is in the


kernel since v · dv vanishes on the sphere. These two vector fields generatethe parametrisation invariance on the curves, exactly as expected. As for themanifolds L[t0,t1], they are all the same; being equal to

(q1, v1, ρ1, q2, v2, ρ2) ∈ TM × TM∣∣∃t : q2 = exp(tv), v2 =

exp(tv)′

‖ exp(tv)′‖

.

This is a submanifold of dimension Dim(TM) + 2. As previously noted the ker-nel of ωL is 2-dimensional, making the kernel of ωL ⊕ (−ωL) 4-dimensional.

The maximal dimension of an isotropic submanifold is thus 2 Dim(TM)+42 =

Dim(TM) + 2. The L[to,t1] are thus actually Lagrangian.

Example 3.1.9. Our second and final example will be that of non-degeneratefirst order Lagrangian mechanics. For our setup we have a manifold M equippedwith a 1-form α. A first order Lagrangian system is then such a manifold withthe Lagrangian

L(q, v) = α(v)− V (q).

The Euler-Lagrange equations for this system are

−dα(γ′, ·) = dV.

This motivates the obvious terminology that the system is non-degenerate pre-cisely if ω = −dα is a symplectic form. In this case the system is actuallyHamiltonian with flow XV . On the other hand the hessian of L is clearlyzero, so the Lagrangian formulation of this system is degenerate. If we haveπ : TM → M for the tangent bundle, then clearly αL = π∗α and ωL = π∗ω.These forms are horizontal and ωL has constant rank equal to Dim(M), mak-ing it a pre-symplectic form. Clearly the admissable curves are precisely thosecurves (γ, γ′) such that γ′ = XV , i.e. the lifts of the integral curves of XV . Ifwe let φt be the flow of XC then the dynamics gives the isotropic submanifold

L[t0,t1] =

(p0, v0, p1, v1) ∈ TM × TM∣∣vi = XV

∣∣pi, p1 = φt1−t0p0

.

This submanifold is not Lagrangian since its dimension is only Dim(M).This is related to the fact that the Euler-Lagrange equations are first order andhence the initial velocity is not a part of the initial data of a solution. Thereforeit is superfluous to formulate the dynamics of this system on the tangent bundle.Instead we can formulate the dynamics in terms of Lagrangian submanifolds ofM ×M .

If we view the Lagrangian as a funtion on curves γ : I → M , instead ofcurves γ : I → TM , we get the Lagrangian submanifold

L[t0,t1] =

(q0, q1) ∈M ×M∣∣q1 = φt1−t0(q0)

.

This is Lagrangian with respect to the form ω ⊕ (−ω) since φt is a symplecto-morphism. It is interesting to note that the boundary term in the variation of


the action is given by α in this case as well; this is completely in line with thegeneral arguments above. That the symplectic structure relevant to the problemlies on M , not on TM , is a general feature of first order Lagrangian theories.We will see more examples of these in the sequel and in all cases we work on M .

Now we have seen the generalisation of the Lagrangian mechanics away fromonly strictly convex Lagrangians, and some examples motivating this generali-sation. We now finish with with defining the remaining structure that we alsotreated in the convex case. Firstly a boundary condition is defined preciselyas before, with exactly the same reasoning behind it. If we now take a fibrefrom a boundary condition and the manifold L[t0,t1] we can take the intersec-tion to obtain the manifold of boundary values of solutions obeying the specifiedboundary conditions.

We have now given the analogues of all the constructions from the strictlyconvex case, except formulating the dynamics as a functor. This can still bedone, except that a generalisation is necessary. In the previous case we used asa symplectic category the symplectic manifolds with the graphs of symplecto-morphisms as morphisms. Since in the general case the dynamics is still givenby an isotropic submanifold, but not necessarily by a graph, we must allow moremorphisms in our category. There are multiple ways of doing this. One way isto allow just enough isotropic submanifolds that the composition law (3.2) stillworks out. An easier way is to allow so many more morphisms that compositioncan be defined in a different way.

It is the second approach we will take and this leads to the Wehrheim-Woodward category [12]. As a preliminary we need a definition.

Definition 3.1.10. A generalised Lagrangian correspondence between pre-symplectic manifolds (M,ω) and (M ′, ω′) consists of two data

• A sequence of pre-symplectic manifolds (Ni, ωi), 0 ≤ i ≤ k such that(N0, ω0) = (M,ω) and (Nk, ωk) = (M ′, ω′).

• A sequence of Lagrangian submanifolds Li ⊂ (Ni ×Ni+1, ωi ⊕ (−ωi+1)),0 ≤ i ≤ k.

We would like to make a category out of the symplectic manifolds by usingthe generalised Lagrangian correspondences as morphisms. To do so there is adegeneracy we need to get rid of however. Let

(L0, ..., Li, Li+1, ..., Lk)

be a generalised Lagrangian correspondence such that the composition Li+1 Liaccording to (3.2) is well-defined. We put the equivalence relation ∼ on thegeneralised Lagrangian correspondences such that

(L0, ..., Li, Li+1, ..., Lk) ∼ (L0, ..., Li+1 Li, ..., Lk).

We can now define the Wehrheim-Woodward category.

3.2. CLASSICAL FIELD THEORIES 41

Definition 3.1.11. The Wehrheim-Woodward category is the category whichhas pre-symplectic manifolds as its objects. It morphisms are generalised La-grangian correspondences under the above equivalence relation ∼. The identityid : M →M is given by the diagonal in M ×M .

Remark. In the above we work with generalised Lagrangian correspondencessince in our applications usually the L[t0,t1] are Lagrangian. To tackle the generalcase not that all the above can be done just as well for isotropic manifolds.

It is clear that the above defines a category. There is a natural functor fromthe category of symplectic manifolds with graphs of Lagrangian submanifoldsto the Wehrheim-Woordward category given by the identity. It is the presenceof the equivalence ∼ that gives the equation F (L L′) = F (L) F (L′) requiredfor a functor. The existence of this functor makes the categorial formulation ofgeneral classical mechanics obvious. For a particle with configuration space Mand Lagrangian L we specify the following:

• A space-time category: objects are subsets of R, morphisms are disjointunions of intervals as in the cobordism category.

• A functor from the space-time to the Wehrheim-Woodward category (ormore generally its isotropic generalisation). It assigns to a point the (pre-)symplectic manifold (TM,±ωL), where the sign is given by the orienta-tion of the point, and takes the union of points to the product. An interval[t0, t1] is then taken to L[t0,t1] := π(ker(EL)).

It is clear that in the case of strictly convex Lagrangians this functorial formu-lation is equivalent to the one given previously.

Remark. In the case of non-degenerate first order mechanics we can chooseanother functor. We send a point t ⊂ R to the symplectic manifold (M,±ω),and the union of points to the product. An interval [t0, t1] is then taken toL[t0,t1] as defined in the example. This functor is clearly preferable to theone from the more general description above since all the spaces involved areproperly symplectic, and all the relevant subspaces are Lagrangian instead ofmerely isotropic.

3.2 Classical Field theories

Now that we have given the functorial formulation of classical mechanics wedirectly generalise this to arbitrary space-time categories according to [13].

Definition 3.2.1. A classical field theory is given by

• A d-dimensional space-time category: a subcategory of the d-dimensionalco-bordism category (see appendix) of oriented manifolds.


• A functor F from the space-time to the Wehrheim-Woodward category(or more generally its isotropic generalisation). It is required to satisfythe following axioms:

– F (∅) = 0.

– F (M tM ′) = F (M)×F (M ′) where M and M ′ are either objects ormorphisms.

– If F (M) = (N,ω) then F (M−) = (N,−ω) where the superscriptminus indicates change of orientation.

– An orientation preserving diffeomorphism f : M →M ′ should lift toa symplectomorphism F (f) : F (M)→ F (M ′).

Clearly the functors defined for classical mechanics satisfy the above axioms.It follows that classical mechanics is a 1-dimensional field theory. The reasonwhy we generalise the functorial description of classical mechanics is that thisis the most bare-bones description that captures the essential structure. Theobjects from the space-time category are the boundaries of the field domain;the functor takes the boundary to the space of possible boundary fields. A mor-phism is a field domain, and this is taken to the Lagrangian (or more generallyisotropic) submanifold of boundary fields that are boundaries of acceptable so-lutions. In classical mechanics it was only the boundary values of admissablecurves that we were interested in, and in general field theories this situationturns out to be no different. That this correspondence of boundary fields tothe boundary of the domain is a functor ensures that solutions glue properly.It is far more difficult to make general statements in the case of arbitrary fieldtheories. We will still see that all our examples come from a Lagrangian definedon a space of fields much as in the case of classical mechanics. The similaritiesin all these different cases was already described by V. Fock as attested in [11].The arguments involved will have to be taylored to the specific situation at handhowever.

3.2.1 scalar field theories

To ease into the discussion we first consider the standard scalar field theoriesfrom physics.

Example 3.2.2. As a start we consider the free scalar field on a compactorientable Riemannian n-manifold M with boundary. We want the manifold tohave positive-definite metric so that the Laplace operator will be elliptic. Wecan use all the known results from Hodge theory, allowing for a nice statement ofthe theory. The theory is defined in terms of a space of fields with a Lagrangianon it, analogous to our previous constructions, and all those that will follow.

The space of fields is

FM := C∞(M).


On this space we use the Lagrangian

S[φ] :=

∫M

1

2dφ ∧ ?dφ.

We calculate the variation

δS(δφ, δφn) =

∫M

dδφ ∧ ?dφ

= −∫M

δφd ? dφ+

∫∂M

?dφδφ

Now the bulk term EL is zero for all δφ precisely if

0 = ?d ? dφ = −∆φ.

Furthermore remember that in the case of classical mechanics the boundary termwas given by the canonical 1-form α. We can define a similar structure in thiscase. The boundary term can be viewed as a 1-form on the space F∂M×F∂M , thespace of Cauchy data for the boundary value problem. The first function givesthe boundary value, and the second the normal derivative on the boundary.Letting a subsript n denote the normal derivative the boundary term of thevariation is given by:

αφ,φn(δφ, δφn) :=

∫∂M

?dφ · δφ =

∫∂M

φn · δφdV.

We can only define this 1-form if we also know the normal derivative of theboundary field, as we see from the explicit formula for it. Note the analogy tothe case of mechanics, where on the boundary we worked with two copies ofTM . Using α we can define a symplectic form on F∂M × F∂M :

ω((δφ, δφn), (δψ, δψn)) := −dα((δφ, δφn), (δψ, δψn))

=

∫∂M

(?dδψ · δφ− ?dδφ · δψ)

=

∫∂M

(δψn · δφ− δφn · δψ) dV

That the above is a symplectic form is obvious. It is interesting to note that theabove symplectic form is a special case of the following general structure: givenan inner-product space (V, 〈·, ·〉) we get the symplectic form ω on V ⊕ V givenby ω(v + w, v′ + w′) = 〈v, w′〉 − 〈v′, w〉. This is precisely the symplectic formused for mechanics on Rn, making this a generalisation to infinite dimensions.Taking the exterior derivative of the formula for δS we again get that ω vanisheson the kernel of EL analogous to the case of mechanics.

Using the map

π : C∞(M)→ C∞(∂M)× C∞(∂M) : φ 7→ (φ∣∣∂M

, φn∣∣∂M

)


we can define the isotropic submanifold

LM := π (ker(EL)) .

We now use the hodge theory to show that this apriori immersed submanifold isactually embedded and Lagrangian. Notice that LM is the subspace of boundaryvalues of φ ∈ C∞(M) satisfying ∆φ = 0. We begin by showing that LM is thegraph of a function ψ 7→ ψn. This function is called the Dirichlet-to-Neumannfunction and it is obtained as follows: we solve the Euler-Lagrange equationsfor φ with φ

∣∣∂M

= ψ for given ψ ∈ C∞(∂M) and then take ψ 7→ φn∣∣∂M

. Whatwe need to show is that the boundary problem

∆φ = 0

φ∣∣∂M

= ψ

has a unique solution. This follows from page 137 of [14]. This gives thatLM is an embedded isotropic submanifold. It is Lagrangian since it has anisotropic complement, namely the subspace 0×F(M). Similarly to the case ofmechanics we can define the concept of a boundary condition to be a Lagrangianfibration of the space of Cauchy data. Solutions lying in a specific fibre willthen lie in the intersection of this fibre with LM . As an example we can usethe foliation by x×C∞(∂M), which is the boundary condition correspondingto the Dirichlet problem. The Von Neumann problem is given by foliating asC∞(∂M) × x. We can formulate this functorially by taking our morphismsto be the compact d-dimensional Riemannian manifolds with boundary, andobjects their boundary (as in the cobordism category). We then have a functortaking ∂M to C∞(∂M) × C∞(∂M) and a morphism M is taken to LM . Itcan be shown that this functor satisfies all the axioms from the definition of afield theory above. This shows that scalar field theory is indeed a field theoryaccording to our definition.

Remark. Compactness of the manifold in question is certainly important, as isthe presence of a boundary. If no boundary is present the boundary problembecomes vacuous and the field theory becomes trivial. Compactness is requiredto keep the problem relatively simple. Otherwise it would be necessary to keeptrack of integrability conditions to ensure good behaviour of the solutions, whichwould take us too far afield. To see that integrability conditions factor in in thenon-compact case solve the Laplace operator with Dirichlet boundary conditionsin the upper half-plane. It will have a unique solution only if integrabilityconditions (such as boundedness or being L2) are imposed.

Example 3.2.3. The above example was essentially the problem of sourcefree electrostatics in an arbitrary compact orientable manifold with boundary.If we want to tackle the problem time evolution of the fields we would need tochange to a d+1-dimensional Minkovski manifold with signature (−1,+d). Thesituation here will be somewhat less satisfying, owing to the greater number of


boundary conditions that must be specified. This is because the wave equationis essentially about time evolution of a state. We use as the space-times d+ 1-dimensional Lorentzian manifolds of the form Ω := [t, t′] ×M where M is a d-dimensional oriented Riemannian manifold. We look at the minkovskian scalarfield theory. The space of fields is

C∞f (Ω) :

smooth functions on Ω that have some time-independent boundary value f on∂M for all time. On this we have the Lagrangian

S[φ] :=

∫Ω

1

2dφ ∧ ?dφ.

