aa r X i v : . [ m a t h . DG ] F e b Hamburger Beiträge zur Mathematik Nr. 886ZMP–HH/21–1
Gerbes in Geometry, Field Theory, and Quantisation
Severin Bunk
Abstract
This is a mostly self-contained survey article about bundle gerbes and some of their recent appli-cations in geometry, field theory, and quantisation. We cover the definition of bundle gerbes withconnection and their morphisms, and explain the classification of bundle gerbes with connectionin terms of differential cohomology. We then survey how the surface holonomy of bundle gerbescombines with their transgression line bundles to yield a smooth bordism-type field theory. Fi-nally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation ofquasi-symplectic groupoids.
Contents
A A glance at 2-categories 31References 32
Bundle gerbes on a manifold M are differential geometric representatives for the elements of H ( M ; Z ) ,in analogy to how line bundles on M represent the elements of H ( M ; Z ) . Originally, gerbes wereintroduced as certain sheaves of groupoids by Giraud [Gir71], and their popularity in geometry andphysics was boosted by Hitchin’s notes [Hit01] and Brylinski’s book [Bry08]. The concept of bundlegerbes goes back to Murray [Mur96], who had learned about gerbes from Hitchin [Mur10], and who1as looking for a more differential geometric way of describing classes in H ( M ; Z ) . Since then, thetheory of bundle gerbes has been developed further, and various applications of bundle gerbes havebeen found and studied in mathematics and physics.The main goal of the present article is to survey the theory of bundle gerbes with connection andsome of its applications in a mostly self-contained fashion. Additionally, we hope that this article mayserve as a modern entry point to the area of bundle gerbes. We assume only basic familiarity withcategory theory, not going beyond the notions of categories, functors and natural transformation. Theonly original contributions of this article are the new presentation of the material, the notion of thecurvature of a morphism of gerbes, and the suggestion to use bundle gerbes with connection to treatshifted symplectic quantisation in the world of differential geometry. We point out that gerbes have alsobeen employed very recently in shifted geometric quantisation in [Saf] in the original algebro-geometriccontext of shifted symplectic structures. Further, we apologise in advance for any incompleteness ofreferences. In particular, we do not attempt to present a full literature review in this introduction, butwe include numerous references and pointers to further literature throughout the main text.Let us provide a very basic idea of what a bundle gerbe is: any hermitean line bundle on a manifold M can be constructed (up to isomorphism) via local U (1) -valued transition functions with respect tosome open covering U = { U a } a ∈ Λ of M . These transition functions are smooth maps g ab : U ab → U (1) ,where U ab = U a ∩ U b , for a, b ∈ Λ , satisfying the cocycle condition g bc · g ab = g ac on each triple overlap U abc . Heuristically speaking, a bundle gerbe is obtained by replacing the tran-sition functions g ab : U ab → U (1) by hermitean line bundles L ab → U ab . However, since line bundlesadmit morphisms between them, we cannot simply demand a strict analogue of the cocycle conditionof the form “ L bc ⊗ L ab = L ac ” over triple overlaps. Instead, we have to specify how the two sides ofthis would-be equation are identified: on each triple overlap U abc we have to give isomorphisms µ abc : L bc ⊗ L ab ∼ = −→ L ab , and these have to satisfy a version of the Čech 2-cocycle condition (see Section 2.2). The idea to replacefunctions by vector bundles gives great guidance for how to pass from the theory of line bundles tothat of bundle gerbes; many rigorous analogies between the two theories can be discovered in this way.The relevance of (bundle) gerbes includes, but is not limited to, the following results: gerbes aregeometric models for twists of K-theory [BCM +
02, CW08], describe the B -field and D-branes in stringtheory [Kap00, Gaw05, Wal07a], and play an important role in topological T-duality [BEM04, BN15].It has been shown that (bundle) gerbes with connection even model the third differential cohomology ofa manifold [Gaj97, MS00], and that they describe various anomalies in quantum field theory [CMM00,BMS]. Bundle gerbes have found additional relevance as sources for twisted Courant algebroids ingeneralised geometry [Hit03, Gua], and certain infinitesimal symmetries of gerbes (and bundle gerbes)correspond to the Lie 2-algebra of sections of their associated Courant algebroids [FRS14, Col]. Further,bundle gerbes with connection on a manifold M correspond to certain line bundles with connectionon the free loop space LM [Wal16b], and they give rise to smooth bordism-type field theories on M (in the sense of Stolz-Teichner [ST11]) in a functorial manner [BWa]. Gerbes as well as bundle gerbeshave been used in -plectic and shifted geometric quantisation [LGX05, Rog, Bun17, Saf], where theyreplace the prequantum line bundle of conventional geometric quantisation. We survey some of theseapplications in the main part of this text. From now on, whenever we use the term ‘gerbe’, we shallmean ‘bundle gerbe’.This article is structured as follows: in Section 2, we first recast the theory of line bundles in alanguage which will allow us to directly obtain Murray’s definition of gerbes with connection through2he above process of replacing functions by vector bundles. In particular, we recall the notion ofa simplicial manifold, which we use throughout this article. Then, we define bundle gerbes withconnections, their morphisms, and their 2-morphisms, and survey the tensor product and duals ofgerbes, before giving a detailed outline of the classification of gerbes with connection in terms ofDeligne cohomology. Along the way, we introduce the curvature of a morphism of gerbes, show howvector bundles on M act on morphisms of gerbes on M , and give an introduction to the Delignecomplex as a model for differential cohomology. We finish this section with the examples of liftingbundle gerbes and cup-product bundle gerbes.Section 3 is an introduction to the parallel transport of gerbes: this is defined not just on pathsand loops, but also on surfaces in M with and without boundary. We start with the most well-knowncase of gerbe holonomy around closed oriented surfaces and introduce the transgression line bundleas a necessary gadget for extending this construction to surfaces with boundary. We illustrate howthis gives rise to a smooth functorial field theory on M in the sense of [ST11, BWa]. We concludethe section with various comments on the inclusion of D-branes into this picture, on the full paralleltransport of gerbes with connection, and how the transgression line bundle arises as its holonomy.Finally, in Section 4 we survey two approaches to geometric quantisation in the presence of higher-degree versions of symplectic forms. There are two such generalisations in the literature, going by thenames of n -plectic forms and shifted symplectic forms . We demonstrate that gerbes play the role of ahigher prequantum line bundle in both cases. In the n -plectic case, we survey Rogers’ theory of PoissonLie n -algebras [Rog] and a recent result by Krepski and Vaughan which relates multiplicative vectorfields on a gerbe to its Poisson Lie -algebra. In the n -shifted symplectic case, we first describe derivedclosed and shifted symplectic forms in differential geometry following Getzler’s notes [Get14]. Then, wedemonstrate how gerbes and their morphisms are perfectly suited to provide higher prequantum linebundles in this setting. This contains the case of symplectic groupoids, where the notion of curvatureof gerbe morphisms introduced here allows us to circumvent the exactness condition on the 3-formpart of the shifted symplectic form from [LGX05]. We finish by relating Waldorf’s multiplicativegerbes [Wal10] to the 2-shifted prequantisation of the simplicial manifold B G for any compact, simple,simply connected Lie group G . Topics not addressed in this survey
The literature and relevance of gerbes is too vast to coverevery aspect of it in this article. However, there are several topics which should not go unmentionedentirely (for the same reason, though, the following list is necessarily still not exhaustive): gerbesand higher gerbes are relevant in index theory; the n -form part of the Atiyah-Singer index theoremfor families arises as the curvature of an ( n − -gerbe [Lot02]. Further, gerbes and 2-gerbes underlievarious smooth models for the string group and control string structures on a manifold (and thus spinstructures on its free loop space) [Wal13, Wal16a, BMS, Buna]. Certain types of equivariant gerbescan be used to describe geometrically the three-dimensional Kane-Mele invariant of topological phasesof matter, see [Gaw17, MT17, BS20] and references therein. Finally, all gerbes that appear in thisarticle are abelian (their transition functions are valued in an abelian group, see Section 2.4). There isalso a theory of non-abelian gerbes —a recent review with further references can be found in [SW]—andgerbes can be defined on geometric spaces more general than manifolds; see, for instance, [Hue, Schb]. Acknowledgements
The author would like to thank Ezra Getzler and Pavel Safronov for insight-ful conversations about shifted symplectic forms in derived geometry, and Vicente Cortes, ThomasMohaupt, and Carlos Shahbazi for various discussions about categorical structures in differential ge-ometry. Further, the author is grateful to Konrad Waldorf for comments on a first draft of this article.This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) under Germany’s Excellence Strategy—EXC 2121 “Quantum Universe”—390833306.3
Bundle gerbes and their morphisms
In this section, we survey the main definitions of gerbes and their morphisms on manifolds, as well astheir tensor product and their duals. We explain the classification of gerbes with connections in termsof Deligne (i.e. differential) cohomology and exhibit two classes of gerbes as examples.
As a warm-up, let us recall the construction of hermitean line bundles from transition functions. If M isa manifold and U = { U a } a ∈ Λ is an open covering of M , local data for a hermitean line bundle consists ofa U (1) -valued Čech 1-cocycle on U , i.e. functions g ab : U ab → U (1) such that g bc g ab = g ac on each (non-empty) triple intersection U abc . (Note that we are using the common notation U a ··· a n := U a ∩ · · · ∩ U a n for a , . . . , a n ∈ Λ .) From these data we can construct a hermitean line bundle P → M given as P = (cid:18) a a ∈ Λ U a × C (cid:19) / ∼ , where the equivalence relation is defined as follows: we denote the elements in ` a ∈ Λ U a × C by ( a, x, z ) ,where a ∈ Λ , x ∈ U a , and z ∈ C . Then, ( a, x, z ) ≃ ( b, x ′ , z ′ ) precisely if x = x ′ and z ′ = g ab ( x ) z . TheČech cocycle relation ensures that this is indeed an equivalence relation.While open coverings of M are the most common device to describe line bundles in term of tran-sition data, they do not provide the most general choice: let π : Y → M be any surjective submersion.Consider a smooth map g : Y × M Y → U (1) , where Y × M Y = { ( y , y ) ∈ Y | π ( y ) = π ( y ) } is themanifold of pairs of points in Y which lie in a common fibre over M . If, for every y , y , y ∈ Y in acommon fibre, we have g ( y , y ) g ( y , y ) = g ( y , y ) , we can define P = ( Y × C ) / ∼ , with ( y , z ) ∼ (cid:0) y , g ( y , y ) − z (cid:1) ∀ ( y , y ) ∈ Y × M Y . (2.1)This, again, defines a hermitean line bundle on M . (For a precise statement of how this generalisesthe open-covering picture, see Example 2.7.)This construction extends to principal G bundles on M : if G is a Lie group and g : Y × M Y → G is a smooth map such that g ( y , y ) g ( y , y ) = g ( y , y ) for every y , y , y ∈ Y in a common fibre, weobtain a principal G -bundle as the quotient P = ( Y × G ) / ∼ , ( y , h ) ∼ (cid:0) y , g ( y , y ) − h (cid:1) . We can reformulate this construction in the following way, which motivates much of our treatmentof bundle gerbes in the later sections. Given a surjective submersion π : Y → M of manifolds, weintroduce the following notation; this may seem unnecessarily cumbersome for the description of linebundles at first, but it opens up a very general and powerful perspective on geometric structures, andwill appear throughout this article. For n ∈ N , we define the manifolds ˇ CY n − = Y [ n ] = Y × M · · · × M Y = { ( y , . . . , y n − ) ∈ Y n | π ( y i ) = π ( y j ) ∀ i, j = 0 , . . . , n − } . We define smooth maps d ni : ˇ CY n → ˇ CY n − , , d ni ( y , . . . , y n ) = ( y , . . . , b y i , . . . , y n ) ,s ni : ˇ CY n → ˇ CY n +1 , s ni ( y , . . . , y n ) = ( y , . . . , y i − , y i , y i , y i +1 , . . . , y n ) , i ∈ { , . . . , n − } and where the hat over an element denotes omission of that element. Inthe following we will write d i and s i instead of d ni and s ni , respectively, leaving the superscript n asunderstood from context. A direct check confirms that the maps d i , s i satisfy the so-called simplicialidentities d i ◦ d j = d j − ◦ d i if i < j , (2.2) d i ◦ s j = s j − ◦ d i if i < j ,d j ◦ s j = 1 ˇ CY n = d j +1 ◦ s j ≤ j ≤ nd i ◦ s j = s j ◦ d i − if i > j + 1 ,s i ◦ s j = s j +1 ◦ s i if i ≤ j . Definition 2.3.
Let C be a category. A simplicial object in C is a collection { X n } n ∈ N of objects X n ∈ C together with morphisms d i : X n → X n − for i = 0 , . . . , n , and s i : X n → X n +1 for i = 0 , . . . , n (for each n ∈ N ), satisfying the simplicial identities (2.2) (replacing ˇ CY n by X n ). The morphisms d i and s i are called the face maps and the degeneracy maps of X , respectively. We will denote a simplicialobject ( { X n } n ∈ N , d i , s i ) simply by X . A morphism X → X ′ of simplicial objects in C is a collectionof morphisms f = { f n : X n → X ′ n } n ∈ N with f n − ◦ d i = d ′ i ◦ f n and f n +1 ◦ s n = s ′ n ◦ f n for all n and i . Example 2.4.
A simplicial object in the category M fd of manifolds and smooth maps is called a simplicial manifold . We see from our arguments above that the data ( ˇ CY, d i , s i ) obtained from anysurjective submersion π : Y → M form a simplicial manifold. We call this simplicial manifold the Čechnerve of π : Y → M . ⊳ Definition 2.5.
Let C be a category. An augmented simplicial object in C is a simplicial object X in C together with an object X − ∈ C and an additional morphism d − = d − : X → X − such that d − ◦ d = d − ◦ d . We will denote an augmented simplicial object by X → X − . A morphism ( X → X − ) → ( X ′ → X ′− ) of augmented simplicial objects in C is a morphism f : X → X ′ ofsimplicial objects together with a morphism f − : X − → X ′− such that f − ◦ d − = d ′− ◦ f . Example 2.6.
