AN INTRODUCTION TO BUNDLE GERBES MICHAELKMURRAY 8 Abstract. Anintroductiontothetheoryofbundlegerbesandtheirrelation- 0 shiptoHitchin-Chatterjee gerbes ispresented. Topicscovered areconnective 0 structures,trivialityandstableisomorphismaswellasexamplesandapplica- 2 tions. n a J 9 1. Dedication Over the years I have had many interesting mathematical conversations with ] G Nigel and regularly came away with a solution to a problem or a new idea. While D preparing this article I was trying to recall when he first told me about gerbes. I . thought for awhile that age was going to get the better of my memory as many h conversations seemed to have blurred together. But then I discovered that the t a annual departmental research reports really do have their uses. In July of 1992 I m attendedthe‘Symposiumongaugetheoriesandtopology’atWarwickandreported [ in the 1992 Departmental Research Report that: 3 I ...had discussions with Nigel Hitchin about ‘gerbes’. These are v a generalisationof line bundles ... 1 Furthersearchingofmy electronicfiles revealedanorderforBrylinski’sbook Loop 5 spaces, characteristic classes and geometric quantization on the 29th April 1993. I 6 1 recallthatthebooktooksomemonthstomakesitsawayacrosstheseatoAustralia . duringwhichtimeIponderedtheadvertisingmaterialIhadwhichsaidthatgerbes 2 were fibrations of groupoids. Trying to interpret this lead to a paper on bundle 1 7 gerbes which I submitted to Nigel in his role as a London Mathematical Society 0 Editoronthe25thJuly1994. TheDepartmentalResearchReportofthesameyear : reports that: v i This year I began some work on a geometric construction called a X bundle gerbe. These provide a geometric realisation of the three r a dimensional cohomology of a manifold. My sincere thanks to Nigel for introducing me to gerbes and for the many other fascinating insights into mathematics that he has given me over the years. 2. Introduction ThetheoryofgerbesbeganwithGiraud(1971)andwaspopularisedinthebook by Brylinski (1993). A short introduction by Nigel Hitchin (2003) in the ‘What is a ...?’ series can be found in the Notices of the AMS. Gerbes provide a geometric realisationofthethreedimensionalcohomologyofamanifoldinamanneranalogous to the way a line bundle is a geometric realisation of two dimensional cohomology. 2000 Mathematics Subject Classification. 55R65. The author acknowledges the support of the Australian Research Council and would like to thanktheorganisersoftheconferencefortheinvitationtoparticipate. Therefereeisalsothanked forusefulremarksonthemanuscript. 1 2 M.K.MURRAY Part of the reason for their recent popularity is applications to string theory in particularthe notion of the B-field. Strings ona manifold are elements in the loop spaceofthemanifoldandwewouldexpecttheirquantizationtoinvolveahermitian line bundle on the loop space arising from a two class on the loop space. That two class can arise as the transgressionof some three class on the underlying manifold. Gerbes provide a geometrisation of this process. String theory however is not the only application of gerbes and we refer the interested readerto the related work of Hitchin (1999; 2006) which applies gerbes to generalised geometry and to reviews suchas(Careyetal.,2000)and(Mickelsson,2006)whichgiveapplicationsofgerbes to other problems in quantum field theory. Aswitheverythingelseinthetheoryofgerbes,therelationshipofbundlegerbes