Gerbes on Lie groupoids
Transcript of 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/??
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)
Christoph Schweigert, Gerbes on Lie groupoids – p.2/??
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)
Christoph Schweigert, Gerbes on Lie groupoids – p.2/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.3/??
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
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
Christoph Schweigert, Gerbes on Lie groupoids – p.3/??
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
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
Christoph Schweigert, Gerbes on Lie groupoids – p.3/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.4/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.4/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.4/??
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
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
Christoph Schweigert, Gerbes on Lie groupoids – p.5/??
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].
Christoph Schweigert, Gerbes on Lie groupoids – p.6/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.7/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.8/??
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 .
Christoph Schweigert, Gerbes on Lie groupoids – p.8/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.9/??
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
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)
Christoph Schweigert, Gerbes on Lie groupoids – p.9/??
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
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
Christoph Schweigert, Gerbes on Lie groupoids – p.9/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.10/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.11/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.12/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.13/??
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 ).
Christoph Schweigert, Gerbes on Lie groupoids – p.14/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.14/??
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 .
Christoph Schweigert, Gerbes on Lie groupoids – p.15/??
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 .
Christoph Schweigert, Gerbes on Lie groupoids – p.15/??
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 .
Christoph Schweigert, Gerbes on Lie groupoids – p.15/??
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 .
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
• Objects are in bijection with 2-forms ω ∈ Ω2(M) and denoted by Iω .
• A 1-morphism Iω → I′ω is a 1-form λ such that dλ = ω′ − ω.
• A 2-morphism λ→ λ′ is a U(1)-valued function g onM such that1idlog g = λ′ − λ.
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
Christoph Schweigert, Gerbes on Lie groupoids – p.16/??
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′)
)
Christoph Schweigert, Gerbes on Lie groupoids – p.17/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.18/??
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′
Christoph Schweigert, Gerbes on Lie groupoids – p.19/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.19/??
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)
Christoph Schweigert, Gerbes on Lie groupoids – p.20/??
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)
Christoph Schweigert, Gerbes on Lie groupoids – p.20/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.21/??
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]
Christoph Schweigert, Gerbes on Lie groupoids – p.21/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.22/??
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].
Christoph Schweigert, Gerbes on Lie groupoids – p.22/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.23/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.23/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.25/??
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.
Christoph Schweigert, Gerbes on Lie groupoids – p.26/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.27/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.28/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.29/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.30/??
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].
Christoph Schweigert, Gerbes on Lie groupoids – p.31/??
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
Christoph Schweigert, Gerbes on Lie groupoids – p.32/??