The Čech nerve of any surjective submersion π : Y → M is even an augmented simplicialmanifold: we set ˇ CY − = M and d − = π . ⊳ Example 2.7.
A particular case of the preceding example arises from open coverings U = { U a } a ∈ Λ ofa manifold M : setting ˇ C U := ` a ∈ Λ U a , we obtain a canonical surjective submersion π : ˇ C U → M .(A point in ˇ C U consists of a pair ( a, x ) of a ∈ Λ and x ∈ U a , and π sends this pair to x ∈ M .) TheČech nerve of π agrees with the usual Čech nerve of the open covering U , i.e. we have ˇ C U n = a a ,...,a n ∈ Λ U a ··· a n , for every n ∈ N . ⊳ Example 2.8.
Let G be a Lie group with neutral element e ∈ G , acting on a manifold M from theright. We define a simplicial manifold M //G as follows: we set ( M //G ) n = M × G n and d i ( x, g , . . . , g n ) = ( x · g , g , . . . , g n ) i = 0 , ( x, g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n ) 0 < i < n , ( x, g , . . . , g n − ) i = n ,s i ( x, g , . . . , g n ) = ( x, e, g , . . . , g n ) i = 0 , ( x, g , . . . , g i , e, g i +1 , . . . , g n ) 0 < i < n , ( x, g , . . . , g n , e ) i = n . M //G is the action Lie ∞ -groupoid associated with the G -action on M . ⊳ Example 2.9.
In the previous example, if M = ∗ is the one-point manifold carrying the trivial G -action, we also write B G := ∗ //G . We call this simplicial manifold the classifying space for (principal) G -bundles . (This is not the classifying space of G -bundles used in algebraic topology; to arrive there,one has to take the geometric realisation of our B G . However, the nomenclature used here receives itsjustification through Proposition 2.10.) ⊳ Proposition 2.10.
Let G be a Lie group and π : Y → M a surjective submersion. Transition datafor a principal G -bundle on M with respect to π is the same as a morphism of simplicial manifolds g : ˇ CY → B G .Proof. Let g : ˇ CY → B G be a morphism of simplicial manifolds. This consists of a collection of smoothmaps g n : ˇ CY n → B G n = G n . We can thus write the map g n as an ( n + 1) -tuple g n = ( g n, , . . . , g n,n ) of maps g n,i : ˇ CY → G . Observe that the map g : Y → B G = ∗ is trivial. The compatibility of g with the face and degeneracy maps has the following consequences: for g = ( g , ) , we obtain thenormalisation condition g ( y, y ) = g ◦ s ( y ) = s ◦ g ( y ) = e for all y ∈ Y . For g = ( g , , g , ) , wehave g , = d ◦ g = g ◦ d , or equivalently g , ( y , y , y ) = g ( y , y ) , for all ( y , y , y ) ∈ Y [3] = ˇ CY .Analogously, using d instead of d we obtain g , ( y , y , y ) = g ( y , y ) . That is, the map g : Y [3] → G is completely determined by g : we have g ( y , y , y ) = ( g ( y , y ) , g ( y , y )) forall ( y , y , y ) ∈ Y [3] = ˇ CY . Finally, using d , we obtain that g , · g , = d ◦ g = g ◦ d , and withour previous findings, this yields d ∗ g · d ∗ g = d ∗ g , (2.11)which is precisely the cocycle condition (2.1) we need to build a principal bundle from g .For g = ( g , , g , , g , ) , we can use that g , = d ◦ d ◦ g = g ◦ d ◦ d , and similarly for theother components, so that g is completely determined by g as well. Inductively, we derive that thisholds true for g n , for any n ≥ , and that the remaining compatibilities with the face and degeneracymaps follow readily from the cocycle condition (2.11).We have thus seen that we can encode principal G -bundles in morphisms of simplicial manifolds,for any Lie group G . Let us now include connections. For simplicity, we will restrict ourselves to U (1) -bundles, i.e. to the case of G = U (1) . In this case, connection forms are represented locally by1-forms valued in i R . To formulate connections in sufficient generality for our purposes, we first needthe following dual notion of a simplicial object: Definition 2.12.
Let C be a category. A cosimplicial object in C is a collection { X n } n ∈ N of objects X n ∈ C together with morphisms ∂ i : X n − → X n for i = 0 , . . . , n , and σ i : X n +1 → X n for i = 0 , . . . , n (for each n ∈ N ), satisfying the cosimplicial identities ∂ j ◦ ∂ i = ∂ i ◦ ∂ j − if i < j , (2.13) σ j ◦ ∂ i = ∂ i ◦ σ j − if i < j ,σ j ◦ ∂ j = 1 X n = σ j ◦ ∂ j +1 ≤ j ≤ nσ j ◦ ∂ i = ∂ i − ◦ σ j if i > j + 1 ,σ j ◦ σ i = σ i ◦ σ j +1 if i ≤ j . The morphisms ∂ i and σ i are called the coface maps and the codegeneracy maps of X , respectively.Often, we will denote a cosimplicial object ( { X n } n ∈ N , ∂ i , σ i ) simply by X . A morphism X → X ′ ofcosimplicial objects in C is a collection of morphisms f = { f n : X n → X ′ n } n ∈ N with f n ◦ ∂ i = ∂ ′ i ◦ f n − and f n ◦ σ n = σ ′ n ◦ f n +1 for all n and i . 6 onstruction 2.14. Let X be a simplicial manifold. For any k ∈ N , we obtain a family ofreal vector spaces { Ω k ( X n ) } n ∈ N . The face and degeneracy maps of X induce pullback maps ∂ i = d ∗ i : Ω k ( X n − ) → Ω k ( X n ) and σ i = s ∗ i : Ω k ( X n +1 ) → Ω k ( X n ) , respectively, which satisfy thecosimplicial identities (2.13). More concretely, (Ω k ( X ) , d ∗ i , s ∗ i ) is a cosimplicial object in the categoryof real vector spaces and linear maps. For each n ∈ N , we define the linear maps δ : Ω k ( X n ) → Ω k ( X n +1 ) , δ ( ω ) = n +1 X i =0 ( − i ∂ i ( ω ) , which make (Ω k ( X ) , δ ) into a cochain complex of R -vector spaces. Note that the same constructionextends to augmented simplicial manifolds X → M , giving a complex (Ω k ( X → M ) , δ ) with Ω k ( M ) in degree − and δ : Ω k ( M ) → Ω k ( X ) given by d ∗− . ⊳ Lemma 2.15. [Mur96, Sec. 8]
Let Y → M be a surjective submersion. For any k ∈ N , the complex (Ω k ( ˇ CY ) , δ ) has trivial cohomology in all non-zero degrees. The complex (Ω k ( ˇ CY → M ) , δ ) has trivialcohomology in all degrees. We also obtain that for any finite-dimensional R -vector space V , the complex (Ω k ( ˇ CY → M ; V ) , δ ) of V -valued k -forms has trivial cohomology, for any k ∈ N . Proposition 2.16.
Let Y → M be a surjective submersion, and let g : ˇ CY → B U (1) be a morphismof simplicial manifolds. Let µ U (1) ∈ Ω ( U (1); i R ) denote the Maurer-Cartan form on U (1) .(1) We have δ ( g ∗ µ U (1) ) = 0 .(2) The data of a connection form A ∈ Ω ( Y ; i R ) for the principal U (1) -bundle defined by g is thesame as a coboundary A ∈ Ω ( ˇ CY , i R ) which trivialises the cocycle g ∗ µ U (1) ∈ Ω ( ˇ CY ; i R ) . Remark 2.17.
If we view g ∗ µ U (1) as the curvature of the transition functions , then a connection onthe bundle defined by g : ˇ CY → B U (1) is precisely a way of witnessing that the curvature g ∗ µ U (1) istrivial up to a specified homotopy in (Ω ( ˇ CY ; i R ) , δ ) . ⊳ In this section we survey the definition of bundle gerbes and their morphisms. Bundle gerbes wereintroduced in [Mur96], and this theory was further developed in particular in [MS00, BCM +
02, Wal07b,Wal07b, BSS18, Bun17]. A short-hand approach to the same theory from a higher sheaf theoreticperspective has been developed in [NS11].In Section 2.1 we used U (1) -valued functions of manifolds to construct hermitean line bundles (or U (1) -bundles). That is, we used objects with little structure— U (1) -valued functions first of all forma set —to obtain new objects with more structure: the collection of line bundles has two layers ofstructure consisting of the objects (i.e. the line bundles) and their morphisms. That is, line bundleson a manifold form a category rather than a set.In order to define bundle gerbes, we iterate this idea: we now aim to use line bundles as transitiondata for new geometric objects. With this goal in mind, we imitate the constructions of Section 2.1.Thus, let π : Y → M be a surjective submersion. The new ‘transition functions’ now consist of a(hermitean) line bundle L → ˇ CY , replacing the function g : ˇ CY → U (1) . The key to constructing U (1) -bundles from transition functions g : ˇ CY → U (1) in Section 2.1 was the cocycle condition (2.11). The collection of U (1) -valued functions on M also has an abelian group structure, which induces the tensor productof line bundles. We come to the analogue of this additional algebraic structure for gerbes in Section 2.3. U (1) by the tensor product of line bundles, here this amounts to choosing anisomorphism µ : d ∗ L ⊗ d ∗ L ∼ = −→ d ∗ L (2.18)of line bundles over ˇ CY . (Since line bundles admit morphisms between them, we can—and haveto—specify how the two sides of the cocycle condition on L are identified, rather than simply sayingthat they are equal.) For consistency, µ has to satisfy the following coherence condition: the diagram p ∗ L ⊗ p ∗ L ⊗ p ∗ L p ∗ L ⊗ p ∗ Lp ∗ L ⊗ p ∗ L p ∗ L ⊗ p ∗ µp ∗ µ ⊗ p ∗ µp ∗ µ (2.19)of line bundles over ˇ CY commutes. Here, p i ··· i k : Y [ n +1] → Y [ k +1] , ( y , . . . , y n ) ( y i , . . . , y i k ) arethe smooth projection maps. Note that these can be written as compositions of the face maps d i innon-unique ways; for instance, p : Y [4] → Y [2] can be written as p = d ◦ d = d ◦ d . Definition 2.20.
Let M be a manifold. A bundle gerbe on M is a tuple G = ( π : Y → M, L, µ ) consisting of a surjective submersion π , a line bundle L → ˇ CY , and an isomorphism of line bundles µ as in (2.18) which satisfies the coherence condition (2.19). Definition 2.21.
Let M be a manifold carrying a bundle gerbe G = ( π : Y → M, L, µ ) . A connectionon G is a pair of a unitary connection on the line bundle L and a 2-form B ∈ Ω ( ˇ CY ; i R ) such that curv( L ) = − δB in Ω ( ˇ CY ; i R ) , where curv( L ) is the curvature of the connection on L . The 2-form B is called the curving of G . By Lemma 2.15, there exists a unique closed 3-form H ∈ Ω ( M ; i R ) with π ∗ H = d B ; we set curv( G ) = H and call this the curvature 3-form of G .As a consequence of Lemma 2.15, every bundle gerbe admits a connection [Mur96]. Remark 2.22.
Compare the condition curv( L ) = − δB to Remark 2.17: here, curv( L ) is the curvatureof the transition data (replacing g ∗ µ U (1) ), and − B is a degree-zero element in Ω ( ˇ CY ; i R ) whichtrivialises curv( L ) in (Ω ( ˇ CY ; i R ) , δ ) . ⊳ We now define morphisms of bundle gerbes. These are again modelled on morphisms of linebundles on M , which are defined in terms of transition functions g, g ′ : ˇ CY → B U (1) : morphisms ofline bundles correspond to functions ψ : ˇ CY → C satisfying d ∗ ψ · g = g ′ · d ∗ ψ over ˇ CY . To obtain morphisms of gerbes, we replace functions by vector bundles and identities by isomorphisms.However, in general two bundle gerbes G , G will be defined over different surjective submersions, sothat we can only compare the gerbes after choosing a common refinement of the surjective submersions. Definition 2.23.
Let G i = ( π i : Y i → M, L i , µ i ) , for i = 0 , , be two bundle gerbes on a manifold M .A morphism of bundle gerbes E : G → G is a tuple E = ( ζ : Z → Y × M Y , E, α ) , consisting of thefollowing data: ζ is a surjective submersion onto Y × M Y = { ( y, y ′ ) ∈ Y × Y | π ( y ) = π ( y ′ ) } , and E → Z is a hermitean vector bundle. The composition Z → Y × M Y → M is a surjective submersionto M with Čech nerve ˇ CZ . Then, α : d ∗ E ⊗ L −→ L ⊗ d ∗ E
8s an isomorphism of hermitean vector bundles over ˇ CZ = Z × M Z . This has to be compatible with µ and µ in the sense that the following diagram of vector bundles over ˇ CZ commutes: p ∗ E ⊗ p ∗ L ⊗ p ∗ L p ∗ L ⊗ p ∗ E ⊗ p ∗ L p ∗ L ⊗ p ∗ L ⊗ p ∗ Ep ∗ E ⊗ p ∗ L p ∗ L ⊗ p ∗ E p ∗ α ⊗ ⊗ µ ⊗ p ∗ α µ ⊗ p ∗ α Note that we have omitted the pullbacks of L i from ( ˇ CY i ) = Y i × M Y i to ˇ CZ along the maps ˇ CZ → ( ˇ CY i ) induced by ζ in order to avoid overly cluttered notation. Definition 2.24. If G , G additionally carry connections, then a morphism of gerbes with connection is a morphism E = ( ζ : Z → Y × M Y , E, α ) : G → G of the underlying bundle gerbes where thehermitean vector bundle E additionally carries a unitary connection and α is connection-preserving.We call a morphism E : G → G parallel (equivalently, it satisfies the fake curvature condition ) if curv( E ) = B − B in Ω ( ˇ CZ ; i R ) .Recall that in Section 2.1 we constructed geometric objects with more structure (line bundles,which form a category) out of geometric objects with less structure ( U (1) -valued functions, which forma set). The same is true here: morphisms of bundle gerbes are built from vector bundles, which areagain geometric objects that admit morphisms between them. This leads to: Definition 2.25.