togerbes isbest understoodby comparisonwith the caseofhermitianline bundles orequivalentlyU(1)(principal)bundles. Therearebasicallythreewaysofthinking about U(1) bundles over a manifold M: (1) A certain kind of locally free sheaf on M. (2) A co-cycle g : U ∩U →U(1) for some open cover U ={U |α∈I} of αβ α β α M. (3) A principal U(1) bundle P →M. In the case of gerbes over M we can think of these as: (1) A certain kind of sheaf of groupoids on M (Giraud, Brylinski). (2) Aco-cycleg : U ∩U ∩U →U(1)forsomeopencoverU ={U |α∈I} αβγ α β γ α ofM oralternativelyachoiceofU(1)bundleP →U ∩U foreachdouble αβ α β overlap (Hitchin, Chatterjee). (3) A bundle gerbe (Murray). Note that we are slightly abusing the definition of gerbe here as what we are considering are gerbes with band the sheaf of smooth functions from M into U(1). There are more general kinds of gerbes on M just as there are more general kinds of sheaves on M beyond those arising as the sheave of sections of a hermitian line bundle. Recall some of the basic facts about U(1) bundles on a manifold M. (1) If P →M is a U(1) bundle there is a dual bundle P∗ →M and if Q→M is another U(1) bundle there is a product P ⊗Q→M. (2) If f: N →M is a smooth map there is a pullback bundle f∗(P)→N and this behaves well with respect to dual and product. That is f∗(P∗) and (f∗(P))∗ are isomorphic as also are f∗(P ⊗Q) and f∗(P)⊗f∗(Q). (3) Associated to a U(1) bundle P → M is a characteristic class, the chern class, c(P) ∈ H2(M,Z), which is natural with respect to pullback, that is f∗(c(P)) = c(f∗(P)) and additive with respect to product and dual, that is c(P ⊗Q)=c(P)+c(Q) and c(P∗)=−c(P). (4) P → M is called trivial if it is isomorphic to M ×U(1) or equivalently admits a global section. P is trivial if and only if c(P)=0. (5) There is a notion of a connection on P → M. Associated to a connection AonP isaclosedtwo-formF calledthecurvatureofAwiththe property A that F /2πi is a de Rham representative for the image of c(P) in real A cohomology. (6) If γ: S1 →M is a loop in M and P → M a U(1) bundle with connection A then parallel transport around γ defines the holonomy, hol(A,γ) of A INTRODUCTION 3 around γ which is an element of U(1). If γ is the boundary of a disk D ⊂M then we have hol(A,∂D)=exp F . A (cid:18)ZD (cid:19) AgerbeisanattempttogeneralisealltheabovefactsaboutU(1)bundlestosome new kind of mathematical object in such a way that the characteristic class is in threedimensioncohomology. Obviouslyforconsistencyotherdimensionsthenhave tochange. Inparticularthecurvatureshouldbeathree-formandholonomyshould be over two dimensional submanifolds. It turns out to be useful to consider the generalcaseofanydimensionofcohomologywhichwecallap-gerbe. Forhistorical reasonsa p-gerbehas a characteristicclassin Hp+2(M,Z) sothe interesting values of p are −2,−1,0,1,... with U(1) bundles corresponding to p=0. A p-gerbe then is some mathematical object which represents p+2 dimensional cohomology. To make completely precise what representing p+2 dimensional co- homology means would take us to far afield from the present topic but we give a sketch here to motivate the behaviour we are looking for in p-gerbes. To this end we will assume our p-gerbes P live in some category G and there is a (forgetful) functor Π: G → Man the category of manifolds. The functor Π and the category G have to