It has the same form as above, but since the metric is Minkovski the star operatoris different. The variation of the fields yields the analogous formula

δS(δφ, δφn) =

∫Ω

dδφ ∧ ?dφ

= −∫

Ω

δφd ? dφ+

∫∂Ω

?dφδφ

A field is in the kernel of EL if and only if

?d ? dφ = −φ = 0,

which means it satisfies the wave equation without sources. This equation willhave a unique solution if we specify φ

∣∣t×M and φ′

∣∣t×M (this essentially

specifies the field and its velocity at time t, analogous to mechanics, see [15]for the details). Here the prime denotes derivation with respect to t. Since thevariations δφ vanish on ∂M by the boundary condition the boundary term isgiven by a 1-form on the space of Cauchy data

C := TC∞f (t ×M)× TC∞f (t′ ×M) ' C∞f (M)2 × C∞0 (M)2

given by

α(δφt, δφ′t, δφt′ , δφ

′t′) :=

∫M

?dφt · δφt −∫M

?dφt′ · δφt′

=

∫M

(φ′t · δφt − φ′t′ · δφt′) dV.

We can turn C into a symplectic manifold by defining

ω ((δφt, δφ′t, δφt′ , δφ

′t′), (δψt, δψ

′t, δψt′ , δψ

′t′))

:= −dα ((δφt, δφ′t, δφt′ , δφ

′t′), (δψt, δψ

′t, δψt′ , δψ

′t′))

=

∫M

(δφt · δψ′t − δφ′t · δψt + δφ′t′ · δψt′ − δψ′t′ · δφt′) dV


which can be checked to be non-degenerate. Analogous to our previous caseswe can define

π : C∞(Ω)→ C : φ 7→(φ∣∣t×M , φ

′∣∣t×M , φ

∣∣t′×M , φ

′∣∣t′×M

).

We then get that the manifold

LΩ := π(ker(EL))

is the graph of a symplectomorphism (since the boundary value problem has aunique solution) and hence a Lagrangian submanifold of C by (C.2.4). We canthen define a boundary condition to be a Lagrangian fibration of C. In termsof functors we now have a space-time category with morphisms given by the Ωand objects by (t′ ×M) × (t ×M)− where the superscript minus denotesreversal of orientation. The functor then sends an object to its correspondingC, and Ω to the Lagrangian submanifold LΩ. Once again this functor satisfiesthe necessary axioms so this example is also a field theory according to ourdefinition.

Remark. The situation here is somewhat analogous to classical mechanics, wherethe initial position and velocity of a particle must be specified. In order to dothis in a consistent way we need to specify a time-direction on our manifold,which we have done by just considering product manifolds (more generality canbe obtained by looking at general Lorentzian manifolds with a time-like vectorfield and a space-like fibration). This sacrifices the covariance of the theory ina rather unpleasant manner that was not necessary in the Riemannian case.

3.2.2 First order field theories

Before we move on to gauge theories in the next section first we say somethingabout first order field theories. Chern-Simons theory is a first order field theoryand by giving an easier example most of the constructions that will be used inChern-Simons theory will be more familiar. First off we give the definition:

Definition 3.2.4. A first order field theory is a field theory coming from aLagrangian that is linear in the derivatives of the fields involved.

This is an obvious generalisation of first order classical mechanics to arbitraryfield theories coming from a Lagrangian function. The reader may recall that thefirst order mechanics could still be formulated in terms of a symplectic structureeven though the pre-symplectic form on the tangent bundle was degenerate.Something similar happens in the case of first order field theories: the space ofboundary data has to be chosen somewhat differently for everything to workout.

Example 3.2.5. As an example before moving on to general gauge theorieswe work through first order d-dimensional Riemannian scalar field theory. As


a physical theory it is equivalent to what we’ve done in the previous section.The symplectic structure is described in a different fashion that shows clearlythe parallels with Chern-Simons theory, making this an excellent example. TheMinkovskian scalar field theory can also be subjected to this treatment, butsince boundary conditions are chosen somewhat differently this example is lessappropriate for our purposes and left as an exercise to the interested reader.In the first-order scalar field theory the space-time is again a compact orientedd-dimensional Riemannian manifold M with boundary. The space of fields is

FM := Ω0(M)⊕ Ωd−1(M).

Note that Ωd−1(M) ' Ω1(M) via the Hodge dual, so the space of fields isessentially the pair of fields and their derivatives considered as separate entities.We denote a function by φ and a form by π. We define the following Lagrangianon the space of fields:

S[φ, π] :=

∫M

π ∧ dφ− 1

2π ∧ ?π.

We find the variation of the action

δS =

∫M

δπ ∧ dφ+ π ∧ dδφ− δπ ∧ ?π

=

∫M

(δπ ∧ (dφ− ?π)− (−1)d−1dπ · δφ

)+ (−1)d−1

∫∂M

πδφ.

The EL term vanishes if and only if

π = (−1)d−1 ? dφ

0 = ∆φ.

This establishes equivalence of the two theories. Now we formulate the symplec-tic structure. Note that in the case of mechanics the structure on the tangentbundle TM was degenerate, but there was a proper symplectic structure on thecoordinate space M . Analogous to this case we define the symplectic structureon the space of fields, not its square. To do this we note that the boundaryterm from the variation of the action comes from a 1-form on F∂M

α(δφ, δπ) := (−1)d−1

∫∂M

πδφ.

This defines a symplectic form

ω((δφ, δπ), (δφ′, δπ′)) := −dα((δφ, δπ), (δφ′, δπ′))

= (−1)d∫∂M

δπδφ′ − δφδπ′.

We can now define the Lagrangian submanifold LM to be the graph of thefunction

φ∣∣∂M7→ φ 7→ π

∣∣∂M


which solves the EL equations with boundary condition φ∣∣∂M

and spits out

π∣∣∂M

. That this is a function defined on all of C∞(∂M) = Ω0(∂M) followsfrom the existence of unique solutions to the laplace equation as was discussedpreviously. We also get a functorial formulation in the by now familiar way.

The crucial point in this example, which we also saw in the first order clas-sical mechanics, is that in this case we still have a symplectic structure, butthat it lies directly on the space of fields, not its tangent bundle (which in thecase of functions on a manifold is just the square of the space of fields). Inthe Chern-Simons theory we will see this again, although in a somewhat moregeneral fashion. The Lorentzian scalar field theory can be turned into a firstorder theory in much the same way as was done in the previous example. Oneneeds to notice that specifying the field π on t ×M is equivalent to givingφ′∣∣t×M , so that specifying φ and π gives the correct boundary conditions for

the wave equation. The details are left to the reader.

3.3 Gauge Theories

Now that we have developed a feeling for the constructions one encounters infield theories we can step up the pace and move on to gauge theories. Thegeneral concept of a gauge theory is somewhat more abstract, but the ideaoriginally came from vector field theories akin to those we discussed above wherethe Lagrangians are invariant under a Lie group action on the fields. Thearchetipal example is classical electrodynamics, which we treat first. Then wereformulate it in the language of general gauge theories (where we will finally useour knowledge of connections from the previous chapter), and finally give thegeneral definition. Finally we close with Yang-Mills theory as another example.

In source-free classical electrodynamics one works with a compact orientable4-dimensional space-time manifold M which is Lorentzian in signature. On thiswe have a space of fields

FM := Ω2(M).

The equations of motion giving the physically admissable fields are

dF = 0

δF = 0.

The top equation of motion means that locally we can write F = dA. Thebottom equation then becomes

−∂dA = 0.

Owing to the Lorentzian nature of the manifold this equation shows wave-like behaviour. Just like the scalar Lorentzian field theory we assume thatΩ = [t0, t1]×M for a compact Riemannian manifold M so we can accomodatethe inherent evolution behaviour in these equations. In accordance with this

3.3. GAUGE THEORIES 49

splitting we let d, δ, ? denote the operators on M from now on. First we exam-ine the kind of boundary conditions we must pose to obtain a well formulatedtheory. To do so write

−F = dt ∧ E + ?B,

defining the electric and magnetic fields. In terms of E,B the equations ofmotion are

δE = 0 δB = 0

∂E

∂t= ?dB

∂B

∂t= −?dE.

Let ν be the dual form to the unit normal on ∂M , then natural boundaryconditions for the problem are given by

ν ∧ E = 0 δE = 0 E∣∣t0×M = E

ν ∧ ?B = 0 δB = 0 B∣∣t0×M = B,

for which we get a unique set of fields on Ω (once more the details are in [15]).The two rightmost conditions specify the initial fields, and the leftmost onesspecify that E is perpendicular to the boundary and B tangent to it. Themiddle two conditions are obviously necessary for consistency. These conditionscorrespond to source-free electrodynamics in a region bounded by a perfectconductor. Let ιt : M ' t ×M ⊂ Ω denote inclusion at time t; in terms ofthe original field tensor F these boundary conditions become

ν ∧ F = 0 F∣∣t0×M

= F

dι∗0 ? F = 0 dι∗0F = 0,

where in this instance restriction to t0 does not mean pull back along the in-clusion; the full form has to be specified since the information about E is inthe dt component of F . We want to formulate the solvability of the equationsof motion in the language of a first order lagrangian field theory. To do so weassume that H1(M) = H2(M) = 0 so that also H(Ω) = H2(Ω) = 0 since Ωretracts onto M . We can then satisfy the first equation of motion by writingF = dA and it turns out that the potential A is the natural field. There is acaveat here in that the potential is not unique. Let us for the moment ignorethis difficulty. For a second order Lagrangian formulation we would then use asthe space of fields

F(Ω) :=A ∈ Ω1(Ω) : ν ∧ dA = 0

,

and for our Lagrangian we would have

S[A] :=

∫Ω

1

2F ∧ ?F.


Calculating the variation to A of this action yields δF = 0, the second equationof motion. We want to have a first order fields theory however, so as our spaceof fields we use instead

F(Ω) :=

(A,G) ∈ Ω1(Ω)× Ω2(Ω) : (ν ∧ dA = 0) ∧ (ν ∧ ?G = 0).

To the action we add a term to fix the field G:

S[A,G] :=

∫Ω

(F ∧G− 1

2G ∧ ?G).

The variation of this action is

δS[A,B] :=

∫Ω

(δG ∧ (F − ?G) + δA ∧ dG) +

∫∂Ω

δA ∧G.

The equations of motion are then

F = ?G dG = 0,

which is equivalent to the remaining equation of motion δF = 0. We havedι∗tA = ι∗tB and ι∗tG = ?ι∗tE. The final step would be to define our space ofCauchy data as those elements of ι∗t1F(Ω) t ι∗t0F(Ω)− with dG = 0, since thenspecifying A,G at t0 allows us to solve for E,B on Ω. This is where we run intotrouble because the potential cannot be chosen uniquely. Specifying ι∗t0A, ι

∗t0G

determines G uniquely but A is only determined up to a differential df ∈ Ω1(Ω)with ι∗t0df = 0. We could try to fix this problem by defining an equivalence re-lation on our choice of F(Ω); unfortunately this approach has the problem thatthe variation of the action does not disappear if δA = df . Instead we solve theproblem by formulating the theory in terms of connections on principal bundles,which turns out to be very natural. It is also this approach that generalises toarbitrary field theories.

We formulate the theory on the trivial bundle P = Ω × R, where R hasthe standard additive structure. It follows that ∂

∂t is a left-invariant vectorfield, and the Maurer-Cartan form θ on R is simply dt. Since R is abelian theadjoint representation is trivial and a function is R-equivariant precisely if itis constant on the fibres. The gauge transformations are thus characterised bysmooth functions on f ∈ C∞(M). Under these transformations a connectionA transforms as A · f = A + df , specifically connections are invariant undertranslations by a constant. It thus follows that the space AP of connections onP consists of those 1-forms A such that

A(∂

∂t) = 1

R∗tA = A.

The projection π : P → Ω thus gives a canonical isomorphism Ω1(M) → AP :A 7→ π∗A+ dt. This isomorphism induces an isomorphism between the 2-formson M and the space of curvatures by

dA 7→ π∗dA.


We can thus view the tensors F as curvatures on P , and the potentials as theircorresponding connections. Similarly the fields G are identified with horizontalG-equivariant 2-forms on P : Ω2

H,G(P ) (a priory they are not curvatures yet,since this only follows from the equation dG = 0). In this formulation of thetheory the gauge freedom in choosing our potential A is expressed by sayingthat A · f = A+ df but F · f = F , together with its inverse: if FA = FA′ thenA′ = A · f for some f ∈ C∞(Ω). For the G field we also have G · f = G.

These considerations lead us to define the space of fields

F(Ω) :=

(A,G) ∈ AP × Ω2H,G(P ) : (ν ∧ dA = 0) ∧ (ινG = 0)

.

Here ιnu denotes contraction with the normal to ∂M . This condition arrisesfrom the fact that on M we have ινG = 0⇔ ν ∧ ?G = 0. On the space of fieldswe use the action

S[A,G] :=

∫Ω

(s∗F ∧ s∗G− 1

2s∗G ∧ ?s∗G).

Since bopth F,G transform trivially under gauge transformations this is inde-pendent of s. It is clear that this is essentially the action we used before. It hasvariation

δS[A,B] :=

∫Ω

(s∗δG ∧ (s∗F − ?s∗G) + δs∗A ∧ s∗dG) +

∫∂Ω

s∗δA ∧ s∗G.

Notice that this is also independent of s since the tangent vectors all transformaccording to Ad. Furthermore in the boundary integral the contribution of[t0, t1] × ∂M vanishes because δA annihilates ∂M by ν ∧ A = 0. We get theEuler-Lagrange equations

F = ?G dG = 0,

just like before. We know that these equations have a unique solution in F(Ω)if we specify ι∗t0A, ι

∗t0G with the condition dι∗t0G = 0. This leads us to consider

the space of Cauchy data

C :=

(A,G) ∈ ι∗t1F(Ω)× ι∗t0F(Ω) : dG = 0.