Let G , G be bundle gerbes on M , and suppose E = ( π : Z → Y × M Y , E, α ) and E ′ = ( π ′ : Z ′ → Y × M Y , E ′ , α ′ ) are two morphisms G → G . A E → E ′ is an equivalenceclass of tuples ( w : W → Z × M Z ′ , ψ ) , where w is a surjective submersion, and where ψ : E → E ′ is amorphism of hermitean vector bundles over W (where we have omitted the pullback maps for E and E ′ ), making the diagram d ∗ E ⊗ L L ⊗ d ∗ Ed ∗ E ′ ⊗ L L ⊗ d ∗ E ′ αd ∗ ψ ⊗ ⊗ d ∗ ψα ′ of hermitean vector bundles over ˇ CW = W × M W commute. Two tuples ( w : W → Z × M Z ′ , ψ ) , ( ˜ w : ˜ W → Z × M Z ′ , ˜ ψ ) are equivalent if there exists a surjective submersion v : V → W × M ˜ W suchthat the pullbacks of ψ and ˜ ψ to V agree.We will usually denote a 2-morphism [ w : W → Z × M Z ′ , ψ ] just as ψ : E → E ′ . The definition of 2-morphisms of bundle gerbes carries over verbatim to bundles gerbes with connection. In that situation, E and E ′ carry connections, and we call ψ : E → E ′ parallel whenever the underlying morphism ψ : E → E ′ of vector bundles is connection-preserving.We thus have three layers of structure for bundle gerbes, given by bundle gerbes, their morphisms,and their 2-morphisms. The main structural result is the following theorem. We refer the reader toAppendix A for a very brief survey of 2-categories. Theorem 2.26.
Let M and N be manifolds, and let f : N → M be a smooth map.(1) The collection of bundle gerbes on M forms a 2-category G rb ( M ) .(2) The collection of bundle gerbes with connection on M forms a 2-category G rb ∇ ( M ) .
3) In both G rb ( M ) and G rb ∇ ( M ) , a morphism E : G → G is invertible (see Definition A.2) if andonly if its underlying hermitean vector bundle E has rank one.(4) The smooth map f induces pullback 2-functors f ∗ : G rb ( M ) → G rb ( N ) and f ∗ : G rb ∇ ( M ) → G rb ∇ ( N ) , satisfying curv( f ∗ G ) = f ∗ curv( G ) , for any G ∈ G rb ∇ ( M ) . The notion of morphisms of gerbes goes back to [MS00], where only line bundles were used. Thiswas extended to general vector bundles in [BCM +
02, Wal07b].
Example 2.27.
The composition of E : G → G and E ′ : G → G is given by E ′ ◦ E = (cid:0) Z ′ × Y Z → Y × M Y , E ′ ⊗ E, α ′ ⊗ α (cid:1) , where we have again omitted pullbacks. For any gerbe G (possibly with connection), the identitymorphism G : G → G is given by G = (1 Y × M Y , L, p ∗ µ − ◦ p ∗ µ ) . ⊳ Remark 2.28.
One can show that any morphism E : G → G is 2-isomorphic to a morphism E ′ whosesurjective submersion is the identity on Y × M Y (see [Wal07a, Thm. 2.4.1] and [Bun17, Thm. A.19]).Further, every 2-morphism ψ = [ W → Z × M Z ′ , ψ ] has a unique representative whose surjectivesubmersion is the identity on Z × M Z ′ [Bun17, Prop. A.16]. However, including the general choices ofsurjective submersions ensures that we have a functorial composition of morphisms in G rb ∇ ( M ) ratherthan a weaker notion of composition. ⊳ We saw in Theorem 2.26 that one can pull back gerbes, their morphisms and their 2-morphisms alongsmooth maps of manifolds. In this section we survey further operations on gerbes and their morphisms.These operations are again motivated by the operations one can perform on the category of hermiteanline bundles. We present all constructions for gerbes with connection; the corresponding versions forgerbes without connections are obtained simply by forgetting all connections.
Definition 2.29.
Let G , G , G ′ , G ′ ∈ G rb ∇ ( M ) be bundle gerbes with connection on M , let E , F : G → G and E ′ , F ′ : G ′ → G ′ be morphisms in G rb ( M ) , and let ψ : E → F and ϕ : E ′ → F ′ be2-morphisms.(1) The tensor product of gerbes G = ( π : Y → M, L , µ , B ) and G ′ = ( π ′ : Y ′ → M, L ′ , µ ′ , B ′ ) isthe bundle gerbe with connection G ⊗ G ′ = ( Y × M Y ′ → M, L ⊗ L ′ , µ ⊗ µ ′ , B + B ′ ) , where we are omitting pullbacks along the projections Y ← Y × M Y ′ → Y ′ .(2) The tensor product of morphisms E = ( Z → Y × M Y , E, α ) and E ′ = ( Z ′ → Y ′ × M Y ′ , E ′ , α ′ ) reads as E ⊗ E ′ = (cid:0) Z × M Z ′ → ( Y × M Y ′ ) × M ( Y × M Y ′ ) , E ⊗ E ′ , α ⊗ α ′ (cid:1) , where we have omitted pullbacks and canonical isomorphisms which rearrange tensor products ofvector bundles, such as E ⊗ L ⊗ E ′ ⊗ L ′ ∼ = L ⊗ L ′ ⊗ E ⊗ E ′ .(3) The tensor product of 2-morphisms ψ = [ W → Z × M X, ψ ] and ϕ = [ W ′ → Z ′ × M X ′ , ϕ ] is givenby ψ ⊗ ϕ = [ W × M W ′ → ( Z × M Z ′ ) × M ( X × M X ′ ) , ψ ⊗ ϕ ] . A ∈ Ω ( M ; i R ) , denote the trivial line bundle with connection given by A as I A = ( M × C , A ) .The trivial line bundle with connection, I = ( M × C , is the monoidal unit (the neutral element)for the tensor product of line bundles. Example 2.30.
Let ρ ∈ Ω ( M ; i R ) . The trivial bundle gerbe on M with connection given by ρ is I ρ = (1 M : M → M, L = I , m, ρ ) , where m : I ⊗ I → I is the morphism given by ( x, z ) ⊗ ( x, z ′ ) ( x, z · z ′ ) for x ∈ M , z, z ′ ∈ C . The trivial bundle gerbe with connection is I ; this is the monoidal unit for the tensor product of bundlegerbes with connection [Wal07b]. ⊳ Theorem 2.31. [Wal07b, Wal07a]
For any manifold M , the tensor product of gerbes, their morphismsand their 2-morphisms turns ( G rb ∇ ( M ) , ⊗ , I ) into a symmetric monoidal 2-category. Given a vector bundle E → M , we denote its dual vector bundle by E ∨ . Given a morphism ψ : E → F of vector bundles, we denote its dual morphism by ψ ∨ : F ∨ → E ∨ . Definition 2.32.
Let G , G ∈ G rb ∇ ( M ) be bundle gerbes with connection on M , let E , E ′ : G → G ,and let ψ : E → E ′ be a 2-morphism.(1) The dual gerbe of G = ( π : Y → M, L , µ , B ) is G ∨ = ( π : Y → M, L ∨ , µ −∨ , − B ) .(2) The dual morphism of E = ( ζ : Z → Y × M Y , E, α ) is E ∨ = ( sw ◦ ζ : Z → Y × M Y , E ∨ , α −∨ ) ,where sw : Y × M Y → Y × M Y is the swap of factors. This defines a morphism E ∨ : G ∨ → G ∨ .(3) The dual 2-morphism ψ ∨ : E ′∨ → E ∨ is obtained by sending the bundle morphism E → E ′ under-lying the 2-morphism ψ to its dual bundle morphism E ′∨ → E ∨ . Remark 2.33.
There are several variations on the notion of duals of morphisms and 2-morphismsof gerbes, some of which only reverse the direction of morphisms but not of 2-morphisms. Certaindefinitions of duals may be more adapted to particular problems, see e.g. [Wal07a, Bun17]. ⊳ Proposition 2.34. [Wal07a]
For any G ∈ G rb ∇ ( M ) , there are canonical parallel isomorphisms G ∨ ⊗ G → I and I → G ⊗ G ∨ , establishing G ∨ as the categorical dual of G . The following observation has important consequences for the existence of morphisms betweengerbes (see Corollary 2.58):
Proposition 2.35. If E : G → G is a morphism in G rb ( M ) (or G rb ∇ ( M )) with underlying vectorbundle E , then the determinant bundle det( E ) induces an isomorphism det( E ) : G ⊗ rk( E )0 ∼ = −→ G ⊗ rk( E )1 .Proof. One can directly see that det( E ) induces a morphism det( E ) : G ⊗ rk( E )0 ∼ = −→ G ⊗ rk( E )1 ; since rk(det( E )) = 1 , this is an isomorphism by Theorem 2.26(3). Construction 2.36.
Let E : G → G be a morphism of bundle gerbes with connection on M , givenas the tuple E = ( Z → Y × M Y , E, α ) . Further, let F be a hermitean vector bundle with connectionon M . We can form a new morphism (omitting the pullback of F along Y × M Y → M ) F ⊗ E : G → G , F ⊗ E = ( Z → Y × M Y , F ⊗ E, F ⊗ α ) . If ϕ : F → F ′ is a morphism of hermitean vector bundles, and ψ = [ W → Z × M Z ′ , ψ ] : E → E ′ is a2-morphism of bundle gerbes (with some E ′ : G → G ), then we set ϕ ⊗ ψ = [ W → Z × M Z ′ , ϕ ⊗ ψ ] : F ⊗ E −→ F ′ ⊗ E ′ . ⊗ : HVB un ∇ ( M ) × Hom G rb ∇ ( M ) ( G , G ) −→ Hom G rb ∇ ( M ) ( G , G ) (2.37)of the symmetric monoidal category HVB un ∇ ( M ) of hermitean vector bundles with connection on M on the category of morphisms from G to G , for any pair of gerbes with connection on M . ⊳ Remark 2.38.
The module action (2.37) is a categorified analogue of the fact that the set of morphisms L → L between any line bundles on M is a module over C ∞ ( M ) . ⊳ Construction 2.39.
There is a dual operation to the action (2.37): let E , E ′ : G → G be morphismsbetween bundle gerbes on M . After choosing a common refinement of the underlying surjective sub-mersion, we may assume that E and E ′ are defined over the same surjective submersion Z → Y × M Y .We then consider the Hom bundle
Hom(
E, E ′ ) , which comes with the isomorphism β defined as thecomposition d ∗ Hom(
E, E ′ ) d ∗ Hom(
E, E ′ ) ⊗ Hom( L , L ) Hom( L ⊗ d ∗ E, L ⊗ d E ′ ) d ∗ Hom(
E, E ′ ) Hom( L , L ) ⊗ d ∗ Hom(
E, E ′ ) Hom( d ∗ E ⊗ L , d ∗ E ′ ⊗ L ) ( − ) ⊗ L ∼ = β ∼ = Hom( α − ,α ′ ) f ⊗ ψ fψ ∼ = ∼ = This satisfies the cocycle identity, and thus the pair (Hom(
E, E ′ ) , β ) induces a unique (up to canonicalisomorphism) hermitean vector bundle with connection on M , which we denote by Hom( E , E ′ ) . Werefer to [BSS18, Sec. 4.5] and [Bun17, Sec. 3.2] for full details for this construction. ⊳ Proposition 2.40. [BSS18, Prop. 4.37], [Bun17, Cor. 3.67]
Let E , E ′ : G → G be morphisms of gerbeswith connection. There is a canonical bijection (which is even an isomorphism of C ∞ ( M ) -modules) G rb ∇ ( M ) ( E , E ′ ) ∼ = Γ (cid:0) M ; Hom( E , E ′ ) (cid:1) (2.41) between 2-morphisms E → E ′ and sections of the bundle Hom( E , E ′ ) . Furthermore, under this isomor-phism, parallel 2-morphisms correspond to parallel sections, and unitary isomorphisms correspond tounit-length sections. Let E : G → G be a morphism of gerbes with connection. Since α preserves connec-tions (see Definition 2.24) the 2-form curv( E ) + ( B − B ) descends to a bundle-valued 1-form curv( E ) ∈ Ω ( M ; End( E )) ; here, we have set End( E ) = Hom( E , E ) . Definition 2.42.
