gerbes on lie groupoids

Gerbes on Lie groupoidsChristoph Schweigert

Department of Mathematics, University of Hamburgand

Center for Mathematical Physics, Hamburg

Joint work with Thomas Nikolaus

CRCG Workshop - Higher Structures in Topology and Geometry IV,Göttingen, June 2010

Transparencies available athttp://www.math.uni-hamburg.de/home/schweigert/transp.html

Christoph Schweigert, Gerbes on Lie groupoids – p.1/??

http://www.math.uni-hamburg.de/home/schweigert/transp.html

Sigma-models

One approach to two-dimensional conformal field theory:Lagrangian approach based on Sigma models

Description of classical theory in terms of two smooth manifolds:

• World sheet ΣTwo-dimensional smooth conformal manifold on which 2d QFT is definedNo physical meaning in string theory

• Target spaceM– Orbifolds, equivariance, local data⇒M is a Lie groupoid– With additional structure (topic of this talk)

Ansatz for configuration space of a sigma-model: space of smooth maps C∞(Σ,M)

Two different types of conformal field theories– Σ oriented– Σ unoriented and even unorientable (e.g. for string theory of type I)


Sigma-models

One approach to two-dimensional conformal field theory:Lagrangian approach based on Sigma models

Description of classical theory in terms of two smooth manifolds:

• World sheet ΣTwo-dimensional smooth conformal manifold on which 2d QFT is definedNo physical meaning in string theory

• Target spaceM– Orbifolds, equivariance, local data⇒M is a Lie groupoid– With additional structure (topic of this talk)

Ansatz for configuration space of a sigma-model: space of smooth maps C∞(Σ,M)

Two different types of conformal field theories– Σ oriented– Σ unoriented and even unorientable (e.g. for string theory of type I)

Lagrangian approach: actionS : C∞(Σ,M)→ R

Part of the integrand in a path integral: exponential of actioneiS(g) for g ∈ C∞(Σ,M)


Sigma-models on group manifolds

Particularly tractable case [Witten, ’84]:Target spaceM is a compact smooth manifold: a connected and simply connected Liegroup G

Kinetic term uses invariant metric on G

Skin :=k

2

∫

Σ〈g∗θ ∧ ?g∗θ〉

with g ∈ C∞(Σ, G) and θ Maurer-Cartan form on G





Skin :=k

2

∫

Σ〈g∗θ ∧ ?g∗θ〉


To restore conformal invariance of quantum theory,1. Choose a 3-manifold B with ∂B = Σ2. Extend the smooth map g : Σ→M to g : B →M3. Add the Wess-Zumino term:

SWZ(g) := k

∫

B

g∗ H

with invariant closed three-form H = 112π〈θ, [θ, θ]〉 on G





Skin :=k

2

∫

Σ〈g∗θ ∧ ?g∗θ〉


To restore conformal invariance of quantum theory,1. Choose a 3-manifold B with ∂B = Σ2. Extend the smooth map g : Σ→M to g : B →M3. Add the Wess-Zumino term:

SWZ(g) := k

∫

B

g∗ H

with invariant closed three-form H = 112π〈θ, [θ, θ]〉 on G

Existence and uniqueness of Wess-Zumino term, since π1(G) = π2(G) = 0

Uniqueness up to 2πZ requires integrality of k


Towards local dataWess-Zumino term expressed in terms of a closed three-form H ∈ Ω3(M)Comparison to holonomy of bundles with connection:

– write as integral of locally defined two-forms over the worldsheet Σ

– identify corresponding geometric objects





Consider for any smooth manifoldM the following bicategory Grbtriv∇(M):

• Objects are in bijection with 2-forms ω ∈ Ω2(M) and denoted by Iω .

• A 1-morphism Iω → Iω′ is a 1-form λ ∈ Ω1(M) such that dλ = ω′ − ω.

• A 2-morphism λ→ λ′ is a U(1)-valued function g onM such that1idlog g = λ′ − λ.

Forms can be pulled back→ Presheaf in bicategories onMan










1. Task:Find a construction that turns a presheaf in bicategories onMan to a stack for anappropriate topology onMan→ Local constructions of bundle gerbes










1. Task:Find a construction that turns a presheaf in bicategories onMan to a stack for anappropriate topology onMan→ Local constructions of bundle gerbes

Conservation laws→ structures equivariant under the adjoint action of G2. Task:Construct a theory for prestacks in bicategories on Lie groupoids

Remark:Our results apply not only to (bundle) gerbes (with connection), but also to otherpresheaves in bicategories like KV 2-vector bundles. Christoph Schweigert, Gerbes on Lie groupoids – p.4/??