We then get that ker(EL) is the graph of the invertible function ι∗t0F(Ω) →ι∗t1F(Ω) obtained by solving the initial conditions. We now construct the sym-plectic structure that makes this graph Lagrangian. For this we use the bound-ary term from δS, which is the 1-form

α(A,G)(δA, δG) :=

∫t1×M

s∗δA ∧ s∗G−∫t0×M

s∗δA ∧ s∗G.

We define the symplectic structure by ω = −dα as usual. We then get thatker(EL) is a Lagrangian submanifold by (C.2.4) since it is isotropic and the


graph of an invertible function.

We thus have a functor sending a compact 3-manifold M to F(M) definedabove, with the symplectic structure

ω((δA, δG)(δA′, δG′)) :=

∫M

(s∗δA ∧ s∗δG′ − s∗δA′ ∧ s∗δG).

It sends a morphism Ω to the Lagrangian submanifold ker(EL) of (F(M) ×F(M), ω ⊕ (−ω)).

Summarising we have a field theory in which the space of fields is the setof connections on a principal bundle over the space-time manifold. We have anaction by integrating pull-backs of the connection over the space-time that isindependent on the section chosen. For physical results one actually looks atthe set of equivalence classes of connections. It is this setup which gives us agauge theory.

Definition 3.3.1. A gauge theory is a field theory where the space of fields isrequired to be AP /GP where P is a principal bundle, AP its set of connectionsand GP its gauge group. We have an action on AP that is of the form

S[A] =

∫M

f(s∗A),

where f is some function on Ω1(M, g). We demand that this action is indepen-dent of the section s chosen (possibly by making the action take values in a linebundle), but it needs not descend to an action on the space of fields AP /GP .

Remark. The action is not required to be invariant under gauge-transformations,but the demand that it takes values in a line-bundle whose trivialisations dependon the chosen gauge means that a gauge transformation takes critical points tocritical points. As a result the Euler-Lagrange equations do descend to the spaceof fields, and a field theory can be formulated in much the same way as we havedone before.

Remark. In regard tot the above definition it should be noted that general rel-ativity is often considered to be a gauge theory as well, since any two isometricmanifolds are considered equivalent. It certainly is true that the theory has aninitial value formulation which interprets the Einstein equations as evolutionequations of a metric on a Riemannian 3-manifold ( [16]). This would makethe theory a field theory in our sense of the word if we assume the time evo-lution exists for a finite time. The space of fields in this case is not a spaceof connections, but the space of metric tensors on the 3-manifold. If we wantto make the equations first-order we could independently vary the Levi-Civitaconnection as well by adding the appropriate fixing term to the Einstein-Hilbertaction. In this light a more general definition of gauge theory is any field theoryin which the action is invariant under some Lie group action. Such generalityis not necessary for our purposes however, so we stick to the definition above.


Before we finish this chapter we quickly make mention of Yang-Mills theory.This is a set of gauge theories containing electrodynamics as a specific example.Their application to the standard model of physics makes them of tremendousinterest, and though we will not go into the details the theory is worth men-tioning. To make the formulation of the theory easier we first introduce a newconstruction. Let f : M → V be a function taking values in a G-representationwhere G is a Lie group and let ω be a g-valued 1-form. Then the derivativeof the G-representation induces a g-representation on V and we can define theinduced derivative dωf := (d+ ω)f . Here ωf(Xp) := ω(Xp) · f(p). We will usethis induced derivative several times in our formulation of Yang-Mills.

Example 3.3.2. In the Yang-Mills theory we work with a principal G-bundleπ : P →M where M is the space-time manifold in question. In the case of thesource-free theory we have a Lagrangian on the space AP /GP

S[A] :=

∫M

〈s∗F ∧ ?s∗F 〉.

This has the same form as the action for electrodynamics, but the curvatureF = dA + 1

2 [A ∧ A] is dependent on the connection in a more complicatedmanner. Also since F is no longer real-valued an Ad-invariant inner-product〈·, ·〉 has been used in the action (if G is simple the only choice is a multiple ofthe Killing form). The Euler-Lagrange equations are now

dA ? F = 0.

Here the G-action on g is given by the adjoint representation, which has theLie bracket as its derivative. These equations are non-linear in A, makinggeneral statements about solvability much more difficult. A treatment of whatis possible would take us too far afield, which is why only a passing mentionis made of Yang-Mills. It is obvious that the action descends to an action onthe equivalence classes of connections under gauge transformations just as inelectrodynamics. There is one striking difference in this instance and that isthat the field tensor F is no longer gauge-invariant: from the previous chapterwe have

F · a = Ada−1 F.

This degeneracy is related to the manner in which matter is incorporated inquantum field theories. For a thorough description of the process of minimalsubstitution the reader is referred to [17]. We state the results for a non-abelianscalar field theory. We still work with a principal bundle π : P → M . Ourmatter fields are functions φ : P 7→ V . Here V is a unitary left G-representationwith inner product (·, ·) and we demand φ(p · a) = a−1φ(p). The Yang-Millsfield is as before. The Lagrangian is now

S[A, φ] :=

∫M

1

2〈s∗F ∧ ?s∗F 〉 − ?V (φ) +

1

2(ds∗As

∗φ ∧ ?ds∗As∗φ).

A calculation will yield that this action is independent of the choice of s and alsothat S[A ·a, a−1 ·φ] = S[A, φ] so it is invariant under the gauge action. Here the


minimal substitution is sending d 7→ dA, which can also be done for the diracequation for example (see [17] for this also). From the Lagrangian we see thatF is gauge variant because the physical fields couple differently to the differentmatter fields in the orbits of G. Since we identify these fields we should alsoidentify the different field tensors. In the standard model the structure group isU(1)× SU(2)× SU(3) and we use the trivial bundle of this group. It is worthnoting that u(1) ' r, so the U(1) part of the structure group gives a complexifiedformulation of electrodynamics, which is equivalent to what we did above.

With this last example we have finished our treatment of general classicalfield theories. The reader should by now have a good feel for the kind of con-structions one expects to encounter when working with a field theory. All thatremains is to give a formulation of the Chern-Simons theory. This will take afair bit effort and the entire final chapter is dedicated to doing so. The generalform of the construction is entirely parallel to the one we have used over andover again in this chapter however.

Chapter 4

Chern-Simons theory

We now turn to a rigorous and in-depth formulation of the Chern-Simons theory.The final goal is to formulate it as a field theory in terms of the framework fromthe previous chapter. To do so we first define the Lagrangian used in the theorytogether with a motivation. In the beginning we will only define the theoryfor bundles with compact 1-connected structure groups. This is for a large partbecause all such bundles are trivialisable, and the action will depend on a choiceof trivialisatoin. We will show how this dependence can be absorbed by makingthe action take values in a line bundle. The boundary terms from this actionwill define a symplectic structure on the space of fields that we will then useto formulate the field theory using functors. Apart from thechnicalities specificto the theory this setup is much like what was repeatedly done in the previouschapter. We will then finish by extending this construction for toral structuregroups Tn by using the results we already had.

4.1 The Chern-Simons form and the action

Let Sk(g) denote the set of all symmetric k-linear functionals ⊗kg→ R. Define

Ik(G) :=f ∈ Sk(g) : f(AdgX, ...,AdgX) = f(X, ...,X) ∀g ∈ G,X ∈ g

;

i.e. it is the set of Ad-invariant symmetric k-linear functionals. If π : P → Mis a principal bundle with structure group G and we have a connection A on itwith curvature F we can define for all f ∈ Ik(G)

f(F )(X1, ..., X2k) :=1

2k

∑σ∈S2k

ε(σ)f(F (Xσ(1), Xσ(2)), ..., F (Xσ(2k−1), Xσ(2k))).

It is clear that f(F ) is a real valued 2k-form. Note the prefactor 12k that cancels

redundancy of permutations switching the arguments of a single F . One of themost beautiful theorems from differential geometry gives the existence of theWeil homomorphism [18].

55

56 CHAPTER 4. CHERN-SIMONS THEORY

Theorem 4.1.1 (The Weil homomorphism). Let π : P → M be a principalG-bundle over M . Let A be a connection on P with curvature F . For everyf ∈ Ik(G) there is a unique 2k-form ΛA in Ω2k(M) such that

π∗ΛA = f(F ).

The form ΛA is closed and its cohomology class [ΛA] is independent of theconnection A chosen. This gives us a mapping

wP : Ik(G)→ H2k(M) : f 7→ wP (f) := [ΛA].

The mapping wP is a homomorphism of algebras; linearity is obvious and

wP (fg) = wP (f) ∪ wP (g).

Here the product on symmetric functions is given by

fg(X1, ..., Xk+l) :=1

(k + l)!

∑σ∈Sk+l

f(Xσ(1), ..., Xσ(k))g(Xσ(k+1), ..., Xσ(k+l))

and the ∪ is the cup product in cohomology

[ω] ∪ [η] = [ω ∧ η]

The cohomology classes wP (f) are characteristic classes of the principalbundle P in the sense that

wφ∗P (f) = φ∗wP (f)

for a bundle isomorphism φ : P ′ → P covering φ : N → M . As a consequencetwo isomorphic bundles over the same manifold induce the same homomor-phism. Note that for groups SO(n) and SU(n) a principal bundle is isomorphicto the bundle of frames of an oriented metrised vector bundle (respectively realand complex). In this case it turns out that the above construction yields thePontryagin and Chern characteristic classes.

The connection of the above with Chern-Simons theory is as follows. As acorollary from the above theorem the forms f(F ) are closed. A more extensiveargument can be used to show that they are actually exact. In their paper [19]Chern and Simons gave a canonical way of exhibiting this exactness by findingforms Tf(F ) such that dTf(F ) = f(F ). In the case of 3-manifolds these Chern-Simons forms can be used to probe the geometry through the use of Chern-Simons theory. We consider a trivial principal G bundle π : P → M , togetherwith a non-degenerate Ad-invariant bilinear form 〈·, ·〉 on g. Since 〈·, ·〉 is anelement of I2(G) the form 〈F ∧ F 〉 is one of the characteristic forms from thetheorem. Its Chern-Simons form is a 3-form and it has specific formula

ch(A) = 〈F ∧A〉 − 1

6〈A ∧ [A ∧A]〉.

It is this form that we use to define the Chern-Simons action on a 3-manifold.

4.1. THE CHERN-SIMONS FORM AND THE ACTION 57

Definition 4.1.2. Let M be a compact 3-manifold, possibly with boundary,and π : P →M a G-bundle over M . Here we also require G to be compact and1- connected. For a global section s of P we define the Chern-Simons action onthe space AP by

Ss[A] :=

∫M

s∗ ch(A).

Remark. The Chern-Simons action is dependent on the trivialisation s andbelow we will calculate in what way. It is for this reason that we only definethe theory in trivial bundles; note that because G is assumed simply connectedevery G-bundle is trivialisable (see appendix). Furthermore we also want non-degeneracy of the bilinear form to avoid degeneracies in the variation of theaction that will make the Euler-Lagrange equations more complicated. Thereason for demanding compactness of G will become clear below. Connectednessis no great sacrifice, since the components of a Lie group are all diffeomorphic.It is worth noting that by requiring G to be simply connected we also get that itis semisimple (a compact Lie group is semisimple if and only if its fundamentalgroup is finite [20] [21]). Since the Ad-invariant bilinear forms on a simpleLie algebra are multiples of the killing form this gives a nice classification ofthe possible brackets we can use. It is for these reasons that both [3] and [4]demand that G be simply connected.

The rest of this chapter will be devoted to deriving properties of this actionand showing that it actually defines a field theory called the Chern-Simons the-ory. Before we do this a note is in order about why we chose the Chern-Simonsform of 〈F ∧F 〉 instead of a different one. A cynical answer would be that Wit-ten showed this theory yields invariants of 3-manifolds generalising the Jonespolynomial from knot theory [2], which certainly explains why people are stilllooking at the theory. Another thing that goes a little way towards explainingit is that the Chern-Simons theory is in a sense the simplest thing one can doon 3-manifolds using Chern-Simons forms. For simple Lie algebras the killingform is the only Ad-invariant bilinear function, so in that case 〈F ∧ F 〉 is thesimplest characteristic 4-form there is. The general theory is a straightforwardgeneralisation of this case.

Now that that’s out of the way we state some properties of ch(A) that wewill need below.

Proposition 4.1.3. The form ch(A) satisfies the following properties.

1. If ix : G→ P : g 7→ x · g then

i∗xch(A) = −1

6(〈i∗xθ ∧ [i∗xθ ∧ i∗xθ]〉)

2.

d (ch(A)) = 〈F ∧ F 〉.


3.R∗g ch(A) = ch(A).

4. If φ is a gauge transformation then

φ∗ ch(A) = ch(φ∗A).

5. Let a be the Ad-invariant map belonging to a gauge transformation then

ch(A · a) = ch(A) + d〈Ada−1 A ∧ a∗θ〉 − 1

6(〈a∗θ ∧ [a∗θ ∧ a∗θ]〉)

Proof. (1) is immediate since i∗xF = 0. (2) was already noted before, and adirect calculation is left to the reader. (3) follows because R∗gA = Adg−1A andR∗gF = Adg−1 F together with Ad-invariance of 〈·, ·〉. (4) is obvious. (5) is acalculation as follows.

ch(A · a) = ch(A) + 〈a∗θ ∧Ada−1 F 〉 − 1

2〈a∗θ ∧Ada−1 [A ∧A]〉

− 1

2〈Ada−1 A ∧ a∗[θ ∧ θ]〉 − 1

6〈a∗θ ∧ [a∗θ ∧ a∗θ]〉

= ch(A) + 〈a∗θ ∧Ada−1 dA〉+ 〈Ada−1 A ∧ a∗dθ〉

− 1

6〈a∗θ ∧ [a∗θ ∧ a∗θ]〉

= ch(A) + d〈Ada−1 A ∧ a∗θ〉 − 1

6〈a∗θ ∧ [a∗θ ∧ a∗θ]〉.