We call curv( E ) ∈ Ω ( M ; End( E )) the curvature of the morphism E : G → G .Note that if rk( E ) = 1 , then End( E ) is trivial; it is a hermitean line bundle on M with a canonicalunit-length section given by the identity 2-isomorphism E : E → E under the isomorphism (2.41). Proposition 2.43. If E : G → G is an isomorphism, then there exists a unique 2-form ρ ∈ ω ( M ; i R ) such that E : G ⊗ I ρ → G is parallel.Proof. By Theorem 2.26(3), we have rk( E ) = 1 . The fact that α is parallel implies that δ (curv( E ) − ( B − B )) = 0 . The claim then follows from Lemma 2.15: there exists (a unique) ρ ∈ Ω ( M ; i R ) such that the pullback of ρ to Z equals − (curv( E ) + ( B − B )) . (Note that we preciselyhave ρ = − curv( E ) and that G ⊗ I ρ can be identified with the gerbe ( Y → M, L , µ , B + ρ ) .)12 emark 2.44. Using a (pseudo-)Riemannian metric on M and the structure on End( E ) induced from End( E ) one can now define a Yang-Mills functional on the connections on E compatible with thoseon G and G , using the curvature curv( E ) . This defines twisted Yang-Mills theory , which, to ourknowledge, has so far not been investigated. Physically, it describes Yang-Mills theory on the world-volume of D-branes in string theory in the presence of non-trivial Kalb-Ramond charge. Note that curv( E ) (or its trace) does not have integer cycles in general. ⊳ Using the above techniques one can prove the following results: for η ∈ Ω ( M ; i R ) with integercycles, let HLB un ∇ η ( M ) be the groupoid of hermitean line bundles on M with connection of curvature η and their unitary parallel isomorphisms. For bundle gerbes G , G ∈ G rb ∇ ( M ) , let Iso par G rb ∇ ( M ) ( G , G ) denote the groupoid of parallel gerbe isomorphisms and their unitary parallel 2-isomorphisms. Finally,recall the trivial gerbe with connection I ρ from Example 2.30. Proposition 2.45. [Wal07b, Bun17]
For any ρ, ρ ′ ∈ Ω ( M ; i R ) , there are equivalences of categories Hom G rb ∇ ( M ) ( I ρ , I ρ ′ ) ≃ HVB un ∇ ( M ) , Iso par G rb ∇ ( M ) ( I ρ , I ρ ′ ) ≃ HLB un ∇ ρ ′ − ρ ( M ) . Consider a manifold M , and suppose we are given a differentiably good open covering U = { U a } a ∈ Λ of M . Here, differentiably good means that each finite intersection of the patches U a is either emptyor diffeomorphic to R m with m = dim( M ) . Recall from Example 2.7 that U induces a surjectivesubmersion ˇ C U → M whose Čech nerve agrees with the usual Čech nerve of the open covering U . In order to give a gerbe with connection defined over this surjective submersion, we have tospecify a line bundle L → ˇ C U . However, since ˇ C U = ` a,b ∈ Λ U ab is a disjoint union of manifoldsdiffeomorphic to R m , we may assume without loss of generality that L = I A , i.e. that L is the trivialline bundle with connection given by some A ∈ Ω ( ˇ C U ; i R ) . We further have to specify a parallelbundle isomorphism µ : d ∗ L ⊗ d ∗ L → d ∗ L ; since L = I A , such an isomorphism is equivalent to a smoothfunction g : ˇ C U → U (1) satisfying d ∗ A + d ∗ A = d ∗ A + g ∗ µ U (1) , or equivalently δA = g ∗ µ U (1) in (Ω ( ˇ C U ; i R ) , δ ) . The coherence condition (2.19) for µ then translates to the condition that d ∗ g · d ∗ g = d ∗ g , or equivalently δg = 0 in ( C ∞ ( ˇ C U ; U (1)) , δ ) , the Čech complex associated to ˇ C U and the sheaf of U (1) -valued functions. (ThisČech complex is obtained in a way analogous to Construction 2.14.) This is precisely the conditionthat g be a U (1) -valued Čech 2-cocycle on M with respect to the open covering U . Finally, we have togive a curving for our bundle gerbe, which consists of a 2-form B ∈ Ω ( ˇ C U ; i R ) such that d A = − δB in (Ω ( ˇ C U ; i R ) , δ ) , where we have used Definition 2.21 and Construction 2.14.A bundle gerbe as described here is often called a local bundle gerbe or a Hitchin-Chatterjeegerbe , after [Cha98]. Observe that every surjective submersion Y → M admits local sections definedover some open covering U of M , and after possibly choosing a refinement we may assume that U isdifferentiably good. The local sections assemble to give a smooth map s : U → Y , commuting with13he maps to M . This further induces a morphism ˇ Cs : ( ˇ C U → M ) −→ ( ˇ CY → M ) of augmentedsimplicial manifolds (see Definition 2.5) with s − = 1 M . This is the key step in proving: Lemma 2.46.
Every bundle gerbe with connection on M is isomorphic (by a parallel isomorphism) toa local gerbe ( g, A, B ) with connection, defined over some differentiably good open covering U of M . (This statement can be refined to allow one to fix U arbitrarily and to also include morphismsand 2-morphisms is [Bun17, Thm. 2.101].) If we hope to classify gerbes with connection on M up toparallel isomorphism, we may thus restrict ourselves to local gerbes. For details, we refer to [Wal07a,Sec. 1.2]. In a similar vein, one can show that parallel isomorphisms ( g, A, B ) → ( g ′ , A ′ , B ′ ) of localgerbes over the same open covering U are equivalently given by a function h : ˇ C U → U (1) togetherwith a 1-form C ∈ Ω ( ˇ C U ; i R ) satisfying g g ′− = δh , A − A ′ = h ∗ µ U (1) − δC and B − B ′ = d C .
Finally, a parallel 2-isomorphism ( h, C ) → ( h ′ , C ′ ) corresponds to a function ψ : ˇ C U → U (1) with h h ′− = − δψ and C − C ′ = ψ ∗ µ U (1) . The local data for gerbes and their isomorphisms are conveniently described via the following enhance-ment of Construction 2.14:
Construction 2.47.
Suppose C is a presheaf of cochain complexes (of abelian groups) on the category M fd of smooth manifolds and smooth maps. That is, C is an assignment M → C ( M ) of a cochaincomplex to each manifold, and a morphism of complexes f ∗ : C ( M ) → C ( M ′ ) to each smooth map f : M ′ → M such that ( f ◦ f ) ∗ = f ∗ ◦ f ∗ for each pair of composable smooth maps, and such that ∗ M is the identity. We further assume that C l ( M ) = 0 for each l > and each manifold M .Suppose that ( X = { X k } k ∈ N , d i , s i ) is a simplicial manifold. We obtain a family of cochaincomplexes { C ( X k ) } k ∈ N , and like in Construction 2.14 the face and degeneracy maps of X inducemorphisms of cochain complexes ∂ i = d ∗ i : C ( X k − ) → C ( X k ) and σ i = s ∗ i : C ( X k +1 ) → C ( X k ) ,satisfying the cosimplicial identities (2.13). We can now apply Construction 2.14 to each level of thesechain complexes: thereby we obtain, for each l ∈ Z , a cochain complex ( C l ( X ) , δ ) (which is trivial foreach l > ). In fact, since this construction is compatible with the differential d on C , we even obtain a double cochain complex ( C ( X ) , d, δ ) . (Note that this means, in particular, that d ◦ δ = δ ◦ d .) Finally,we take the total complex (Tot( C ( X )) , D) of ( C ( X ) , d, δ ) : this is the (ordinary) cochain complex ofabelian groups with Tot (cid:0) C ( X ) (cid:1) l = M p + q = l C p ( X q ) and D | C p ( X q ) = d + ( − p δ . This construction allows us to define not only Čech hypercohomology, but also derived closed forms(see Section 4.2). ⊳ Definition 2.48.
Let M be a manifold and n ∈ N . The degree- n Deligne complex of M is the cochaincomplex of abelian groups ˆB n ∇ U (1)( M ) = (cid:0) C ∞ (cid:0) M, U (1) (cid:1) Ω ( M ; i R ) · · · Ω n ( M ; i R ) 0 d log d d (cid:1) where d log( g ) = g ∗ µ U (1) , and where C ∞ (cid:0) M, U (1) (cid:1) lies in degree − n . The degree- n Čech-Delignecohomology groups of M are given by ˇ H k ( M ; ˆB n ∇ U (1)) = colim U H k (cid:0) Tot( ˆB n ∇ U (1)( ˇ C U )) , D (cid:1) , U runs over all open coverings of M . The n -th differential cohomology group of M is ˆH n ( M ; Z ) = ˇ H ( M ; ˆB n − ∇ U (1)) . Remark 2.49.
There is a way of defining Deligne cohomology without using Čech nerves; this usesthe concept of hypercohomology of complexes of sheaves of abelian groups; for details and background,see e.g. [Bry08, Voi07]. ⊳ Note that there are slightly different conventions in some of the literature, amounting to a degree-shift of one in the definition of ˆH n ( M ; Z ) . The following general statement allows us to computeČech-Deligne cohomology groups: Proposition 2.50. [EZT14, Thm. 2.8.1]
Let C be a complex of sheaves of abelian groups on M fd such that C k = 0 for each k < . Let U be an open covering of a manifold M such that each C k induces an acyclic sheaf on each finite intersection U a ··· a m of patches of U (i.e. the complex C ( U a ··· a m ) has non-trivial cohomology at most in degree zero). Then there is a canonical isomorphism between H n (Tot( C ( ˇ C U )) , D) and the hypercohomology of C on M . Corollary 2.51.
Let U be a differentiably good open covering of a manifold M . Then, there is acanonical isomorphism H k (cid:0) Tot( ˆB n ∇ U (1)( ˇ C U )) , D (cid:1) ∼ = ˇ H k ( M ; ˆB n ∇ U (1)) . Proof.
Each of the level sheaves of ˆB n ∇ U (1)[ − n ] (where [ − n ] denotes the degree shift by − n ) is acyclicon each U a ··· a m by the Poincaré Lemma and the long exact sequence in cohomology associated to theshort exact sequence Z → R → U (1) . Thus, the claim follows from Proposition 2.50 and the fact thaton paracompact spaces Čech hypercohomology agrees with hypercohomology [Bry08, Thm. 1.3.13].From our investigation of local bundle gerbes, we see that a local gerbe with connection definedwith respect to a differentiably good open covering U of M is the same as a 0-cocycle ( g, A, B ) ∈ Z (cid:0) Tot( ˆB ∇ U (1)( ˇ C U )) , D (cid:1) , and two such local gerbes are isomorphic (via a parallel isomorphism) precisely if their cocycles differby a coboundary. One can check that restricting cocycles along refinements of differentiably goodopen coverings does not change their class in Čech-Deligne cohomology; thus we arrive a the followingcrucial classification theorem for gerbes with connection on a manifold M : Theorem 2.52. [MS00]
Let M be a manifold. There are isomorphisms of abelian groups D : (cid:0) G rb ∇ ( M ) , ⊗ (cid:1) / {par. iso} ∼ = −→ ˇ H ( M ; ˆB ∇ U (1) (cid:1) ∼ = ˆH ( M ; Z ) , D : (cid:0) G rb ( M ) , ⊗ (cid:1) / {iso} ∼ = −→ H (cid:0) M ; U (1) (cid:1) ∼ = H ( M ; Z ) . The second isomorphism is obtained in complete analogy with the first, but replacing the com-plex ˆB n ∇ U (1) by the complex ˆB n U (1) = U (1)[ n ] ; for a manifold M , the complex ( ˆB n U (1))( M ) has C ∞ ( M, U (1)) in degree − n and is trivial in all other degrees. An analogous theorem for gerbes in thesense of Giraud has been proven in [Gaj97]. Definition 2.53.
For G ∈ G rb ( M ) , we call the class D( G ) the Dixmier-Douady class of G . For G ∈ G rb ∇ ( M ) a gerbe with connection, we call the class D ( G ) associated to G in ˆH ( M ; Z ) the Deligneclass of G . We also write D( G ) for the Dixmier-Douady class of G with its connection forgotten. That is, each C k is a sheaf with respect to open coverings of any manifold.
15e summarise the key technical properties of Čech-Deligne cohomology for the context of gerbes:
Proposition 2.54. [Bry08]
For any manifold M and n ≥ there are the following exact sequences ofabelian groups: n − , Z ( M ; i R ) Ω n − ( M ; i R ) ˆH n ( M ; Z ) H n ( M ; Z ) 00 H n − ( M ; U (1) δ ) ˆH n ( M ; Z ) Ω n cl , Z ( M ; i R ) 0 triv c curv (2.55)Here, U (1) δ is the sheaf of locally constant U (1) -valued functions. Let us look at the se-quences (2.55) for gerbes, i.e. for n = 3 . In the first sequence, the map triv sends a 2-form ρ tothe Deligne class D ( I ρ ) of the trivial gerbe with connection ρ . The second map c , also called the characteristic class or the Chern class , takes the Deligne class D ( G ) of a gerbe G with connection andsends it to the Dixmier-Douady class D( G ) of the underlying gerbe without its connection. The firstmap in the second sequence takes a Čech 2-cocycle g ∈ Z (Tot( ˆB U (1) δ ( ˇ C U )) , D) of locally constant U (1) -valued functions and sends it to the Deligne class of the local gerbe ( g, , . The map curv sends D ( G ) to the 3-form curv( G ) . Proposition 2.56.
For any n ∈ N , the exact sequences (2.55) fit into the differential cohomologyhexagon [SS08], which is the commutative diagram of abelian groups Ω n ( M ;i R )Ω n cl , Z ( M ;i R ) Ω n +1cl , Z ( M ; i R )H n ( M ; i R δ ) ˆH n +1 ( M ; Z ) H n +1 ( M ; i R δ )H n ( M ; U (1) δ ) H n +1 ( M ; Z )0 0 dtriv f dRdRexp ∗ curv c β (i · ( − )) ∗ The morphisms dR and f dR arise from the de Rham isomorphism H • ( M ; R δ ) ∼ = H • dR ( M ; R ) from sheafto de Rham cohomology, and where β is the usual Bockstein homomorphism. The top-left-to-bottom-right short exact sequence implies:
Corollary 2.57.
Suppose G , G ∈ G rb ∇ ( M ) are two bundle gerbes with connection such that D( G ) = D( G ) ; that is, G and G are isomorphic as gerbes without connection (see Theorem 2.52).Then, there exists an isomorphism E : G → G of gerbes with connection. In particular, there is a parallel isomorphism E : G ⊗ I − curv( E ) −→ G (by Proposition 2.43). Corollary 2.58.
If there exists any morphism E : G → G between two gerbes in G rb ∇ ( M ) , then thedifference D( G ) − D( G ) is a torsion element in H ( M ; Z ) .Proof. This is a direct consequence of Lemma 2.35 and Theorem 2.52.16 efinition 2.59.
Let G ∈ G rb ∇ ( M ) . A trivialisation of G is a parallel isomorphism T : I ρ → G forsome ρ ∈ Ω ( M ; i R ) . We call a bundle gerbe G ∈ G rb ∇ ( M ) trivialisable if it admits a trivialisation.If T ′ : I ρ ′ → G is a second trivialisation, an isomorphism of trivialisations of G is a unitary parallel2-isomorphism ψ : T → T ′ in G rb ∇ ( M ) . We let Triv( G ) denote the groupoid of trivialisations of G . Proposition 2.60.
Let G ∈ G rb ∇ ( M ) be a gerbe with connection on M .(1) G is trivialisable (i.e. Triv( G ) = ∅ ) precisely if D( G ) = 0 in H ( M ; Z ) .(2) If T : I ρ → G and T ′ : I ρ ′ → G are trivialisations, then there exists a canonical isomorphism oftrivialisations Hom( T , T ′ ) ⊗ T ∼ = −→ T ′ (where we have used Construction 2.39), and we have curv (cid:0) Hom( T , T ′ ) (cid:1) = ρ ′ − ρ ∈ Ω , Z ( M ; i R ) . Proof.