Lie groupoidsDefinitionA groupoid in the categoryMan or a Lie-groupoid consists of two smooth manifolds Γ0and Γ1 together with the following collection of smooth maps:

• Source and target maps s, t : Γ1 → Γ0.

For compositions, need the existence of the pullback Γ1 ×Γ0 Γ1⇒ Require s and t to be surjective submersions.Other structural maps:• A composition map ◦ : Γ1 ×Γ0 Γ1 → Γ1

• A neutral map ι : Γ0 → Γ1 providingidentities

• A map in : Γ1 → Γ1 giving inverses

s.t. the usual diagrams commute


Lie groupoidsDefinitionA groupoid in the categoryMan or a Lie-groupoid consists of two smooth manifolds Γ0and Γ1 together with the following collection of smooth maps:

• Source and target maps s, t : Γ1 → Γ0.

For compositions, need the existence of the pullback Γ1 ×Γ0 Γ1⇒ Require s and t to be surjective submersions.Other structural maps:• A composition map ◦ : Γ1 ×Γ0 Γ1 → Γ1

• A neutral map ι : Γ0 → Γ1 providingidentities

• A map in : Γ1 → Γ1 giving inverses

s.t. the usual diagrams commute

2) Examples:

1) Lie groupoid BG for any Lie group G: Gt

spt pt

ι→ G

Composition = Multiplication in G

2) Action groupoidM//G: G×Mt

s

M M → G×M

with s(g,m) = m and t(g,m) = g.m.

3) Cech groupoid C(Y ) for a cover Y = ti∈IUα �M : Y ×M Yt

s

Y

Composition (y1, y2) ◦ (y2, y3) = (y1, y3) Christoph Schweigert, Gerbes on Lie groupoids – p.5/??

Presheaves in bicategoriesDefinitionA presheaf in bicategories X is a weak (bi)functor

X : Manop → BiCat.

Technical condition: X preserves products: for a disjoint unionM =⊔i∈IMi of

manifolds indexed by a set I,we require the equivalence of bicategories

X(M) ∼=∏

i∈I

X(Mi)

RemarkAny presheaf X in categories can be considered as a presheaf in bicategories with trivial2-morphisms. Special cases of our more general results on presheaves in bicategoriesapply to sheaves in categories.We recover part of the results of [Metzler, Heinloth].


Equivariant objectsDefinitionLet X be a presheaf in bicategories onMan and Γ a Lie groupoid or, more generally, asimplicial manifold

∙ ∙ ∙

∂0

∂3

Γ2

∂0

∂2

Γ1∂0

∂1

Γ0

A Γ-equivariant object of X consists of

(O1) An object G of X(Γ0)

(O2) A 1-isomorphism P : ∂∗0G → ∂∗1G in X(Γ1)

(O3) A 2-isomorphism μ : ∂∗2P ⊗ ∂∗0P ⇒ ∂

∗1P in X(Γ2),

where we denote the horizontal product by ⊗;

(O4) A coherence condition∂∗2μ ◦ (id⊗ ∂

∗0μ) = ∂

∗1μ ◦ (∂

∗3μ⊗ id)

on 2-morphisms in X(Γ3)

1-morphisms and 2-morphisms are defined analogously.

Proposition

This construction provides for any Lie groupoid Γ a bicategory X(Γ). The bicategoriesform a presheaf in bicategories on the category LieGrpd of Lie groupoids.


Example: Action groupoids

Action groupoid N//G with notation XG(N) := X(N//G)

1. X a presheaf in bicategories and G a discrete groupA G-equivariant object on a G-manifold N consists of

• An object G ∈ X(N)

• A morphism g∗Gϕg→ G. for every group element g ∈ G

• A coherence 2-isomorphism for every pair of group elements g, h ∈ G,

g∗h∗Gg∗ϕh

ϕhg

g∗G

ϕg

G

• A coherence condition


Example: Action groupoids

Action groupoid N//G with notation XG(N) := X(N//G)

1. X a presheaf in bicategories and G a discrete groupA G-equivariant object on a G-manifold N consists of

• An object G ∈ X(N)

• A morphism g∗Gϕg→ G. for every group element g ∈ G

• A coherence 2-isomorphism for every pair of group elements g, h ∈ G,

g∗h∗Gg∗ϕh

ϕhg

g∗G

ϕg

G

• A coherence condition

2. Equivariant bundles on a G-manifold N with action w : N ×G→ N

• A bundle π : P → N on NSimplicial maps: ∂0 : N ×G→ N is projection, ∂1 = w is the action⇒ ∂∗0P = P ×G and ∂∗1P = w

∗P .• A morphism P ×G→ w∗P = (N ×G)×N P , i.e. a commuting diagram

P ×Gw

π×idG

P

π

N ×Gw

N

• Coherence condition:w is a G-action that covers the G-action w onN .