satisfy: (1) If P is a p-gerbe there is a dual p-gerbe P∗ and if Q is another p-gerbe there is a product p-gerbe P ⊗Q. In other words G is monoidal and has a dual operation. (2) If f: N → M is a smooth map and Π(P) = M there is a pullback p- gerbe f∗(P) and a morphism fˆ: f∗(P) → P such that Π(f∗(P)) = N and π(fˆ) = f. Pullback should behave well with respect to dual and product. That is f∗(P∗) and (f∗(P)∗ should be isomorphic as also should be f∗(P ⊗Q) and f∗(P)⊗f∗(Q). (3) Associated to a p-gerbe P is a characteristic class c(P) ∈ Hp+2(Π(P),Z), which is naturalwith respect to pullback, that is f∗(c(P))=c(f∗(P)) and additive with respect to product and dual, that is c(P ⊗Q)=c(P)+c(Q) and c(P∗)=−c(P). (4) As well as the notion of P and Q being isomorphic there is a possibly weaker notion of equivalence where P and Q are equivalent if and only if c(P)=c(Q). We say P is trivial if c(P)=0. (5) There is a notion of a connective structure A on P. Associated to a con- nective structure A on P is a closed (p+2)-form ω on Π(P) called the (p+2)-curvature of A with the property that ω/2πi is a de Rham repre- sentative for the image of c(P) in real cohomology. (6) IfX ⊂Π(P)isanorientedp+1dimensionalsubmanifoldofΠ(P)weshould beabletodefinetheholonomyoftheconnectivestructurehol(A,X)∈U(1) overX. MoreoverifY ⊂Π(M)isanorientedp+2-dimensionalsubmanifold with boundary then we want to have that hol(A,∂Y)=exp F . ω (cid:18)ZY (cid:19) Clearly by construction the categoryof U(1) bundles, with the forgetful functor whichassignstoaU(1)bundleitsbasemanifold,isanexampleofa0-gerbe. Before weconsiderotherexamplesweneedsomefactsaboutbundleswithstructuregroup 4 M.K.MURRAY an abelian Lie group H. If P → M is an H bundle on M then by choosing local sections of P for an open cover U = {U | α ∈ I} we can construct transition α functions h : U ∩U → H and in the usual way this defines a class c(P) ∈ αβ α β H1(M,H)wherehereweabusenotationandwriteH forwhatisreallythesheafof smooth functions with values in H. It is a standard fact that isomorphism classes of H bundles are in bijective correspondence with H1(M,H) in this manner. If P → M is an H bundle we can define its dual as follows. Let P∗ be isomorphic to P as a manifold with projection to M and for convenience let p∗ ∈ P∗ denote p ∈ P thought of as an element of P∗. Then define a new H action on P∗ by p∗h=(ph−1)∗. Itisoviousthatifh aretransitionfunctionsforP thenh∗ =h−1 αβ αβ αβ are transition functions for P∗. In particular we have that c(P∗) = −c(P) if we write the group structure on H1(M,H) additively. If Q is another H bundle we can form the fibre product P × Q and let H act on it by (p,q)h = (ph,qh−1). M Denote the orbit of (p,q) under this action by [p,q] and define an H action by [p,q]h = [p,qh] = [ph,q]. The resulting H bundle is denoted by P ⊗Q → M. If h are transition functions for P and k are transition functions for Q then αβ αβ h k aretransitionfunctionsforP⊗Qandthusc(P⊗Q)=c(P)+c(Q). Notice αβ αβ that these constructions will not generally work for non-abelian groups because in such a case the action of H on P∗ is not a right action and the action on P ⊗Q is not even well-defined. Example 2.1. Thesimplestexampleisthatoffunctionsf: M →Zwherewedefine the functor Π by Π(f) = M. The degree of f is the class induced in H0(M,Z) so functions from M to Z are −2 gerbes over M. Product and dual are pointwise