Claim (1) follows readily from Proposition 2.56. The isomorphism in claim (2) is either obtaineddirectly from the construction of
Hom( T , T ′ ) , or, more abstractly, from the fact that Hom provides aninternal hom functor for the tensor product of gerbe morphisms [BSS18, Bun17]. The curvature identityis again a direct consequence of Proposition 2.56, or can be seen explicitly from the construction of
Hom( T , T ′ ) and the fact that T and T ′ are parallel morphisms of gerbes. Remark 2.61.
Because of Corollary 2.58, attempts to allow for infinite-dimensional bundles have beenmade (see e.g. [BCM +
02, Sec. 7] and [CW08]), but Kuiper’s Theorem (the contractibility of the unitarygroup of an infinite-dimensional separable Hilbert space) prevents this from yielding good categories ofmorphisms [Bun17, Prop. 4.91]. One can consider Hilbert bundles of reduced structure group instead,but this leads to conflicts with tensor products. Presumably, circumventing the torsion constraint willinvolve passing from bundles to sheaves of modules over C ∞ ( M ) , but we leave this to future work. ⊳ Remark 2.62.
Morphisms of gerbes are also called bundle gerbe (bi-)modules [Wal07a], or twisted vec-tor bundles [BCM + +
02, CW08],at least when the bundle gerbe represents a torsion class in H ( M ; Z ) ; otherwise, one has to use( Z -graded) ∞ -dimensional Hilbert bundles (with reduced structure groups) in place of hermiteanvector bundles as morphisms of bundle gerbes, which works fine for the purposes of twisted K-theory [BCM +
02, CW08, AS04, Kar12]. ⊳ We conclude this section with a couple of remarks on Deligne complexes, homotopy theory, andhigher gerbes. These remarks are not relevant for the remaining sections of this article, but we hopethat they provide an entry point to the extensive works on higher gerbes by Schreiber and collaborators(see, for instance, [Schb, FSS15, FSS12]).
Definition 2.63. An n -gerbe on a manifold M is a cocycle ( g, A , . . . , A n +1 ) ∈ Z (cid:0) Tot( ˆB n ∇ U (1)( ˇ C U )) , D (cid:1) . It would be tedious to define morphisms and higher morphisms of n -gerbes by hand. However,there exists a general construction, called the Dold-Kan correspondence , which turns a complex ofabelian groups into a simplicial set (i.e. a simplicial object in the category of sets and maps in thesense of Definition 2.3). The simplicial set we obtain under this correspondence from the complex τ ≥ (Tot( ˆB n ∇ U (1)( ˇ C U ) , D) can be viewed as an ( n +1) -groupoid of n -gerbes with connection on M and their parallel morphisms. ( τ ≥ denotes the truncation to non-negative degrees.) This is made17recise by the fact that a particular type of simplicial sets, called Kan complexes , provide a modelfor ∞ -groupoids. Note that it is essential to not evaluate ˆB n ∇ U (1) on the manifold M itself, butinstead on the Čech nerve of a differentiably good open covering U of M . There is a formal reasonfor this, stemming from homotopy theory (one needs to perform a cofibrant replacement of M in alocal projective model category of simplicial presheaves). Abstract homotopy theory and ∞ -categoriesare the basis for the theory of higher gerbes developed in [Lur09, Schb]; good references on homotopytheory and higher categories include [Qui67, DS95, Hov99, Lur09, Rie14, Cis19]. Remark 2.64.
For 2-gerbes, a hands-on definition in the spirit of Section 2.1 and 2.2 is still feasible;one follows the same principle as in those sections, using gerbes with connection as local transitionfunctions to define 2-gerbes with connection. For background and examples, we refer the readerto [Ste04, CJM +
05, Wal10]. ⊳ In this section, we exhibit two examples of bundle gerbes which appear frequently. For various furtherexamples of gerbes, we refer the interested reader to [Hit01, Cha98, MS00, GR04, Mei03, Wal10,Wal16b, BS17] and references therein.
Lifting bundle gerbes
Let U (1) → G → H be a central extension of Lie groups. In particular, G → H is a principal U (1) -bundle. The fact that it is also a group extension can be rephrased asfollows: there exists an isomorphism µ G : d ∗ G ⊗ d ∗ G ∼ = −→ d ∗ G of U (1) -bundles over H , which satisfies a verbatim analogue of equation (2.19) over H . Observe thatthese data look suspiciously like a bundle gerbe defined using the simplicial manifold B H in place ofthe Čech nerve of a surjective submersion Y → M . Remark 2.65.
If we were able to establish the simplicial manifold B H as the the Čech nerve of acertain morphism π : ∗ = B H → N , we would see that a U (1) -extension of H is the same as a gerbeon N with respect to π . This can be made precise using a theory of principal ∞ -bundles and theirclassifying spaces [Lur09, NSS15, Buna]; in this theory, a U (1) -extension of H is the same as a gerbe onthe classifying space of H -bundles, and it is a shadow of this fact that we are observing here. However,introducing principal ∞ -bundles would lead us too far afield here. ⊳ Suppose, P → M is a principal H -bundle on a manifold M . This gives rise to a Čech nerve,which is the augmented simplicial manifold ˇ CP → M . By the principality of the H -action on P , thiscomes with a canonical smooth map h : ˇ CP → B H . In particular, h is fully determined by its level-onecomponent h : ˇ CP → H , satisfying d ∗ h · d ∗ h = d ∗ h (cf. Section 2.1). Let L = h ∗ G be line bundleassociated to the pullback of the U (1) -bundle G → H along the map h , and let µ = h ∗ µ G . Then, G = ( Y → M, L, µ ) defines a bundle gerbe on M , called the lifting bundle gerbe of P . Theorem 2.66. [Mur96]
The principal H -bundle P → M is a reduction of a principal G -bundle ifand only if the gerbe G is trivialisable. If the principal H -bundle P → M carries a connection, one can extend the construction of thelifting gerbe to also include a connection [Gom03].18 up product bundle gerbes We now consider a manifold M , a hermitean line bundle L → M withconnection, and a smooth map f : M → S . We can understand L as representing a class in ˆH ( M ; Z ) and f as representing a class in ˆH ( M ; Z ) (since S ∼ = U (1) as manifolds). From these data one canconstruct a gerbe with connection on M which represents the cup-product [ L ] ∪ [ f ] ∈ ˆH ( M ; Z ) [Joh].This construction proceeds as follows: let R → Z , r exp(2 π i r ) denote the canonical Z -principalbundle over S . Let p : Y = f ∗ R → M denote the pullback of this bundle along f . Then, there is aninduced Z -action on Y , and there is a canonical diffeomorphism of simplicial manifolds (i.e. a morphismof simplicial manifolds which is a diffeomorphism in each degree) ( f ∗ R ) // Z −→ ˇ C ( f ∗ R ) , where we have used the notation from Example 2.8. An element in (( f ∗ R ) // Z ) n = ( f ∗ R ) × Z n is atuple ( x, r, k , . . . , k n ) , where x ∈ M , r ∈ R , and k , . . . , k n ∈ Z , and where f ( x ) = exp(2 π i r ) . Theabove map sends ( x, r, k , . . . , k n ) to (( x, r ) , ( x, r + k ) , . . . , ( x, r + k n )) ∈ ( f ∗ R ) [ n +1] . We define a linebundle with connection ( p ∗ L ) − Z on (( f ∗ R ) // Z ) = ( f ∗ R ) × Z by ( p ∗ L ) − Z | ( f ∗ R ) ×{ k } = ( ( p ∗ L ∨ ) k , k > , ( p ∗ L ) | k | , k ≤ , where for any line bundle with connection J , we understand J to be the trivial line bundle with trivialconnection. Further, we define an isomorphism of line bundles over (( f ∗ R ) // Z ) : µ : d ∗ ( p ∗ L ) − Z ⊗ d ∗ ( p ∗ L ) − Z −→ d ∗ ( p ∗ L ) − Z , consisting of the canonical isomorphisms L k | x ⊗ L k | x −→ L k + k | x , for x ∈ M , k , k ∈ Z . Finally, wedefine B ∈ Ω ( f ∗ R ; i R ) as B = r · p ∗ curv( L ) (recall that (( f ∗ R ) × Z ) = f ∗ R ). Then, we have curv (cid:0) ( p ∗ L ) − Z (cid:1) ( x,r,k ) = − k p ∗ curv( L ) | ( x,r ) = B | ( x,r ) − B | ( x,r + k ) = − ( δB ) | ( x,r,k ) . Thus, G = ( p : f ∗ R → M, ( p ∗ L ) − Z , µ, B ) defines a bundle gerbe with connection on M , called the cupproduct gerbe of L and f . Using the identification S ∼ = U (1) , its curvature is curv( G ) = f ∗ µ U (1) ∧ curv( L ) . Vector bundles with connection have a parallel transport along smooth paths, producing holonomiesaround smooth loops. Locally, parallel transports are built directly from connection 1-forms. Con-nections on gerbes consist of a 1-form and a 2-form (see Section 2.4); one could therefore ex-pect connections on gerbes to have holonomies around one- and two-dimensional objects. Thisis indeed the case; such holonomies and parallel transports have been investigated, for instance,in [Gaw88, GR02, CJM02, MP02, BS07, SW09, MP10, Wal18, BMS]. Here, we provide a modern,field-theoretic perspective on these holonomies on surfaces with and without boundaries, using theresults of Sections 2.
Let M be a manifold carrying a gerbe G with connection. Let Σ be a closed oriented surface, and let σ : Σ → M be a smooth map. By Theorem 2.26, we can pull G back along σ to obtain a gerbe σ ∗ G A manifold N is closed if it is compact and satisfies ∂N = ∅ . Σ . As H (Σ; Z ) = 0 , by Proposition 2.60 there exists a trivialisation T : I ρ → σ ∗ G (see also Definition 2.59). Again by Proposition 2.60, for any other trivialisation T ′ : I ρ ′ → σ ∗ G , wehave ρ ′ − ρ ∈ Ω , Z (Σ; i R ) . Thus, the following is well-defined: Definition 3.1.
Let M be a manifold and let G ∈ G rb ∇ ( M ) . Let Σ be a closed, oriented surface,let σ : Σ → M be a smooth map, and let T : I ρ → σ ∗ G be any trivialisation of σ ∗ G . The (surface)holonomy of G around (Σ , σ ) is hol ( G ; σ ) = exp (cid:18) − Z Σ ρ (cid:19) ∈ U (1) . (3.2)This definition of surface holonomy goes back to [CJM02], which succeeded earlier construc-tions [Wit83, Alv85, Gaw88, GR02]. Definition 3.1 makes full use of the 2-categorical theory of gerbesand does not rely on extensions of σ to 3-manifolds N with ∂N = Σ , or on combinatorial decompo-sitions of Σ . In physics, it describes the Wess-Zumino-Witten action , which is part of the action ofbosonic and fermionic strings. One now readily proves:
Proposition 3.3.
Let M be a manifold and let G ∈ G rb ∇ ( M ) .(1) If N is a compact oriented 3-manifold with ∂N = Σ and f : N → M is a smooth map with f | ∂N = σ , then hol ( G ; σ ) = exp (cid:18) − Z N curv( G ) (cid:19) . (2) G has trivial holonomy around every closed surface (Σ , σ ) if and only if G is flat, i.e. curv( G ) = 0 . Remark 3.4.
Finding a geometric origin for the Wess-Zumino-Witten action in conformal field theorywas one of the driving forces behind the development of gerbes with connection, going back to [Wit83,Gaw88]. For further treatments of gerbes and D-branes from the perspective of conformal field theory,see, for instance, [GR02, Wal07a, FNSW10] and references therein. Finally, defects in conformal fieldtheories and gauging of sigma-models are also described by gerbes [FSW08, GSW11, GSW13]. ⊳ Remark 3.5.
There exists an extension of surface holonomy of gerbes to unoriented surfaces. Thisrequires a certain type of compatibility of the gerbe with the canonical involution on the orientationdouble cover of a surface; this compatibility is additional data on the gerbe called a
Jandl structure .For references on Jandl gerbes, unoriented surface holonomy and applications to Wess-Zumino-Wittentheory, we refer the reader to [SSW07]. ⊳ If Σ is an oriented surface with ∂ Σ = ∅ , the surface holonomy of G ∈ G rb ∇ ( M ) around a smooth map σ : Σ → M is no longer well-defined as an element in U (1) . Instead, if T : I ρ → σ ∗ G and T ′ : I ρ ′ → σ ∗ G are two trivialisations, we have hol ( G ; σ, T ′ ) = hol ( G ; σ, T ) exp (cid:18) − Z Σ ρ ′ − ρ (cid:19) . Here, hol ( G ; σ, T ) is the surface holonomy of G around σ as in (3.2), computed with respect to T .Observe that the error term on the right-hand side still vanishes if there is an isomorphism T ∼ = T ′ oftrivialisations (by Proposition 2.40).The connected components of ∂ Σ are circles (up to orientation-preserving diffeomorphism). To bet-ter understand what happens at the boundary, we thus consider smooth maps γ : S → M . By Proposi-tion 2.60 there exist trivialisations T : I → γ ∗ G . (Here, we must have ρ = 0 , since ρ ∈ Ω ( S ; i R ) = 0 .20t follows from Construction 2.36, Proposition 2.56, and Proposition 2.60 that any choice of trivialisa-tion T : I → γ ∗ G induces a canonical bijection π (cid:0) Triv( γ ∗ G ) (cid:1) := { trivialisations of γ ∗ G } / iso. ∼ = −→ HLB un ∇ ( S ) / iso. =: π (cid:0) HLB un ∇ ( S ) (cid:1) ∼ = U (1) , ( T ′ : I → γ ∗ G ) hol (cid:0) Hom( T , T ′ ) (cid:1) , (3.6)where the holonomy on the right-hand side is that of a line bundle with connection on S . At thesame time, the tensor product from Construction 2.36 induces an action of U (1) ∼ = π ( HLB un ∇ ( S )) on π (Triv( γ ∗ G )) . In fact, using Proposition 2.60, we derive Proposition 3.7. [Wal16b]
The set π (Triv( γ ∗ G )) is a torsor over U (1) ∼ = π ( HLB un ∇ ( S )) . To any U (1) -torsor P , we can associate a complex line by viewing P as a principal U (1) -bundleover the point and performing the associated bundle construction. In our situation, this yields thecomplex line T G | γ := π (cid:0) Triv( γ ∗ G ) (cid:1) × U (1) C . (3.8)An element in T G | γ is thus an equivalence class [[ T ] , z ] of an isomorphism class [ T ] ∈ π (Triv( γ ∗ G )) and a number z ∈ C . Under the isomorphism π ( HLB un ∇ ( S )) ∼ = U (1) , the equivalence relation readsas (compare Section 2.1) (cid:2) [ T ] , z (cid:3) = (cid:2) [ T ⊗ J ] , hol ( J ) − z (cid:3) , for [ J ] ∈ π ( HLB un ∇ ( S )) .Let Σ be a compact oriented surface with a smooth map σ : Σ → M . Suppose the boundary of Σ is partitioned into incoming circles S in,a ⊂ ∂ Σ , for a = 1 , . . . , N in , and outgoing circles S out,b ⊂ ∂ Σ ,for b = 1 , . . . , N out . We set γ in,a := σ | S in,a and γ out,b := σ | S out,b . Suppose further that the incoming circles are endowed with the opposite orientation of that inducedfrom Σ , and that the outgoing circles carry the orientation induced from Σ . Consider a vector ξ = N in O a =1 (cid:2) [ T a ] , z a (cid:3) = z · N in O a =1 (cid:2) [ T a ] , (cid:3) ∈ N in O a =1 T G | γ in,a , where T a : I → γ ∗ in,a G is a trivialisation . Since σ ∗ G is trivialisable over Σ , we may even choose atrivialisation T : I ρ → σ ∗ G and assume that T a = T | S in,a , for a = 1 , . . . , N in , where S in,a ⊂ Σ is the a -th incoming boundary circle. Define a new vector Z G [Σ , σ ; T ]( ξ ) := z · exp (cid:18) − Z Σ ρ (cid:19) · N out O b =1 (cid:2) [ T b ] , (cid:3) ∈ N out O b =1 T G | γ out,b , (3.9)where we have set T b = T | S out,b . Proposition 3.10.