In the last step the identity

d〈Ada−1 A ∧ a∗θ〉 = 〈Ada−1 dA ∧ a∗θ〉+ 〈Ada−1 A ∧ a∗dθ〉

was used, whose proof is not altogether obvious. The critical step in the proofis establishing the equality

d (Ada−1 A) = Ada−1 dA− [Ada−1 A ∧ a∗θ]

which we will prove, leaving the rest to the reader. Let X,Y be tangent vectorsat a point p ∈ P and let X, Y := a∗θ(X), a∗θ(Y ) then

X (Ada−1) =d

dt

∣∣∣∣t=0

Ad(a exp(tX))−1 =d

dt

∣∣∣∣t=0

Adexp(tX)−1 Ada−1

= − ada∗θ(X) Ada−1

and similarly for Y . We use this on

d (Ada−1 A) (X,Y ) = X(Ada−1)A(Y )− Y (Ada−1)A(X) + Ada−1 dA(X,Y )

= + ada∗θ(Y ) Ada−1 A(X)− ada∗θ(X) Ada−1 A(Y )

+ Ada−1 dA(X,Y )

= Ada−1 dA(X,Y ) + [a∗θ(Y ),Ada−1 A(X)]

− [a∗θ(X),Ada−1 A(Y )]

= Ada−1 dA(X,Y )− [a∗θ ∧Ada−1 A](X,Y ).

4.2. CHERN-SIMONS AS A LAGRANGIAN THEORY 59

We will now use the properties from the above proposition to rigorously con-struct the action independently of sections s and find the equations of motion.

4.2 Chern-Simons as a Lagrangian theory

First we treat the case where M is a compact manifold without boundary, sincethen the theory is simpler. Furthermore the results from this case will furnishthe general theory.

4.2.1 The theory without boundary

In this case it follows easily from statement 5 of 4.1.3 that

exp (2πiSs·a[A]) = exp (2πiSs[A]) exp

(−2πi

1

6

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉).

This means we can use exp(2πiS) as an action provided that

1

6

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉

is an integer. The form 16 〈θ ∧ [θ ∧ θ]〉 is closed since any bi-invariant form is

closed [22], and hence defines an element of H3(G,R). If we demand that it isactually an integer class, i.e. that

∫c

16 〈θ∧ [θ∧ θ]〉 ∈ Z for all singular 3-chains c

(with integer coefficients), then the correction term will vanish. This is becauseM is a singular 3-chain (it can be triangulated) and thus

1

6

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉 =1

6

∫a(M)

〈θ ∧ [θ ∧ θ]〉 ∈ Z

since a(M) is a singular 3-chain in G.

Hypothesis 4.2.1. In what follows we will always assume that

1

6〈θ ∧ [θ ∧ θ]〉 ∈ H3(G,Z).

This has as a consequence that

exp (2πiSs·a[A]) = exp (2πiSs[A])

and thus thatCS := exp(2πiS) : AP /GP → S1

is a well defined action on the space of fields for manifolds without boundary.

Remark. One can wonder how restrictive the demand that 16 〈θ ∧ [θ ∧ θ]〉 be

integer is. Specifically can a bracket 〈·, ·〉 always be chosen such that the formis integer, and if so are there many different choices if we do not distinguish


constant multiples? The answer is that for semisimple Lie groups (which we aredealing with) such a bracket always exists and that it is not unique. In [20] it isproven that for G compact Hk(G) is isomorphic to the space of bi-invariantk-forms. These in turn are uniquely determined by their value on g, andan alternating k-linear function f on g determines a bi-invariant k-form pre-cisely if

∑i f(X1, ..., [Xi, X], ..., Xk) = 0,∀X,X1, ..., Xk. Define B([X,Y ], Z) :=

f(X,Y, Z), then

f([X,W ], Y, Z) + f(X, [Y,W ], Z) + f(X,Y, [Z,W ])

= B([[X,Y ],W ], Z)−B([X,Y ], [W,Z]).

In a semi-simple Lie algebra g = [g, g] so f is the restriction of a bi-invariant formif and only if B is an Ad-invariant bilinear form. Clearly B 7→ 1

6B([θ∧ θ]∧ θ) isthe bi-invariant extension of an Ad-invariant bilinear form B. We thus need toanswer the question whether we can choose an Ad-invariant inner product thatcorresponds to an integer class. Now the space of Ad-invariant bi-linear forms isisomorphic to Rn where n is the number of simple factors in the Lie algebra sinceany Ad-invariant bilinear form on a simple Lie algebra is a multiple of the Killingform. A form is then non-degenerate if it does not have any coordinates equal to0. The inclusion H3(G,Z) 7→ H3(G,R) embeds the integer cohomology into Rnas a lattice, and all we have to do is choose a lattice point with all coordinatesnon-vanishing. Clearly such a choice is not unique, even up to a constant.

We thus see that for closed 3-manifolds the Chern-Simons theory behavesnicely with only a mild extra assumption.

4.2.2 The theory with boundary

The case of a theory with boundary can be treated using the knowledge thatthe theory without boundary is well-defined. The only difference is that nowwe must make the action take values in a line bundle over AP . How this worksout will be treated here. First of all we notice that because of the boundary thetransformation properties of Ss under changes of section are more complicated.Once again using statement 5 from 4.1.3 we see that

exp (2πiSs·a[A]) = exp (2πiSs[A]) exp

(−2πi

1

6

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉)

· exp

(2πi

∫∂M

〈Ada−1 s∗A ∧ a∗θ〉).

The correction term now contains an extra part, and generally will not be integervalued anymore. The surprising thing is that the correction term is actually aboundary term. That this is so follows from the following lemma.

Lemma 4.2.2. Because of 4.2.1 the functional

W∂M (a) := −1

6

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉 (mod 1)

depends only on the restriction of a to the boundary.


Proof. Let M denote the orientation reversed version of M . We can create aclosed manifold N by gluing M and M along their common boundary. If a anda are two G-equivariant functions that agree on the boundary then they glue toa G-equivariant function a+ a on N . Now since M has orientation reversed weget

W∂M (a)−W∂M (a) = −1

6

(∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉+

∫M

〈a∗θ ∧ [a∗θ ∧ a∗θ]〉)

=1

6

∫N

〈(a+ a)∗θ ∧ [(a+ a)∗θ ∧ (a+ a)∗θ]〉 ∈ Z.

Remark. The functional W is the action of a 2-dimensional field theory calledthe Wess-Zumino-Witten action. The interested reader can find more about itsproperties in [3].

We thus see that the action is invariant under changes of section up to aboundary term. This already implies that the equations of motion will be in-variant under changes of section; we will explicitly see this invariance later whenwe actually calculate the EL equations. Before doing this we will take a momentto formulate the action as a section of a line bundle called the Chern-Simonsline bundle. This has the easthetic benefit of describing the action as an objectthat is independent of sections s. More importantly, however, it will allow usto formulate the symplectic structure allowing us to view Chern-Simons theoryas a field theory according to our definition.

In order to define the Chern-Simons line look at the function

c∂M (s∗A, a) := exp

(2πi

∫∂M

〈Ada−1 s∗A ∧ a∗θ〉+ 2πiW∂M (a)

).

It is precisely the transformation factor of S under changes of section.

Lemma 4.2.3. The function c∂M obeys a cocycle condition

c∂M (s∗A, a1 · a2) = c∂M (s∗A, a1)c∂M (s∗A · a1, a2).

Proof. Observe that

(a1 · a2)∗ = La1∗a2∗ +Ra2∗a1∗

by the product rule, or

(a1 · a2)∗ = a∗2L∗a1 + a∗1R

∗a2 .

Also the equality

d(Ada a∗θ) = −Ada a

∗dθ,


holds, the proof of which is similar to the proof of the identity for changing dand Ad from 4.1.3. Using these equalities we get that

W∂M (a1 · a2) = W∂M (a1) +W∂M (a2) +

∫∂M

〈a∗1θ ∧Ada2 a∗2θ〉

Also we calculate

c∂M (s∗A, a1 · a2) = exp(2πi

∫∂M

〈Ada−11s∗A ∧ a∗1θ + Ad(a1·a2)−1 ∧a∗2θ〉

+W∂M (a1 · a2))

c∂M (s∗A, a1) = exp(2πi

∫∂M

〈Ada−11s∗A ∧ a∗1θ〉+W∂M (a1))

c∂M (s∗A · a1, a2) = exp(2πi

∫∂M

〈(Ada−12

Ada−11s∗A+ Ada−1

2a∗1θ) ∧ a∗2θ〉

+W∂M (a2))

Combine the bunch of formulas and we get the required equality.

We can now describe the Chern-Simons line.

Definition 4.2.4. We have a line bundle π : L → AP with trivialisationsφs : AP → L depending on sections s : ∂M → P restricted to the boundary ofM . The transformation rule between the sections is

φs·a(A) =φs(A)

c∂M (s∗A, a).

Because c∂M satisfies the cocycle condition this correctly defines a line-bundle.This line bundle is called the Chern-Simons line. The function

CS[A] := exp(2πiSs[A])φs[A]

is a section of the Chern-Simons line and we call it the Chern-Simons action (orChern-Simons invariant) for the Chern-Simons theory with boundary. In orderfor the theory without boundary to be formulated in terms of Chern-Simonslines as well we define

L∅ := C

and for completenessCS∅ = 1.

Remark. It is important to realise that the change in trivialisations is given byc∂M which only depends on the section and the connection restricted to theboundary. If πP : P →M is the projection in P denote ∂P := π−1(∂M). usingthis notation the above construction already gives us a line-bundle π : L∂M →A∂P . If r : AP → A∂P is the restriction of connections to the boundary thenthe above line-bundle L = r∗L∂M . The line over a connection A thus depends


only on the restriction ∂A := r(A) to the boundary and we shall denote it byL∂A to emphasise this fact.

Note also we can metrise the Chern-Simons line by demanding that all thetrivialisations φs have norm 1 (since the bundle is complex this metric is Her-mitian). In this case the Chern-Simons action is also a section of norm 1. Inwhat follows we will always assume the presence of this natural metric, whichwill make the statement of several theorems neater.

Finally all the asignments above are smooth, so L is a smooth vector bundle.

For completeness we want to end the section with a theorem which givesthe behaviour of CS under some natural operatrions on manifolds like revers-ing orientation and taking disjoint unions. Basically we want to formulate thefact that the original action Ss was additive over disjoint unions (ignoring somecomplications with the choice of s), changed sign under orientation reversal, andbehaves nicely under pull-back to different bundles(i.e. it is a functor betweenproperly chosen categories). Since CS is a section of a line-bundle the formula-tion of these behaviours becomes somewhat more cumbersome, but the generalidea is straightforward. Before we formulate this theorem we introduce a newconcept that we will need in what follows.

Definition 4.2.5. Given a compact 3-manifold M we introduce the categoryCM of connections over M . Its objects are connections A on a principal G-bundlewith G compact and 1-connected (the category CM clearly depends on the choiceof G, but G is usually clear from the context and hence is surpressed from thenotation). A morphism between connections φ : A 7→ A′ on bundles P, P ′ isa bundle isomorphism φ : P → P ′ such that A = φ∗A′. The above categoryinduces the set CM whose elements are objects in CM modulo an equivalencerelation ∼. Here A ∼ A′ precisely if there is a morphism A 7→ A′.

Note that since all G-bundles are isomorphic by the assumptions on G thereis an isomorphism CM ' AP /GP where P is any G-bundle. These isomorphismsare clearly dependent on the choice of P so we work with CM which is invariantlydefined. Having gotten this definition out of the way we are ready for ourtheorem.

Theorem 4.2.6. Let M be a closed 3-manifold and G a compact 1-connectedLie group with Ad invariant non-degenerate bilinear form 〈·, ·〉 on g satisfying4.2.1. The asignments

∂A 7→ L∂A ∂A ∈ C∂MA 7→ CS[A] ∈ L∂A A ∈ CM

defined above satisfy:

Functoriality If ∂φ : ∂P → ∂P ′ is a bundle isomorphism covering an orienta-tion preserving diffeomorphism ∂φ : ∂M → ∂M ′ and ∂A′ is a connectionin P ′ then there is an induced isometry

∂φ∗ : L∂A′ → Lφ∗∂A′ .


These isometries compose properly. Furthermore if φ : P → P ′ is a bundleisomorphism covering φ : M →M ′ and A′ is a connection on P ′ then ∂φ∗

is the unique isometry such that

∂φ∗CS[A′] = CS[φ∗A′].

Orientation Orientation reversal ∂M 7→ −∂M of a surface induces a naturalisometry

L−∂M,∂A ' L∂M,∂A.

Here L denotes the complex conjugate bundle. If ∂M is actually the bound-ary of M and ∂A the restriction of A to ∂M then this isometry is theunique one satisfying

CS−M [A] = CSM [A]

Additivity If ∂M = ∂M1t∂M2 and ∂Ai are connections over ∂Mi then thereis a natural isometry

L∂A1+∂A2' L∂A1

⊗ L∂A2.

If these things are actually boundaries then this isometry is the unique onesatisfying

CS[A1 +A2] ' CS[A1]⊗ CS[A2].

Gluing We have a trace map Tr : L∂M ⊗ L∂M → C given by the hermitianmetric on L. Let M be obtained from M c by gluing two of its boundarycomponents together using a map π: i.e. ∂M c = ∂M t S t −S. Let Abe a connection over M and Ac := π∗A the induced connection. Then thetrace map induces an isometry

CS[A] ' Tr(CS[Ac]).

Here we used the previous properties to write CS[Ac] ∈ L∂A⊗LA∣∣∣S

⊗ LA∣∣∣S

Proof. First we do the functoriality. For a section ∂M → P ′ the isometryshould clearly take a trivialisation φs to the trivialisation φφ∗s. That this iswell defined follows from the properties of W∂M under pull-back. The isometryfor orientation reversal is given by φs 7→ φs, which is well defined becauseall integrals involved change sign under orientation reversal. For the additivityproperty let si be trivialisations of Pi over ∂Mi, then the isometry should clearlysend φs1ts2 7→ φs1 ⊗ φs2 . This is well defined since all integrals involved areadditive under disjoint unions. The proof of the gluing law is a combination ofthe previous properties together with the fact that

∫M

=∫Mc π∗.