addition and negation. There is no sensible notion of connective structure. Example 2.2. Consider next principal Z bundles P → M. Clearly we want the functor Π to be Π(P)=M andpull-backs arewell knownto exist. As Z is abelian the constructions aboveapply andthere are duals andproducts. The isomorphism class of a bundle is determined by a class in H1(M,Z) so Z bundles are p = −1 gerbes. A Z bundle is trivialas a −1 gerbe if and only if it is trivialas a Z bundle. ItisnotimmediatelyobviouswhataconnectivestructureonaZbundleisbutit turns out that the correct notion is that of a Z equivariant map φˆ: P →iR where theactionofn∈ZoniRisadditionof2πinsothatφˆ(pn)=φˆ(p)+2πin. Themap φˆ then descends to a map φ: M →S1 and the class of the bundle is the degree of this map. The pull-back of the standard one-form on R, that is dφˆ is a one-form onP which descends to a one-formφ−1dφ onM. The de Rham class (φ−1dφ)/2πi is the image of the class of the bundle in real cohomology. We expect holonomy to be over a −1 + 1 = 0 dimensional submanifold. If m ,...,m is a collection ofpoints in M with eachm orientedby some ǫ ∈{±1} 1 r i i let us denote by ǫ m their union as an oriented zero dimensional submanifold i i of M. Then we define P r hol φˆ, ǫ m = φ(m )ǫi. i i i (cid:16) X (cid:17) Yi=1 Inthecaseofanorientedone-dimensionalsubmanifoldX ⊂M withends−X and 0 +X the fundamental theorem of calculus tells us that 1 hol φˆ,X −X =exp dφ 1 0 (cid:16) (cid:17) (cid:18)ZX (cid:19) INTRODUCTION 5 Notice that if we express a Z bundle locally in terms of transitions functions these are maps of the form f : U ∩U → Z. That is, over each double overlap αβ α β we have a −2 gerbe. Example 2.3. It is clear from the above example that maps φ: M →U(1) are also −1 gerbes with a connective structure. The dual and product are just pointwise inverseandpointwiseproduct. Theclassisthedegreeandtheconnectivestructure is included automatically as part of φ. We can also forget that there is a natural connective structure and just regard maps φ: M →U(1) as −1 gerbes. In that case the natural notion of isomorphism betweentwomapsφ,χ: M →U(1)wouldbeequality. Howevertwosuchmapshave the same degree if and only if they are homotopic. So the notion of equivalence of mapsφ: M →U(1),thoughtofas−1gerbes(withoutconnectivestructure)should be homotopy and is different to the notion of isomorphism. Example 2.4. As we have remarked U(1) bundles are, of course, p = 0 gerbes. Notice that locally a U(1) bundle is given by transition functions g : U ∩U → αβ α β U(1),thatisoneachdoubleoverlapwehavea−1gerbe(withconnectivestructure). Wewillseebelowthatthispatternofapgerbebeingdefinedasap−1gerbeon double overlaps of some open cover is exploited by Hitchin and Chatterjee to give a definition of a 1 gerbe. But first we need some additional background material. 3. Background We will be interested in surjective submersions π: Y → M which we regard as generalizations of open covers. In particular if U = {U | α ∈ I} is an open cover α we have the disjoint union YU ={(x,α)|x∈Uα}⊂M ×I with projection map π(x,α) = x. The surjective morphism π: YU → M is called the nerve of the open cover U. A morphism of surjective submersions π: Y → M and p: X → M is a map ρ: Y → X covering the identity, that is p ◦ρ = π. Any surjective submersion π: Y → M admits local sections so there is an open cover U of M and local