The vector Z G [Σ , σ ; T ]( ξ ) in (3.9) is independent of the choice of trivialisation T : I ρ → σ ∗ G over Σ . Technically, in order to match the definition of
T G | γ , we have to choose diffeomorphisms S in,a ∼ = S and then checkthat the constructions are independent of that choice. We will not go into these details here, but instead refer the readerto [BWa]. roof. Let T ′ : I ρ ′ → σ ∗ G be another trivialisation. By (3.6) and (3.8), we have ξ = z · N in O a =1 (cid:2) [ T a ] , (cid:3) = z · N in Y a =1 hol (cid:0) Hom( T a , T ′ a ); S in,a (cid:1) − · N in O a =1 (cid:2) [ T ′ a ] , (cid:3) = z ′ · N in O a =1 (cid:2) [ T ′ a ] , (cid:3) . Applying the construction (3.9) to this, using the trivialisation T ′ in place of T , we obtain Z [Σ , σ ; T ′ ]( ξ ) = z ′ · exp (cid:18) − Z Σ ρ ′ (cid:19) · N out O b =1 (cid:2) [ T ′ b ] , (cid:3) = z · exp (cid:18) − Z Σ ρ ′ (cid:19) · N in Y a =1 hol (cid:0) Hom( T a , T ′ a ); S in,a (cid:1) − · N out Y b =1 hol (cid:0) Hom( T ′ b , T b ); S out,b (cid:1) − · N out O b =1 (cid:2) [ T b ] , (cid:3) = z · exp (cid:18) − Z Σ ρ ′ (cid:19) · N in Y a =1 hol (cid:0) Hom( T a , T ′ a ); S in,a (cid:1) − · N out Y b =1 hol (cid:0) Hom( T b , T ′ b ); S out,b (cid:1) · N out O b =1 (cid:2) [ T b ] , (cid:3) = z · exp (cid:18) − Z Σ ρ ′ (cid:19) · hol (cid:0) Hom( T , T ′ ); ∂ Σ (cid:1) · N out O b =1 (cid:2) [ T b ] , (cid:3) = z · exp (cid:18) − Z Σ ρ (cid:19) · N out O b =1 (cid:2) [ T b ] , (cid:3) . Here we have used that
Hom( T , T ′ ) is the dual line bundle to Hom( T ′ , T ) (which one can see fromits construction, for instance—see Section 2.2), Proposition 2.60(2), and the particular choices oforientations on the incoming and outgoing boundary circles. Corollary 3.11.
Let G ∈ G rb ∇ ( M ) . For any compact, oriented surface Σ whose boundary is parti-tioned into incoming and outgoing boundary components as above, and which is endowed with smoothmap σ : Σ → M , we obtain a linear map Z G [Σ , σ ] : N in O a =1 T G γ in,a −→ N out O b =1 T G | γ out,b . If ∂ Σ = ∅ , this reproduces the surface holonomy of G from Section 3.1 as a linear map C → C . One can show that the linear maps Z G [Σ , σ ] have various useful properties: for instance, theydepend only on the thin homotopy class of σ , but at the same time depend on σ smoothly in a precisesense. Most importantly, they are compatible with gluing of surfaces along boundary components, andthus assemble into what is called a smooth functorial field theory on M (in the sense of [ST11]). Werefer the reader to [BWb, BWa] for the full constructions and details; here, we shall restrict ourselvesto stating the main result: Theorem 3.12. [BWa]
Any bundle gerbe G ∈ G rb ∇ ( M ) with connection on M gives rise to a two-dimensional, oriented, smooth functorial field theory on M . This field theory depends functorially on G ,and it admits an extension to an open-closed field theory in the presence of D-branes (see Section 3.3). This extends earlier results in this direction in [Gaw88, BTW04, Pic04]. Let us illustrate someaspects of the results in Theorem 3.12. First, consider the complex line
T G | γ from (3.8) we assigned to Two smooth maps of manifolds f, g : N → M are thinly homotopic if there exists a homotopy h : [0 , × N → M between them whose differential h ∗ has rank at most dim( N ) everywhere on [0 , × N . γ in M . Varying the curve γ , we have the following statement: let LM be the space ofsmooth maps γ : S → M . This is no longer a manifold, but one can describe it very conveniently as a diffeological spaces [IZ13, BH11] (see also [Wal12, Bunb] for background on diffeological vector bundles).The main idea behind diffeological spaces is to study spaces N not by locally defined diffeomorphismsto euclidean space, but instead to use all maps from euclidean spaces to N which satisfy a certainsmoothness condition. Many infinite-dimensional spaces which appear in geometry, such as mappingspaces and diffeomorphism groups of manifolds, can be naturally described as diffeological spaces. Theorem 3.13. [Wal16b]
The complex lines { T G | γ } γ ∈ LM assemble into a diffeological hermitean linebundle T G → LM over the free loop space LM of M . Definition 3.14.
The line bundle
T G on LM is called the transgression line bundle of G ∈ G rb ∇ ( M ) .The transgression line bundle T G → LM comes with a natural parallel transport, which can beseen as induced from the above field-theory construction: a smooth path Γ : [0 , → LM is equivalentto a smooth map Γ ⊣ : [0 , × S → M . By Corollary 3.11 it gives rise to an isomorphism pt G Γ := Z G (cid:2) [0 , × S , Γ ⊣ (cid:3) : T G | Γ(0) −→ T G | Γ(1) . One can show that this map is compatible with concatenation of paths ; that is, for concatenatablepaths Γ , Γ : [0 , → LM , we have pt G Γ ◦ pt G Γ = pt G Γ ∗ Γ . The maps pt G endow the line bundle T G with a connection [Wal16b]. The parallel transport pt G has several additional properties; it is compatible with fusion of loops, depends only on the thinhomotopy class of Γ , and for any two thin paths Γ , Γ : [0 , → LM which agree at and , wehave pt G Γ = pt G Γ [Wal16b, BWb]. Here, a path Γ : [0 , → LM is thin if Γ ⊣∗ has rank at most oneeverywhere on [0 , × S . Let HLB un ∇ fus ( LM ) denote the symmetric monoidal groupoid of hermiteanline bundles with connection on LM which have the above additional properties. Theorem 3.15. [Wal16b]
There is an equivalence of symmetric monoidal groupoids h (cid:0) G rb ∇ par iso ( M ) , ⊗ (cid:1) ≃ (cid:0) HLB un ∇ fus ( LM ) , ⊗ (cid:1) , where on the left-hand side G rb ∇ par iso ( M ) is the 2-groupoid of gerbes with connection on M and onlytheir parallel isomorphisms and 2-isomorphisms. The h denotes the identification of 2-isomorphicisomorphisms, making the left-hand side into a groupoid. Remark 3.16.
It is possible to obtain from the transgression of gerbes a formalism of dimensionalreduction, which turns a gerbe with connection over an oriented circle bundle P → M into a principal U (1) -bundle P ′ → M [Bun17, BS17] (see [BWa] for the equivariance of T G with respect to diffeomor-phisms of S ). We expect that this is part of an explicit construction of topological T-duality in thepresence of generic H -flux (see e.g. [BEM04, NW20, BN15]). ⊳ In this section we briefly survey some further aspects of the surface holonomy and transgression con-structions for gerbes with connection. The phrase surface holonomy of a gerbe (3.2) suggest that thisis the holonomy of a certain parallel transport. Technically, one needs to demand sitting instants normal to the boundary, or talk about cutting paths instead ofconcatenating. Details for the first approach can be found in [BWb, Wal16b].
23n some sense, this is indeed the case: if
Σ = T = ( S ) is a 2-torus, then a smooth map σ : T → M is equivalent to a loop σ ⊢ : S → LM , and we have hol ( G ; σ ) = hol ( T G ; σ ⊢ ) . Here, the holonomy on the left-hand side is the surface holonomy of the gerbe G ∈ G rb ∇ ( M ) , whereason the right-hand side we have the (ordinary) holonomy of the transgression line bundle T G → LM .However, a full parallel transport on a gerbe G ∈ G rb ∇ ( M ) should assign an isomorphism pt G ,γ : G | γ (0) → G | γ (1) to every smooth path γ in M , such that pt G depends smoothly on γ . Furthermore, for any smoothhomotopy h : γ ⇒ γ ′ of paths, we should specify how the parallel transport changes; that is, any suchhomotopy should induce a 2-isomorphism pt G ,h : pt G ,γ → pt G ,γ ′ . These constituents of the parallel transport need to satisfy various further conditions regarding con-catenations and thin homotopies. This notion of parallel transport for gerbes with connection has beendeveloped in [BMS], extending the ideas in [BMS19].In particular, let γ : S → M be a smooth loop in M . It induces an automorphism hol ( G ; γ ) = pt G ,γ of G | γ (0) , which is a gerbe on the one-point manifold. The category of automorphisms of G | γ (0) iscanonically equivalent to the groupoid of one-dimensional hermitean vector spaces. On can show thatthere is a canonical isomorphism [BMS, Prop. 4.19] hol ( G ; γ ) ∼ = T G | γ , of such vector spaces, which depends smoothly on γ and functorially on G . Thus, we can interpret thetransgression line bundle of G simply as the holonomy of the parallel transport pt G of G .Another extension of the transgression formalism and surface holonomy facilitates the inclusion of D-branes . The idea that D-branes and Chan-Paton bundles in string theory with non-trivial B -field arerelated to gerbes and their morphisms goes back to [Kap00]. Further work on this geometric perspectiveon D-branes has been carried out in [CJM02, GR02, Gaw05, FSW08, FNSW10], for instance. Definition 3.17.
Let G ∈ G rb ∇ ( M ) . A D-brane for G is a pair ( Q, E ) of a submanifold Q ⊂ M anda morphism E : I → G | Q .Let ( Q i , E i ) i ∈ Λ be a collection of D-branes for G . Using the 2-categorical theory of gerbes fromSection 2, one can show that the transgression formalism extends from loops to open paths in thefollowing way: for each i, j ∈ Λ , let P ij M be diffeological space P ij M of smooth paths in M withsitting instants which start on Q i and end on Q j . From G and ( Q, E ) one can construct a hermiteanvector bundle with connection R ij → P ij M [BWb]. These vector bundles come with various structuremorphisms related to operations on the level of paths: for instance, there is a canonical linear map R jk | γ ⊗ R ij | γ −→ R ik | γ ∗ γ for each γ ∈ P ij M and γ ∈ P jk M such that their concatenation γ ∗ γ is defined. These linear mapsassemble into a smooth morphism of diffeological vector bundles.In fact, for any fixed collection of submanifolds { Q i ⊂ M } i ∈ Λ , the vector bundles R ij and theirstructure morphisms depend functorially on the D-branes E i supported on Q i , and one can even24econstruct the gerbe G and the D-branes E i —up to canonical isomorphism—from just knowing thetransgression line bundle T G → LM , the bundles R ij → P ij M , and their structure morphisms. Werefer the reader to [BWb] for the full statement and proof. From a physical perspective, this makesprecise how closed strings in M and open strings stretched between D-branes in M can detect the B -field and the twisted Chan-Paton bundles on the D-brane world volumes. Let ( M, ω ) be a symplectic manifold. Geometric quantisation of ( M, ω ) relies, first of all, on a real-isation of i ω as the curvature of a connection on a hermitean line bundle L on M . The (compactlysupported) square-integrable sections of L then form the prequantum Hilbert space of the system.Kostant-Souriau prequantisation sends functions f ∈ C ∞ ( M ) to operators O f on this Hilbert space,acting as O f = i ~ ∇ LX f ( − ) + f · ( − ) . Here, X f is the Hamiltonian vector field of f . This, however, doesnot represent the commutative algebra C ∞ ( M ) on the prequantum Hilbert space, but rather C ∞ ( M ) endowed with the (rescaled) Poisson bracket i ~ {− , −} induced by ω .In many geometric situations, symplectic forms are absent (for instance, on any odd-dimensionalmanifold). However, related features might still be present. For example, while the 2-sphere is sym-plectic, the 3-sphere is not, but instead it carries a closed 3-form which is non-degenerate in a certainsense (see below). The curvature of G ∈ G rb ∇ ( M ) is a closed 3-form curv( G ) on M with integer cycles,and an analogue of the first step in geometric prequantisation of 3-forms should be to realise a 3-formas the curvature of a gerbe with connection, instead of a line bundle. However, there are two differentconcepts of what a higher symplectic form should be; we will survey both of these and show how gerbesfit into both frameworks. In this section, we recall parts of the theory of n -plectic forms , focussing on the case of n = 2 . Forbackground and details we refer the reader to [CIdL99, Rog, Rog13, FRS14, FRS16]. Definition 4.1. An n -plectic form on a manifold M is a closed ( n +1) -form ω which is non-degenerate,meaning that the map ι ( − ) ω : T M → Λ n T ∗ M , X ω ( X, − , · · · , − ) is injective. Definition 4.2.