Prestacks and stacksGoal: Generalizations of the sheaf conditions→ specify a collection τ of morphisms inMan: a Grothendieck (pre-)topologyτ is closed under compositions, pullbacks and contains all isomorphisms

Examples:

• The family τsub of surjective submersions

• The family τopen of morphisms obtained from an open covering (Ui)i∈I of amanifoldM by taking the local homeomorphismπ : Y �M with Y := ti∈IUi



Examples:



Morphism π : Y �M in the topology τ (“covering”)

⇒ Descent bicategory DescX(Y �M) := X(C (Y )

)

Cech cover Y �M gives Lie functor ΠY : C(Y )→MApplication of presheaf functor X⇒ functor of bicategories

τY : X(M)→ X(C(Y )) = DescX(Y �M)



Examples:



Morphism π : Y �M in the topology τ (“covering”)

⇒ Descent bicategory DescX(Y �M) := X(C (Y )

)

Cech cover Y �M gives Lie functor ΠY : C(Y )→MApplication of presheaf functor X⇒ functor of bicategories

τY : X(M)→ X(C(Y )) = DescX(Y �M)

DefinitionX be a presheaf in bicategories onMan and τ a topology onMan.

1. A presheaf X is called a τ -prestack, if for every covering Y �M in τ the functorτY is fully faithful, i.e. if all functors on Hom categories are equivalences

2. A presheaf X is called a τ -stack, if for every covering Y �M in τ the functor τYof bicategories is an equivalence of bicategories


Morita equivalences

Grothendieck topology τ onMan⇒ single out morphisms of Lie groupoidsDefinition• A morphism F : Γ→ Λ of Lie groupoids is called fully faithful, if the diagram

Γ1F1

s×t

Λ1

s×t

Γ0 × Γ0F0×F0

Λ0 × Λ0

is a pull back diagram

• A morphism of Lie groupoids Γ→ Λ is called τ -essentially surjective, if the smoothmap

Γ0 ×Λ0 Λ1 → Λ0

induced by the target map of Λ is in the topology τ .• A τ -equivalence of Lie groupoids is a fully faithful and τ -essentially surjective Liefunctor

Remark• If we omit the prefix τ , we refer to τsub-equivalences.• Lie functors which are τsub-equivalences are sometimes called Morita equivalences.

Example

The Lie functor ΠY : C(Y )→M is a τ -equivalence for all τ -covers.


The first theoremThe stack axiom asserts that for all τ -equivalences coming from τ -covers, the inducedfunctor on bicategories τY : X(M)→ X(C(Y )) is an equivalence of bicategories.Claim: All τ -equivalences of Lie groupoids yield equivalences of bicategories:

Theorem 1Suppose, Γ and Λ are Lie groupoids and Γ→ Λ is a τ -equivalence of Lie groupoids.

1. Let X be a τ -prestack on LieGrpd. Then the functor

X(Λ)→ X(Γ)

given by pullback is fully faithful.

2. Let X be a τ -stack on LieGrpd. Then the functor

X(Λ)→ X(Γ)

given by pullback is an equivalence of bicategories.


An application of theorem 1

Open covers are in particular surjective submersions⇒A τsub-(pre)stack is in particular a τopen-(pre)stack.

PropositionA presheaf in bicategories on LieGrpd is a τopen-(pre)stack, iff it is a τsub-(pre)stack.

ProofFix a surjective submersion π : Y �M ⇒ Functor

τY : X(M) → DescX(Y �M) = X(C (Y ))

Glue together local sections for π:

ti∈IUis

Y

π

M

Commuting diagram of Lie groupoids:

C (ti∈IUi)s

C (Y )

π

M

X a τopen-stack⇒ diagram commuting up to a 2-cell

DescX(ti∈IUi) DescX(Y )s∗

X(M)

π∗

X a τopen-stack ⇒ lower left arrow isequivalence of bicategoriess is a τopen-equivalence of Liegroupoids, theorem 1⇒ s∗ is an equivalence of bicategories.


An application of theorem 1 (continued)Corollary• A presheaf in categories on LieGrpd is a τopen-(pre)stack, iff it is a τsub-(pre)stack.• A presheaf on LieGrpd is a τopen-separated presheaf if and only if it is aτsub-separated presheaf.• A presheaf on LieGrpd is a τopen-sheaf if and only if it is a τsub-sheaf.

