L'6 "os
Daniel Stevenson
Thesis submitted for the degree of Doctor of Philosophy in the Department of Pure Mathematics IJniversity of Adelaide
9 February 2000 Abstract
This thesis reviews the theory of bundle gerbes and then examines the higher di mensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced. and it is shown that there is a class in Ha(M;Z) associated to any bundle 2-gerbe. n
IV Acknowledgement
I would like to thank my supervisor Michael Murray for his patience, generosity and expert guidance in the writing of this thesis. He has greatly enhanced my under- standing and enjoyment of mathematics over the past few years and I would like to say that it has been an honour to work \Mith him during this time. It is my pleasure also to thank AIan Carey for some useful conversations and also Ann Ross for help in ad- ministrative matters. I would also like to gratefully acknowledge the support provided by an Australian Post Graduate Award.
v vl Contents
1 Introduction 1 2 Review of some simplicial techniques b 2.7 Simplicial spaces 5 2.2 Geometric realisation 8 2.3 Classifying sPaces 11 2.4 BPC" bundles 11 3 Review of bundle gerbes 15 3.1 Properties of C* bundles 15 3.2 Bundle gerbes and the Dixmier-Douady class ' 18 3.3 Bundle gerbe connections and the extended Mayer-Vietoris sequence 23 3.4 Bundle gerbe morphisms and the 2-category of bundle gerbes . 25 4 Singular theory of bundle gerbes 35 4.1 Construction of a classifying map 35 4.2 The singular theory of C" bundles 37 4.3 The singular theory of bundle gerbes 4I
5 Open covers 51 5.1 Bundle gerbes via oPen covers 51 5.2 Bundle gerbes and Deligne cohomology 52 6 Gerbes and bundle gerbes Ðl 6.1 Torsors 57 6.2 Sheaves of categories and gerbes 58 6.3 Properties of gerbes 61 64 6.4 The gerbe HOM(P,Q . ' 6.5 The gerbe associated to a bundie gerbe 66 6.6 Higher gluing laws 69 7 Definition of a bundle 2-gerbe 77 7.1 Simplicial bundle gerbes and bundle 2-gerbes 77 7.2 Stable bundle 2-gerbes 80 8 Relationship of bundle 2-gerbes with bicategories 83 8.1 Bicategories 83 8.2 Bundle 2-gerbes over a point and bicategories 92
vll I Some examples of bundle 2-gerbes 97 9.1 The lifting bundle 2-gerbe 97 9.2 The tautological bundle 2-gerbe 98 9.3 The bundle 2-gerbe of a principal G bundle 100
10 Bundle 2-gerbe connections and 2-curvings 103
10.1 Definitions and some preliminary lemmas . 103 10.2 Proof of the main result 106
11 Comparison of the Cech, Deligne and de Rham classes associated to a bundle 2-gerbe 111 11.1 A Õech three class 111 1I.2 A Deligne hypercohomology class 115 11.3 The first Pontryagin class IL7
12 Trivial bundle 2-gerbes 119 12.1 Tfivial bundle 2-gerbes 119 L2.2 T)ivial stable bundle 2-gerbes I28
13 The relationship of bundle 2-gerbes with 2-gerbes 131 13.1 Simplicial gerbes and 2-gerbes 131
vnl Chapter 1 Introduction.
There has been much recent interest in applying the theory of gerbes to differential geometry l22l and mathematical physics ([Za] and [13]). Gerbes were originally in- troduced by Giraud (see [19]) and were extensively studied in [7]. Loosely speaking a gerbe is a 'sheaf of categories' except that the¡e is an extra level of complication when one comes to discuss sheaf-iike glueing laws for objects. Much of the importance of gerbes in the previously cited applications stems from the fact that equivalence classes of gerbes over a manifold M are in a bijective correspondence wifh H3(M;V'). This is entirely similar to the fact that H'(M;Z) lsin abijective correspondence with isomorphism classes of principal C" bundles on M. One can think of Ct bundles and gerbes as a geometric 'realisation' or 'represen- tation' of the cohomology of the manifold in dimensions two and three respectively. Flom this perspective it is natural to look for such objects in other dimensions. In dimension zero we can consider the set of all maps f , M + Z which are, in fact, in bijective correspondence with Ho(M;Z). In dimension one there are various kinds of objects we can consider which are in bijective correspondence with H'(M;Z). For exampie we could consider (isomorphism classes of) principalZbundles over M. Or, because the classifying space of. Z is C" we could equally well consider the space of homotopy classes of maps from M into C'. A similar story is true lor H2(M;Z): one could. consider H'(M;Z) to be the set of all homotopy ciasses of maps Í , M - BC', where BCt is the classifying space for the group C', or, we could think of an element of H2(M;Z) corresponding to a homotopy class of maps M -+ BC' as giving rise to a principal Cx bundle on M. In the case of H'(M;Z) we can go one step further and think of a principal C* bundle on M as a principal BZ bundle on M. The crucial fact here, as pointed out in [36], is that BZ, can be chosen to be an abelian topological group since Z is. This last fact is exploited in the paper [1S] by Gajer where the author introduces a notion of a principal BnCx bundle on M and proves that there is a bijection be- tween ¡{n+z(M;Z) and the set of all principai BPC'bundles on M (here BrC' is the pfotd iterated classifying space B...BC" of the group C"). Therefore, when one comes to consider H3(M;Z) there are several possibilities: one option is to think of Ht(M;Z) as the set of all homotopy classes of maps f , M --+ BBC, another is to think of H3(M;Z) as the set of equivalence classes of gerbes on M. Yet another is to follow Gajer and think of. H3(M;Z) as the set of all principal BC" bundles on M or, alternatively, the set of all principal BBZ bundles on M. gerbes give rise In f33] the notion oî. a bundle gerbe was introduced. Bundle also to
1 classes in H3(M;Z). A bundle gerbe on M is a tripie (P,X,M) where ry : X -+ M is a surjection which locally admits sections and where P is a principai CX bundle over thefibre product ylz): X x¡r,tX. P is required tohave aproduct, that is aC" bundle isomorphism which on the fibres of P takes the form
P(r",r")Ø Pçqp2¡ - P(rr,rr) for ø1, 12 aîd z3 in the same fibre of X. It can be shown that every bundie gerbe on M gives rise to a gerbe on M. There is a characteristic class associated to bundle gerbes, the Dixmier-Douady class. This is a class in H3(M;Z) which measures the obstruction to the bundle gerbe being 'trivial' in a certain sense. In [7] the notions of a connectiue structure on a gerbe and a curuing for the con- nective structure were introduced. It was shown that one could associate to a gerbe equipped with a connective structure and curving a closed integral three form ø on M - the three curvature of the curving. In [33], it is shown that by considering a connection on P which is compatible with the product above, it is possible to introduce the notion of a bundle gerbe connection and a curving fo¡ a bundle gerbe connection and that, furthermore, a bundle gerbe on M equipped with a bundle gerbe connection and curving gives rise to a closed integrai three form on M. This three form is a repre- sentative of the image of the Dixmier-Douady class of the bundle gerbe in Ë13(M;R.). It can be shown that a bundle gerbe connection and curving on a bundle gerbe induces a connective structure and curving on the gerbe associated to the bundle gerbe. In the series of papers [S] and [9] the authors considered a certain Z-gerbe associated to a principal G bundle P --+ M where G was a compact, simpie, simply connected Lie group. 2-gerbes, introduced in [5], are higher dimensional analogues of gerbes. Broadly speaking, a2-gerbe on M is a'sheaf of bicategories' on M. Bicategories, defined in [1], are higher dimensional analogues of categories. A bicategory consists of a set of objects and for each pair of objects, a category. The objects of this category are then l-arrows of the bicategory and the arrolvs of the category are 2-arrows of the bicategory. As a result, the giueing iaws involved in the definition of a sheaf of bicategories become quite intricate. One can show that if one considers a certain class of 2-gerbes (2-gerbes bound by C") then one can associate an element of. Ha(M;Z) to each 2-gerbe and moreover it can be shown that H (M;Z) is in a bijective correspondence with 'biequivalence' classes of 2-gerbes. In [14] a related object, a bundle 2-gerbe, was introduced. In the latter part of this thesis we will study bundle 2-gerbes using a variation of the definition given in [t4]. A bundle 2-gerbe is a quadruple (8, Y, X, M) where r : X -> M is a surjection admitting local sections and where (Q,Y,¡tzJ) is a bundle gerbe on the fibre product X[2]. One also requires that there is a bundle 2-gerbe product. This consists of a'product' onY, ie a map which on the fibres of Y takes the form
ffi : Y(rr,rr) x Y(rr,rr) --+ Y61,rs), where rt, t2 and ø3 are points of X lying in the same fibre. There is also a product on Q covering the induced product on Yl2l. The product on Q is a C' bundle morphism rîz given fibrewise by
ñ t Q (nrr,oLr) Ø Q (n,,",v'rr) - Q (rn(orr'orr),^@Ls,v'rz))' This is required to commute with the bundle gerbe product on Q and satisfy a certain associativity condition. One can show that there is a Ct valued Õech 3-cocycle g¿¡¡,¡
2 representing a class ín Ha(M;Z) associated to a bundle 2-gerbe. In [33] it was shown that the complex ao(M) \ ar6) \ snçYtz)) g, was exact, where 5 , çnlyløl) --+ f-¿P(X[ø+t]¡ '. the differential formed by adding the projection maps zri with an alternating sign (here zr¿ ' yÍd + Xls-tl deletes the ith factor). By exploiting this fact one can define the notion of. a bundle 2-gerbe connect'ion and a 2-curving for a bundie 2-gerbe. This leads to a closed integral four form @ on M which is a representative in Ha(M;R) for the image, in real cohomology, of the class defined by the cocycle g¿¡¡¡. Associated to a principal G bundle P --> M f.ot G a compact, simple, simply connected Lie group, is a bundle 2-gerbe (Q,Þ,P,M). If one calculates the cocycle g¿j¡,¿ associated to the bundle 2-gerbe Q then one recovers the result of [8] giving an explicit formula for the first Pontryagin class of P. Finally one can also show that every bundle 2-gerbe on M gives rise to a 2-gerbe on M. In outline then this thesis is as follows: In Chapter 2 we review the definitions of simplicial spaces and the construction of the universal G bundle EG - BG, we give a description of the group law on EG and we discuss some of the results of Gajers paper [1S]. In Chapter 3 we begin with some remarks on principal C'bundles before going on to review the definition of a bundle gerbe and some of the main results from [33]. We then discuss the notion of stable isomorphism of bundle gerbes and the result from [34] that the set of all stable isomorphism classes of bundle gerbes on M is in a bijective correspondence with Ht(M;Z). Next we discuss bundle gerbe morphisms and transformations of bundle gerbe morphisms as well as the related notion of stable morphisms and stable transformations. Finally we show that the collection of all bundle gerbes on M forms a 2-category with bundle gerbe morphisms as 1- arro\r¡s and transformations as 2-arrows. In Chapter 4 we discuss the singular theory of principal C' bundles in a form which readily generalises to give a singular theory of bundle gerbes. Chapter 5 discusses bundle gerbes from a local viewpoint along similar lines to the presentation in l22l for gerbes and Deligne cohomology for bundle gerbes. Chapter 6 is mostly devoted to a review of the material presented in [7]. We also discuss the construction from [34] of a gerbe associated to a bundle gerbe. The iast section of Chapter 6 contains a discussion of higher glueing laws. The remainder of the thesis is concerned with 2-gerbes and bundle 2-gerbes. In Chapter 7 we define the notions of simplicial bundle gerbe and bundle 2-gerbe. Chap- ter 8 starts with the definitions of a bicategory and bigroupoid and goes on to discuss in some detail the homotopy bigroupoid II2(X) associated to a space X. We also show that a bundie 2-gerbe on a point is the same thing as a bicategory in which the auto- morphism groups of each 1-arrow are isomorphic to Cx. In Chapter 9 we give some examples of bundle 2-gerbes, namely the tautological bundle 2-gerbe from [t4] and the bundle 2-gerbe associated to a principal G bundle P ---+ M. In Chapter 10 we define the notions of bundle 2-gerbe connections and 2-curvings for bundle 2-gerbe connections and show how a bundle 2-gerbe connection with a 2-curving on a bundle 2-gerbe gives rise to a closed, integral four form O on the base manifold M. We also show that bun- dle 2-gerbe connections exist, something that is not immediately obvious. Chapter 11 begins with the construction of a C* valued Cech S-cocycle 9¿j¡¿ associated to a bundle 2-gerbe and then goes on to show how a bundle 2-gerbe equipped with a 2-curving and bundle 2-gerbe connection gives rise to a class in the Deligne cohomology group
3 Ht(M;Ci* A!* As a consequence, v/e see that the four form O - - -,2u - Air) represents the image in Ha(M;R.) of the class in Ha(M;Z) p,ssociated to the cocycle g¿¡ta. Finally we show the four class of the bundle 2-gerbe Q arising from a principal G bundle P + M is the first Pontryagin class of P. In Chapter 72 we discuss the notion of a trivial bundle 2-gerbe and prove that if the four class in Ha(M;Z) of abundle 2-gerbe vanishes then the bundle 2-gerbe must be trivial. The final Chapter 13 discusses the reiationship of bundle 2-gerbes to 2-gerbes.