Let ( M, ω ) be a -plectic manifold. A prequantum bundle gerbe for ( M, ω ) is a gerbe G ∈ G rb ∇ ( M ) with connection on M such that curv( G ) = i ω . We call the choice of such a gerbe withconnection a prequantisation of ( M, ω ) . Example 4.3.
Let G be a compact, simple, simply connected Lie group with Lie algebra g . TheKilling form h− , −i and commutator on g induce a closed 3-form ω = h− , [ − , − ] i on G . This form is2-plectic and admits a prequantisation, given by the so-called basic gerbe [Mei03], or the tautologicalgerbe [Mur96]. ⊳ It follows from Theorem 2.52 and Proposition 2.56 that a 2-plectic manifold ( M, ω ) admits aprequantisation if and only if i ω ∈ Ω cl, Z ( M ; i R ) . The construction of the Poisson algebra of functions inthe symplectic case relies on the notion of Hamiltonian vector fields: given f ∈ C ∞ ( M ) , a Hamiltonianvector field for f is a vector field X f ∈ Γ( M ; T M ) such that ι X f ω = d f . In the n -plectic case,Hamiltonian vector fields cannot be associated to functions, but to (certain) ( n − -forms: Definition 4.4.
Let ( M, ω ) be an n -plectic manifold. A Hamiltonian n -form is an ( n − -form η ∈ Ω n ( M ) such that there exists a vector field X η ∈ Γ( M ; T M ) with ι X η ω = d η . We denotethe vector space of Hamiltonian ( n − -forms on M by Ω n − ( M ) .25ote that for n > the map ι ( − ) ω : T M → Λ n T ∗ M is generally not surjective, so that Ω n − ( M ) is,in general, a proper subspace of Ω n − ( M ) . However, since the map is injective, it follows that, for each η ∈ Ω n − ( M ) , the vector field X η with ι X η ω = d η is unique. We hence call X η the Hamiltonian vectorfield of η . The n -plectic version of the Poisson algebra of smooth functions on M is given as follows:we first recall the definition of L ∞ -algebras (also called strongly homotopy Lie algebras) [LS93, LM95],see also [Rog, Def. 3.7]. Definition 4.5. An L ∞ -algebra is a Z -graded vector space L with a collection { l k : L ⊗ k → L | k ∈ N } of skew-symmetric linear maps of degree | l k | = k − , satisfying the identity X i + j = m +1 X σ ∈ UnSh( i,m − i ) ( − σ ǫ ( σ ) ( − i ( j − l j (cid:0) l i ( v σ (1) , . . . , v σ ( i ) ) , v σ ( i +1) , . . . , v σ ( m ) (cid:1) = 0 for every m ∈ N . Here, UnSh( i, j ) is the set of ( i, j ) -unshuffles, where ǫ ( σ ) is the Koszul sign arisingfrom applying the permutation σ ∈ UnSh( i, j ) to the vectors v , . . . , v i + j , and where ( − σ is thedegree of the permutation σ . A Lie n -algebra is an L ∞ -algebra whose underlying graded vector spaceis concentrated in degrees , . . . , n − (in that case we obtain l k = 0 for all k > n + 1 ).One can check that l =: d is a differential, turning L into a chain complex, and that it is a gradedderivation with respect to the bracket l =: [ − , − ] . This bracket, however, does not satisfy the Jacobiidentity; instead the Jacobi identity is violated up to a coherent set of homotopies. Morphisms of L ∞ -algebras are more intricate to define; instead of doing this directly on the level of L ∞ -algebras,one usually passes to the coalgebra description of L ∞ -algebras: if L is a graded vector space, an L ∞ -algebra structure is equivalent to a choice of a codifferential on the (non-unital) coalgebra W • L [1] ofsymmetric tensor powers, and morphisms of L ∞ -algebras are most elegantly described as morphismsof the associated codifferential coalgebras; see [JRSW19, Appendix A] for a fully detailed account. Example 4.6.
A Lie 2-algebra consists of a 2-term chain complex L → L , a skew-symmetricbracket l = [ − , − ] : L ⊗ L → L and the skew-symmetric Jacobiator l = J ( − , − , − ) : L ⊗ → L ,which is precisely a chain homotopy of maps L ⊗ → L from x ⊗ y ⊗ z [ x, [ y, z ]] to the map x ⊗ y ⊗ z [[ x, z ] , z ] + [ y, [ x, z ]] , for x, y, z ∈ L . Finally, there is a compatibility relation between [ − , − ] and J (c.f [Rog, Eq. 3.10]).A morphism of Lie 2-algebras L → L ′ is a morphism of complexes φ : L → L ′ and a chain homotopy Φ of maps L ⊗ L → L ′ from x ⊗ y φ ([ x, y ]) to x ⊗ y [ φ ( x ) , φ ( y )] , satisfying a compatibility relation(see [Rog, Eq. 3.11]). Such a morphism is called a quasi-isomorphism if φ induces isomorphisms betweenthe homology groups of the complexes L and L ′ . For full details, see [BC04] or [Rog, Sec. 3.2.1], forinstance. ⊳ Theorem 4.7. [Rog, Thm. 3.14]
Let ( M, ω ) be an n -plectic manifold. There exists a Lie n -algebra L ∞ ( M, ω ) with • underlying graded vector space given by L = Ω n − ( M ) , L i = Ω n − − i ( M ) for i = 1 , . . . n − , and L i = { } otherwise, • differential l = d given by the de Rham differential on L i with i > , and • higher brackets given by l k ( α , . . . , α k ) = , | α ⊗ · · · ⊗ α k | > , ( − k +1 ι X α ∧···∧ X αk ω , | α ⊗ · · · ⊗ α k | = 0 , k even , ( − k − ι X α ∧···∧ X αk ω , | α ⊗ · · · ⊗ α k | = 0 , k odd . efinition 4.8. For an n -plectic manifold ( M, ω ) , we call L ∞ ( M, ω ) the Poisson Lie n -algebra asso-caited to ( M, ω ) .In the case where ( M, ω ) is a -plectic manifold which admits a prequantisation G ∈ G rb ∇ ( M ) ,one can associate to it the Lie 2-algebra of infinitesimal symmetries of the gerbe with connection G .It has recently been proven by Krepski and Vaughan [KV] that this Lie 2-algebra is equivalent to thePoisson Lie 2-algebra of ( M, ω ) , thus giving L ∞ ( M, ω ) a geometric description in terms of vector fieldson the prequantum gerbe. Similar results, though in a less geometric and more homotopy-theoreticflavour, have been obtained in [FRS14, FRS16].The explicit description of infinitesimal symmetries of gerbes in terms of local data goes backto [Col, FRS14, FRS16] and has recently been recast in global terms and the language of bundlegerbes in [KV]. This uses the theory of multiplicative vector fields on Lie groupoids, introducedin [MX98]. In particular, given a bundle gerbe G = ( π : Y → M, L, µ ) , we will from now on tradethe line bundle L for its underlying U (1) -bundle, which we denote by P (note that this neither losesnor adds information). The structure of the bundle gerbe gives rise to smooth maps s, t : P → Y and s : Y → P , and together with µ : d ∗ P ⊗ d ∗ P → d ∗ P , we obtain a Lie groupoid ( P ⇒ Y ) from thesedata. We shall not describe multiplicative vector fields on Lie groupoids in full generality here, butrestrict ourselves to the specific case of multiplicative vector fields on gerbes. Definition 4.9. [KV, Prop. 3.9, Cor. 3.18] Let G = ( π : Y → M, P, µ ) be a gerbe on M . A multiplica-tive vector field on G is a pair ξ = ( ξ , ξ ) , where ξ ∈ Γ( Y ; T Y ) and ξ ∈ Γ( P ; T P ) satisfying that ξ is U (1) -invariant, that d i ∗ ξ = ξ for i = 0 , , and that µ ∗| ( y ,y ,y ) ( ξ | ( y ,y ) ⊗ ξ | ( y ,y ) ) = ξ | ( y ,y ) for all ( y , y , y ) ∈ Y [3] = ˇ CY .Note that ξ = ( ξ , ξ ) is denoted ( e ξ, ˇ ξ ) in [KV]. We now consider connection-preserving multiplica-tive vector fields on G ∈ G rb ∇ ( M ) . The connection on G consists of a connection 1-form A ∈ Ω ( P ; i R ) and a curving B ∈ Ω ( Y ; i R ) . A multiplicative vector field ξ on G is connection preserving if thereexists α ∈ Ω ( Y ; i R ) such that ( £ ξ A, £ ξ B ) = (d α, p ∗ δα ) =: D α , where £ denotes the Lie derivative. This can be seen as the requirement that ( £ ξ A, £ ξ B ) be exact inthe complex obtained by applying Construction 2.47 to the two-term complex of sheaves Ω → Ω andthe simplicial manifold ˇ CY . Krepski and Vaughan then define an appropriate notion of morphismsbetween such connection-preserving vector fields, following [BEL]; these are obtained as certain sectionsof the Lie algebroid associated to the Lie groupoid ( P ⇒ Y ) ; for details we refer to [KV]. Proposition 4.10. [KV, Cor. 3.18, Prop. 4.8]
Let G ∈ G rb ∇ ( M ) . There exists a Lie 2-algebra X ∇ ( G ) whose level-zero part is the vector space of connection-preserving vector fields on G , with bracket (cid:2) ( ξ , ξ , α ) , ( ζ , ζ , β ) (cid:3) = (cid:0) [ ξ , ζ ] , [ ξ , ζ ] , £ ξ β − £ ζ α (cid:1) . The geometric interpretation of the Poisson Lie 2-algebra associated to the 2-plectic manifold ( M, ω ) is the following result. It uses the notion of butterflies between Lie 2-algebras. These providea weak notion of morphisms of Lie 2-algebras which describes the localisation of the 2-category ofLie 2-algebras at the quasi-isomorphisms; essentially, the existence of an invertible butterfly L → L ′ L ← J → J ← J → · · · ← J n → L ′ of quasi-isomorphisms of Lie2-algebras. For details, see [Noo13], where this theory was developed. Theorem 4.11. [KV, Thm. 5.1]
Let ( M, ω ) be a 2-plectic manifold with prequantisation G ∈ G rb ∇ ( M ) .Then, there is an invertible butterfly of Lie 2-algebras L ∞ ( M, ω ) → X ∇ ( G ) . Remark 4.12.
There is also a relation between certain infinitesimal symmetries of gerbes with con-nection and the Lie 2-algebra of sections of the Courant algebroid associated to ( M, ω ) . This was firstdescribed in works of Rogers [Rog] and worked out later in more detail in [Col, FRS14, FRS16]. Sucha relation is expected from the link between gerbes and Courant algebroids in generalised geometryobserved already in [Hit01, Hit03, Gua], for instance. ⊳ Remark 4.13.
In this section, we have focussed purely on the observables in geometric prequantisationof 2-plectic manifolds. It is a different—but related—problem to describe the states in this theory.Since the line bundle of geometric prequantisation is replaced by a gerbe with connection G on M ,one should expect the states to consist of sections of G . This idea was first investigated in [Rog], anda categorified Hilbert space of sections was constructed in [BSS18, Bun17]. However, these sections—and even more so how the observables act on them—are still not understood well enough, and thereis broad scope for further research. Let us point out the recent paper [Saf], which also makes progressin this direction. ⊳ Finally, we illustrate another approach to higher-degree generalisations of symplectic manifolds, goingby the name of shifted symplectic structures . Their introduction in [PTVV13] has lead to significantadvances in (derived) algebraic geometry. In differential geometry, (1-)shifted symplectic forms have sofar mostly appeared in the study of quasi-symplectic groupoids [BCWZ04, Xu04, LGX05], but see [Pri]for a perspective from derived differential geometry.Here, we consider shifted symplectic forms on simplicial manifolds [Get14]. Let X = { X k , d i , s i } be a simplicial manifold (cf. Section 2.1). If a 2-form ω on X k is not closed, its failure to be so couldbe an exact term with respect to the Čech differential, i.e. there could be a 3-form ω on X k − suchthat d ω = δω . The 3-form ω could now again fail to be closed up to a Čech-exact term, and soon. The central idea for shifted symplectic forms is to replace closed 2-forms by 2-forms closed up toa coherent chain of such higher-degree forms. Making this rigorous relies simply on Construction 2.47.In the presentation of this material, we heavily draw from [Get14].For k ∈ N , consider the cochain complex of sheaves of abelian groups τ ≥ k Ω • [ k ] = (cid:0) k Ω k +1 Ω k +2 · · · d d d (cid:1) . Note that Ω k sits in degree zero in this complex. We remark that τ ≥ k Ω • [ k ] is an injective resolutionof the sheaf Ω k cl of closed k -forms. Definition 4.14. [PTVV13, Get14] The complex of closed k -forms on a simplicial manifold X is A k cl ( X ) = Tot (cid:0) τ ≥ k Ω • [ k ]( X ) (cid:1) . A closed k -form of degree p on X is a degree- p cocycle ω ∈ Z p ( A k cl ( X )) .Explicitly, we obtain from Construction 2.47 that a closed k -form of degree p on X is a p -tuple ω = ( ω p + k , ω p + k − , . . . , ω k +1 , ω k ) with ω k + i ∈ Ω k + i ( X p − i ) , satisfying D ω = 0 , i.e. d ω p + k = 0 , ω k + i +1 + ( − i δω k + i = 0 , for i = 0 , . . . , p − ,δω k = 0 . Example 4.15.