Application

U(1) principal bundles form a stack onMan with respect to the open topology τopen.Corollary of Theorem 1⇒ U(1) bundles also form a stack with respect to surjectivesubmersions⇒We can glue bundles also with respect of surjective submersions

Proposition

Free action groupoidM//G such that the quotient spaceM/G has a natural structure ofa smooth manifold and the canonical projection is a submersion.(This is, e.g., the case if the action of G onM is proper and discontinuous.)Then the category of smooth U(1)-bundles onM/G is equivalent to the category ofG-equivariant U(1)-bundles onM .

Proofπ : M →M/G is a surjective submersion and thus induces a τsub-equivalence of Liegroupoids.U(1)-bundles form a τsub-stackTheorem 1⇒ π induces an equivalence of categories.


The plus construction

Goal: Obtain a 2-stack X+ onMan from 2-prestack X onMan

Idea: Complement bicategory X(M) by adding objects in descent bicategories

DefinitionAn object of X+(M) consists of a covering Y �M and an object G in the descentbicategory DescX(Y ).


The plus construction

Goal: Obtain a 2-stack X+ onMan from 2-prestack X onMan

Idea: Complement bicategory X(M) by adding objects in descent bicategories

DefinitionAn object of X+(M) consists of a covering Y �M and an object G in the descentbicategory DescX(Y ).

MorphismsTo define 1-morphisms and 2-morphisms between objects with different coveringsπ : Y �M and π′ : Y ′ �M , need common refinements: A covering ζ : Z �M withcoverings s : Z � Y and s′ : Z � Y ′ such that the diagram

Y

π

Zs s′

ζ

Y ′

π′

M

commutes.

Example : Fibre product Z :=Y×M Y ′ �M is the canonical common refinement


The plus construction (continued)Definition• A 1-morphism between objects G = (Y,G) and G′ = (Y ′, G′) of X+(M) consists of– A common refinement Z �M of Y �M and Y ′ �M– A 1-morphism A : GZ → GZ′ of the two refinements in DescX(Z).

• A 2-morphism between 1-morphisms m = (Z,A) and m′ = (Z′, A′) consists of– A common refinementW �M of the coverings Z �M and Z′ �M

(respecting the projections to Y and Y ′)– A 2-morphism β : mW ⇒ m′W of the refined morphisms in DescX(W ).– Identify two such 2-morphisms (W,β) and (W ′, β′) iff there exists a further commonrefinement V �M ofW �M andW ′ �M , compatible with the other projections,such that the refined 2-morphisms agree on V .





• Define compositions and identities, check that this defines a bicategory X+(M)• Use the pullback of covers and the pullback functors of the prestack X to define thepullback functors f∗ : X+(N)→ X+(M) for all smooth maps f :M → N .






Theorem 2If X is a prestack, then X+ is a stack. The canonical embedding X(M)→ X+(M) isfully faithful for each smooth manifoldM .






Theorem 2If X is a prestack, then X+ is a stack. The canonical embedding X(M)→ X+(M) isfully faithful for each smooth manifoldM .

Remarks

• One can show X+(M)sub ∼= X+open(M) for each smooth manifoldM .

• Theorem 1 enters in the proof (via an explicit description of descent objects).Christoph Schweigert, Gerbes on Lie groupoids – p.15/??

Application: bundle gerbes with connection

Definition of holonomy→ Bicategory for any smooth manifoldM


• A 1-morphism Iω → I′ω is a 1-form λ such that dλ = ω′ − ω.


Close homomorphism categories under descent→Prestack Grbtriv∇of trivial bundle gerbes with connection with bicategories Grbtriv∇(M):

• An object Iω is a 2-form ω ∈ Ω2(M), called a trivial bundle gerbe with connection.

• A 1-morphism Iω → Iω′ is a U(1) bundle L with connection of curvature ω′ − ω.

• A 2-morphism φ : L→ L′ is a morphism of bundles with connection.