Remark. The functoriality property states that the assignment of Chern-Simonslines and the action is a functor F to the category of metrised lines with isome-tries. The naturality of all the isometries discussed in the theorem means thatthey are obtained from F by a natural transformation.

4.3. THE E-L EQUATIONS AND THE FIELD THEORY 65

4.3 The E-L equations and the field theory

Now that we have seen how to rigorously define the Chern-Simons action, andthat it behaves in a nice way, we can tackle the equations of motion for theChern-Simons theory and show that it actually defines a field theory. We willdo so in this section. First we derive the equations of motion. To do so fixa section s; relative to the trivialisation φs the Chern-Simons action takes thevalue

ln(CS[A]) =

∫M

〈dAs ∧As +1

3As ∧ [As ∧As]〉.

Here a subscript s denotes pullback along it. From this we get the variation

d ln(CSA[a]) =

∫M

〈das ∧As + as ∧ dAs + as ∧ [As ∧As]〉

=

∫M

〈d(as ∧As) + 2as ∧ dAs + as ∧ [As ∧As]〉

= 2

∫M

〈as ∧ Fs〉+

∫∂M

〈as ∧As〉

We would like to split this variation up into a bulk term and a boundary termand have the boundary term define a canonical 1-from α on A∂P . The problemis that things depend on the section s chosen, so we cannot directly proceed.For the bulk term there is no problem. Let A be a connection, then it is easilychecked that a form A+ ta is a connection for all t precisely if a · u = Adu−1 afor u the G-equivariant map of a gauge transformation. Since AP is convex thismeans that a is a tangent vector precisely if it transforms like a · u = Adu−1 a.Thus the form 〈as ∧Fs〉 is independent of s and we get the equations of motion

F = 0

since the bracket was assumed non-degenerate (in general the equations of mo-tion are F ∈ Ω2

G(P, g⊥)). We see that the equations of motion are gauge-invariant, so we have a well-defined dynamics on the space of fields AP /GP . Onthe other hand the boundary term is dependent on s, but we can work aroundthis. It transforms as∫

∂M

〈as·u ∧As·u〉 =

∫∂M

〈as ∧As〉+

∫∂M

〈Adu−1 as ∧ u∗θ〉.

The correction term is just d( 12πi ln(c∂M )) as might be expected. We thus get a

family of 1-forms αs :=∫∂M〈as ∧ As〉 on A∂P . In a beautiful twist of fate the

identity d2 = 0 implies that ω = −dαs is well defined. It has formula

ω(a1, a2) = −2

∫∂M

〈s∗a1 ∧ s∗a2〉

which is clearly non-degenerate, closed and independent of s. This means thatit defines a symplectic form on A∂P , which is clearly the same one we have been


using all this time. Note that just like in the first order mechanics and firstorder scalar field theory the symplectic structure lies on the space of fields, notits tangent bundle.

There are two things we have yet to tackle before we are done. The phys-ical space of fields is actually the moduli space AP /GP and we must find aLagrangian submanifold in the space of boundary values. The first step to tack-ling these problems is finding the space of boundary values, and to do so we mustknow how G∂P relates to GP . Clearly there is a restriction map r : GP → G∂P ;this map is certainly not injective, but it is surjective which is the importantproperty, which we will make a lemma.

Lemma 4.3.1. The restriction mapping

r : GP → G∂P

is surjective.

Proof. Choose a section s of P ; a gauge transformation on P is completelydetermined by its image on s, and a gauge transformation on P by its imageon ∂s. The section s thus induces isomorphisms GP ' C∞(M,G) and G∂P 'C∞(∂M,G). We must thus prove that any smooth G valued function on ∂Mcan be extended to a function on M . The proof of this uses the fact that acompact 1-connected Lie group is 2-connected (use that H1(G) = H2(G) = 0for G semisimple [20] and the Hurwicz theorem). Since ∂M is a closed connectedsurface it is homeomorphic to a polygon with an equivalence relation on its 4gedges where g is the genus of ∂M . Since G is simply connected a function on∂M can be smoothly homotoped into a function on S2 by contracting the imageof the polygon boundary to a point. Every function on S2 can then be smoothlycontracted since π2(G) = 0. This homotopy contracting the map f∂M → Gallows us to extend f to a tubular neighborhood [0, 1) × ∂M such that f onlytakes the value e for t > 1

2 . This extension can clearly be extended smoothlyby defining it to be e outside of the tubular neighborhood.

Since the restriction on gauge transformations is surjective we get a restric-tion map

r : AP /GP → A∂P /G∂P ,or in terms of the category C a restriction map

r : CM → C∂M .

The space C∂M is not the correct space of boundary data however. The reason forthis is that the restriction of a flat connection is flat, so a boundary connectioncan only have a hope of being a boundary value of a solution to the equationsof motion if it is flat.

Definition 4.3.2. We define

MM :=A ∈ CM : FA = 0

.


ClearlyMM is the set of solutions to the Euler-Lagrange equations ker(EL).We also have a restriction map

r :MM →M∂M .

We will useM∂M as the space of Cauchy data for our field theory. We need toshow that the form ω on A∂P descends to a symplectic structure onM∂M , andthen show that the map restricting to the boundary is Lagrangian. The structureofM∂P is somewhat more complicated than that of an ordinary manifold. Thisshould come as no surprise since the action of G∂P isn’t free. Fortunately wecan use symplectic reduction to show thatM∂P is a stratified symplectic space.The reader wanting to see the details of the symplectic reduction process isreferred to [23]. To use the symplectic reduction we assume that A∂P is givensuch a topology that the action of G∂P is proper. It now remains to show thatthe action of G∂P is Hamiltonian with a co-Adjoint equivariant moment map,which has the flat connections as its zero-set. We state the result as a lemma.

Lemma 4.3.3. The action of G∂P on A∂P is Hamiltonian with co-Adjointequivariant moment map

Jξ(A) = 2

∫∂M

〈s∗FA ∧ s∗ξ〉,

which specifically is independent of the chosen s. Here the Lie algebra G∂P isnaturally identified with the Ad-equivariant functions C∞G (P, g).

Proof. Since tangent vectors to A∂P transform with Ad under the action of G∂Pit is clear that the action is symplectic. Also clearly Jξ(A · a) = JAda ξ, which isthe co-Adjoint equivariance. It thus suffices to show that

d(Jξ)(a) = ω(σ(ξ), a),

where σ is defined in A.3.2. We have

d(Jξ)(a) = 2

∫∂M

〈(da+ [A ∧ a]) ∧ ξ〉 = 2

∫∂M

〈a ∧ (dξ + [A, ξ]).〉

We are done if we show that

σ(ξ) = dξ + [A, ξ].

But we have that

σ(ξ)A =d

dt

∣∣∣∣t=0

A · (exp(tξ)) =d

dt

∣∣∣∣t=0

(Adexp(−tξ)A+ exp(tξ)∗θ)

= [A, ξ]] +d

dt

∣∣∣∣t=0

exp(tξ)∗θ.


Finally let γ be a curve with γ′(0) = X, so that we can write

d

dt

∣∣∣∣t=0

exp(tξ)∗θ(X) =d

dt

∣∣∣∣t=0

exp(tξ)∗θ(X)

=d

dt

∣∣∣∣t=0

d

ds

∣∣∣∣s=0

exp(−tξ(γ(0))) exp(tξ(γ(s)))

=d

dt

∣∣∣∣t=0

d

ds

∣∣∣∣s=0

exp(t(ξ(γ(s))− ξ(γ(0))) +O(t2))

= X(ξ) = dξ(X).

Using this lemma we can use the reduction to give the symplectic structureon M∂M . The result deserves to be a proposition.

Proposition 4.3.4. Let J be the moment map from the preceding lemma; weclearly have

M∂M = J−1(0)/G∂M .

Assume that A∂P has a topology such that the action of G∂P is proper. We canuse symplectic reduction and obtain that M∂M is a stratified symplectic spacewith the symplectic form induced from

ω(a1, a2) = −2

∫∂M

〈s∗a1 ∧ s∗a2〉.

We now finish by showing that the map

r :MM →M∂M .

is Lagrangian with respect to the symplectic structure from the proposition.To do so it is convenient to give an alternative description of the symplecticstructure ω in terms of a connection on the Chern-Simons line. It is in terms ofthis formulation that both [3] and [4] introduce the structure.

Lemma 4.3.5. Let At be a smooth path of connections over I = [t0, t1] inπP : ∂P → ∂M . This At defines a connection A on πI×P : I × ∂P → I × ∂M .There is a unique connection in L such that

φ(t) = CS[A∣∣[t0,t]×∂P

]

is a horizontal section of L over the curve At. The 1-forms

−2πiαs(a) = −2πi

∫∂M

〈s∗a ∧ s∗A〉

defined previously are the connection forms of this connection. It has curvature2πiπ∗ω, where π : L→ AP is the bundle projection and we indentify L with itsU(1) bundle of frames.


Proof. From our previous calculations we know that

−2πiαs·a = −2πiαs − d(ln(c∂M )) = −2πiαs − c∂M (·, s)−1d(c∂M (·, s))= −2πiαs + c∂M (·, s)d(c∂M (·, s)−1).

Since the Ad-equivariant map for a gauge transformation is equal to c∂M (·, s)−1

by definition of the trivialisations φs this is precisely the transformation be-haviour of a connection. Clearly the curvature forms of the −2πiαs are allequal to 2πiω; the statement about the curvature follows. It now remains toshow that φ(t) is horizontal. For this we use that FA = FAt

− At∧dt. It followsthat

ch(A) = −〈At ∧ At〉 ∧ dt,

since everything without a dt component vanishes. Choosing a section s we thensee that

φ(t) = exp

(2πi

∫ t

0

dt′∫∂M

s∗〈At′ ∧At′〉)φs.

Finally then:

∇Atφ = 2πi

∫∂M

s∗〈At ∧At〉φ(t)− 2πiαs(At)φ(t) = 0

Remark. Note that viewing the forms αs as connection forms is nothing trulynew. We were given a set of 1-forms that transforms nicely under a groupaction, and we try to glue them together into a connection. The real content ofthe lemma lies in the fact that this connection has the action as its horizontalsections. It is this characterisation of the connection that we need.

Theorem 4.3.6. The restriction

r :MM →M∂M

is Lagrangian

Proof. We must show two things, that the symplectic form r∗ω vanishes and thatDim(Im(r)) = 1

2 Dim(M∂M ). For the first claim we show that the connectionfrom the previous lemma has no holonomy for curves lying in Im(r). Then itscurvature disappears there. Let ∂At be a loop in M∂M with extension At. Itdefines a connection A on S1 ×M . The holonomy around the loop is

CSS1×∂M (∂A) = exp

(2πi

∫S1×M

d cs(A)

)= exp

(2πi

∫S1×M

〈FA ∧ FA〉)

= exp

(2πi

∫S1×M

〈At ∧ dt ∧ At ∧ dt〉)

= 1.

We used FA = FAt− At ∧ dt. The first term in this disappears because At is a

loop of flat connections. To find the dimension of the image one uses arguments


from cohomology. The argument is somewhat non-standard, and to treat thenecessary background would be too large a detour. The interested reader isreferred to [3] and [4] for more details.

We have thus found an assignment

∂M 7→ M∂M

M 7→ r(MM )

from the category of cobordisms to a generalised Wehrheim-Woodward categorythat includes stratified symplectic spaces. The statements from 4.2.6 give thatthis assignment is a functor since it behaves correctly under gluing. We havethus shown that Chern-Simons theory is a field theory as we originally set outto do.

4.4 The generalisation to toric bundles

We finish the chapter on Chern-Simons by extending the theory to principalbundles with structure group Tn. Since the n-torus is not simply connectednot all toric bundles are trivialisable. This problem can be circumvented byembedding the bundle in the SU(2)n bundle. In this section we give the detailsof this procedure for the circle group T and show that it once again defines a fieldtheory. The extension to arbitrary tori by taking direct products everywherewill be obvious at the end. The construction is modelled after that in [24].

4.4.1 Bundle extensions

In order to construct the SU(2)-bundle and the embedding of our circle bundlewe use a very general construction for extending a principal bundle to one witha larger structure group.

Definition 4.4.1. Let π : P →M be a principal G bundle and h : G→ H andinclusion of Lie groups. We define the bundle

P ×h H := P ×H/G.

Here g ∈ G g′ ∈ H and p ∈ P , and G acts on P ×H via

(p, g′) · g =(p · g, h(g−1) · g′

). The action of H on this bundle is by ordinary multiplication (p, g′) · g′′ :=(p, g′ · g′′). The projection map is defined by πh : (p, g′) 7→ π(p).

Clearly the projection πh is well defined on equivalence classes and the fibresare diffeomorphic to H. Furthermore the H action is free and transitive on eachfibre, so P ×h H is a principal H-bundle. There is a natural inclusion

ih : P → P ×h H : p 7→ [p, e].

4.4. THE GENERALISATION TO TORIC BUNDLES 71

If we have an inclusion of bundles i : P → P ′ then any connection on P canbe uniquely extended to one in P ′. For the extension of a connection on P toP ×h H we have an explicit formula.

Lemma 4.4.2. Let P be a principal bundle and P ×h H the induced bundlecoming from the inclusion h : G → H. If A is a connection on P then theextended connection Ah has the form

Ah∣∣[p,g]

= h∗(Adg−1 Ap) + θH∣∣g.

In this formula h∗ is the Lie algebra homomorphism coming from h, A acts onthe first factor and θH is the Maurer-Cartan form on H acting on the secondfactor.