4 Chapter 2 Review of some simplicial techniques
2.L Simplicial spaces pn+l : Recali, see [29], the standa¡d n-simplex A' in it defined by A' {(ro, ... ,rn)'. ro*...*rn: t) If.n> l there are face maps 5i ' 6n-r -t A' 1.or i:0,1,... ,n given by õo(ro,... ,r,.-t): ("0, ... ,ti-r,O,r¿,... ,r,,,-t). For the same n we have degeneracy maps oi : Ln + [n-]','i : 0,1,...,n - 1 with on(ro,... rrn) : (ø0, .. .,ri-Lrr¿ * r¿¡1,1i+2,...,rn)' Definition 2.L. A simpli,cial space X is a sequence {X"} of spaces Xn f.ot n : 0,I,2,... together with face operators d,¡ : Xn --+ Xn-1,'i : 0,1,..',n for n : 1,2,3,... and degeneracy operators s¿ Xn+ Xn+t,'i:0,1,... , nf'or n:0,I,2,... which together satisfy lhe simpli,cial identi'ttes
d¡d¡ : d¡-ñt, i
Notice that there is a definition of simplicial manifold, simplicial group, simplicial ring and so on; we simply repiace the topological category by the smooth category or the category of groups and so forth. One then requires that the face and degeneracy operators are morphisms in the appropriate category. Thus a simplicial manifold is a sequence of manifolds {X"} together with smooth maps d,¿: Xn + Xn-r for i : 0,. . .n and s¿ ; Xn -+ Xn+r for i, : L,, . . .fl satisfying the same conditions as in Definition 2.1 above. Similarly, a simplicial group is a sequence of groups {G"} with face and degen- eracy operators d,¿ and s¿ as above which are required to be homomorphisms between the various groups G,,. A more concise definition is given as follows (see [27] or [29]). Let A be the category whose objects are all finite ordered sets [n] : {0, 1, . . . ,n} and whose morphisms
5 l*l -- [n] are all nondecreasing monotonic functions lml -- fnl. One then restricts attentiontothe morphisms 6i:[n- 1] * [rz] and oi:ln]-ln- 1] foli:0,L,...,n given by
rf j ö'(j): J
j rtt L o'(j): j-I iÎ t ?,
One can then prove a Lemma which states that every morphism p t lm) + [n] in Hom(A) can be written uniquely as
þ - 6'"6u ... StcoJz7Js ... oJ", where n ) 'io ) 'i6 ) .'. > n'L- z *c. (See for instance [27]). Now we can define a sr,mpli,cial object in the category of topological spaces, a simplicial space in other words, as being a functor F : Lo + C where C is the category of topological spaces. Here A" denotes the category opposite to A - recall the category C" opposite to C is the category with the same objects as C but with the source and target of each arrorü/ in C interchanged. The simplicial identities arise because the morphisms ó'' and o' satisfy the analogous identities in A. Changing C to the category of smooth manifoids or the category of groups recovers the definitions of simplicial manifolds and simplicial groups. A good source of examples of simplicial spaces is the nerve of a category. (see [35]) Let C be a category. We defi.ne a simplicial set ,A/C called the nerve of the category C as follows. ,À/C¡ is defined to be the set of objects, Ob(C), of C (so we had better only consider small categories) , NCt is defined to be the set of arrows or morphisms Mor(C) of C and in general NC^ is the subset Mor(C) o . . . o Mor(C) of Mor(C) x . . . x Mor(C) (n factors) consisting of all n-tuples (fi ,. . .fò of morphisms of C with source(f¿) : target(f¿..,.1). The face operators are defined by the source and target maps and composition of morphisms while the degeneracy operators are defined by including an identity morphism. More specifically, d,; : NCn -. NCn-t for n ) 1 is defined by (f2,...,f-) fori:0 d¿(h,...,f,"): ("ft,.. ., f¿o fn+r,..., f-) forl(i s¿(h,..., Í,) : (Íu..., f¿,id,f+r, ..., f,"), where id is the identity arrow at target(f¿) in C. We will be especially interested in categories C f.or which the sets of objects and arrows have topologies such that the source and target maps and the operation of composition of arrows are continuous in these topologies. Segal in [35] calls these 6 topologi,cal categories. Segal shows that there is a topological category Xy associated to a coveringl,l : {Un)n¿ of a topological space X (see [17]). The objects of. Xu consist of pairs (r,Ur) where r € [J¿. There is a unique morphism (t,Un) - (A,Ui) if and only if. r: A e LI¡¡: U¿l(J¡. The objects of. X¿1 are topologised as Ob(Xu):IfrqUn and the arrorvs of. Xu are topologised as Arr(Xs) : If.¿,jet,ur,¡øUoi. Clearly Xq is an example of a topological category. Of more interest are the two topological categories G and G defi.ned by Segal which are associated to a topological group G. Here G is the standard category associated to a group, so G has one object and the morphisms of G are just the set of points of G. Clearly G is a topoiogical category. Thus the nerve of the topological category G is the simplicial space l[G with NG- - G x .. . x G (n factors) and face and degeneracy operators given by (s, , 9n), i,:0 dn(gr,...,gn): (g,. ,9¿9¿+t, 9n), !3i' dn(g0,..., g,") : (go,...,gi-rtT¿+t,...,9n) and s¿(go, . . . , gn) : (go, . . . , 9i, 9¿, 9¿+t, . . . , 9n)' We shall return to these examples in the next section. Finally, we wouid like to define the notion of a simplicial map between simplicial spaces. Definition2.2. Let X and Y be simplicial spaces. A simplici'al map f , X + Y is a sequence of maps Ín : Xn -+ Yn which commute with the face and degeneracy operators. Thus if d,¿, s¿ a,Íd d!u, sl denote the face and degeneracy operators of X' and Y, respectivel¡ then we have Ín-tod,¡:dtoofn and ln+roS¿:stoofn' If we view simplicial spaces, simplicial groups and so on as functors F : Lo + C for an appropriate choice of the category C, then a simplicial map corresponds to a natu¡al transformation between functors. More concretely, if Fr, F2 : L - C represent simplicial objects in some category C, then a simplicial map from ,Fr to F2 is a natural transformation r : F1 è Fz. The definition above can be recovered by recalling that 7 anaturaltransformationþ: F + Gbetween functors F,G:C -D assigns to each object c of. C an arro\r¡ ó(") , F(c) + G(c) in D such that the diagram F(')' ,F'(cr) F("r) ó(ct) ó("2) G(o) G("t) G("r) commutes for all arro\Ms a: cy + czin C. Thus to each object [n] of Ao, T associates an arrov/ rn : F1(["]) - Fr(|"D, which, by the commutativity of the diagram above, must commute with the face and degeneracy maps 4(ôu), Fr(on), Fr(6n) and .F'2(øi). Notice that a functor F : C1 -- Cz induces a simplicial map ,^rIF : NC1 ---+ NC2 between simplicial sets by definition of a functor. If. Q and Cz are topological categories then we of course require that the maps on objects and arrows induced by F are continuous. As an example, take the topological categories G and G associated to a topological group G. Then 'ú/e can define a functor p : G G by defining p(h, gz) : _1 -' gl'gz. It is easy to check that this defines a continuous functor and hence a map of simplicial spaces Np : NG -' .^/G. 2.2 Geometric realisation There is a procedure by which one can obtain a single topological space from a simplicial space known as geometri,c reali,sation. Definition 2.3. Lei X be a simpìiciaÌ space. The geomeiri,c reaiisaüon oi X, cienoteci lXl, is the quotient x xn/ f[n" - , where - is the equivalence reiation (6i(t),r¡ - (t,d¿(r)) for ú € A'-1, r €. Xn and ,i:0,1,...,T1, TL: I,2,... and (oi(t),r) - (ú,s¿(ø)) for ú € Ln, r €. Xn_1, ,i: 0,1,.. ,nandn:0,7,2,.... lxl is made into atopological space in the following way as explained in [30] . Let FnlX I denote the image of IJI:o Li x X¿ in lxl and give Fnlxl the quotient topology. Then Fnlxl is a closed subset of 4*t lXl and lXl is given the topology of the union of the Fnlxl. We remark that there is another realisation associated to a simplicial space X called Ihe fat ¡ealisation of X and denoted llXll. llxll is formed from the same disjoint union LI A" x Xn as lxl but the equivalence relation - now only identifies (õi(t),r¡ - (t,d¿(r)). See [37] for a comparison of the two methods of realisation. In particuiar, Segal proves that under certain conditions on the degeneracy operators of a simplicial spaceX, there is ahomotopy equivalence llxll = lxl. If X is asimplicial manifold, then it is sufficient that the degeneracy operators s¿ embed Xn-1 a,s a submanifold of X", see [31]. Also note that a simplicial map "f : X + Y between simplicial spaces X and Y induces a map l/l , lXl ---> lyl on the geometric realisations of X and Y. We now examine the effect of realisation on our examples so far. Let EG : lNGl and BG : l¡/Gl EG is then the space consisting of all equivalence classes l(tg,...,rr)(g0,...,9òl with (r0,...,rp) e Ap and g¿ € G under the appropriate equivalence relations. These are 8 ((0,"0, ... ,rp-t),(go,' -. ,gp)) - ((r0,. .. ,rp-t), ((gt,' " ,9ò) (("0,. ..,tp-rtO), (go, ...,g)) - ((r0,. ..,rp-l),(gy, "',go-)) ((r0,. ..,,ri-r,O,ri,.'.,rp-t),(go,'. ., gp)) - ((t0,. ..,rp-t),(go, "',0¿, "',9ò) ((r0,. ..,ti-r,r¿ -l ri+!,ti+2,...,rp),(go,...,gò) - ((to,' ",rp),(go, "',9i-r,9¡, 9¡, "',9ò) BG \s then the space consisting of all equivalence classes l(r0,.',rr),lgr,'..,gÅl with the equivaience relations given by ((0,"0, . -.,rr-t),lgr.,'. -,gp]) - ((t0,. ..,rp-t),lgr, "', 9ol) (("0,. .. ,tp-rt0), [gt, ... ,gof) - (("0,. " ,rp),lgt', " ' ,go-l) ((t0,. ..,ti.-!,O,r¿,.'.,rp-t),lgr,.