Pullback is induced by pullback on differential forms and pullback on U(1)-bundles⇒Grbtriv∇ is a prestack. By theorem 2, the plus construction yields a stack

Grb∇ :=(Grbtriv∇

)+

of bundle gerbes with connection onMan and even a stack on the category of Liegroupoids


Corollaries and comparison to other approaches to gerbes

Corollary of theorem 1• For an equivalence F : Γ→ Λ of Lie groupoids, the pullback functor

F ∗ : Grb∇(Λ)→ Grb∇(Γ)

is an equivalence of bicategories.• In particular, for a free, proper and discontinuous action of a Lie group G on a smoothmanifoldM we have the equivalences of bicategories

Grb∇G(M) ∼= Grb∇(M/G)

Comments• Spelling out the data explicitly⇒ Objects are bundle gerbes in the sense of Murray• Special case of open cover Y :=

⊔Ui: object in DescGrbtriv∇(Y ) is a Chaterjee-Hitchin

gerbe.

Definition• A morphism A : (Y,G)→ (Y ′, G′) in X+(M) is called a stable isomorphism, if it isdefined on the canonical common refinement Z := Y ×M Y ′.

• Similarly, stable 2-isomorphisms are defined.

• Two objects (Y,G) and (Y ′,G′) are called stably isomorphic if there is a stableisomorphism (Y,G)→ (Y ′,G′).For bundle gerbes (Y,G) and (Y ′, G′), stable morphisms are a subcategory,

HomStab

((Y,G), (Y ′, G′)

)⊂ HomGrb∇

((Y,G), (Y ′, G′)

)


Comparison of morphism categories

For bundle gerbes (Y,G) and (Y ′, G′), stable morphisms are a subcategory,

HomStab

((Y,G), (Y ′, G′)

)⊂ HomGrb∇

((Y,G), (Y ′, G′)

)

Proposition

For any two objects O = (Y,G) and O′ = (Y ′,G′) in X+(M), the 1-categoryHom(O,O′) is equivalent to the subcategory of stable isomorphisms and stable2-isomorphisms.In particular, two objects are isomorphic in X+(M), iff they are stably isomorphic.

Remarks

• The bicategory with stable isomorphisms (Murray, Stevenson) is equivalent to ourbicategory. With our definition of morphisms, composition has a simpler structure.

• Waldorf has made a further different choice of common refinements. Hisbicategories have morphisms categories that are contained in our morphismcategories and contain the morphism categories of Murray.⇒ All three bicategories are equivalent.


Holonomy for gerbes with connection

Σ closed oriented surface.1. The holonomy of a trivial bundle gerbe Iω on the surface Σ with ω ∈ Ω2(Σ)

HolIω := exp(2πi∫Σ ω)∈ U(1)

1-isomorphism Iω → Iω′ , i.e. a U(1) bundle L⇒

∫Σ ω

′ −∫Σ ω =

∫Σ curv(L) ∈ Z ⇒ HolIω = HolIω′


Holonomy for gerbes with connection

Σ closed oriented surface.1. The holonomy of a trivial bundle gerbe Iω on the surface Σ with ω ∈ Ω2(Σ)

HolIω := exp(2πi∫Σ ω)∈ U(1)

1-isomorphism Iω → Iω′ , i.e. a U(1) bundle L⇒

∫Σ ω

′ −∫Σ ω =

∫Σ curv(L) ∈ Z ⇒ HolIω = HolIω′

2. Bundle gerbe G with connection over smooth manifoldM , and a smooth map

Φ : Σ→M

H3(Σ,Z) = 0 ⇒ Pullback gerbe Φ∗G is isomorphic to a trivial bundle gerbe Iω .Choose a trivialization: a 1-isomorphism

T : Φ∗G∼−→ Iω

and define the holonomy of G around Φ by HolG(Φ) := HolIω .

3. Independence of choice of trivialization:

if T ′ : Φ∗G∼−→ Iω′ is another trivialization, transition isomorphism of gerbes on Σ

L := T ′ ◦ T −1 : Iω∼−→ Iω′

The independence then follows by integrality of the curvature


Unoriented surfaces:

Jandl bundlesmanche meinenlechts und rinkskann man nicht velwechsern.werch ein illtum!

DefinitionA Jandl bundle over a smooth manifoldM is a pair:• A U(1)-bundle P with connection overM• A smooth smooth map σ :M → Z/2 = {1,−1}

Morphisms of Jandl bundles (P, σ)→ (Q,μ) only exist if σ = μ. In this case, they aremorphisms P → Q of bundles with connection.⇒ Category of Jandl bundles JBun∇(M)


Unoriented surfaces:

Jandl bundlesmanche meinenlechts und rinkskann man nicht velwechsern.werch ein illtum!