Proof. A priori the above formula only defines a form on P × H. We need tocheck that it descends to the quotient. This follows because

h∗A

(d

dt

∣∣∣∣t=0

p · exp(tX)

)= Adg θH(

d

dt

∣∣∣∣t=0

exp(tX) · g),

and Ah is invariant under the G-action on P ×H. The above equality impliesthat Ah vanishes along the orbits of the equivalence relation; this, togetherwith G-invariance, implies that Ah descends to the quotient. That Ah satisfiesthe axioms for a connection is an easy check. Finally, it is obvious from thedefinition that we have

i∗hAh = h∗A;

in other words Ah restricts to A if we identify g with its image in h. By unique-ness of the extension it must be equal to Ah.

4.4.2 The Lagrangian formulation

Now that we know how to extend the structure group of a principal bundle wecan use this to define a Lagrangian for our theory. For this we use that there isa natural inclusion

h : T→ SU(2) : a 7→(a 00 −a

).

The inclusion h allows us to embed every T-bundle P into an SU(2) bundleP ′ := P ×h SU(2). Every connection A ∈ AP extends uniquely to a connectionA′ ∈ AP×hSU(2) according to the formula in lemma 4.4.2.

Definition 4.4.3. Let π : P → M be a principal T-bundle, h the inclusiondefined above, and A a connection on P with extension A′. For a section s ofthe trivialisable bundle P ′ := P ×h SU(2) we define

Ss[A] :=

∫M

s∗ ch(A′).


To simplify the notation in the sequel we make a specific choice of Chern-Simons form. The Lie group SU(2) is simple, so all its Ad-invariant innerproducts are a multiple of the Killing form. Specifically we have

− 1

48π2

∫SU(2)

Tr(θ ∧ [θ ∧ θ]) = 1.

The generator of H3(SU(2),Z) ' Z is therefore given by the Ad-invariant innerproduct

〈·, ·〉 = − 1

8π2Tr(··)

con forma the remark following 4.2.1. Every other allowable Ad-invariant innerproduct is an integer multiple of this one.

Hypothesis 4.4.4. Without loss of generality we make the assumption

〈·, ·〉 = − 1

8π2Tr(··).

In this case we have

ch(A′) = − 1

8π2Tr(F ∧A− 1

6A ∧ [A ∧A]).

As a kind of sanity check on our definition of S above assume for the momentthat P = M ×T is trivial. In this case a section s of P induces the section ih sof P ′. In this situation we have

Sihs[A] = − 1

8π2

∫M

s∗i∗h Tr(F ′ ∧A′ − 1

6A′ ∧ [A′ ∧A′])

= − 1

8π2

∫M

s∗ Tr(i∗hF′ ∧ i∗hA′ −

1

6i∗hA

′ ∧ [i∗hA′ ∧ i∗hA′])

= − 1

8π2

∫M

s∗ Tr(h∗F′ ∧ h∗A′ −

1

6h∗A

′ ∧ [h∗A′ ∧ h∗A′])

= − 1

4π2

∫M

s∗(F ∧A− 1

6A ∧ [A ∧A]) = − 1

4π2

∫M

s∗F ∧ s∗A.

This is precisely the expected form for the action in the trivial T-bundle. If weformulate the theory without boundary we see that exp(2πiSs) is independentof s because of 4.2.1. Formulating the theory with boundary we need to havethe action take values in a line-bundle again to absorb the dependence on thetrivialisation of P ′. The line-bundle we use is the pull-back of the Chern-Simonsline on AP ′ . Specifically let

ε : AP → AP ′

denote the extension of connections according to 4.4.2. We can then pull-backthe Chern-Simons line, obtaining a line bundle ε∗L on AP (which we also callthe Chern-Simons line for obvious reasons). It has trivialisations ε∗φs dependingon trivialisations s of P ′. It is worth noting that if P is trivialisable and has


trivialisation s then we get trivialisations ψs := ε∗φihs of the Chern-Simonsline. If a : M → T is a gauge transformation then we have

ψs·a = ε∗φihs·h(a) =ψs

c∂M (ih s∗A′, h(a))=

ψsc∂M (s∗A, a)

= exp

(1

2πi

∫∂M

Tr(s∗A ∧ a∗θ))ψs.

This is precisely the transformation behaviour if we had contructed the Chern-Simons line directly for a trivial T-bundle.

Remark. An oddity in the current construction is that the trivialisations Chern-Simons line in this case depend on sections of ∂P ′ in stead of sections of ∂P(which don’t necessarily exist). If the boundary ∂P is trivialisable we can makethe Chern-Simons line depend on trivialisations of ∂P in the straightforwardway [24]. This is the case if the first Chern-class c1(P ) lies in the torsion partTor

(H2(M,Z)

)of the Cohomology. However, this restriction is not necessary

to give a field theory formulation of the theory. For this reason we work withthe most general case where we work with sections of P ′.

The correct definition of the Chern-Simons action is now straightforward.

Definition 4.4.5. Let s be a section of the principal bundle P ′, then we definethe Chern-Simons functional by the formula

CS(A) := Ss(A)φs.

Of course this Chern-Simons functional behaves nicely because the one foran SU(2)-bundle does.

Theorem 4.4.6. The Chern-Simons functional for toral bundles has all theproperties from 4.2.6.

Proof. This true because the Chern-Simons functional on an SU(2)-bundle hasall these properties and extending connections respects all these properties.

4.4.3 The Hamiltonian theory

We compute the variation of the action Ss relative to a section s of P ′ to obtainthe equations of motion and the symplectic structure on AP . We have

Ss[A] :=

∫M

s∗ ch(A′),

so its variation has the form

δSs(A) =1

8π2Tr

(2

∫M

s∗a′ ∧ s∗F ′ +∫∂M

s∗a′ ∧ s∗A′).

Here a′ is the variation in A′ caused by the variation a in A. Now the form

Tr(a′ ∧ F ′)


is invariant under gauge transformations and hence has a transgression form ηobeying π′∗η = Tr(a′ ∧F ′), where π′ is the projection on P ′. Similarly a, F areinvariant under gauge transformations in P and hence have transgression formsa′′, F ′′. Clearly

η = 2a′′ ∧ F ′′.

The bulk term thus reads1

2π2

∫M

a′′ ∧ F ′′.

This vanishes for all a, and hence all a′′, precisely if

F = 0.

These are the same equations of motion as before, and they descend to AP /GPby invariance under GP . The boundary term αs depends on a trivialisation of∂P ′, just like in the previous case. Again its differential

ω := −dαs

does not. It has the explicit form

ω(a1, a2) =−1

4π2

∫∂M

Tr(s∗a′1 ∧ s∗a′2).

Once again using transgression forms a′′1 , a′′2 for the variations a1, a2 we get

ω(a1, a2) =−1

2π2

∫∂M

a′′1 ∧ a′′2 .

This defines the symplectic structure on A∂P . It is clearly invariant under theaction of G∂P . In fact, just as before, it is Hamiltonian.

Lemma 4.4.7. The action of G∂P on A∂P is Hamiltonian with co-Adjointequivariant moment map

Jξ(A) = 4

∫∂M

F ′ ∧ ξ

Here the Lie algebra G∂P is naturally identified with the functions C∞(∂M, iR),and F ′ is the transgression form for the curvature F of A.

Proof. Exactly the same calculation as before.

We can now use symplectic reduction to get a symplectic structure onM∂M

just as before.

Proposition 4.4.8. Let J be the moment map from the preceding lemma; weclearly have

M∂M = J−1(0)/G∂M .


Assume that A∂P has a topology such that the action of G∂P is proper. We canuse symplectic reduction and obtain that M∂M is a stratified symplectic spacewith the symplectic form induced from

ω(a1, a2) =−1

2π2

∫∂M

a′′1 ∧ a′′2 .

Remark. According to 2.3.11 we have MM ' Hom(π1(M), G)/G for any M1-connected. Since T is abelian, and H1(M) is the abelianisation of π1(M) weget for our torus bundles M∂M ' H1(∂M,T). The isomorphism is explicitlygiven by

∂A 7→ [∂A′].

Here ∂A′ is the unique form on ∂M such that ∂A = π∗∂A′+ vertical part. Thespace H1(∂M,T) has a canonical symplectic form [24]. It has formula

ω(α, β) =

∫∂M

α ∧ β.

Clearly this differs from the symplectic form we derived above only by a con-stant.

In the case of compact 1-connected Lie groups before we argued that M∂M

was the proper space of Cauchy data by showing that the map r : GP → G∂Pwas surjective. By constructing the symplectic structure onM∂M just as beforewe have implicitly identified it as the space of Cauchy data in the abelian caseas well. That this choice is correct might have been guessed from the closedanalogy we have so far seen with the compact 1-connected case. Certainly wewill see in a bit that the restriction map of flat connections has Lagrangianimage. In the abelian case it is, however, generally untrue that r : GP → G∂Pis surjective. As a counter-example one might consider M to be the solid toruswith ∂M = T2, and choose P = M × T2. Gauge transformations are thenparametrised by functions T2-valued functions since Ad is trivial. If we choosethe gauge transformation on the boundary given by the identity function this hasno extension to the solid torus since then the torus would retract onto the circle.This has the immediate consequence that the restriction r : MM → M∂M isnot injective since the equivalence classes inM∂M are too big. In fact it can beshown that the fibres of r are principal fibre bundles over M∂M [24].

We now finish by showing that

r :MM →M∂M

is Lagrangian. We might argue in a manner analogous to the previous caseby showing the αs define a connection in the Chern-Simons line. However inthe abelian case a straight-forward proof that r is isotropic can be given. Wewill skip the construction of the connection given by the action and perform astraight calculation.


Theorem 4.4.9. The restriction mapping

r :MM →M∂M

is Lagrangian.

Proof. The proof that the dimension of Im(r) is maximal is the same as before.As for its isotropy let a1, a2 be two variations of a connection with transgressionforms a′1, a

′2, then

ω(r∗a1, r∗a2) =

∫∂M

a′1 ∧ a′2 =

∫M

d(a′1 ∧ a′2) = 0.

The last equality holds because the variations are amongst flat connections:dai = 0.

Using this theorem we have an assignment

∂M 7→ M∂M

M 7→ r(MM )

from the category of cobordisms to a generalised Wehrheim-Woodward categorythat includes stratified symplectic spaces, just like before. That this assignmentis actually a functor is true by exactly the same reasoning.

4.5 Closing remarks

It shall come as little surprise that there is much of Chern-Simons theory that wehave not covered in this chapter. As was already mentioned in the introductionFreed [3] constructs a generalisation of Chern-Simons theory to manifolds withcorners. In this case the boundary surface of the space-time has a boundary itselfand the Chern-Simons line will also depend on a trivialisation of ∂∂P . Anotherpossible generalisation is to define the Chern-Simons theory for general principalbundles over 3-manifolds without boundary. This is done in [25] by extendingthe bundle P to one over a 4-manifold with boundary M . One can then use theLagrangian 〈F ∧ F 〉, which gives our action by stokes theorem in the case of atrivial bundle over a closed 3-manifold. There are some topological obstructionsto the necessary extension of P , bu they can be resolved using group cohomology.Finally there is the matter of quantisation of the Chern-Simons theory. Due tomatters of scope we have decided only to treat classical field theories, and henceonly classical Chern-Simons, in this thesis. There is, however, a sizeable bodyof results on quantisation of Chern-Simons. For example Witten has succeededin quantising the theory using Feynmann path integrals [2]. Another approachis the use of geometric quantisation. Lemma 4.3.5 gives a natural starting pointfor this approach. More on it can be found in [26] for example.

Bibliography

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[2] E. Witten. Quantum field theory and the Jones polynomial. Communica-tions in Mathematical Physics, 121:351–399, September 1989.

[3] D.S. Freed. Classical chern-simons theory, part 1. 1992.

[4] B. Himpel. http://www.math.uni-bonn.de/people/himpel/himpel_

cstheory.pdf. Lecture notes on Chern-Simons theory from a course givenat Aarhus University.

[5] M. Spivak. A comprehensive introduction to differential geometry vol. 2.Publish or perish, 3rd edition, 1999.

[6] L.W. Tu. An introduction to manifolds. Springer, 1st edition, 2008.

[7] S. Gallot, D. Hulin, and J. Lafontaine. Riemannian geometry. Springer,3rd edition, 2004.

[8] A. Hatcher. Algebraic Topology. Cambridge University press, 1st edition,2001.

[9] J.E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry.2nd edition, 1998.

[10] J. Kijowski and W.M. Tulczyjew. A Symplectic Framework for Field The-ories. Springer Verlag, 1st edition, 1979.

[11] A.S. Cattaneo, P. Mnev, and N. Reshetikhin. Classical and quantum la-grangian field theories with boundary. 2007.

[12] K. Wehrheim and C.T. Woodward. Functoriality for lagrangian correspon-dences in floer theory. 2007.

[13] N. Reshetikhin. Lectures on quantization of gauge systems. 2010.

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Appendix A

Lie groups

In this appendix some facts about Lie groups will be reviewed. There will beno proofs, as what is treated here can be found in most books on Lie groups,for example [27]. This appendix is mainly intended to allow referencing somefacts whose proof would be too distracting from the main text. We begin withthe necessary definition.

A.1 Lie groups and the exponential map

Definition A.1.1. A Lie group G is a group together with a smooth manifoldstructure such that the multiplication and inversion maps are smooth.

It immediately follows that the maps Lh : g 7→ h · g and Rh : g 7→ g · h arediffeomorphisms. The inversion map is a diffeomorphism as well. The maps Land R allow us to distinguish special tangent vector fields.

Definition A.1.2. A tangent vector field X on G is left invariant if Lg∗X = Xfor all g ∈ G.

Remark. Since this definition only makes sense for tangent vector field, we willfrom now on just speak of left invariant vector fields. It will be understood thatthe vector fields are then sections of TG.

Remark. In the following we will focus on left-invariance. Everything treatedin this context can of course also be done for right invariant vector fields whereRg∗X = X.

The set of left invariant vector fields is in bijection with TeG. The bijectionis given by X 7→ Xe with the inverse being extension by left translation Xg =Lg∗Xe. It can be shown that the flow of the left invariant vector fields is definedfor all time. This gives us the following definition.

Definition A.1.3. Let X ∈ TeM be a tangent vector, and X its left invariantextension. If φtX is the flow during time t along X then we de define

exp(X) := φ1X(e).