-.,gr]) - ((r0,. ..,rp),lgr',"',9¿-t9¿,"',9r1) (("0,. ..,ri-L,r¿* r4Ltri+2,...,rp),lgt, -.',Zp-tl) - (("0,. ",rp),lgr, "', 9i-r,7,9¿, "',Zp-l) These coordinates are called the homogenous coordinates on EG and BG. Recall the simplicial map l/p : NG + lúG defrned above. .l/p induces a map, also denoted p, on geometric realisations p : EG --. BG. This map is given in the homogenous coordinates by p(l(ro, ... ,rp),(90,.. - ,9òl) : l(ro, '.' ,rp),lgotgr,"' ,gl]rgo]l' We will come back to this later. Of particular interest to us is the following fact: the geometric realisation of a simpliciai group is a group. For a proof of this see [30]. It is not hard to see that the simplicial space l/G is a simplicial group while l/G is a simplicial group if and only if G is abelian. Thus we have the foliowing result (see [36]). Proposition 2.L (136]). EG is 0, group and contains G as a closed subgroup' To make sense of the group structure on EG it is convenient to introduce nerù/ (see If (to, . . . € coordinates - the so-called non-homogenous coordinates - [1S]). ,xp) Ap thenlet ú¿: ro*"'*r¿-tfoti':1,...,P.Then 0 ( úi < "' equivalence class l("0,. .. ,rp),(gg,.. . ,gòl ín EG is written in the non-homogenous cáordinates as lú1,.. . ,to,hoihtl"'lho]l where we have set hs :9o, h¿:9-¿]t9¿, ¿>7. Given an element ( of. EG written in the non-homogenous coordinates as ( - Itr,... ,te,hsfhyl...lhr]l we may recover the description of { in the homogenous coor- dinates by setting r¡ : t¡+t - ú¿ (definê ts:0 and tp+r:1), 9o : ho, gt: hohr, gz: hohth2, etc. Then € : l("0, ... ,rp),(go,. .. ,gò1. When \¡/e use non-homogenous coordinateson EG the equivalence relations above take the following form (see [t8]). (0,úr, ...,t.p-L,holhl"' läo]) - (út,.. .,tp-r,hoh]h2l"'lhrD (úr,.. .,în,...,tp,holhl...läo]) - (úr,.. . ,t,p,hslh1l"'länlllhn+tl "'lhr]) (úr,. .,ti-r,t¿rt¿,t¿¡1,. ..,t,p-r,holhrl' " lho]) - (út, . . .,tp-r,holhl. . .lhohn+l.l''' lhr]) o Similarly an element a € BG is expressed in the non-homogenous coordinates as a : Itt, . . . ,to,fh.tl lhr]l where ú¿ and h¡ are as above. The above equivalence relations then take the form (see [ia]) (ú,, . . .,te,fhtl . . lh"]) forú1 :0orht:7 "lhuhn+| ' lhr]) for t¿ : ú¿a1 or lL¿ : I t]) fot to: I ot ho:1' The map p: EG + BG is expressed in the non-homogenous coordinates as . . Itr, . . .,tp,hsfh1l lho| - l¿r, . . .,to,lhrl. lhr]l We can now write down the group law on EG. We have Itr,... ,tþhsfhll lhrll.lto*r,. . . ,tp+q,h'olho+tl" .lhr*oll . .. : lúo(r) ,. . . ,to(p+q¡,hsh!olk"ç1ll |k"ø+rl]1, Q I) where ko?):fh"¡'¡ifo(r)>P' | {nn.r h"<,¡)-thïth"rohto(D,¿es,ho(i)) tf- o(r) < p. Here,S": {i < r: o(i.) > p} and where o € Sr¡n is the permutation onp*q letters such that 0 ( ú,11¡ < ... < One can check that this is well deflned. It is worth mentioning that Segal gives a much more eiegant ciescription of ihis group iaw in [36]. Segai takes the viewpoini that one may regard EG as the space of step functions on [0,1] with values in G, ie a G valued function constant on each half open interval (tn,t¿+tl for some partition 0: úo ( ú1 ( "'1t, ltr*r: 1 of [0,1]. The group law on EG is then the pointwise multiplication of these step functions. The unit of EG is the obvious one and an element €: lút, ... ,tp,holhl'..lho]l has inverse given by €-1 : lúr,. . . ,tp,hlLlkrl .. .lko]|, where k¿ : hoht. . . he-thrr hi]r. . . h,,' hlt. IL is worth noting that there is a method of calculating the cohomology of the geo- metricreaiisation lxl of asimplicialspace X,at leastinthecasewherethehomotopy equivalence llXll = lXl is available, from the following theorem. Theorem 2.L. Let X be a simpl'ic'ial space. Then there is an i,somorphism H'(l R) .Fr'(S'''(X); IR), lXl I ; = where H'(S"(X);R) denotes the total cohomology of the double compler 9n'e(X) : Sn(Xr;lR) u'iúh boundary operators d andõ. Here d is the s'ingular coboundary operator and 6 : Sq(X,p;R) - Sq(Xpt"t;R) is th,e sim,pl,ici,al cobou,ndary formed bg addi,ng the pullback face operators di wi,th an alternating si,gn. For a proof see either [tZ], or [3]. As explained in both of these papers, the singular functor ,9 can be replaced by the de Rham functor f) in the above theorem if X is a simplicial manifold. 10 2.3 Classifying spaces group Recall [23] that a topological principal G bundle for G a topological is defined as follows. Definition 2.4 (1231). A topologi.cal pri,nczpal G-bundle is a triple (n, P, M) where P and M are topological spaces such that 1. There is an effective action of G on the right of P (effective means that p' g : p imPlies g : L). 2. r : P - M is a surjection whose fibres are the orbits of the G-action on P. 3. Let plzl -- P x u P denote the fibre product of P with itself. Then the natural map 1 plzl -- G is continuous. We say that the action of G on P is stronglg free. 4. There exist local sections of ø, ie there is an open cover U : {U¿}¿¿¡ of. M together with sectiors ,e¿ : U¿ --+ P of r. Recall, see for example [23], that one way of defining a uniuersal G bundle is to say that it is a principal G bundle whose total space is contractible. The base space of such a bundle is called a classifying space and is unique up to homotopy equivalence. The classifying space has the property that isomorphism classes of topological principal G bundles are in a bijective correspondence with homotopy classes of maps from M into the classifying space. A choice of representative / of the homotopy class corresponding to the bundle P has the property that the pullback of the universal G bundle by / is isomorphic to P. It is not to difficult to see that the space -ÐG constructed above is contractible with contracting homotoPy hr(l@g,. ..,np), (gr,..., 9p)l) : l(1 - t,trs,...,trp),(L, gr,. .', 9ò1, as given in [17]. Since G is a subgroup of. EG, there is a right action of G on EG. By the formula 2,1 above, we have that the right action of G on EG is given by lút; ... ,te,hslh1l"'läo]l'g: lút,... ,tphoglhrl ' lh"]l Clearly p is invariant under this right action of G- provided that G is a In [36], it is shown that p : EG - BG has local sections, locally contractible group, and so p : EG -- BG is a principal G bundle and therefore a universat G bundle. \üy'e shall list some properties of. EG and BG for G abelian in the next section. 2.4 BPC" bundles In this section we are interested in some properties of. EG and BG for G : C". Because C" is abelian, both ,EC" and BC" are groups in fact they are abeiian - 'We groups. The formula of equation 2.1 now takes a particularly simple form. have 11 for lú1,...,t,20121...1+]l and lúoa1 ,...,tp*Qty'olzo+t|..'l"o*n]l elements of ,EC* the following formula for their product (see [18]). Itr.,. . .,to, zsfzll. . . l"r)1. ltr*r, . . .,tp*qt zolzp+tl. . . Ito*Ål lú,(r), . . . tto(p+q), zsz'slLzrly¡l' . . lrrø+nil, where o € Sr¡n is a permutation on p -l q letters such that 0(ú"11¡<"'< The group law on BC" is given as follows (see [1S]). If l¿1,...,trfrtl...l+| and Itp+t,... ,tr¡nfzp+t|..-lto*nll are elements of BCX then their product is given by Itr,. . .,to[rrl " . l+]1. ltp+t,. . .,,tpaqlzo+1| . .' lro*Àl : lúo(r),. . -,to(p*s)lzoe)l''' lr"ø*nll, where o is as above. Ciearly the projection p : EC" --, BC is now a group homomor- phism. In fact there is a short exact sequence of groups 1 -* C" -- EC,* -> BCx ---+ 1. This is a generalisation of the exponential short exact sequence 1 -+ V, + (l --+ (lx --+ l Since BC" is now an abelian topological group, r,¡¡e can iterate the classifying space construction to form the abelian topological group BBC" : B2C' and in general B ... 8C" : BpCx. Again we have the iterated exponential short exact sequence ! ---+ BPC." - EBPC' --; fiP+1Çx + l. In [18], Gajer develops a theory of smooth principal BpC" bundles. To do this he introduces a differentiable space structure on EBpC,' and BpCX and defines the notion of a smooth (in this differentiable space sense) BPC* bundle. It is important to note that this differentiable space structure is not the same as a differentiable structure on a smooth manifold. Indeed, this differentiable space structure on a Hausdorff space M is defined in [16] (see also l3Z]) by requiring that the¡e is a family of continuous maps 1 : U + M called plots, each of whose domain is a convex set in some Euclidean space (the dimension of which need not be fixed), which satisfy o if.1 : U + M is a plot, V a convex set and 0 : V --. U a smooth map, then 1o0:V+Misalsoaplot, o ev€ry constant map from a convex set into M is also a plot, o lf. U is a convex set and {Un}nq is a covering of U by convex sets U¿ each of which is open in [/, and 'y : U + M is a map such that each restriction 7¿ : 7l¿ is a plot, then 7 is also a plot. 72 Clearly every smooth manifold has a differentiable space structure, but not every Haus- dorff space with a differentiable space structure is a smooth manifold. Gajer then goes on to show how every BpC* bundle is induced by pullback from the universal BPC" bundle EBvC" --+ BP+LCX and shows that there is an bijection between isomorphism classes of. BpC" bundles and HP+2(M;Z). For us the most important thing about all of this is that BzC" provides us with an explicit realisation of an Eilenberg-Maclane space K(2,3) 'We see this as follows. Flom the exact homotopy sequence of the fibering BpC' ---+ EBPC" --+ BP+tçx and the contractibility of EBpC" we get rn(Bn+LCx):Ts-t(BeCx). Hence by induction we getrn(BeC*) : zrn-o(C') 1f q >p and rn(Bnc,") : 0 if q < P. Hence Zifq:p*I tro(BeC.x) : 0 otherwise. So BpC* is a K(Z,p+ 1). A theorem of algebraic topology asserts that for a given space M and, a cohomology class u e H\(M;Z) therc is a map Í, M + K(2,3), unique up to homotopy, such that ø is the pullback f * t of. the fundamental class r" e HT(K(Z;3);Z):V, (see for example [38]). Thus every class ø e H3(M;Z) isthe putlbaci< of the generator of H3(82C";Z)by some map M -+ B2C" which is unique up to homotopy. 13 74 Chapter 3 Review of bundle gerbes 3.1 Properties of C" bundles Let G ¡a denotes the sheaf of smooth G valued functions on M for G an abelian Lie group. For the definitions and basic properties of sheaves see Chapter 1 of Brylinski's book [7]. For us, it is sufficient to know that a sheaf 5 of abelian groups on a manifold M gives rise to sheaf cohomology groups HI(M;S) foli:0,1,... and that given an exact sequence of sheaves of abelian groups 0--+A+B--+C+0, on M , there is a long exact sequence of sheaf cohomology groups g ... H^(M;A) - H'(M;þ) - H"(M;q L gn*r@;A) -) ... The homomorphism 6 : H"(M;C) - H^+L(M;.