sections sα: Uα → Y of π. These local sections define a morphism s: YU → Y by s(x,α) = sα(x). Indeed any morphism YU → Y will be of this form. If V = {Vα | α ∈ J} is a refinement of U, that is there is a map ρ: J → I such that for every α∈J we have Vα ⊂Uρ(α), we have morphism of surjective submersions YV →YU defined by (α,x)7→(ρ(α),x). Given a surjective morphism π: Y →M we can form the p-fold fibre product Y[p] ={(y ,...,y )|π(y )=···=π(y )}⊂Yp. 1 p 1 p The submersion property of π implies that Y[p] is a submanifold of Yp. There are smooth maps π : Y[p] → Y[p−1], for i = 1,...,p, defined by omitting the ith i element. We will be interested in two particular examples. Example 3.1. If U is an open cover of M then the p-th fibre product Y[p] is the U disjoint union of all the ordered p-fold intersections. For example if U = {U ,U } 1 2 is an open cover of M then Y[2] is the disjoint union of U ∩U , U ∩U , U ∩U U 1 2 2 1 1 1 and U ∩U . 2 2 6 M.K.MURRAY Example 3.2. If P → M is a principal G bundle then P → M is a surjective submersion. It is easy to show that P[p] = P ×Gp−1. In particular P[2] = P ×G andwe shallneed later the relatedfact that there is a mapg: P[2] →G defined by p g(p ,p )=p . 1 1 2 2 Let Ωq(Y[p]) be the space of differential q-forms on Y[p]. Define p δ: Ωq(Y[p−1])→Ωq(Y[p]) by δ = (−1)p−1π∗. i i=1 X These maps form the fundamental complex 0→Ωq(M)→π∗ Ωq(Y)→δ Ωq(Y[2])→δ Ωq(Y[3])→δ ... and from (Murray, 1996) we have: Proposition 3.1. The fundamental complex is exact for all q ≥0. Note thatifY =YU thenthis Propositionisawell-knownresultabouttheCˇech de Rham double complex. See, for example Bott and Tu’s book (1982). Finally we need some notation. Let H be an abelian group. If g: Y[p−1] → H we define δ(g): Y[p] →H by δ(g)=(g◦π )−(g◦π )+(g◦π )··· . 1 2 3 If P →Y[p−1] is an H bundle we define an H bundle δ(P)→Y[p] by ∗ ∗ ∗ ∗ δ(P)=π (P)⊗(π (P)) ⊗π (P)⊗··· . 1 2 3 It is easy to check that δ(δ(g))=1 and that δ(δ(P)) is canonically trivial. 4. Bundle gerbes Definition 4.1. A bundle gerbe (Murray, 1996) over M is a pair (P,Y) where Y →M is a surjective submersion and P →Y[2] is a U(1) bundle satisfying: (1) There is a bundle gerbe multiplication which is a smooth isomorphism ∗ ∗ ∗ m: π (P)⊗π (P)→π (P) 3 1 2 of U(1) bundles over Y[3]. (2) Thismultiplicationisassociative,thatis,ifweletP denotethefibreof (y1,y2) P over(y ,y )thenthefollowingdiagramcommutesforall(y ,y ,y ,y )∈ 1 2 1 2 3 4 Y[4]: P ⊗P ⊗P → P ⊗P (y1,y2) (y2,y3) (y3,y4) (y1,y3) (y3,y4) ↓ ↓ P ⊗P → P (y1,y2) (y2,y4) (y1,y4) Weremarkthatforany(y ,y ,y )∈Y[3] thebundlegerbemultiplicationdefines 1 2 3 an isomorphism: m: P ⊗P →P (y1,y2) (y2,y3) (y1,y3) of U(1) spaces. We can show using the bundle gerbe multiplication that there are natural iso- morphisms P ∼=P∗ and P ≃Y[2]×U(1). (y1,y2) (y2,y1) (y,y) We can rephrase the existence and associativity of the bundle gerbe multipli- cation to an equivalent pair of conditions in the following way. The bundle gerbe INTRODUCTION 7 multiplication gives rise to a section s of δ(P) → Y[3]. Moreover δ(s) is a sec- tion of δ(δ(P)) → Y[4]. But δ(δ(P)) is canonically trivial so it makes sense to ask that δ(s) = 1. This is the condition of associativity. The family of spaces {Y[p] | p = 1,2,...