DefinitionA Jandl bundle over a smooth manifoldM is a pair:• A U(1)-bundle P with connection overM• A smooth smooth map σ :M → Z/2 = {1,−1}

Morphisms of Jandl bundles (P, σ)→ (Q,μ) only exist if σ = μ. In this case, they aremorphisms P → Q of bundles with connection.⇒ Category of Jandl bundles JBun∇(M)

We need the covariant involutive functor

(?)−1 : Bun∇(M)→ Bun∇(M)

with P 7→ P ∗ and sending f : P → Q to(f∗)−1

: P ∗ → Q∗.

For every smooth map σ :M → Z/2, we get a functor(?)σ : Bun∇(M)→ Bun∇(M)

with the power of (?)−1 on each connected component given by the value of σ.Monoidal structure on category JBun∇(M) of morphisms of trivial objects:

(P, σ)⊗ (Q,μ) := (P ⊗Qσ , σμ)


Jandl gerbes

Guiding principle for definition of trivial Jandl gerbes: holonomy for unoriented surfaces

DefinitionPrestack JGrbtriv∇of trivial Jandl gerbes:

• An object Iω is a 2-form ω ∈ Ω2(M), a trivial Jandl gerbe with connection

• A 1-morphism Iω → Iω′ is a Jandl bundle (P, σ) of curvature curvP = σ ∙ ω′ − ω

• A 2-morphism φ : (P, σ)→ (Q,μ) is a morphism of Jandl bundles with connection

Composition of 1-morphisms is defined as the tensor product of Jandl bundles.

Jandl gerbes are defined by applying the plus construction:

JGrb∇ :=(JGrbtriv∇

)+

By theorem 2, this defines a stack onMan.


Jandl gerbes

Guiding principle for definition of trivial Jandl gerbes: holonomy for unoriented surfaces

DefinitionPrestack JGrbtriv∇of trivial Jandl gerbes:

• An object Iω is a 2-form ω ∈ Ω2(M), a trivial Jandl gerbe with connection

• A 1-morphism Iω → Iω′ is a Jandl bundle (P, σ) of curvature curvP = σ ∙ ω′ − ω

• A 2-morphism φ : (P, σ)→ (Q,μ) is a morphism of Jandl bundles with connection

Composition of 1-morphisms is defined as the tensor product of Jandl bundles.

Jandl gerbes are defined by applying the plus construction:

JGrb∇ :=(JGrbtriv∇

)+

By theorem 2, this defines a stack onMan.

Remark : A Jandl gerbe in terms of descent data:• A cover Y �M• A two-form ω ∈ Ω2(Y )

• A Jandl bundle (P, σ) on Y [2] such that σ∂∗1ω − ∂∗0 = curv(P )

• A 2-morphismμ : ∂∗2 (P, σ)⊗ ∂

∗0 (P, σ)⇒ ∂

∗1 (P, σ)

of Jandl bundles on Y [3]. Such a morphism only exists, if ∂∗2σ ∙ ∂∗0σ = ∂

∗1σ

⇒ Data on Y [3] reduce to a morphism of U(1)-bundles

μ : ∂∗2P ⊗ ∂∗0P ⇒ ∂

∗1P + Associativity condition of ordinary gerbes on Y [4]


Jandl gerbes vs. gerbes

• Inclusion j : Bun(M)→ JBun(M)P 7→ (P, 1)

Bun(M) is a full monoidal subcategory of JBun(M) ⇒

• Inclusion Grbtriv∇(M)→ JGrbtriv∇(M) of bicategories⇒• Inclusion functor

J : Grb∇(M)→ JGrb∇(M)

In terms of descent data

(Y, ω, P, μ) 7→ (Y, ω, (P, 1), μ) .

Inclusion functor J is faithful, but neither full nor essentially surjective.


Jandl gerbes vs. gerbes

• Inclusion j : Bun(M)→ JBun(M)P 7→ (P, 1)

Bun(M) is a full monoidal subcategory of JBun(M) ⇒

• Inclusion Grbtriv∇(M)→ JGrbtriv∇(M) of bicategories⇒• Inclusion functor

J : Grb∇(M)→ JGrb∇(M)

In terms of descent data

(Y, ω, P, μ) 7→ (Y, ω, (P, 1), μ) .

Inclusion functor J is faithful, but neither full nor essentially surjective.

To describe essential image, introduce functor

O : JGrb∇(M)→ BunZ/2(M)

In terms of descent data(Y, ω, (P, σ), μ) 7→ (Y, σ) .

Cocycle condition on Y [3] ⇒ Pair (Y, σ) describes local data of Z/2-bundle.Note: a section of the bundle (Y, σ) in local data: a function s : Y → Z/2 such that

σ = ∂∗0s ∙ ∂∗1s on Y [2].