Á) is called the connecting homomor- phism. In [7][Corollary 1.1.9], Brylinski proves that the sequence of sheaves of abelian groups on M, 0-Zu-Crø-CTt-0 is exact. Hence there is a long exact sequence in sheaf cohomology g ...4 H"(M;Øu)-H'(M;Çà-H"(M; îr) H^+t(M;Øà-"' (3.1) In Chapter 1 of his book [7], Brylinski defines the notion of a soft sheaf of abelian groups and proves that the sheaf cohomology groups associated to a soft sheaf are all zero. It is a fact that C¡r, is a soft sheaf. Since the sequence of sheaf cohomology groups 3.1 is exact, we have in particular that Ht(M;Air) = H'(M;Z*), the iso- morphism being provided by the connecting homomorphism ô. Brylinski in [7] proves that if M is a paracompact manifold, as we will always assume, then Cech coho- mology is canonically isomorphic to sheaf cohomo phisms H'(u;Aio) = H'(M;C|,.) and n'(u;v'à = H own in [ ] that tüe Õech cohomology groups ¡t^(¡WtZnò u groups H*(M;Z) coíncide. Thus we have an isomorphism Recall that there are several ways of characterising principal C" bundles. One such way is to describe a C' bundle P -+ M by its transition functions gii : U¿¡ + Çx relative to some open cover î,1 :,{U¿}¿et of M. These transition functions are then representatives of a class in the Õech cohomology group Ht@;Ai)' By the above 15 discussion, the transition functions gij define a class in H2(M;Z). This is the Chern class of the CX bundle P. It is a standard fact, see [26], that given smooth maps g4: U¿, _-t C* satisfying the cocycle condition g¡ng*tg¡¡: 1, one can construct a CX bundle P - M with Chern ciass represented by g¿¡ in F/r @;Ait) If Pr and P2 are two C* bundles on M described by transition functions p¿¡ : gØ - C' and h¿¡ : U¿¡ - C' respectively, where the transition functions are both defined relative to the same cover U: {Un}nq of. M, then we can form a C* bundie with transition functions equal to g¡¡h¿¡. The reason this is possible is because C' is abelian. The resulting bundle we call the contracted product of. P1 and P2 and denote by h8R. Another way of constructing this bundle, see [7][Chapter 2], is to first form the fibre product Pt x v P2 and then factor out by the subgroup of Ct x C" consisting of all pairs (r, t-') f.or z eC'. The resulting C' bundle is Pt ø P2 andone can check that P1 S P2 has transition functions gt¡h¿t. Given a C" bundle P --+ M wifh transition functions g4 then ìve can construct a C" bundle P* + M with transition functions OÇt. Again the reason this is possible is because C' is a commutative group. This bundle is easily seen to be the one obtained from P by changing the action of C" on P to its inverse, that is the CX bundle associated to P via the isomorphism C" -+ C', z ¡-+ z-r. Notice that if we have a C" bundle P on M, then the CX bundle P Ø P* is canonically trivial (look at the transition functions). It is often more convenient to work with complex line bundles than with C* princi- pal bundles. Given a principal Ct bundle P -+ M we may form a complex line bundle Lp on M by the associated bundle construction. Recall, see [26], that Lp is formed from the product P x C by factoring out by the equivalence relation - defined by (p,u') - (q,u) if anci oniy if there is a z € C' such that q :p.z anci'u :uz-T. Con- versely, given a complex line bundle L on M we can form a principal Ct bundle on M by forming the frame bundle associated to Z (see [26]). This correspondence gives an equivalence of categories between the category of principal CX bundles on M and the category of complex line bundl es on M , see [7] [Chapter 2]. Under this correspondence, the operation of contracted product on principal C' bundles becomes the operation of tensor product on compiex line bundles while the 'inverse' of a principal C" bundle corresponds to taking the dual of the associated complex line bundle. Recall, see for example [26], that the pullback of a C" bundle r : P ---+ M by a smooth map ,f : Iú ---+ M is the C" bundle f-Ln t f-rp ---+ ly' where f-'p is the submanifold of N x P consisting of all pairs (n,p) such that l(n): r(p). Every C" bundle P on M is induced by pullback f¡om the universal bundle EC,x --+ BCX with a map g : M - BC', the classifying map of P. This classifying map g is unique up to homotopy. If P has transition functions g4 : U¿t -, Ct relative to some open cover U : {Uo},r¡ of. M, then a choice of the classifying map g : M --+ BCX is given by g(m): lór(m),... ,ór(m),fguon,(*)1"'lso,-,no]|, (3.2) see for example [18]. Here ó^(m) : tþno(m) +. . .+ tþ;.(m), where {rþn}oq is a partition of unity subordinated to the cover U. We have the following proposition. Proposition 3.1. Let P and Q be pri,ncipal C" bundles on a smooth manifold M. Supposethat P has a classi,fyi,ngrnap f : M + BC" andthatQ has a class'ify'ingmap g : M - BC" . Then the contracted product PØQ has a choice of class'ify'ing rnap gzuen by Í'g: M + BC" andP* hasachoiceof classi,fyingn'10,p giuenbA T-r: M - BC". Here f-r rnel,ns the map i,nu o f , where tnu : BC* -r BCt ts tL¿e rnl,p €r. €-1. 16 Proof. First of all, choose an open covering y : {U.i,}tç¡ of. M such that P and Q have transition functions 9l¡ : U., _' C" and h¿¡ : U¿¡ + C' respectively, both relative to the same open cover Ll. Then, as we have already seen, choices for / and 9 are given by Í(*) : lór(^),.. . ,óo(*),lsoon,(-)l ' ''19n,-,,,(-)ll, and g(m) : lór(m), . . ., óp(m),lhnon,(-)l''' lh6-,n,(m))1, where the ó, are defined as above. If one performs the calculation, using the group Iaw in the abelian group BC* given in Section 2.4, then one finds that f @)g(*) : lôt(m), "' ,ör(*),lgnoo,(*)hrou,(-)l ' "19n,-,uo(m)h¿,-,¿,(rn)ll' But this is precisely the classifying map we would have obtained for P Ø Q using the recipe of equation 3.2. The other statement of the proposition is proved similariy. ¡ principal bundle Recall, see [26], that one way of defining a connection orL a Ct p M is to specify a family of one forms with ø¿ € f-¿t(t/,), for some open -, {rn}nr, * cover tt : {U¿}¡ç¡ of. M relative to which P has transition functions 94 : U" C*, and to require the a¿ to satisfY d,o,^ uj:tt¿* * each overlap ¡J;i: U¿lU¡. A connection on a complex line bundle L M is a on - : Iinear map V :l(M,L) -l(M,T*M Ø,L) which satisfies the Liebniz rule V(/s) d/ A s + /V(s) for all s €l(M,.L) and all / € C*(M,A). If .L1 and L2 are complex line bundles on M wilh connections V1 and V2 respectively, then the tensor product bundle LtØ Lz has a connection Vr f Vz defined by (see [7][Chapter 2]) (vr + vz)(sr I s2) : vr(st) I sz + sr I vz(sz)' Recall also that if V is a connection on a complex line bundle L on M, then given any complex valued one form Aon M, r'¡/e can define a new connection V + A on 'L by (V+A)(s) :V(s) *A8s for a section s of .L. Connections on complex line bundles are easier to get a handle on than connections on principal C* bundles and so in most calculations we will work with connections on complex line bundles. This is justified by Proposilion2.2.5 of [7]. Let P -- M be a C* bundle, and let /' lü -- M and I : O lü be smooth p - p, maps. There is a canonical natural isomorphism rp¡,, , u o g)-L + g-t ¡-t where by natural we mean natural with respect to isomorphisms tþ , Pt - Pz. It has the property that if V¿ is a connection on the associated line bundle L : Lp over M, then g)-tV o g-t o (/ " r: Ør,s ¡-tO r Øt,s, means where Q¡,n denotes the isomorphism of line bundles induced by g¡,n. This that : g-r p for any calcuiations' we can fréely identify (l " ù-'P ¡-t 77 3.2 Bundle gerbes and the Dixmier-Douady class Suppose V 3 tW is a surjection admitting local sections. We form the fibre product Y xy [ : ylz] in the usual way. That is, Y[2] is the set consisting of all pairs (At,aù €Y x Y such thaf r(g): r(Uz). We denote by Ytsl the triple fibre product Y x ¡aY x uY and in general we put Yþl equal to the p-fold fibre produ ctY x ¡a. . .x uY. Notice that we have projection maps r¿ : fb) --+ YIP-rl fori : 1,. . . ,p, obtained by omitting lhe i.th factor. It is easy to see that in fact we have a simplicial space Y : {Yo} with ye : yln+tl and the obvious face and degeneracy operators. Definition 3.1 ([33]). A bundle gerbe consists of a triple of manifolds (f, X, M) with a surjection r : X + M admitting local sections and a Ct-bundle P + X[2], such that P has a product; that is an isomorphism of C*-bundles rn : r¡r P Ø n;r P - r;IP over X[3], covering the identity and such that the following diagram of C* bundle isomorphisms over Xlal commutes: r;Ir¡tP Øral(n;rP ør;LP): r¡1(tr;tP A 7rt1P) Ør;rr;LP rø.;'- j J'l'-ør n;rr¡rP Ør;rr;rP r¡Lr;rP I zrflzr;lP r;1(tr;lP A 7rt1P) r¡r(n¡tP ØlhrP) T2 'fn lÍ"'rnt_, v r;rr;r P r;rr;r P- Here we have used the remarks made at the end of Section 3.1 and the simplicial identities satisfied by the ø'¿. This last condition gives an 'associativity' constraint on the product rn,in the sense that we havem(uØm(u 8r)) : m(m(u ø,u) I u) for LL € Pçr",ra), u e P@r,r") and u € P61,rz), where (rr,rz,rs, r+) ç yln). We will sometimes denote a bunclle gerbe (P, X, M) hV a, single letter P. A bunclle gerbe is typically depicted as in the diagram below. P J lfi --+ ytz) --+ Y lr M We will need the definition of. a simpli,cial line bundle - see [8]. Definition 3.2 ([8]). Let X : {Xr} be a simplicial manifold. Let L be a line bundie on X, Denote by ô(Z) the line bundle d;rLØd,.'L.Ød;rLØ... on Xoa1, where the last factor in the above tensor product line bundle is d;]rL if p is even and d;]rL* If p is odd or zerol where d"¡ : Xp+t - Xn are the face operators. 