79

80 APPENDIX A. LIE GROUPS

If we give TeG the standard manifold structure then exp is a local diffeomor-phism. The map is surjective if G is compact [22] but in general it is neithersurjective nor injective. For every X ∈ TeG we have a smooth group homomor-phism t 7→ exp(tX). It can be proven that all smooth homomorphisms R→ Gare of this form.

Example A.1.4. The groups GLn, SLn, On, SOn and their complex coun-terparts are all Lie groups. If A,B are elements of one of these groups thenLA : B 7→ AB. If X is a tangent vector then LA∗ : X 7→ AX; here X is seenas a matrix. Choose a vector X ∈ TeG, then its left invariant extension hasthe form XA = AX. We thus have that d

dt exp(tX) = exp(tX)X. The uniquesolution to this differential equation is the matrix exponential:

exp(tX) =

∞∑i=0

(tX)i

i!.

The details about convergence of this series are left to the reader.

A.2 The Lie algebra of a Lie group

An important concept within the theory of Lie groups is that of its correspondingLie algebra. We now define these objects and show how every Lie group givesrise to one.

Definition A.2.1. A Lie algebra is a pair (A, [·, ·]). Here A is a vector spaceand [·, ·] : A×A→ A a bilinear map that

• is alternating [X,X] = 0

• obeys the Jacobi identity [X, [Y,Z]] + [Z, [X,Y ]] + [Y, [Z,X]] = 0

A Lie algebra homomorphism L : A→ B is a linear map such that

L ([X,Y ]A) = [L(X), L(Y )]B

Remark. The property of being alternating is equivalent to [X,Y ] = −[Y,X]over the reals.

The vector space we want to use for our associated Lie algebra is TeG, wejust need to define the bracket. For this we need to introduce some new concepts.

We have a diffeomorphism Cg : G → G : h 7→ g · h · g−1. Clearly Ce = idand Cg Ch = Cg·h so we have an action of G on itself. The maps Cg are alldiffeomorphisms taking e to itself. They allow us the following definition:

Definition A.2.2. Define Adg := Cg∗∣∣e. The map g 7→ Adg is called the

adjoint representation.

A.3. SMOOTH ACTIONS ON A MANIFOLD 81

Since the Cg are all diffeomorphisms it is easy to show that the above actuallydefines a representation of G on TeG. We can use the adjoint representationto define a bilinear operator on TeG by adX(Y ) := d

dt Adexp(tX)(Y ). It can bechecked that the map ad satisfies all the properties of a Lie bracket. We arenow ready for our definition.

Definition A.2.3. To each Lie group G we associate its Lie algebra g :=(TeG, ad).

Example A.2.4. Look once again at the matrix Lie groups from the previousexample. We wish to see what the corresponding vector spaces are. We merelystate what the results are; a neat derivation would enlarge the appendix beyondreason. The base vector spaces are:

• For gln(R) it is End (Rn)

• For gln(C) it is End (Cn)

• For sln(R) it is the traceless part of End (Rn)

• For sln(C) it is the traceless part of End (Cn)

• For on and son it is the antisymmetric part of End (Rn)

• For un and sun it is the antisymmetric part of End (Cn)

We calculate the algebra structure by following the above trail of definitions.In all cases we see that CA(B) = ABA−1. It follows that AdA(Y ) = AY A−1.Taking once more the derivative to A we see that [X,Y ] = adX(Y ) = XY −Y X. This is just the normal commutator; the above construction is thus ageneralisation of well known structures to more general groups where the notionof derivative can be defined.

Example A.2.5. Another important example of a Lie algebra is the space oftangent vector fields Γ(TM) on a manifold. The Lie bracket is in thise casegiven by the Lie derivative: [X,Y ] := LXY . That the Lie derivative satisfiesthe requirements of a bracket is left as an exercise.

A.3 Smooth actions on a manifold

The study of Lie group actions is a very comprehensive area. We will only needa single result in what follows. We give the definition of a smooth action andthen state the needed result.

Definition A.3.1. The Lie group G is said to have a smooth right action on amanifold M if there is a smooth map · : M ×G→M such that

• x · e = x

• (x · g) · h = x · (g · h).


Observe that the map Ψg : M →M : x 7→ x ·g is a diffeomorphism for everyg ∈ G. An important fact about actions that we will need is that they give amapping σ : g→ Γ(TM). It is defined by

σ(X)x =d

dt

∣∣∣∣t=o

x · exp(tX). (A.1)

Proposition A.3.2. The mapping σ defined above is a Lie algebra homomor-phism. The algebra structure on Γ(TM) is given by the Lie derivative.

The mapping σ is clearly linear. The importance, and perhaps surprise,of the preceding proposition lies in the conservation of the algebra structure:σ([X,Y ]) = [σ(X), σ(Y )]. As a final remark we mention that in the case of aleft action the map σ would still exist, but it would be an antihomomorphism.This is our reason for choosing right actions in this thesis.

A.4 The Maurer-Cartan form

The final topic in this appendix is a treatment of the Maurer-Cartan form. It’suse in the main text will make notation and proofs considerably less cumber-some. First we will need some notation.

If A,B are W -valued functions for a vector space W , then A⊗B is a W⊗W -valued function. Now a linear map [·, ·] : W ⊗W →W induces a W -valued map[A⊗B] : (x, y) 7→ [A(x), B(y)].

Example A.4.1. As an example of this we take ω to be a gln-valued 1-formand we take [·, ·] to be the Lie bracket. Using the above notation we see

[ω ∧ ω](X,Y ) = [ω ⊗ ω](X,Y )− [ω ⊗ ω](Y,X) = 2[ω(X), ω(Y )]

= 2(ω(X) · ω(Y )− ω(Y ) · ω(X)).

The · in the last line is matrix multiplication. In components we thus get

[ω ∧ ω]ij = 2

n∑k=1

ωik ∧ ωkj ,

where ∧ is the ordinary wedge product of 1-forms.

Definition A.4.2. The Maurer-Cartan form θ on a Lie group G is the g-valued1-form that assigns to Xg the value Xe. Here X is the left invariant extensionof Xg.

By the definition of θ it follows that

θ (Xg) = Lg−1∗Xg.

Using this aformula we can check the behaviour of θ under left and righttranslations.

A.4. THE MAURER-CARTAN FORM 83

L∗hθ(Xg) = θ(Lh∗Xg) = Lg−1·h−1∗Lh∗Xg = Lg∗Xg = θ(Xg).

R∗hθ(Xg) = θ(Rh∗Xg) = Lh−1·g−1∗Rh∗Xg = Adh−1Lg∗Xg

= Adh−1 θ(Xg)

Proposition A.4.3. The Maurer-Cartan form θ satisfies the equation

dθ +1

2[θ, θ] = 0

Proof. The left invariant vector fields extending a basis of g form a global frame.It is thus sufficient to check the statement on any two left invariant vector fieldsX,Y . In this case we have

dθ(X,Y ) = Xθ(Y )− Y θ(X)− θ[X,Y ] = X(Ye)− Y (Xe)− [X,Y ]e

= −[Xe, Ye] = −[θ(X), θ(Y )]

Appendix B

Fibre bundles

B.1 Definition and examples

Throughout this thesis the concept of a fibre bundle is very prominent. It isfor this reason that in this appendix follows a quick treatment of the conceptof a fibre bundle, together with the specific kinds of fibre bundle that we willencounter. These are vector bundles and principal bundles. First we give thedefinition of a fibre bundle.

Definition B.1.1. A Fibre bundle E over a manifold M with fibre F is a tripleπ : E → M where M,E are manifolds and π is a submersion. Furthermorefor every x ∈ M there must be an open U 3 x such that there exist localtrivialisations φU : π−1(U) ' U×F which are of the form φU (p) = (π(p), ψU (p))for some ψU .

The maps φU above give that locally E looks like a product U ×F , and thatπ is just the projection onto the first factor. In general there is no trivialisationE ' M × F , meaning that the fibres may be attached to M with a twist. IfE ' M × F then E is called trivialisable, and if E = M × F then E is calledtrivial.

An important concept when one has a fibre bundle is that of a section.

Definition B.1.2. A section of a fibre bundle π : E →M is a map s : M → Esuch that π s = id. A local section is a section defined on some open subset ofM The set of smooth sections of E is denoted by Γ(E).

In the case of a trivial fibre bundle sections of E are just functions s : M → F .Under the trivialisations φU any section can locally be identified with suchfunctions, which will be done in the sequel without fuss. Related to sections isthe concept of a vertical vector field on a fibre bundle.

Definition B.1.3. A vector field v on a fibre bundle E is vertical if π∗v = 0.The set of vertical vector fields at p ∈ E defines a vector space denoted by VpEand their union forms the vertical bundle V E.

85

86 APPENDIX B. FIBRE BUNDLES

This condition means that v is everywhere tangent to the fibres of the fi-bre bundle. The relation between vertical vector fields and sections is that ifst : M → E is a smooth family of sections then d

dtst is a smooth vertical vectorfield. And similarly if φt is the flow of a smooth vertical vectorfield st := φt sis another section.

The final construction we introduce for general fibre bundles is that of abundle map.

Definition B.1.4. Given a map f a bundle map f covering f is a map suchthat the following diagram commutes:

E1 E2

M1 M2

f

π π

f

In the case that f and f are diffeomorphisms we speak of a bundle isomorphism,where if M1 = M2 we also require f to be the identity.

Example B.1.5. Given a fibre bundle π : E → M2, and a smooth mappingf : M1 →M2 we can define the pull-back fibre bundle f∗E over M1. As a set wechoose f∗E = (x, e) ∈M1 × E : f(x) = π(e). This set is clearly an embeddedsubmanifold; It has the unique smooth structure such that f : f∗E → E :(x, e) 7→ (f(x), e) is a bundle map.

Example B.1.6. A common example of a fibre bundle that is encounteredin basic differential geometry is a vector bundle (now denoted by V ). In thiscase the fibre is Rn and we have an extra requirement. Namely that π−1(x)has a vector space structure such that φU

∣∣π−1(x) : π−1(p) → Rn is a linear

isomorphism for each x ∈ M . In this case sections of V are called vector fieldsand a smooth section will be a smooth vector field etc. Notice that the sum oftwo smooth vector fields is smooth since addition is smooth in Rn. We can alsolook at bundle maps between vector bundles. In this case we usually demandsome extra structure. We speak of a vector bundle homomorphism if we have abundle map that is linear in each fibre. When the homomorphism is invertiblewe speak of an isomorphism. As an example for a vector bundle isomorphismtake any Riemannian manifold (M, g). Then the raising and lowering of indicesgives a bundle isomorphism between TM and T ∗M . Notice that a differentchoice of metric yields a different isomorphism. We can also apply the abovepull-back contruction to vector bundles. Let π : V → M2 be a vector bundleand f : M1 → M2 a smooth map. Then as a fibre bundle f∗V is the bundleconstructed in the previous example. We can give it a unique vector bundlestructure such that f is a vector bundle homomorphism.

B.2. PRINCIPAL BUNDLES 87

B.2 Principal bundles

Another type of fibre bundle that will be used extensively is a principal bundle(P ). These fibre bundles are in fact so ubiquitous that they merit their ownsection. In this case the fibre will be a Lie group G, called its structure group.There is now the extra demand that π−1(p) has a free transitive right G-action (which is required to be smooth in this thesis) that commutes with φU .We thus want that if φU (p) = (x, g) ∈ U × G then φU (p · h) = (x, gh). This isextended to functions f : M → G by letting f work on π−1(x) with f(x). Ifwe then choose f to be smooth and also take a smooth local section s then it isclear s · f is also a smooth local section. Also every other smooth local sections′ is of the form s · f for some smooth f : U → G. In the previous remark localsections were used because of a striking difference between vector and principalbundles. In a vector bundle there is always the zero section, and the interestingquestion is the existence of nowhere zero sections. In a principal bundle globalsections need not exist; the existence of a global section is equivalent to thebundle being trivialisable. To see this take a section s : M → P . Take anyp ∈ P and let x = π(p); there is then a unique gx ∈ G such that p = s(x) · gx.We can then use p 7→ (x, gx) as a trivialisation. It turns out, though, that incertain nice situations every principal bundle is trivialisable.

Theorem B.2.1. Let G be a simply-connected Lie group and P a principalG-bundle over M . If Dim(M) ≤ 3 then P is trivialisable.

A proof of this theorem can be found in [4] and will not be repeated here.The necessary background in obstruction theory is too large to cover here.

It should also be remarked that in a principal bundle the space of verticalvector fields has a very specific form. There is a canonical isomorphism VpP ' ggiven by the map σ from (A.1). Since the orbits of the G-action lie in a singlefibre the vectors obtained in this way are vertical. That the map is a linearisomorphism follows because the action is free and transitive.

Example B.2.2. There is a specific kind of principal bundle that will be used inthe text and thus bears mentioning. This is the bundle of frames πF : F (V )→M corresponding to a vector bundle πV : V → M . Recall that a local framefor V is a set of smooth vector fields defined over some open U such that ateach point p ∈ U they form a basis of TpM . As a set F (V ) is the union of allbases of π−1

V (p) for all p ∈ M . The map πF sends a basis of π−1V (p) to p. The

structure group in this case is GLn where n = Dim(V ). The action sends abasis vi into

∑nj=1 vjA

ji or in matrix notation (v,A) 7→ vA. Any local section

s : U → F (V ) now gives a bijection π−1F (U) ' U ×GLn, and we demand this to

be a diffeomorphism if s defines a local frame of V . It is easily checked that thiscorrectly defines a smooth structure. We thus have a principal bundle F (V )whose smooth sections are the frames of V . This bundle is thus trivialisable ifand only if V is. It is clear that if V comes with an orientation we can choosethe structure group to be the connected component of the identity and we get

88 APPENDIX B. FIBRE BUNDLES

the oriented frame bundle SF (V ) of positively oriented frames. Similarly if Vcarries a metric we can obtain O(V ) the orthonormal frame bundle. Combiningthe two we get SO(V ), the bundle of positively oriented orthonormal frames.