Jandl gerbes vs. gerbes (continued)Definition

1. We call O(G) the orientation bundle of the Jandl gerbe G.

2. A global section s of O(G) is called an orientation of the Jandl gerbe G.

3. A morphism ϕ : G → G′ of oriented Jandl gerbes is called orientation preserving, ifthe morphism O(ϕ) of Z/2-covers preserves the global sections, O(ϕ) ◦ s = s′.

4. With all 2-morphism of Jandl gerbes⇒ Bicategory JGrb∇or(M) of oriented Jandlgerbes.


Jandl gerbes vs. gerbes (continued)Definition

1. We call O(G) the orientation bundle of the Jandl gerbe G.

2. A global section s of O(G) is called an orientation of the Jandl gerbe G.

3. A morphism ϕ : G → G′ of oriented Jandl gerbes is called orientation preserving, ifthe morphism O(ϕ) of Z/2-covers preserves the global sections, O(ϕ) ◦ s = s′.

4. With all 2-morphism of Jandl gerbes⇒ Bicategory JGrb∇or(M) of oriented Jandlgerbes.

Proposition

1. For any gerbe G, the induced Jandl gerbe J (G) is canonically oriented.For any morphism ϕ : G → G′ of gerbes, the induced morphismJ (ϕ) : J (G)→ J (G′) is orientation preserving.

2. The functor J induces an equivalence of bicategories

Grb∇(M)→ JGrb∇or(M) .

⇒ Choice of an orientation reduces a Jandl gerbe to a gerbe


About the proof

1. For G an ordinary gerbe with connection: the bundle O(J (G)) is given by thetrivial Z/2 cocycle on the covering of G⇒ Canonical section sG , preserved by O(J (ϕ)) for any morphism ϕ : G → G′ ofgerbes

2. Step 1 (in local data):Data and conditions of a Jandl gerbe (Y, Iω , (P, σ), μ) with σ : Y [2] → Z/2 theconstant map to 1↔ Local data of a gerbe with connection.Orientation bundle (Y, 1) of such a Jandl gerbe is trivial⇒ Choose the trivialsection 1 : Y → Z/2 as the canonical orientation⇒ Such a Jandl gerbe is oriented

Similarly, 1-morphisms have the same local data; 2-morphisms are the same bydefinition.⇒ Functor J is an isomorphism from the bicategory Grb∇(M) to the fullsubbicategory of JGrb∇or(M) with trivial map σ

Step 2 (in local data)Claim: Any oriented Jandl gerbe with connection is isomorphic to an object in thefull subbicategory with trivial map σ:

Apply to a general Jandl gerbe (Y, Iω , (P, σ), μ) with orientation s : Y → Z/2 theisomorphismm = (Y, (triv, s), id)

The target of this isomorphism is a Jandl gerbe of the form (Y, Isω , (P∂∗0 s, 1), μ)

and thus in the full subbicategory of JGrb∇or(M) with trivial map σ.Christoph Schweigert, Gerbes on Lie groupoids – p.24/??

Unoriented surface holonomyFramework for unoriented surface holonomy :Pair consisting of a smooth map ϕ : Σ→M and an isomorphism of Z/2-bundles

O(ϕ∗J )∼

Σ

Σ

In particular, the orientation bundle of pulled back gerbe ϕ∗J must be isomorphic to theorientation bundle of the surface.

Recover oriented holonomy

Orientation of Σ⇔ Global section of the orientation bundle Σ→ Σ.Isomorphism⇒ Global section Σ→ O(ϕ∗J ), i.e. an orientation of the Jandl gerbe ϕ∗J⇔ Gerbe on Σ

Goal : relate to definition of [Schreiber, CS, Waldorf]Smooth manifold N , together with an involution k“Orientifold planes”: ⇒ Action not necessarily free

⇒ Action groupoid N//(Z/2) ⇒ Extend stack of Jandl gerbes to LieGrpd


Some (auxiliary) facts aboutZ/2-bundles on quotients

For transparency, formulation for the action of an arbitrary Lie group G

Motivation :Free G-action on N such that N/G is smooth manifold and the canonical projectionN → N/G is a surjective submersionThen N → N/G is a smooth G-bundle

Non-free action: Replace quotient N/G by Lie groupoid N//GIf action is free, the Lie groupoids N/G and N//G are τ -equivalent.Theorem 1⇒ Categories of G-bundles over N/G and N//G are equivalent

Question:Existence of a natural G-bundle CanG on N//G generalizing the G-bundle N → N/G• As a bundle over N , it is the trivial bundle N ×G• Carries a non-trivial G-equivariant structure: g ∈ G acts on N ×G diagonally

g ∙ (n, h) := (gn, gh)

The G-bundle CanG is the desired generalization:

LemmaN a smooth G-manifold with a free G-action such that N/G is a smooth manifold andsuch that the canonical projection N → N/G is a surjective submersion.Then the pullback of the G-bundle N → N/G to the action Lie groupoid N//G is CanG.