18 Notice that the line bundl e õ6(L) on Xp¡z is canonically trivial. This is a resuit of the commutation relations satisfred by the d¿ which allow us to contract off pairs of line bundles in the tensor product õ6(L). Clearly ð gives rise to a functor from the category of line bundles on Xo to the category of line bundles on Xp+r. If s is a section of .L, then it induces a section ô(s) of ô(I) defined by tr* do-ls I d, Ø d2rs. .' where, as above, the last factor in this induced section is dfi1s if p is even and dl]rt. if p is odd or zero. A si.mpli,cial line bundle on X consists of a line bundle .L on X1, together with a non-vanishing section s of ô(I) on X2 such that ð(s) equals the canonical non-vanishing section 1 of ðð(¿) on X3. It is easy to see that the bundle gerbe product Ttrp on a bundle gerbe (P,X,M) forces the C' bundie nr'P Ø n;t P* Ø r;L P -' ¡lri to be triviai, and indeed one can define a section s of this bundle as follows. Choose u € Pç,2,rs, and u € Pçr,,2¡, then *p(u8'u) e P@r,,r). We put s(r1,r2, rt) : u Ø nxp(u6 u)* I o. One can check that this is well defined and one can also show that the associativity of mp is equivalent to the following condition on s: 1s ls 1s* z"f I nî's* I zrf I zr. : 1. It follows that the bundle gerbe P can be viewed as a simplicial line bundle on the simpiicial manifold f - {Xo} with Xp : XIP+LI. Simpliciai line bundies on the simpiicial manifold IúG associated to the classifying space of a Lie group G take a particularly simple form. We have the following result. Proposition 3.2 (t8]). A si,mpli,ci,al li,ne bundle on the s'implici.al manifold NG i's a central ertension of G bg C". Recall from [33] that a bundle gerbe (P,X,M) t,as two important algebraic struc- tures; an identity and an inverse. The identity is a section e of the pullback C' bundle A-1P over X, where L,: X - ylzl is the inciusion of the diagonal. The inverse is an isomorphism given fibrewise by Pç,r,,2¡ --+ P@2,æ) and denoted p r- p-1. The identity section and inve¡se satisfy the properties one would expect; for instanc" mp(pØ e) : p and mp(p Ø p-r) : e. It is useful sometimes, particularly when we discuss bundle gerbe connections and curvature, to have a slightly different formulation of the definition of a bundle gerbe (see [13]). In Definition 3.1 above we replace the CX bundle P ---+ Xlzl by a line bundie ¿. We ihen require that there is a line bundle isomorphism IrLL : r;L LØrlt L '- n;'L' The product on -L defi.ned by m¡, is then required to be associative in the sense described above. The two definitions are easily seen to be equivalent. Erample 3.1. The basic example of a bundle gerbe is the tautologi,cal bundle gerbe (see point suppose [33] and also [t4]). Lel M be a smooth manifold with a base rne and ø is a closed, integral three form on M. Assume that M is 2-connected. Form the path frbration zr : PM - M, where PM is the path space of. M soPM consists 19 of all piecewise smooth paths 7 : [0, l) ---+ M with 7(0) : rno. The projection n is given by r(l) :7(1). The fibre product PMt2l consists of ail pairs of paths (.yr,lù ín PM which have the same endpoint. There is an identification of p¡4lz) with QM, the based, piecewise smooth loop space of. M, which is given by sending a pair of paths (lr,lù to the loop obtained by going along 71 at double speed and then going along the path 72 at double speed but in the reverse direction. Define a closed integral two form F on P Ml2) : (lM by pulling back ø to QM x [0, 1] via the evaluation map ev : {lM x [0, 1] --+ M and integrating out [0, 1]. Here the evaluation map ev is defined by ev(7, t) : 1(t) for 7 e QM and ú € [0, 1]. We have rs(PMlz)) : ro(QM) : n{M): g. Note also rharPMl2l is simply connected. We would like to construct the tautological C' bundle onPMl2l with Chern class .t'. So let (lt,lù be a point inPMl2l. A path from (rn6,rn6) to (lt,lù is the same thing as a piecewise smooth .yr. homotopy p, : [0,1] x [0, I] -+ M with endpoints fixed from 71 to This means ¡.r satisfies p,(O,t) : 'yr(t), p,(I,t) : 'yz(t), ¡.¿(s,0) : Tno and ¡.r(s, 1) : 7t(1) : lz(7). Introduce an equivalence relatiofi ru on the space P(PUtzl¡ X Ct, where we identify paths inPMÍ21with homotopies ¡.r as above, by saying (p,r) is equivalenlto (p,',w) if. and only if, for all piecewise smooth homotopies .F/ : [0, 1] x [0, t] x [0, 1] - M with endpoints fixed between ¡.1 and ¡.r' we have w: zexp( H.r) t,, The homotopy H is required to satisfy H(0,s,t): tt\,t)., H(I,s,ú) : tt'(s,t), and for each fixed r, the map (s, t) e H(r,s, ú) is a piecewise smooth homotopy with endpoints fixed between 11 and 72. We let Q: (P(P¡14tzt) x C")l - Qis a C' bundle over PMl2l. The triple (Q,PM,M) defines a bundle gerbe with bundle gerbe product given by Itt, t) ø lu,w) -, lp, o u,, zwl whe¡e the square brackets denote equivalence classes and where p, is a piecewise smooth homotopy with endpoints fixed between paths 72 and'ys, u is a piecewise smooth homotopy with endpoints fixed between paths l1 and 12, and where p, o u denotes the piecewise smooth homotopy given by u(2s,t), s e ,)(s, ú) : l},Llzl, 0t " p,(2s-I,t),s€lLl2,Il The proof that this is well defined and associative can be found in [14]. A particularly important example of this construction is when M is G, a simple, simply connected compact Lie group. In this case rü¡e can identify pçIz) with the group of based, piecewise smooth loops in G, f)G, by taking a pair of paths (lr,lù in PGl2l and forming the loop f o ?1, by which we mean the loop which follows 71 at double speed and then f'oliows the path 12 from its endpoint to its start point at double speed. We can also identify the C" bundie Q + QG with the Kac-Moody group f-lG which forms part of the central extension of the loop group Ct -r QG + QG. This is an example of the following general method of constructing bundle gerbes from central extensions. 20 Erample 3.2. frequently occurring example of a bundle gerbe is the so-called li'fii'ng bundle gerbe associated^ to a principal G bundle r : P - M where G is a Lie group with a central extension C" --* G -, G. The lifting bundle gerbe is then formed by pulling back the C' bundle G ø Pl2l via the natural map r : PÍ21 -' G defined by pz : pt ' r(h,p2) f.or p1 and p2 in the same fibre of P. If we let this bundle be P then it is easy to see that the triple (P, P, M) is a bundle gerbe. For more details see [33]. Various operations are possible with bundle gerbes. For example, given a map Í , N -+ M v/e can pullback a bundle gerbe P: (P,X,M) to obtain a new bundle gerbe f-tp : (Í-rP,l-'X,,//) called the pullback of P bv f . Thus /-1X is the manifold obtained by pulling back zr : X -- M wilh / as shown in the diagram below. I Í-rxrl 'xn ¡_r"l I ü L *i,r Then f-rp -, U-rX)[2] is the pullback of P --+ Xt2l by the map /t2J. Ciearly Í-'p inherits a bundle gerbe product from P. Suppose we have a second map I : O - N. Then we have two bundle gerbes g-r f-rP and (/ o g)-rP over O. We will make the convention once and for all that these two bundle gerbes are the same. Given two bundie gerbes (P, X, M) and (Q,Y, M) we can form their product. This is a bundle gerbe over M which is described as follows. We first form the fibre product Xx¡aY -, M. There are two projection maps Pt.:^.XxvY -+ X.and Pz: Xx*tY -Y: We use these to form the pullback C" bundles (p?\-'p ana þ!21)-1Q over (X x¡aY)Iz) and then form the contracted product PØQ: çozrl¡-teø@*l)-lQ over (X x¡¡Y)lzl. This is the bundle whose fibre at ((rr,A),(r2,g2)) rs P(,r,,") Ø Qfur,n"). The triple (P Ø Q,X xv Y, M) is a bundle gerbe on M which we call the product of the bundle gerbes (P,X,M) and (Q,Y,M). If. X :Y then we can form the product above in a slightly different way. If P and Q are as above, then we define a C' bundle P Ø Q -- ylzl by forming the contracted product. The triple (P Ø Q, X, M) is then a bundle gerbe over M. Given a bundle gerbe (P,X,M) we can construct a bundle gerbe (P*,X,M) called the,inverse, of. p. p* ---+ X[2] is the C" bundie,inverse'to P -r ¡121, see Section 3.1. P* inherits a bundie gerbe product via the bundle gerbe product on P using the isomorphism P8 P* = Ylz) vçx. Recall from [33] that there is a characteristic class associated to a bundle gerbe (P,X,M) calted the Dixmier-Douady class of P. This is a class in H3(M;Z). A Õech representative for this class (or rather a representative for its image in the Cech cohomology group it'(¡W;Cfr.) under the isomorphism H'(U;Aiò Ht(M;Z) see = : - Section 3.1) can be constructed as follows - see [33]. Choose an open cover U {Un)nrt o1. M, all of whose intersections are either empty or contractible and such that there exist local sections s¿: (I¿'--+ X oL r. Form maps ("n,s¡) : (J¿¡ -+ X[2] which map m € U¿¡ to (s¿(rn), t¡(m)) ç ylz). Denote by Pn¡ the pullback C" bundle (s¿, s¡)-1P on 2L U,;¡. Since Ø¡ is contractible, there exists a section o¿¡ : U¡¡ -. Pti of. P¿¡ -. Ur.t. Define a map 7t¡u : U4u -- C* by mp(o¡* Ø o¿¡) : oik' gijk. The associativity of the bundle gerbe product rnp implies that gl¡r satisfies the cocycie condition g¡ug¿nTg¿¡tgu'^: L on (J¿¡¡¡. g¿¡t, ís then a representative for a ciass in the Õech cohomology group H'(U;Cio) and it therefore defines a class lgn¡rl in Ht(M;Z) under the isomorphism n'(U;Aià = Ht(M;Z) mentioned in Section 3.1. This class rn H'(M;Z) is rhe Dixmier-Douady class of the bundle gerbe P and is usually denoted by DD(P) or D D(P, X). Suppose we have our usual map 7r : Y -+ M admitting local sections and also a C" bundle Q '--+ Y. Then we can construct a bundle gerbe (ô(Q),Y,M) where ô(8) * Y[2] is the Cx bundle 6(Q) : úrq Ø n;'Q. of Definition 3.2. ô(Q) has an associative product because rü¡e can regard such a product as a section of ôô(Q) which satisfies the coherency condition 3.2 in ôðô(Q). Such a section is provided by the canonical trivialisation I of 6õ(Q) and one can check that ô(1) matches the canonical trivialisation of ôôó(8). Definition 3.3 (133]). A bundle gerbe (P,X,M) is said to be triui,al if there exists a C' bundle Q - X together with an isomorphism ó : 6(Q) -+ P such that the foilowing diagram commutes: I ?rtlô(Ç) ------n;'6(8) "t'Ò(8) t ,¡1øø';1øJ l*;,r { nr'P Ø n;r P ^' ' rlr P, where the map El ?rt1ô(8) --+ r;tõ(Q) denotes the bundle gerbe product in "rr6(Q) the bundle gerbe (d(8), X, M). The pair (Q,Ð is called a triuiølisation of the bundle gerbe P. The Dixmier-Douady class behaves as one would expect under operations such as pullbacks and the iike. It aiso has the property that a bundle gerbe P is trivial if and only if its Dixmier-Douaciy cla,ss is zero. More precisely, we have the foilowing proposition. Proposition 3.3 ([33]). Let (P, X, M) and (Q,Y, M) be bundle gerbes wi,th Diun'ier- Douady classes DD(P) and DD(Q) respectiuely. Then we haue 1. The product bundle gerbe (P Ø v Q, X x u Y, M) has Di,rm'ier-Douadg class DD(P Ø¡r Q) : DD(P) + DD(Q) 2. If P. denotes the bundle gerbe 'inuerse to P then 'it has D'irmi,er-Douady class DD(P.) equal to -DD(P). 3 If Í: .ly' + M i,s a smooth map then the Di,rm'ier-Douady ctass DD(f-rP) of the pullback bundle gerbe U-'P, f-rX,,N) zs equal to f.DD(P). 22 /t. The Dirmi,er-Douady class DD(P) is the obstructzon to P being tri.ui.al. Proposition 3.4 (t33]) . Suppose (P,X, M) i,s the lifting bundle gerbe for some prin- cipal G bundle X -, M where G is part of a central ertension C'--G--G, see er¿nxpte 3.2. Then the Di,rmier-Douady class of P i,s the obstruct'ion to lifting the structure group of X to G. Finally we note that it is possible to consider ',4 bundle gerbes' where A is an abelian Lie group. For this we need to know that there is a well defined notion of contracted product of principal A bundles. To define the contracted product PtØePz of two principal A bundles Pr and P2 we simply mimic the procedure for the case A:C', ie we set Pr ØtPz to bethe quotient of the Ax A bundle Ptxut P2 by the anti-diagonalsubgroupof AxAconsistingof allpairs (o,o-'). Nowwecandefinean A-bundle gerbe to be a triple (P,X,M) where P + Xlz) is a principal A bundle with an associative product r;L P ØAr.lr P - r;r P. Such an A bundle gerbe will then give rise to class in the Õech cohomology group n'(U; A¡) rather than H3(M;V'). 