} is an example of a simplicial space (Dupont, 1978) and by comparing to (Brylinski and McLaughlin, 1994) we see that a bundle gerbe is the same thing as a simplicial line bundle over this particular simplicial space. Example 4.1. If we replace Y in the definition by YU for some open cover U of M we obtain the definition of gerbe given by Hitchin (1999) and by his student Chatterjee (1998). This consists of choosing an open coverU of M and a family of U(1) bundles P: U ∩U such that over triple overlaps we have sections α β ∗ s ∈Γ(U ∩U ∩U |P ⊗P ⊗P ) αβγ α β γ βγ αγ αβ and we require that δ(s)=1 in the appropriate way. Example 4.2. The simplest example of a line bundle is given by the clutching construction on the two sphere S2. If U and U are the open neighbourhoods of 0 1 thenorthandsouthhemisphereswetakethetransitionfunctiong: U ∩U →U(1) 0 1 to have winding number one. As there areonly two open sets there is no condition on triple overlaps and we obtain the U(1) bundle over S2 of chern class one. In a similar fashionwe can consider U andU to be open neighbourhoodsof the north 0 1 and south hemispheres of the three-sphere S3. Their intersection is retractable to the two-sphere so we can choose over this the U(1) bundle P of chern class one. Againtherearenoadditionalconditionsandweobtainthegerbeofdegreeoneover S3. Example 4.3. Hitchin and Chatterjee also consider a gerbe as in Example 4.1 but with the added requirement that each P is trivial in the form P =U ∩U × αβ αβ α β U(1). Writing elements of the disjoint union Y[2] as (α,β,x) where x ∈ U ∩U U α β we see that the bundle gerbe multiplication must take the form ((α,β,x),z)⊗((β,γ,x),w)7→((α,γ,x),zwg (x)) αβγ forsome g : U ∩U ∩U →U(1)and willbe associativeprecisely wheng is αβγ α β γ αβγ a co-cycle. We willreferto gerbesofthe formsinExamples4.1or4.3asHitchin-Chatterjee gerbes. The connection with bundle gerbes is simple. For clarity we define: Definition 4.2. A bundle gerbe (P,Y) over M is called local if Y = YU for some open cover U of M. We then obviously have: Proposition 4.3. A Hitchin-Chatterjee gerbe is the same thing as a local bundle gerbe. If(P,Y)isabundlegerbeoverM thenassociatedtoeverypointmofM wehave agroupoidconstructedasfollows. Theobjectsaretheelements ofthe fibreY and m the morphisms between y and y in Y are P . Composition comes from the 1 2 m (y1,y2) bundlegerbemultiplication. IfwecallagroupoidaU(1)groupoidifitistransitive and the group of morphisms of a point is isomorphic to U(1), then the algebraic conditionsonthe bundle gerbe(that is the multiplication andits associativity)are 8 M.K.MURRAY capturedpreciselybysayingthatabundlegerbeisabundleofU(1)groupoidsover M. We nowconsiderthe propertiesgiveninSection2whichwewouldlikea1-gerbe to satisfy and show how they are satisfied by bundle gerbes. 4.1. Pullback. If f: N → M then we can pullback Y → M to f∗(Y) → N with a map fˆ: f∗(Y) → Y covering f. There is an induced map fˆ[2]: f∗(Y)[2] → Y[2]. Let f∗(P,Y)=(fˆ[2]∗(P),f∗(Y)). To see this is a bundle gerbe notice that all this is doing is pulling back the U(1) groupoid at f(n) ∈ M and placing it at n ∈ N so we have a bundle of U(1) groupoids over N and thus a bundle gerbe. 