Example B.2.3. Also in the case of a principal bundle we can look at bundlemaps. We now define a principal bundle homomorphism to be a map betweenprincipal bundles that is a fibre bundle map and intertwines the group actions:

f(p · g) = f(p) · g.

Related to this concept is the pull-back of a principal bundle. Let π : P →M2

be a principal bundle with structure group G. If f : M1 →M2 is a smooth mapthen we define f∗P as a fibre bundle using the familiar construction. There isnow a unique G-action on f∗P such that f is a principal bundle homomorphism.This action is clearly smooth, free and transitive, making f∗P into a principalbundle.

Appendix C

Symplectic geometry

Throughout this thesis concepts from the area of symplectic geometry will beused. The number of results needed for our purposes is but small, so for comple-tion this appendix gives an account of the relevant results. For a more thoroughexposition the reader is referred to [23] or [9]. Of all the appendices this is theonly one that actually gives proofs, since not all of the results that we use areproven in the references above.

C.1 Symplectic linear algebra

We start with the concept of a symplectic structure on a vector space. Asymplectic manifold will then be defined in terms of symplectic structures onthe tangent spaces.

Definition C.1.1. A symplectic vector space is a pair (V, ω). Here V is abanach space and ω an anti-symmetric bilinear function satisfying one of thefollowing non-degeneracy conditions:

• Weak non-degeneracy: the mapping v 7→ ω(v, ·) is an injective mappingV → V ∗.

• Strong non-degeneracy: the mapping v 7→ ω(v, ·) is an isomorphism V →V ∗

Remark. A symplectic form can be seen as an anti-symmetric inner product.The demand that (v, v) > 0 if v 6= 0 is replaced by the condition ω(v, ·) 6= 0since of course ω(v, v) = 0.

Remark. Every finite dimensional symplectic vector space is strongly symplectic.This is essentially the reason why field theories are so much more complicatedthan ordinary mechanics.

In light of the above remark a symplectic form allows for a notion of orthog-onality to a linear subspace.

89

90 APPENDIX C. SYMPLECTIC GEOMETRY

Definition C.1.2. Let U ⊂ V be a linear subspace. We define the orthogonalcomplement of U by

U⊥ := v ∈ V : ω(v, u) = 0 ∀u ∈ U

It shall come as little surprise that this definition was exactly the sameas would have been used for inner products. Since symplectic forms are anti-symmetric the behaviour of these orthogonal complements is entirely differenthowever. For example

Span(v) ⊂ Span(v)⊥

for any v ∈ V since ω(v, v) = 0.

Definition C.1.3. Let (V, ω) be a symplectic vector space. There are now fourdifferent kinds of interesting subspaces U ⊂ V that we can distinguish between.

• U ⊂ V is called isotropic if U ⊂ U⊥.

• U ⊂ V is called co-isotropic if U⊥ ⊂ U .

• U ⊂ V is called Lagrangian if it is both isotropic and co-isotropic.

• U ⊂ V is called symplectic if ω∣∣U

is a symplectic form.

It will be the lagrangian subsets that we are most interested in. In the finitedimensional case everything is quite well-behaved. This is related to the fact thata gram-schmidt like procedure is possible in this case to put every symplecticform into a block diagonal form. In the infinite dimensional case things arefar less tangible; showing that a set is isotropic is in our applications often notdifficult, but co-isotropy much so. Luckily we have the following proposition tomake our life easier.

Proposition C.1.4. Let (V, ω) be a symplectic vector space and U ⊂ V anisotropic subspace. If U has an isotropic complement then U is Lagrangian.Note that having an isotropic complement means ∃W ⊂ V that is isotropic suchthat V = U ⊕W .

Proof. We must show that if ω(v, u) = 0 for all u ∈ U then v ∈ U . This provesthat U is co-isotropic and hence Lagrangian. Since V = U ⊕W we can writev = u+w for u ∈ U and w ∈W . Our assumption is thus that ω(u+w, u′) = 0for all u′ ∈ U . Since U is isotropic this implies that ω(w, u′) = 0 for all u′ ∈ U .But since W is also isotropic and complements V this implies that ω(w, v′) = 0for all v′ ∈ V . Since ω is non-degenerate then w = 0 and v ∈ U .

C.2 symplectic manifolds

Now that we know some necessary things about symplectic vector spaces wecan define symplectic manifolds.

C.2. SYMPLECTIC MANIFOLDS 91

Definition C.2.1. A strong (weak) symplectic manifold is a pair (M,ω) whereM is a manifold and ω a closed 2-form (dω = 0) such that (TxM,ωx) is a strong(weak) symplectic vector space for all x ∈M .

Using our classification of linear subspaces of a symplectic vector space weobtain a classification of embedded submanifolds.

Definition C.2.2. Let (M,ω) be a symplectic manifold, and N ⊂ M an em-bedded submanifold. We distinguish the following four cases:

• N ⊂M is called isotropic if all its tangent spaces are.

• N ⊂M is called co-isotropic if all its tangent spaces are.

• N ⊂M is called Lagrangian if all its tangent spaces are.

• N ⊂M is called symplectic if all its tangent spaces are.

Finally we come to the symplectic analogue of an isometry.

Definition C.2.3. Let (M,ω) and (M ′, ω′) be symplectic manifolds. A sym-plectomorphism between (M,ω) and (M ′, ω′) is a diffeomorphism φ : M →M ′

such that ω = φ∗ω′.

The following proposition gives an important characterisation of symplecto-morphisms.

Proposition C.2.4. Let φ : M 7→ M ′ be a diffeomorphism between symplecticmanifolds (M,ω) and (M ′, ω′). It is a symplectomorphism if and only if thegraph of φ

(x, y) ∈M ×M ′ : y = φ(x)

is a Lagrangian submanifold of (M ×M ′, ω ⊕ (−ω′)).

Proof. Points in the graph of φ are of the form (x, φ(x)) for x ∈ M . Vectorstangent to the graph are of the form X + φ∗X for X ∈ TxM . Now the graph isisotropic if and only if for all X,X ′ ∈ TxM

0 = ω ⊕ (−ω′)(X + φ∗X,X′ + φ∗X

′) = ω(X,X ′)− ω′(φ∗X,φ∗X ′)= ω(X,X ′)− φ∗ω′(X,X ′).

Thus the graph is isotropic if and only if φ is a symplectomorphism. Now wemust show that if φ is a symplectomorphism we get the co-isotropy for free. Tothis end let (x, y) be a point on the graph and X + Y ∈ T(x,y)M ×M ′ withY 6= φ∗X. Then X + Y is not tangent to the graph and we must show there isan X ′ ∈ TxM such that

ω ⊕ (−ω′)(X + Y,X ′ + φ∗X′) 6= 0.


Now we have

ω ⊕ (−ω′)(X + Y,X ′ + φ∗X′) = ω(X,X ′)− ω′(Y, φ∗X ′)

= ω(X,X ′)− ω′(φ∗X,φ∗X ′)− ω′(Y − φ∗X,φ∗X ′)

= − ω′(Y − φ∗X,φ∗X ′)

By assumption Y − φ∗X 6= 0 so by non-degeneracy of ω′ and surjectivity of φ∗we can find X ′ so that this is non-zero. This yields co-isotropy and finishes theproof.

Before moving on to the next section we give as an example a broad classof symplectic manifolds that is used in the Hamiltonian formulation of classicalmechanics. We will see the basics of this int he next section.

Example C.2.5. Let M be a finite dimensional manifold and (q, U) a coordi-nate system on it. With respect to this coordinate system any 1-form over Ucan be expressed as

∑i pidq

i. As a result the (qi, pj) are coordinates on T ∗U ;they are called canonical coordinates. By choosing coordinate systems coveringM we can show that the 1-form expressed locally as α :=

∑i pidq

i is well de-fined on all of T ∗M . This 1-form is called the canonical form. Define ω = −dα.Then clearly dω = 0 furthermore locally ω =

∑i dq

i ∧ dpi which is easily seento be non-degenerate (strongly since T ∗M is finite dimensional). We thus seethat the cotangent bundle to every manifold is a symplectic manifold.

C.3 Hamiltonian geometry

Now that we have defined the basic structures in symplectic geometry we cansee how a symplectic structure allows us to couple differential equations tofunctions. Both the Lagrangian and Hamiltonian formulation of mechanics areessentially a coupling of differential equations to functions, and both can beformulated in terms of symplectic geometry using the ideas from this section.

Definition C.3.1. A tangent vector field X ∈ Γ(TM) is called Hamiltonian,with Hamiltonian function H, if

ω(X, ·) = dH.

The tangent vector field X is then usually denoted by XH .

Note that if ω is a strong symplectic form every smooth function H inducesa tangent vector field XH since the mapping ω : V → V ∗ can be inverted. Inthe weak case this is not necessarily true, causing complications. As remarkedpreviously this is the reason why general field theories are more complicatedmathematically than classical mechanics.

The flows of Hamiltonian vector fields are particularly well-behaved withrespect to the symplectic structure.

C.3. HAMILTONIAN GEOMETRY 93

Proposition C.3.2. Let Xf be a Hamiltonian vector field. Its flow φt is asymplectomorphism for all t where it is defined.

Proof. We have thatd

dtφ∗tω = φ∗tLXf

ω.

Clearly φ∗0ω = ω so it is enough to show that LXfω = 0. But if we let ιX denote

contraction with the first index then

LXfω = d ιXf

ω + ιXf dω = ddf = 0

Remark. In the proof we used that dω = 0, which is one of the demands on asymplectic form. This demand was made mainly to facilitate the above proof.

We end this appendix with an application of the above to the cotangentbundle T ∗M of a manifold.

Example C.3.3. From the previous example T ∗M is a symplectic manifoldwhere in canonical coordinates we have ω =

∑i dq

i ∧ dpi. Now let H be afunction, and XH its corresponding vector field (which exists since ω is stronglynon-degenerate). In coordinates we can write XH =

∑i qi ∂∂qi + pi

∂∂pi

. In thecanonical coordinates the flow of XH is then the solution of the differentialequation d

dt (q, p) = (q, p). On the other hand we have

qi =∑j

dqj ∧ dpj(XH ,∂

∂dpi) = dH(

∂

∂dpi) =

∂H

∂dpi

and

pi = −∑j

dqj ∧ dpj(XH ,∂

∂dqi) = −dH(

∂

∂dqi) = − ∂H

∂dqi.

These are clearly Hamiltons equations. In this situation the base manifold M iscalled the configuration space, and the cotangent space T ∗M is called the phasespace.

This example shows that symplectic geometry provides a very geometricway of describing Hamiltonian mechanics. The necessity of looking at generalsymplectic manifolds arises when one tries to quotient symmetries out of thephase space. The precise way in which this is done can be found in the referencesstated at the start of this appendix. Suffice it to say that these quotients willbe general symplectic manifolds, not necessarily cotangent bundles.

Appendix D

Category theory

Many of the concepts we will encounter throughout this thesis are more neatlyexpressed in terms of category theory and functoriality. In this appendix thenecessary definitions are given to allow for this formulation.

We begin with the definition of a category, around which all else is built.

Definition D.0.4. A Category C consists of a set ob(C) of objects and a sethom(C) of morphisms between the objects. The morphisms are required tosatisfy the following axioms:

• Composition: Given objects a, b, c and morphisms f : a→ b and g : b→ cwe should also have the composition morphism g f : a→ c.

• Associativity of composition: We require f (g h) = (f g) h

• Existence of identity: For every object a there is a morphism 1a : a → asuch that for every morphism f : a→ b we have f 1a = f = 1b f .

A morphism f that has an inverse f−1 such that f f−1 and f−1f are identitiesis called an isomorphism.

Remark. It should be noted that the objects and morphisms of a category arein actuality classes. In the spirit of readability, and to avoid a lot of formal settheory, this detail is swept under the rug. For those familiar with the necessarybackground it shall be clear that this poses no problems in the situations weencounter.

Example D.0.5. It is easy to write down a few examples of categories.

• The category Set with sets as objects, and mappings between them asmorphisms.

• The category Top of topological spaces and continuous mappings betweenthem.

• The category C∞ of smooth manifolds with smooth mappings.

95

96 APPENDIX D. CATEGORY THEORY

• The category Grp of groups with homomorphisms.

Example D.0.6. An example of a category that we will use quite a bit inthe main text is the category of d-dimensional co-bordisms. The objects inthis category are smooth d − 1-dimensional manifolds without boundary. Amorphism from M to M ′ is a d-dimensional manifold N such that MtM ′ = ∂N .If the manifolds M and M ′ are oriented we require that −M tM ′ = ∂N , wherenow N is also oriented and we use the induced orientation on the boundary. Thecomposition of morphisms is obtained by the gluing of the morphisms manifoldsalong the common boundary component. In the case of oriented manifoldsthe requirement on orientation of the boundary ensures that the gluing workscorrectly. If we assign more structure to the objects in the gategory (such asmetrics or symplectic forms) then we ask that the morphisms also have suchstructures, and that they properly restrict to boundary.

If we have two categories C,D we can now define the notion of a functorbetween them, which is analogous to the notion of a function between sets.

Definition D.0.7. A functor F between categories C and D is a mapping froma category C to D such that

• Every object a of C is sent to an object F (a) of D.

• Every morphism f : a→ b of C is sent to a morphism F (f) : F (a)→ F (b)such that F (1a) = 1F (a) and F (f g) = F (f) F (g).

Example D.0.8. There are obvious functors from Top, Grp and C∞ to Setgiven by the identity. These are called the forgetful functors since they merelyforget the extra structure imposed.

There is one final definition that we can make before closing this appendix.

Definition D.0.9. A category C ′ is called a subcategory of a category C if allthe objects and morphisms from C ′ are in C and the identity F : C ′ → C is afunctor.

The claim that the identity is a functor means that the morphism structurein C ′ is essentially the same as that of C. The only difference is that C ′ ismissing some objects and morphisms, hence the name subcategory.

gauge theories with an application to chern-simons theory · 2020. 9. 15. · of a gauge theory,...

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