3.3 Bundle gerbe connections and the extended Mayer-Vietoris sequence Let r : Y -+ M be a surjection admitting local sections. Then we have seen that we can yln+t) define a simplicial space Y : {Yo} with Yo : and face operators ri : Yþ+t) - Yþl given by omitting the ith factor as mentioned above. Then for p - 0,1,. . . we can form a complex K' with Ks :gr1ytøl¡ and ô : Kq + Úço+L given by the alternating sum of pullbacks zri. The simplicial identities satisfied by the 7r¿ eDSür€ that ð2 : 0. Proposition 3.5 ([33]). The compler Q+eP(M) gCIe(y) g 3eelytøl¡ 4 (33) is eract lor p: 0, 1, 2,. . . . We also have an analogue of this proposition with the de Rham functor 0 replaced by the singular functor S. For the definitions and basic properties of singular cohomol- ogy see for example Again we make Se(Ylsl;Z) a complex for fixed p by defining [3S]. : 5 1 5nçyld;Z) -, SP(yts+l1 ;Z) by the alternating sum of pullbacks zri. As before ó'2 0 and we have the following proposition. Proposition 3.6. For fired P -- 0,1,2,. .. the complex Q + SP(M;Z) \ So(Y;z) ! \ }nlvtøt;z) \ ... (3 4) i,s acyclic. The following proof is due to M. Murray. 23 Proof. Considerfirst thecase of amap ¡r: Z +X which isonto. Here Z andX only have to be sets and z is a function. Denote as usual by /lnl the pfold fibre product over zr. Then define õ : Map(ZP1,Z) ---+ Map(ZlP+Ll,Z) by ô: Dl:o(-l)ozrf where r¿(2t,... ,zp): (2r,... ,2n,... ,zp). In the case that X is a point there is a standard proof that this complex has no cohomology; see for example [38]. This can be adapted completely to the present case by choosing a section s of z'. Notice that because we are dealing with sets the question of existence of a section depends only on the fact that n is onto. Indeed if f e Map(Zlnl,Z) and 6(/) :0 then define F eMap(Zþ-r\,Z) by F("r,.. ., zp_r) : f(sþr(zt)), 4,..., zp_I) and it is straightforward to check that d(F) : /. Notice that tf f : Z --. Zthen d(/) : 0 means that / is constant on the fibres of zr so / descends to a map F : M -, Z. Clearly f : r* F. Now return to the case at hand and define X to be the space of all singular simplices in M and Z tobe the space of all singular simplices in Y. ¡r : Y + M induces a surjective map 7r", : Z + X. Notice that the space of all singular simplices in Yþl is just Zlpi with respect to zr*. To prove the proposition we just have to notice that the space of all singular cochains in Yþl is just Map(Zlnl,Z¡. ! Note 3./. One could think of the complexes 3.3 and 3.4 above as generalisations of the Mayer-Vietoris exact sequence of an open cover (see [a]). One sees this by choosing an open cover {Uo}n¿ of M and letting Y be the disjoint union TJ.u, ieI Then the fibre products Yl2l and Y[3] correspond to disjoint unions of double intersec- tions f[ U¿¡ and disjoint unions of triple intersections I_l U¿¡¡ and so on. Thus we can identify the exact sequences 3.3 and 3.4 with the de Rham and singular Mayer-Vietoris sequences of an open cover respectively. In fact we can think of surjections 7r : Y ---+ M admitting local sections as being 'generalised open sets' of M. This is the viewpoint adopted in Chapter 5 of [7] where open sets are replaced by local homeomorphisms. In this framework fibre products correspond to intersections of open sets and the relationship between the extended Mayer-Vietoris seqr.rences 3.3 a,nd 3.4 and the more familiar Mayer-Vietoris sequences associated to an open covering of M discussed in [a] becomes more apparent. Recall from [33] that there is a notion of. bundle gerbe connection and curai,ng for a bundle gerbe connection. We will briefly recali these definitions . Let (P, X, M) be a bundle gerbe and let tr - ylzJ denote the iine bundle et1 -)¡[zì associated to P. \Me let m¡ denote the iine bundle isomorphism induced by the bundle gerbe product rmp on 'We P. As we noted earlier, rn¿ defines a product on ,L which is associative. make the following definition. Definition 3.4 ([33]) . A bund,Ie gerbe connection on .L is a connection V¿ on .L such that nllVt*rrLY7- rnl'orrrVTomT (3 5) on the line bundle r;t L Ø r;r L over X[3]. 24 V¿ induces a connection Vp on P and we say that Vp is a bundle gerbe connection on P. Bundle gerbe connections always exist since we can choose a connection V¿ on .L and. then notice that the two connections r r-1V¿ -+ rrrV y and mrL o r;rY 7 o ÍrLL ort nrt L Ø r;t L differ by the pullback of a one form A on the base X[3] . The associativity condition on rnL forces 6(A) : 0 and so 'ù/e conclude by Proposition 3.5 that there exists aone form B on X[2] such that 6(B):,4. The connection Vt - B on L defines a bundle gerbe connection on P. It is shown in [33] that the space of all bundle gerbe connections on P is an affi.ne space for f)1(X). If we let Fv, denote the curvature of V¿ then equation 3.5 implies that we must have ô(Fer) :0, and so by Proposition 3.5 again see that there is a two form / on "¡/e X such that Fy, : d(/). Note that / is unique only up to pullbacks of two forms on M. A choice of / is called a curu'ing for the bundle gerbe connection V¿. Therefore we have ô(d/) : 0 and hence there exists a closed three form ø on M such that df : ¡r*u. F\rrthermore it can be shown that ar is an integral three form, ie the integral of ø over any closed three manifold in M ís 2r'i times an integer. We call ø the three curaature of the bundle gerbe connection and curving. In [33] it is shown that ø is a de Rham representative for the image of the Dixmier-Douady class in ø3(ø;n). Note that a bundle gerbe can have a non-zero Cech representative for its Dixmier-Douady class but, due to the phenomenon of torsion, the de Rham representative for the image of the Dixmier-Douady class inside H3(M;lR) can be zero. Erample 3.3. Recall from example 3.1 the tautological bundle gerbe (Q,PM,M) as- sociated to a closed integral three form ø on a 2-connected manifold M. In [33] it is shown that there is a connection V on the C' bundle Q --.' PMl2l which is compatible with the bundie gerbe product on Q and whose curvature equals ô(/) where / is the two form onPM defined by f : Jrev*u (hereev :PM x.I -- M istheevaluationmap sending (7, ú) to ?(ú)). Therefore V and / is an example of a bundle gerbe connection and curving on 8. 9.4 Bundle gerbe morphisms and the 2-category of bundle gerbes Recail the following definition from [33]. Definition 3.5 ([33]). Let (P, X,M) and (Q,Y,¡ú) be bundle gerbes on M and.lr/ respectively. AUundlie gerbemorphi,s,m f , P + Q consists of atriple f : G,l,f') of smoothmaps f',M *lú, Í,X -+Y covering/'and Î, P -+Qcoveringthe induced map /tzl ' ylz) -, ylz). The map / is required to commute with the bundle gerbe products on P and Q. As an example, consider the situation in which we have a smooth map ,f : N - M and a bundle gerbe (P, X, M) We can pullback the bundle gerbe P to a bundle gerbe (f-'P,Í-tx,.n/) on Iú. By definition of the pullback construction there is a bundle gerbe morphism f , f -'P -- P. We will almost always be interested in the case where tW : N and /' is the identity. In this case, a bundle gerbe morphism Í, P + Q is a triple U, f,id¡ø). Definition 3.6 ( [34]). Let (P, X, M) and (Q, Y, M) be bundle gerbes. A. bundle gerbe : such 'isomorph'ism f : P - Q is abundle gerbe morphism f ff,Í,idu) from Pto Q 25 rhal f :X --- Yisadiffeomorphismand Í, P --+Qcovering/tzJ :Xl2l - Y[2] isaC" bundle isomorphism. Unfortunateiy, bundle gerbe isomorphisms seldom appear in practice. Given bundle gerbes (P, X, M) and (Q,Y, M), it is not clear when a bundle gerbe morphism between them exists. Obviously, a necessary condition is that DD(P): DD(Q). One way of dealing with this problem is to consider stable isomorphism classes of bundle gerbes on M. Definition 3.7 (1121, 134]). W" say that bundle gerbes (P, X, M) and (Q ,Y, M) are stably'isomorph'ic if there exist CX bundles fi and T2over X xu Y such that there is an isomorphism p @?))-' s ô("1) = õ(r2) ø @v\-' Q covering the identity on (X x M Y)l2l and commuting with the bundle gerbe products. (Here p1 and p2 arethe two projections of X x ¡tY onto X and Y respectively). Suppose bundle gerbes (P, X, M) and (Q,Y, M) have DD(P) : DD(Q). Then the product bundle gerbe (P Ø Q. , X x tr Y, M) has DD(P I 8-) : 0. Therefore there is a C' bundle 7 -- X xtt Y together with an isomorphism P I Q* - d(?) covering the identity on (X x M Y)Í21 and commuting with the respective bundle gerbe products. Now form the bundle gerbe (P8Q.) ø(plr'))-'Q - (Xx¡¡)lzl. The isomorphism above translates into (p 8 g-) ø (pt'))-'Q --' ô(T) ø @t]t)-'Q. We have (p Ø e.) ø @t]t)-' e : 6llyt eø þl'r)-'( e. Ø e). The bundle gerbe Q. Ø Q -+ [i2l has zero Dixmier-Douady class, hence it is trivial. Therefore its puilback to (X x M Y)121 is trivial and so P and Q arc stabiy isomorphic. Conversely, Lf. P and Q are stably isomorphic, then it is easy to see thar DD(P) : DD(Q).Thus we have the proposition (see [34]): Proposition 3.7 (t34]). Two bundle gerbes (P,X,M) and (Q,Y,M) haue the same D'irm'ier-Douady class if and only if they are stably i,somorphic. This shows that the notion of stable isomorphism is an equivalence relation and so we can form the set of equivalence classes. Taking products of bundle gerbes ciearly respects stable isomorphism and indeed there is a group structure on the set of stable isomorphism classes of bundle gerbes. In Chapter 4 we shall see that there is an isomorphism between stable isomorphism classes of bundle gerbes and H3(M;Z). There is another way of looking at the question of whether a bundle gerbe morphism exists between two given bundle gerbes or not. Before we explain this, we need to digress for a moment. Let P + X be a principal C" bundle on X and let r : X -, M be an onto map admitting local sections. We have the following lemma (see [7][pages 1s5-1s61 ). Lemma 3.1 (l7l). A necessary and suffic'ient condition for P to descend to a bundle on M is that there 'is an i,somorphi,sm þ : r¡L P - r;t P couering the i,dentitU on Xlzl whi,ch sati,sfi,es the glui,ng law ¡rlLÓ o rtrÓ : ¡r;rÓ (3 6) ouer Xl3). ó is known as a descent isomorph'ism. P descends to a bundle on M nxel.ns that there i,s a C,' bundle Q - M together with an 'i,somorphism P - n-rQ. 26 Note that another way of looking at this is that a necessary and sufficient condition ylz) for P to descend to a bundl e on M is that there is a section $ of õ(P) -, such that õ(ó) :1, where 1 is the canonically trivialising section of ô6(P), see Definition 3.2' We have the foilowing slight generalisation of this lemma. Suppose rsç : X --+ M and ry '. Y -+ M are surjections admitting local sections. Then X x¡,t Y is also a surjection admitting local sections. Given a C' bundle P on X xu yþ1, let 6v(P) denote the C" bundle on X xMYIP+L) whose fibre at a point (r,Ar,U2,... ,Ap+\) of. X x¡a YÞ+tl it P(,,0ù Ø P(,,or) 8 ." , where the last factor is P(",vo*r) if p is even and P(,,oo*r) if p is odd - compare with Definition 3.2. As in Definition 3.2 it is easy to check that 6võv(P) is canonically trivialised. We will denote this canonical trivialisation by 1. Lemma 3.2. Suppos€lty andry are 0,s aboue andthat P i,s aC," bundle on X x¡aY such that there i,s a section ó of 6v@) on X x¡aflzl which sati,sfies the coherency cond,i,ti,onOv@)-I onX x¡¿flz). ThenP descendsto abundleQ onX, ietherezs an i,somorphism P = plrQ, whereh: X x¡tY - X i,s the projecti'on onto the f,rst factor. Proof. Choose an open cover Lt : {U.}oeÐ of M such that there exist locai sections to:(Jo+Y of.rv. Let Xo:Txr(tl,) Define aC* bundle Po on Xoby Po:î;LP, where îo: Xo + X x¡rtY sends r € Xu to (r,t.(m)), where m: rx(r) e U". The section $ of. 6v(P) induces a section ôoB of. PB ø PI and hence an isomorphism on ensures that the satisfy the glueing óaB : Po - Pp. The coherency condition @ dop condition ôB.rórp : óot. Hence we can use the standard clutching construction to form a Ct bundle Q from the P. and one can check that there is an isomorphism ¡ P = prtQ. As an example of this, suppose_r,¡¡e have a closed, integral two form F on a non- simply connected manifold M . If. M denotes the simply connected_ universal covering spac_e of. M then we can lift F to a closed integral two form on M which we denote bV F_ \Me ca¡rnot construct the tautological line bundle on M directly, but we can on fu. Let L denote the line bundle on M with curvature F constructed by the tautological method. \Me know fhaf L descends to a line bundte L on M . In [14] it is shown that if has no non-trivial extensions by C", then there is an isomorphism "r(M) þ : rlLL - r21L satisfying 3.6 above. Suppose rù/e are given two bundle gerbes_(P,4,M) and (Q,Y,M) together with bundle gerbe morphisms Í,9 t P: Q. Lel f : ff,f ,idv) and 9 : (g,g:idn).Form a C" bundle D --+ X by ietting D: U,ù-rQ, where U,ù, X ---+ Yt2l is the map which sends r e X to (/(ø), g(r)) e y[21. We have the following series of lemmas. that is, we haue þ r*r Df where Lemma 3.3. D d,escend,s to a bund,le Df ,a - M, = ,s, ry denotes the projection X -> M. Proof. We fi¡st define a map þ: nltD -- r;LD.If € € r1tD6r,,"): QU@r),e('2¡¡ then choose p € P61,,2¡ and define d(€) : rnatna?(p).8€) siþ)). 27 Here mq is the bundle gerbe multiplication in Q. Clearly þ is independent of the choice of p. We need to show that / is a descent isomorphism, ie that the following diagram commutes Tz'Q-1 r¡tr¡LD: rll¡rtrD r;Lr;I0 : ¡rlLrzr D 1f-1 ô rrrr2lD:¡rll¡rtrb ";1ö So let € e QU@ù,s(,s)), p €. Pçq,,2.¡, and p' € P1,r,,r;. Then ¡rltóo"rrôG) : r{þ(mq(meb@). s€) s f@'))) : rna?naþ(p)- I ma?na?(p'). I €) s iþ'))) s iþ)) rna1naþ(^p(p' Ip)).s{) s f@r@' sp))) ";'ó(€). Therefore D descends to a bundle D¡,u. n Lemma 3.4. Suppose we_are g'iuen bundle gerbes (P,X,M) , (Q,Y,M) and three bundle gerbe morphi,sms f¿ : (Í¿, Í¿,idu) : P + Q Íor i, : L,2,3. Then we haue the following i,somorphism of C" bundles on M D D D ¡-",¡-, Ø î.,¡; / f,,fr. Proof. The bundle gerbe product in Q defines an isomorphism ry' t (fz, Íù-tQ ø Ut,Íù-rQ - (.fi, fù-'Q We need to check that this isomorphism is compatible with the descent isomorphism defined in Lemma 3.3. That is, we need to check that the following diagram over Xl2l commutes: ,"l(fz, fs)-'Q8 711-1(.f1, Ír)-1e 4nr'(ft, Ír)-'e ózzØón[I lr,. -,, ,l, n|(fz, fs)-'Q ø n;'ff,., ,)-'Q "1'! n;'(fr, fr)-'Q, where ôr¡ : r¡rç¡ , f )-'Q -- T;r(fi, Í)-'Q is the descent isomorphism constructed in Lemma 3.3. Let €zs e rtrUr, fù-'Q(r1,a2)t €rz e r¡1( h, Íz)-rQ(a7,22)t and p € Pç1,,2). Then r;r1þ " (ózs Ø óp) (€zs8 €rz) : r;tú?na(*ej'@). a €zs) ø iz(ù) Ø mq(mq(ir@)" a €rz) s Âþ))) : mq(m,q(mejz@). I €zs) ø iz@D Ø mq(mq(ir@). I €rz) ø /, (p))) *q(*qjrþ). ø mq(€zs8 6rz)) ø /rþ)) óno¡rtrtþ({zsA€rz). 28 Lemma 3.5. Giuen three bundle gerbes (P,X,M) and (Q,Y,M) and (R,Z,M) to- gether wi,th pairs of bundle gerbe morphi'sms fu: (Îr, f¿, idu) : P - Q g¿:(g¿,g¿,idtø):Q-R for i,: I,2. Then we haue the followi,ng 'isomorphi,sm of C" bundles on M D srofr,sroÍ, / D fr'¡-, Ø Dgr,g"' Proof. First of all deflne a map 1 gt,gz >: Yl2l --. 2lzl by mapping a point (Ut,U) of Yl2) rc the point (gtfut),g2(a2)) of. Zl2l. We then have (gro fr,gzo fz):1 9t,92 ) o(h,rù.Note that 19t,,gz >-1 Æ = çnf,l¡-tnØnl(gr,gz)-rR.We therefore have the following series of isomorphisms of C' bundles (g, o fr, gz o fù-r R : (< gr,9z ) o(ft, Ír))-t R : Ur, Ír)-' 1 gu 9z >-r R : ut, rù-t(@l\-, n Ø r;LØr, gr)-, R) Ur, f z)_L (gf\_, n Ø Í;Lr¿L Du,,u" : (fr, Ír)-t(gLq)-, n Ø r¡r Du,,u,, where zrx denotes the projection ry : X -> M and so on. The C* bundle map gz: Q --+ E pulls back to define an isomorphism ,þ : (fr, fz)-'gz t (Ír, lz)-tQ - (f,', fr)-'bl:l)-'n' We need to show that d is compatible with the descent isomorphisms: t trt (rt, rz)-'Q 53 iL (lt, ,r¡-t çnfl¡-r R *l -,, J" rl(f,,, fz)-'Q % n;'(Í,,, Ír)-'bl:\-'R. This is clearly true, since T/ is induced by the bundie gerbe morphism !2. Therefore there is an isomorphism r*7Dgro¡r,aroîr? nxt(DÌr,hØDgr,a) which is compatible with the descent isomorphisms. Therefore we must have Daroir,arof-r/ Df,.,frØ D-sr,Er. ! Motivated by the results above, we make the foliowing definitions. Definition 3.8. Let (P,X,M) and (Q,Y,M) be bundle gerbes and suppose / : G,f ,iar) and g : (A,g,ídv) are bundle gerbe morphisms f ,9: P + Q. A transfor- mation of bundie gerbe morphisms 0 : f + p is a section of the C' bundle D¡,9 on M. Definition 3.9. Let (P, X, M) and (Q, Y, M) be bundle gerbes. A bundle gerbe equiu- alence consists of a pair of bundle gerbe morphisms.f : U,f ,idv) and g : (g,g,idu) with /-: P - Q and g,Q - P such that there exist transformations of bundle gerbe morphisms 01 : P o f- +idp and 0z : f o g + idq. 29 Now we can look at bundle gerbe morphisms in a different light. We can view the collection of ail bundle gerbes on a manifold M as objects of. a 2-category B (for the definition of a 2-category rve refer to Section 8.1). For given two bundle gerbes P and Q on M we can define a category Hom(P, Q) whose objects consist of all bundle gerbe morphisms from P to Q. Given two objects fi and /2 of Hom(P,Q), that is l-arrows of. ß,lhe arrows from fi to fz, ie 2-arrows of B, consist of all transformations 0¡r,¡, from Ít to Í2. Composition of arrorv¡/s in Hom(P,8) is bV tensor product, with the identifications of Lemma 3.4. (For details on 2-categories and bicategories see Chapter 8). We have the foliowing proposition. Proposition 3.8. ß as defined aboue i,s a 2-category. All we really need to do is show that there is composition functor, in other words if P,8 and R are three objects of 6, then there is a functor m : Hom(Q, R) , Hom(P, Q) - Hom(P, R) which is associative. If we define the action of. m on objects to be composition of bundle gerbe morphisms and the action of n¿ on 2-arrows to be tensor product, with the identifications of Lemma 3.5, then this condition is clearì.y satisfied. As mentioned in [33], an annoying feature of bundle gerbes is that two bundle gerbes can have the same Dixmier-Douady class and yet faii to be isomorphic by a bundie gerbe isomorphism nor even equivalent by a bundle gerbe equivalence. There is a way to circumvent this problem by introducing the notion of. a stable morphism between bundle gerbes. The following definition is due to M. Murray. Definition 3.10. Let (P, X, M) and (Q, Y, M) be bundle gerbes. A stable morphi,sm f :P--, Qisachoiceof trivialisation (L¡,ó¡) of thebundlegerbe P"ØQ,ieachoice of C'bundle LÍ - X xu Y and achoice of isomorphism /¡ : õ(L¡) --+ P* 88 such that the diagram below commutes: n;r6¡Øn;Ló¡ itõ@Ð Ø rsLõQr) nL'(P* s Ç) s r;L (P. ø Q) îz'QÍ-l r;r6@î) n;L(P* I O), see Definition 3.3. Suppose we have two stable morphisms f , g , P -> Q corresponding to choices of trivialisations (L¡,ó¡) and (trr, ó) of P* I Q respectively. Then we have an isomorphis^ ó;' o ó¡ : 6(L¡) -- õ(Ls) which corresponds to a section Ór,n , f(ô(¿i Ø Lò). Because the isomorphisms þ¡ and de commute with the bundle gerbe products, the section þ¡,n wtII satisfy the coherency condition ô(i¡,r) : 1. Therefore there is a C' bundle D¡,n on M and an isomorphis^ Lj Ø Ln = D.¡,n. We define a transformation d : Í + g to be a section 0 of L.¡8 tr, such that ô(A) : ô¡,n. Thus á will descend to a section 0 of D¡,n. Note 3.2. 1. Note that a bundle gerbe morphism f : (i, f ,idv) : P'+ Q gives rise to a stable morphism f t P + Q because rve can define a t¡iviaiisation L¡ of. P. ØQ by letting L¡have fibre .L¡(ø,U) at (r,a) e X xv Y equal to QU@),ù - the bundle gerbe product on Q and the isomorphism P = (¡lzl¡-lQ provides an 30 isomorphism QU@ù,sr¡ Ø Qi¡ç,r),yr) = Q@r,ar) Ø P(,r,,r). Suppose ye have stable morphisms f : P --+ Q and g : P - Q arising from bundle gerbe morphisms f,g, P - Q. Then the c'bundle D¡,n of. Defrnition 3.10 above is the c" bundle D¡,u of Lemma 3.3. 2. Note that a stable morphism exists if and only if P and Q have the same Dixmier- Douady class. 