4.2. Dual and product. If(P,Y)isabundlegerbethen(P,Y)∗ =(P∗,Y)isalso a bundle gerbe called the dual of (P,Y). If(P,Y)and(Q,X)arebundlegerbeswecanformthefibreproductY × X → M M, a new surjective submersion and then define a U(1) bundle P ⊗Q→(Y × X)[2] =Y[2]× X[2] M M by (P ⊗Q) =P ⊗Q . ((y1,x1),(y2,x2)) (y1,y2) (x1,x2) We define (P,Y)⊗(Q,X)=(P ⊗Q,Y × X). M 4.3. Characteristic class. Thecharacteristicclassofa bundle gerbeis calledthe Dixmier-Douady class. We construct it as follows. Choose a good cover U of M (Bott and Tu, 1982) with sections s : U →Y. Then α α (s ,s ): U ∩U →Y[2] α β α β is a section. Choose a section σ of P =(s ,s )∗(P). That is some αβ αβ α β σ : U ∩U →P αβ α β such that σ (x)∈P . Over triple overlaps we have αβ (sα(x),sβ(x)) m(σ (x),σ (x))=g (x)σ (x)∈P αβ βγ αβγ αγ (sα(x),sγ(x)) forg : U ∩U ∩U →U(1). Thisdefinesaco-cyclewhichistheDixmier-Douady αβγ α β γ class DD((P,Y))=[g ]∈H2(M,U(1))=H3(M,Z). αβγ Example 4.4. If U is an open cover and g a U(1) co-cycle then we can build a αβγ Hitchin-Chatterjee gerbe or local bundle gerbe of the type considered in Example 4.3. It is easy to see that this has Dixmier-Douady class given by the Cˇech class [g ]. αβγ Notice that this example shows that every class in H3(M,Z) arises as the Dixmier-Douady class of some Hitchin-Chatterjee gerbe or of some (local) bun- dle gerbe. It is straightforward to check that if f: N → M and (P,Y) is a bundle gerbe over M then f∗(DD(P,Y))=DD(f∗(P,Y)). Moreover we have (1) DD((P,Y)∗)=−DD((P,Y)), and (2) DD((P,Y)⊗(Q,X))=DD((P,Y))+DD((Q,X)). INTRODUCTION 9 We will defer the question of triviality of a bundle gerbe until the next section and consider next the notion of a connective structure on a bundle gerbe. 4.4. Connective structure. As P → Y[2] is a U(1) bundle we can pick a con- nection A. Call it a bundle gerbe connection if it respects the bundle gerbe multi- plication. That is if the section s of δ(P) → Y[3] satisfies s∗(δ(A)) = 0, i.e is flat for δ(A). We would like bundle gerbe connections to exist. This is a straightfor- wardconsequenceofthe factthat the fundementalcomplex isexact. IndeedifA is anyconnectionconsiders∗(δ(A)); wehaveδ(s∗(δ(A)))=δ(s)∗(δδ(A))=0 because δδ(A) is the flatconnectiononthe canonicallytrivialbundle δδ(P). Hence there is a one-form a on Y[2] such that δ(a) = s∗(δ(A)) and thus A−a is a bundle gerbe connection. If A is a bundle gerbe connection then the curvature F ∈ Ω2(Y[2]) satisfies A δ(F ) = 0. From the exactness of the fundamental complex there must be an A f ∈ Ω2(Y) such that F = δ(f). As δ commutes with d we have δ(df) = dδ(f) = A dF = 0. Hence df = π∗(ω) for some ω ∈ Ω3(M). So π∗(dω)= dπ∗(ω)= ddf = 0 A andω is closed. In fact it is a consequence of standardCˇechde Rham theory that: 1 ω =r(DD((P,Y)))∈H3(M,R). 2πi (cid:20) (cid:21) We call f a curving for A, the pair (A,f) a connective structure for (P,Y) and ω is called the three-curvature of the connective structure (A,f). In string theory the two-form f is called the B-field. We can give a local description of the connective structure as follows. Assume we have an open cover U of M with local sections s : U → Y and sections over α α double overlaps σ of (s ,s )∗(P)→U ∩U . We define αβ α β α β A =(s ,s )∗(A)∈Ω1(U ∩U ) αβ α β α β and f =s∗(f)∈Ω2(U ). α α α These satisfy A −A +A =g−1 dg βγ αγ αβ αβγ αβγ f −f =df β α αβ and the three-curvature ω restricted to U is df . α α Example 4.5. We can use this local description of the connective structure to calculate the Dixmier-Douady class of the Hitchin-Chatterjee gerbe on the three sphere defined in Example 4.2. Stereographic projectionfrom either pole identifies S3 −{(1,0,0)} and S3 −{(−1,0,0)} with R3 and maps the equator to the unit sphere