Orientifold backgroundsDefinitionAn orientifold background consists of• An action groupoid N

//(Z/2),

• A Jandl gerbe J on N//(Z/2)

• An isomorphism of equivariant Z/2-bundles

O(J )∼

CanZ/2

N//(Z/2)

PropositionAn orientifold background is the same as a gerbe with Jandl structure from [SSW].More precisely: Equivalence of bicategoriesOrientifold backgrounds over Lie groupoid N

//(Z/2)

↔ Gerbes over manifold N with Jandl structurewith involution k : N → N given by −1 ∈ Z/2


Two-dimensional unoriented surface holonomy

Formula for holonomy of an orientifold background over Σ//(Z/2)

Jandl gerbe described by triples (ω, η, φ)

• 2-form ω ∈ Ω2(Σ)

• 1-form η ∈ Ω1(Σ) such that σ∗ω = −ω + dη• Smooth function φ : M → U(1) such that η = σ∗η−i dlog φ• 2-isomorphism must satisfy the equivariance relation σ∗φ = φ−1

Definition of holonomy

• Choose a dual triangulation Γ of Σ and a preimage for on Σ for each of its faces.• Orientation-reserving edges in Γ: adjacent faces have been lifted to opposite sheets.

→ Disjoint union of piecewise smooth circles c⊂Σ

• For each of these circles, choose preimage cor on Σ.It may not be possible to choose cor to be closed→ Point pc ∈ Σ with two preimages in cor• Choose one of these preimages, denoted pcor. Define

Holω,η,φ := exp

2πi(∑

f

∫

for

ω +∑

c

∫

cor

η)

∏

c

φ(pcor)

– Independent of the choice of the lifts for, cor and por– Independent of the choice of the triangulation


A picture

Assignment of local data.Middle layer: world sheet Σ and subordinated indices

Top and lower layer: of the two sheets of double cover Σ.

i

j

k n

l

ηl

ω

ω

ω

m

ω

ω

φlm

ηm

ηj

φj

ω


Unoriented surface holonomyDefinitionM a smooth manifold, J a Jandl gerbe onM ; Σ an unoriented closed surface.

Given a smooth map ϕ : Σ→M and a morphism f : O(ϕ∗J )→ Σ of Z/2-bundles overΣ, define the surface holonomy

HolJ (ϕ, f) := Hol(ϕ∗J )(f)

Remarks

1. Holonomy gives (exponentiated) Wess-Zumino term

2. Generalization:• Target spaces is a Lie groupoid Γ• Hilsum-Skandalis morphism Φ : Σ→ Λ, i.e. span of Lie groupoids

Λ

∼

Σ Γ

with Λ→ Σ a τ -equivalence.

Theorem 1⇒ Pullback along Λ→ Γ is an equivalence of bicategories⇒ Can pull back a Jandl gerbe over Γ along Φ to Σ.⇒ Notion of holonomy HolJ (Φ, f) for a Hilsum-Skandalis morphism Φ

and an isomorphism f of Z/2-bundles over Σ


Orientifold backgrounds

Orientifold background Γ = N//(Z/2)

Each Z/2-equivariant map ϕ : Σ→ N provides a Hilsum-Skandalis morphism

Σ//(Z/2)

Σ N//(Z/2)

Pullback of CanZ/2 on N//(Z/2) to Σ//(Z/2) is canonical bundle on Σ//(Z/2)

Lemma→ mapped to orientation cover Σ→ Σ.

Pull back the isomorphism of Z/2-bundles

O(J )∼→ CanZ/2

on N//(Z/2) in the orientifold background to an isomorphism

O(ϕ∗J )∼→ Σ

of bundles on Σ needed to define unoriented holonomy

⇒ Holonomies HolJ (ϕ) introduced in [SSW].


Summary and outlook

SummaryGeneral results and tools for presheaves in bicategories on Lie groupoids

• τ -equivalences induce equivalences of bicategories for stacks

• The plus construction gives a 2-stackification

• Results about equivariant descent

Applications

• Unoriented surface holonomy

• KV 2-vector bundles• Non-abelian bundle gerbes

• In fact, Jandl gerbes are non-abelian bundle gerbes for a certain 2-group


gerbes on lie groupoids

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