Let G be a compact, simple, simply connected Lie group. Recall the simplicialmanifold B G from Example 2.9. Further, recall the closed 3-form ω ∈ Ω ( G ) from Example 4.3. Let µ G be the (left-invariant) Maurer-Cartan form on G , let µ G be the right-invariant Maurer-Cartan formon G , and define the 2-form ω = 12 h d ∗ µ G , d ∗ µ G i (4.16)on G = B G . Here, d and d are the face maps of the simplicial manifold B G . Then, ω = (0 , ω , ω ) ∈ Z (cid:0) A (B G ) (cid:1) is a closed 2-form of degree two on B G (see, for instance, [Wal10]). ⊳ For a simplicial manifold X , we can further define a tangent bundle (or tangent complex) in thefollowing sense: the tangent bundles { T X k → X k } k ∈ N each pull back to X along the compositions s ◦ · · · ◦ s : X → X k . As a consequence of the simplicial identities (2.2), the differentials of the faceand degeneracy maps of X induce on the collection of these pullbacks the structure of a simplicialvector bundle T s X on X ; that is, T s X is a collection { T sk X → X } k ∈ N of vector bundles on X ,endowed with morphisms of vector bundles ∂ i : T sk X → T sk − X and σ i : T sk X → T sk +1 X which satisfythe simplicial identities (2.2) (i.e. T s X is a simplicial object in the category of vector bundles on X ).We can now apply a dual version of Construction 2.14 to T s X (dual in the sense that cosimplicialobjects are replaced by simplicial ones, and cochain complexes by chain complexes) to obtain a chaincomplex T X = (cid:0) T s X T s X T s X · · · ∆ ∆ ∆ (cid:1) , where ∆ : ( T X ) k → ( T X ) k − is given by ∆ = P ki =0 ( − i ∂ i . We remark that the construction of T X given in [Get14] is not isomorphic to our construction here, but it is canonically quasi- isomorphic toour definition (this is due to the quasi-isomorphism between the normalised chain complex and theMoore complex associated to a simplicial abelian group [GJ09, Thm. III 2.4]). Definition 4.17.
Let X be a simplicial manifold. The chain complex ( T X, ∆) of vector bundles on X is called the tangent complex of X .Let ω = ( ω p , . . . , ω ) be a closed 2-form of degree p on a simplicial manifold X . Consider twoelements ξ, ξ ′ in the (fibre of the) tangent complex T | x X of X at x ∈ X whose degrees | ξ | =: a and | ξ | =: b satisfy a + b = p . We define the pairing ω ( ξ, ξ ′ ) := X ̺ ∈ Sh( a,b ) ( − ̺ ω (cid:0) ( σ ̺ ( n − ◦ · · · ◦ σ ̺ ( a ) ) ∗ ξ, ( σ ̺ ( a − ◦ · · · ◦ σ ̺ (0) ) ∗ ξ ′ (cid:1) , (4.18)where Sh( a, b ) is the set of ( a, b ) -shuffles. One can now show that this pairing is (graded) antisymmetric,and that ∆ is (graded) self-adjoint with respect to ω . In particular, the pairing (4.18) induces a pairingof degree two on the homology H • ( T X | x , ∆) for each x ∈ X [Get14]. Remark 4.19.
The explicit form for the pairing (4.18) and its properties arise form the Eilenberg-Zilber map, which induces a quasi-isomorphism T X ⊗ T X → T X ˜ ⊗ T X between the usual tensorproduct T X ⊗ T X of chain complexes and the level-wise tensor product, whose level- k vector space is ( T X ˜ ⊗ T X ) k = T k X ⊗ T k X ; for background on the Eilenberg-Zilber map and its properties, see, forinstance, [May92, Sec. 29]. ⊳ efinition 4.20. Let X be a symplectic manifold. A p -shifted symplectic form on X is a closed 2-form ω of degree p on X for which the pairing (4.18) induces a non-degenerate pairing on the homology of T X (at every point x ∈ X ). A p -shifted symplectic simplicial manifold is a pair ( X, ω ) of a simplicialmanifold X and a p -shifted symplectic form ω on X . If Y → M is a surjective submersion, we saythat a p -shifted symplectic form ω on the Čech nerve ˇ CY is a p -shifted symplectic form on M . Example 4.21.
Consider the simplicial manifold B G and its closed 2-form ω = (0 , ω , ω ) of degreetwo from Example 4.15. The manifold (B G ) = ∗ consists of a single point. Thus, the tangent complex T B G is a chain complex of vector bundles on the point; that is, it is simply a chain complex of realvector spaces. We find that ( T B G ) k = g k − , where g is the Lie algebra of G . It remains to understandthe differential on T B G . Since the differential of the multiplication G → G at the neutral element e ∈ G is simple the addition in g , one obtains the following explicit expressions: ∆ : g → g , ( ξ , ξ ) ξ − ( ξ + ξ ) + ξ = 0 , ∆ : g → g , ( ξ , ξ , ξ ) ( ξ , ξ ) − ( ξ + ξ , ξ ) + ( ξ , ξ + ξ ) − ( ξ , ξ ) = ( − ξ , ξ ) , ∆ : g → g , ( ξ , ξ , ξ , ξ ) (0 , ξ + ξ , and so on. Let g [1] denote the chain complex with g in degree one and all other degrees trivial. Themorphism g [1] → T B G , ξ ξ is a quasi-isomorphism, inducing H • ( g [1] , ∼ = −→ H • ( T B G, ∆) . Finally, consider the pairing induced by ω . Since we are interested in the on homology, it suffices towork with g [1] instead of T B G . The only non-trivial case where we have to check its non-degeneracyis for two tangent vectors of degree one, i.e. ξ, ξ ′ ∈ g . There, we obtain ω ( ξ, ξ ′ ) = ω | ( e,e ) ( σ ∗ ξ, σ ∗ ξ ′ ) − ω | ( e,e ) ( σ ∗ ξ, σ ∗ ξ ′ )= ω | ( e,e ) (cid:0) ( ξ, , (0 , ξ ′ ) (cid:1) − ω | ( e,e ) (cid:0) (0 , ξ ) , ( ξ ′ , (cid:1) = h ξ, ξ ′ i − h , i , where in the last step we have used the explicit form (4.16) of ω . Since the Killing form h− , −i on g is non-degenerate, it follows that ω is a 2-shifted symplectic form on B G . (This example is by nomeans new; it can be found in [PTVV13, Saf16, Get14], for instance.) ⊳ Remark 4.22.
Observe the crucial difference from the 2-plectic point of view: in the 2-shifted sym-plectic case, the 2-form ω is responsible for the non-degeneracy, whereas in the 2-plectic case it is the3-form ω on G . The role of ω in the 2-shifted symplectic setting is completely different: it is purelyto establish the (derived) closedness of ω . ⊳ The reason we have included the shifted symplectic perspective is that gerbes provide a promisingtool for geometric quantisation in this context as well. This extends [LGX05] and follows Safronov’srecent article [Saf], which also proposes (higher) gerbes as a replacement of line bundles in shiftedgeometric quantisation. We propose the following definition, adapted from [Saf]:
Definition 4.23.
Let ( X, ω = ( ω , ω )) be a 1-shifted symplectic manifold. A of ( X, ω ) is a triple ( G , E , ψ ) of a gerbe G ∈ G rb ∇ ( X ) with curv( G ) = ω , an isomorphism E : d ∗ G → d ∗ G over X with curv( E ) = ω and a parallel 2-isomorphism ψ : d ∗ E ◦ d ∗ E → d ∗ E over X ,which satisfies an associativity condition over X .30his provides prequantisations for quasi-symplectic groupoids even when ω is not exact, thuscircumventing the exactness constraint in [LGX05]. If X = M //G for some action of a Lie group G ona manifold M (cf. Example 2.8), the data ( G , E , ψ ) are precisely an equivariant gerbe with connectionas defined in [BMS], whose curvatures coincide with ω .For p -shifted symplectic simplicial manifolds with p > , we would, in general, have to pass tohigher gerbes in order to prequantise these simplicial manifolds. However, in the case of the 2-shiftedsymplectic simplicial manifold (B G, ω ) from Example 4.21 we are lucky: since B G = ∗ , any highergerbe on B G is necessarily trivial (see Definition 2.63 and Proposition 2.56), and we can define: Definition 4.24. A (B G, ω ) is a triple ( G , E , ψ ) of a gerbe G ∈ G rb ∇ (B G ) with curv( G ) = ω , an isomorphism E : d ∗ G ⊗ d ∗ G → d ∗ G over B G with curv( E ) = ω , and a parallel2-isomorphism ψ : d ∗ E ◦ d ∗ E → d ∗ E ◦ d ∗ E over B G , satisfying a further coherence condition over B G .We can identify such 2-shifted prequantisations of (B G, ω ) as certain known structures for gerbes,which have not yet been connected with the theory of shifted geometric quantisation: Theorem 4.25.
Let G be a compact, simple, simply connected Lie group with 2-shifted symplecticform ω as in Example 4.21. Then, a 2-shifted prequantisation of (B G, ω ) is precisely the same as amultiplicative bundle gerbe as defined and shown to exist in [Wal10]. In particular, (B G, ω ) admits a2-shifted prequantisation by [Wal10, Ex. 1.5]. A A glance at 2-categories
We give a very brief overview of basic notions of , or bicategories . (We warn the readerthat we use these terms interchangeably here, which is not standard; bicategories are often understoodto be the more general concept, where 2-categories are strict bicategories.) We attempt in no wayto be complete here; we refer readers interested in full definitions and more background to [Lei] for aconcise introduction, and to [Scha] for a detailed and comprehensive account of 2-categories, includingsymmetric monoidal 2-categories.
Definition A.1. A C consists of• a collection of objects, for which we write x ∈ C ,• for each pair of objects x, y ∈ C a morphism category C ( x, y ) = Hom C ( x, y ) , whose objects are called (1-)morphisms f : x → y , and whose morphisms ψ : f → g are called (the compositionin Hom C ( x, y ) is called vertical composition , and we denote it by ( − ) • ( − ) ),• for each x, z, y ∈ C a composition functor ( − ) ◦ ( − ) : Hom C ( y, z ) × Hom C ( x, y ) → Hom C ( x, z ) ,• for each x ∈ C a specified identity morphism x ∈ Hom C ( x, x ) , and• natural isomorphisms α f,g,h : ( h ◦ g ) ◦ f ∼ = −→ h ◦ ( g ◦ f ) ,λ g : 1 y ◦ g ∼ = −→ g ,ρ g : g ◦ x ∼ = −→ g , for all morphisms h : y → z , g : x → y , and f : w → x in C . The natural isomorphisms α , ρ , and λ are called the associator and left and right unitor , respectively.These data have to satisfy the pentagon and triangle axioms , which are, respectively, the commutativity31f the following diagrams: (cid:0) ( k ◦ h ) ◦ g (cid:1) ◦ f (cid:0) k ◦ ( h ◦ g ) (cid:1) ◦ f ( k ◦ h ) ◦ ( g ◦ f ) k ◦ (cid:0) ( h ◦ g ) ◦ f (cid:1) k ◦ (cid:0) h ◦ ( g ◦ f ) (cid:1) α g,h,k ◦ f α f,g,k ◦ h α f,h ◦ g,k α f ◦ g,h,k k ◦ α f,g,h ( g ◦ x ) ◦ f g ◦ (1 x ◦ f ) g ◦ f α f, x,g ρ g ◦ f g ◦ λ f Note that because of the functoriality of the composition the interchange law ( ψ ′ • ψ ) ◦ ( ϕ ′ • ϕ ) = ( ψ ′ ◦ ϕ ′ ) • ( ψ ◦ ϕ ) holds true for any collection of 2-morphisms for which either side is defined. Definition A.2.
A morphism f : x → y in a 2-category C is called invertible if there exists a morphism g : y → x and 2-isomorphisms x → g ◦ f and f ◦ g → y . Example A.3.
The collection of categories naturally assembles into a 2-category C at : its objectsare the categories, and for categories C , D , the morphism category Hom C at ( C , D ) is the category offunctors F : C → D , with natural transformations η : F → G as morphisms. In this case, the associatorand unitors happen to be identity morphisms; one says that C at is a strict 2-category . ⊳ Example A.4.
The collection of gerbes (resp. gerbes with connection) on a manifold M form a 2-category G rb ( M ) (resp. G rb ∇ ( M ) ); see Section 2.2. Here, composition of morphisms relies on formingpullbacks of vector bundles and surjective submersions. This operation is not strictly compatible withcomposition; for two smooth maps g : M ′′ → M ′ and f : M ′ → M , and a vector bundle E → M , thepullback bundles g ∗ f ∗ E and ( f ◦ g ) ∗ E are not equal, but there exists a canonical isomorphism betweenthem, natural in E . These isomorphisms induce the associator in G rb ( M ) (resp. G rb ∇ ( M ) ). ⊳ One can also define monoidal 2-categories , which are 2-categories C endowed with the additionaldata of a tensor product 2-functor ⊗ : C × C → C , together with various 1- and 2-isomorphisms whichestablish its associativity and unitality. Further, there is a hierarchy of different levels of commu-tativity for such products; each of these levels corresponds to further choices of isomorphisms andcoherence conditions. Writing out these data and conditions requires considerable work; a full treat-ment can be found in [Scha, Appendix C]. For the symmetric monoidal 2-categories ( G rb ( M ) , ⊗ ) and ( G rb ∇ ( M ) , ⊗ ) , the tensor product is constructed from pullbacks of submersions and bundles, as wellas the tensor product of vector bundles (see Section 2.3). Therefore, all additional coherence data ariseas the standard canonical isomorphisms which relate different ways of pulling back the same geometricstructures and which establish the associativity of the tensor product of vector bundles.32 eferences [Alv85] O. Alvarez. Topological quantization and cohomology. Comm. Math. Phys. , 100(2):279–309, 1985.[AS04] M. Atiyah and G. Segal. Twisted K -theory. Ukr. Mat. Visn. , 1(3):287–330, 2004. arXiv:math/0407054 .[BC04] J. C. Baez and A. S. Crans. Higher-dimensional algebra. VI. Lie 2-algebras.
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