S2 ⊂R3. Let U and U be the pre-imagesof the interiorof a ball ofradius 0 1 two in R3 under both stereographic projections. We can identify U ∩U with 0 1 S2×(−1,1). Pull back the line bundle of chern class k on S2, with connection A and curvature F, to U ∩U . Because there are no triple overlaps this is a bundle 0 1 gerbe connection. If we choose a partition of unity ψ and ψ for U and U then 0 1 0 1 f = −ψ F and f = ψ F define two-forms on U and U respectively satisfying 0 1 1 0 0 1 F = f −f on U ∩U . These two forms define a curving for the bundle gerbe 1 0 0 1 connection. The curvatureis thegloballydefinedthree-formω whoserestrictionto U and U is −dψ ∧F and dψ ∧F respectively. The integral of ω over the three 0 1 1 0 10 M.K.MURRAY spherereduces,by Stokestheorem,to the integralofF overthe two-sphere. Hence this bundle gerbe has Dixmier-Douady class k ∈H3(M,Z)=Z. Holonomy will need to wait until we have a considered the notion of triviality which we turn to now. 5. Triviality Recall that a U(1) bundle P → M is trivial if it is isomorphic to the bundle M ×U(1) or, equivalently has a global section. This occurs if and only if P →M has zero Chern class. If s : U → P are local sections then P is determined by a a α transition function g: U ∩U →U(1) given by s =s g and P →M is trivial α β α β αβ if and only if there exist h : U →U(1) such that α α g =h h−1. αβ β α In an analogous way Hitchin and Chatterjee (1998) define a gerbe P →U ∩U αβ α β tobetrivialifthereareU(1)bundlesR →U andisomorphismsφ : R ⊗R∗ → α α αβ α β P on double-overlapsin such a way that the multiplication becomes the obvious αβ contraction ∗ ∗ ∗ R ⊗R ⊗R ⊗R →R ⊗R . α β β γ α γ In the bundle gerbe formalism this idea takes the following form (Murray and Stevenson, 2000). Let R→Y be a U(1) bundle and let δ(R)→Y[2] be defined as above. Note that δ(R) has a natural associative bundle gerbe multiplication given by ∗ ∗ ∗ δ(R) ⊗δ(R) =R ⊗R ⊗R ⊗R ≃R ⊗R =δ(R) . (y1,y2) (y2,y3) y1 y2 y2 y3 y1 y3 (y1,y3) Definition 5.1. A bundle gerbe (P,Y) over M is called trivial if there is a U(1) bundle R →Y such that (P,Y) is isomorphic to (δ(R),Y). In such a case we call a choice of R and the isomorphism δ(R)≃P a trivialisation of (P,Y). Example 5.1. Let (P,Y) be a bundle gerbe and assume that Y → M admits a global section s: M → Y. Define R → Y by R = P . Then we have an y (s(π(y),y) isomorphism ∗ δ(R) =P ⊗P (y1,y2) (s(π(y2),y2) (s(π(y1),y1) =P ⊗P (s(π(y2),y2) (y1,s(π(y1)) ≃P (y1,y2) using the bundle gerbe multiplication and the fact that s(π(y )) = s(π(y )). It is 1 2 easytocheckthatthisisomorphismpreservesthe respectivebundle gerbemultipli- cationsandwehaveshownthatifY admitsaglobalsectionthenanybundle gerbe (P,Y) is trivial. Notice thatthe converseis nottrue. Justtakeanopen coverwith more than one element and g = 1 to obtain a Hitchin-Chatterjee gerbe which αβγ has zero Dixmier-Douady class but for which YU →M has no global section. ConsidertheDixmier-Douadyclassofδ(R). Ifs : U →Y arelocalsectionsfor α α a good cover choose local sections η of s∗(R). Then we can take as local sections α α of(s ,s )∗(δ(R)) the sections σ =η ⊗η∗ and it follows that the corresponding α β αβ α β g = 1 and δ(R) has Dixmier-Douady class equal to zero. The converse is also αβγ true. Consider a bundle gerbe with Dixmier-Douady class zero. So we have an open cover and σ such that αβ g =h h−1h . αβγ βγ αγ αβ