3. Suppose we have bundle gerbes (P,X,M) and (Q,Y,M) and stable morphisms f ,g,h: P + Q corresponding to triviaiisations (L¡,ó¡), (Ln,ó) and (L¡,þ¡,) respectively. Suppose aiso that we have transformations 9¡,, : I + g and ?n,n : g + h corresponding to sections âr,n , f(¿ï S ,Lr) and ên,n € l(Lî Ø Lh) to define a composed transformation 0n,n0¡,s : h. respectively. We would like f ^+ We do this as follows. Let 0n,n0¡,.n denote the image of the section 0¡,nØîs,nof L] Ø Ls I ¿; 8.L¿ under the isomorphism can: L, Ø Ln Ø LI ø L¡, -' L.¡ Ø L6 induced by cbntraction. To show that 0n,n0¡,s defines a transformation we need to show that 6(0n,nl¡,): ô¡,n.W"have 6(9¡,, ø0n,n): Ó¡,gØÔs,h and' $¡,nØÓs,h is mapped to ó¡,n under the isomorphism 6(L¡). I ô(¿r) A ô(¿s)- I ô(¿h) -- õ(L¡).I ô(¿/.) : õ(L] g Ln). Therefore we can compose transformations. This operation of composition is clearly associative. Note that there is an identity transformation 1¡ : f + / and so it foilows that we have defined a category Hom"(P, Q) whose objects consist of the stable morphisms P --+ Q and whose arrows are the transformations between these stable morphisms. Finally note that all the arrows in Hom"(P, Q) are invertible. Suppose we have bundle gerbes (P,X,M), (Q,Y,M) and (-R, Z,M) and stable morphisms f : P -r Q and g : Q + R corresponding to trivialisations (tr¡, @¡) and 8.8rq respectively. So ØQ = 6(Lr) : is an isomorphism Q* I h isomorphisms be products. Fibrewise, P* (rr., rr) Ø Q(ar, gz) = L ¡ (rr, az) Ø L](a, uù which can be rewritten as P(rr,, rr) Ø L ¡(rr,az) = Q(ar,aù Ø L¡(rr,a). Similarly we can write Q(at,az) Ø Ln(ar, rr) = R("t, rr) Ø Ls(ab z). We want to define a trivialisation (treo¡, ôs.¡) of. P* I .R corresponding to a stable morphism go f : P --+ R. Consider thã d' bundle L on X xu Z x¡aYlzl whose fibre at a point (r,r,yr,gr) € X xu Z xu Ylzl ¡t L(r, z,at,Uz) : Lt(r,gr) A Ln(ar, z) ø Q(uraz). We will show there is a section þ of. õy1r1 (I) on X x ¡a Z x ¡a(Yl2l)lzJ satisfying 6tpt(Ó) : t. 31 We construct / as an isomorphism given fibrewise by L(r,z,At,Uz) --+ L(r,z,Uz,As). L¡(",Er) 8 Ln(ar, z) ø Q(uuuz) N L¡(r,g/r) a Ln(ar, z) Ø e(u+,uz) Ø e(ur,aò Ø e(ar,as) N L¡(r,gs) I p(r,r) Ø Ln(ar, z) Ø e(ua,az) Ø e(az,an) æ L¡(r,g¡) I Ln(an, z) ø R(2, z) ø Q@s,aa) One can check that this isomorphism is coherent given a third point (At,aa) of.Yl2l. Hence by Lemma 3.2 L descends to a C'bundle Lso¡ on X xy Z. We now need to show that trro¡ trivialises P* 8Æ. Consider ô(I) on (X x¡ø Z xMYl2l)[2]. It has fibre at a point ((rr, rr,AuAz),(rr, rr,Ut,U'z)) of (X xu Z x¡afl2))[2] eqnal to 6 (L) ((q, zt, U t, Az), (rz, rr, A'r, A'r)) : L(rz, 22, a't, a'z) Ø L* (rr, zt, al, a2) : L¡(rr,a't) Ø Ln(u'r, tr) Ø Q(ar,a;) Ø L](a,g/r) a L[(uz, rr) Ø e*(at,az) N P*(rr,rr) Ø Q(ur,a') Ø Q(az,û Ø R(rr, Ø Q(a',.,g'z) Ø "r) Q*(a¡az) = P*(rr.,rr) Ø R(21, z2). This isomorphism commutes with the bundle gerbe products on ô(I) and P* I Æ and is also compatible with the descent isomorphism for ð(Z) induced from the descent isomorphism for L. Hence this isomorphism descends to (X xM Z)t2) to provide a r--: :-l:--r:--- f C/r \ - n* ñ rrr r n ,r r r r ¡ tIIVTaIIöaurotl {psoJ : o\tJsol ) = r Q9^ ft. VVe Oenne Ine COmpOSrf,e SIAOI€ morpnßm g o f : P -- R to be the triviaiisation (Ls.¡,ós"¡). Given bundle gerbes (P,X,M), (Q,Y,M) and (.r?, Z,M) and stable morphisms --+ ft,fz: P - Q and 91 ,gz: Q -R together with transformations d I h+ f2 and p: gt è gz, one can define a composed transformation po0 : ho Ít è gzo /2 using a similar technique to that in the preceding paragraph (except that we now descend a morphism). One can show that this operation of composing transformations is com- patible with the operations of composing transformations in the categories Hom"(P, Q) and Hom"(Q,R). It follows that we have defined a functor o:Hom"(P,Q) x Hom"(8,R) -' Hom"(P,Æ). We would like to know whether or not this operation of composition is associative. Suppose we have bundle gerbes (P, X, M), (Q,Y, M), (R, Z, M) and (5,W, M) together withstablemorphisms f : P-Q,g:Q---+ ¡?and h: R+^9. Let ptX x¡aY[2) x, Zl2l x¡,,tW --+ X xu lutrl denote the natural projection obtained by omitting the factors Y[2] and 7lz) in the fibre product. Then there is an isomorphism p-rLto(soÐ(r,Ar,U2,zr,zz,w) - Ln(zz,w)øLn@2,2t)ØL¡(r,A)ØQ(Ur,gz)ØR(21,22) of fibres at apoint (r,At,Uz,zr,zz,w).At the same point there is an isomorphism p-rL6on¡o¡(r,Ar,U2, zt, zz,w) - Ln(zz,w)øLn(y2, zùØLf (r,a)ØQ(At,Az)ØR(zt, zz). It follows from this that p-lLh.o(sof) - p-r Lç,oo¡.¡ and hence that L¡o6o¡¡ = L6os¡"¡. Furthermore tkris isomorphism is compatitrle with the isomorphisms cÞno6o¡1 and þ6"n¡o¡ 32 and hence defines a transformation î¡,s,n: h" (g o /) + (ho g) o f . One can show that this transformalion 0¡,s,n satisfies a coherency condition given a fifth bundle gerbe (7,V, M) and a fourth stabie morphism lt : S - 7. This coherency condition is (0n,n,* o I¡)0¡,n"s,¡(1¡ o 0¡,s,n) : 0¡,s,*onîsoJ,h,te . Given a stable morphism f , P ---+ Q between bundle gerbes P : (P, X, M) and Q : (Q,Y.,M), there is a natural stable morphism Í , Q -- P. If / is given by a trivialisation (.L¡, ô¡) of P* I Q, then / corresponds to the trivialisation (Lf ,Ó.r) the stable of 8. I P. _Let us calculate the trivialisation of P* I P corresponding to First we form the C'bundle L on X xu X x¡aflz) whose fibre at a morphism i " Í point (rr,*r,At,Az) of. Xl2l x¡¡Ylzl is (A L(r r., t2, A r, Az) : L ¡ (r r,g/r ) I L.¡ (rz, aù Ø Q r, Aù. Since (L¡,ó¡) is a trivialisation of. P* I 8, we see there is an isomorphism L(rt, r,2, Ut, Az) - P (n1, r.2). This isomorphism preserves the descent isomorphism f.or L and so we see there is an isomorphism L¡"¡ - P. This is a transformation Í " f è Lp, where lp is the stable morphism P -, P induced by the identity bundle gerbe morphism P - P. Similarly are compatible with the there is a transformation f " f + 1q. These transformations transformations d¡,e,r, in the sense of Defrnition 8.2. We have the following Proposition. Proposition 3.9. G,iuen a manifold M, we can form a bi,categorg ß" whose objects are the bundle gerbes on M and where the l-arrous are the objects of the categories Hom"(P,Q) and where the Z-arrows are the o,rTows of the categories Hom,(P,Q). Thi,s bicategory i,s 'in fact a bigroupoid. We refer to Definitions 8.1 and 8.2 for the precise meanings of the terms b'icategorE and b'igroupoi,d. 33 34 Chapter 4 Singular theory of bundle gerbes 4.L Construction of a classifoing map Definition 4.1. The uniuersal bundle gerbe (EBC" , EBC , BBC* ) is the lifting bun- dle gerbe associated to the principal BC' bundle EBC" -, BBC" via the short exact sequence of abelian groups C' --- EC' --- BC'. So EBC' - (EBC")t'l h the pullbackr-LEC" of the universal C* bundle EC" -- BC'- via the canonicai map r : (EBC";tzl -* BC". Let EBC" have Dixmier-Douady class u in H3(BBC";Z) : Z. Suppose (P,Y, M) is a bundle gerbe on M with Dixmier-Douady class DD(P) in H3(M;V'). We will review a construction of [1S] which shows how to construct a map g : M ---+ BBC from a Õech cocycle g¿¡¡" representing the Dixmier-Douady class of P. It wiil follow from this that the two classes DD(P) and g*u are equal. Suppose the class DD(P) is represented by a Õech 2-cocycle hr¡t : (J;¡x - C", relative to a Leray covel tl : {(l¿}¿ç¡ of. M, in the Õech cchomology group n2çU;C';¡ via the isomorphism induced by the short exact sequence of sheaves of abelian groups onM 1 - Zu - Cv --+ Cf;o --+ 7, see Section 3.1. Thus gr¡r satisfies the Õech 2-cocycle condition 9¡u9¡t"Ig¡¡tg;'r: I (4 1) on (J¡¡¡,¿. We will construct maps gt¡ : U..-- BC' which satisfy the Õech l-cocycle condition g¡ng¿tr g¿¡ : 1 on U¡¡¡". These will be transition functions for a BC,' bundle on M, whose classifying map will be the map I '. M --+ BBC" referred to above. We will make the assumption that every manifold M we deal with has a localiy fi- nite, countable Leray cover U such that there is a smooth partition of unity {di}¿€i sub- ordinate to the open cover l,l. For each m e M fotm the subset I,n: {io,'it,. . ' ,in@)} of -I consisting of those i e I such that ón(^) is not zero. Since our Leray covering is locally finite, this subset wilt aiways be frnite. For r:0,1, -. ' ,n(m),Iet ,þ,(m): ó6(m) +'..+ þr,-,(*), 35 Then we wili have þ6(m) < . . . 1 tþn@) (rn) and using the non-homogenous coordi- nates on BC", see Section 2.2 and [18], we can define g;¡ : U¿t -- BC" by 94(m) : l1þt(*), . . . ,ú.(m),lgn¡no(*)-'9r¡;,(m)l' ' 'lgo¡u^-,(^)-'go¡n^(*)ll where m € U¡¡, n: n(rn) and {i6, ...,i,@)} is the set I,n above. Notice that this makes sense, that is respects the face and degeneracy operators, precisely because each gul.g¿¡¡" defines a trivial cocycle. By Lemma 7.2 of [18] we have that 9;¡ : U '-* BC" is smooth - that is smooth in the differentiable space sense of [16] and [32] - see also Section 2.4 - and hence is continuous. Lemma 4.1. The maps g;¡ : U --, BC' sati.sfy the l-cocgcle condition g¡Ám)g¿t"(m)-r g¿¡(m) : 1 on U¿¡¡. Proof. To show this we need to use the group multiplication in BC" using the non- homogenous coordinates on BC' as reviewed in [18] and see also Section 2.2. We get 9¿¡9¡t" , r rr tÞ -1 -l ' ' g -l -1 ' , ¿ 9 j j t"¿r l 9 t t^ 9 9 j t"t^ lrþo,. ", 19 4îo9 i t r¡o9 l''' ;¡i^- r ¡ ¡ ¡,'n^- r Jl . rþ,, g;t lrþo, ' , lg n*Iogot"nrl''' lg¿n¿^-, ¡^]l 9¿t", where we have used the 2-cocycle condition 4.1 to write gojl"g¿¡¿rgjt:.iogj¡,¿, equal to gntlogonr, and so on. ¡ Hence the g¿¡ represent transition functions for a BC" bundle Xp on M. We want to show that the class in Ht(M;Z) determined by Xp and the Dixmier-Douady class DD(P) of the bundle gerbe P arc the same. Lemma 4.2. Let (ftp, Xp, M) d,enote the li,fting bund,le gerbe associ,ated to Xp uia the short eract sequence of groups 1 -- C* -, EC^ ---+ BCx --+ 1 Then DD(Xp): DD(P). Proof. We will calculate the Dixmier-l)ouady class of Íp relative to the same open cover U of M used above. We choose sections s¡ of. Xp + M above each U¿ and form the pullback C* bundies (s¿,s¡)-r*p over (J¿¡. Since the sections s¿ and s¡ are related by the transition cocycles g4, we see that a classifying map for (s¿, t¡)-'Xp is given by 9;¡:U *BC* (Jü Thus choosing sections o¿¡ of (s¿, t¡)-'Xp - corresponds to choosing lifts g¿¡ : U¿j - EC" of. the g¡¡. We have a canonical choice for the g¿¡ given in the non- homogenous coordinates by \t¡(m) : lrþo(*),' . .,rþ^(*), g¿¡¡o(rn)lgo¡no(m)-' 7r¡;,(m)l''' t g (or)- g I o¡ o--, o¡ o^(-) ] l, 36 where again n : n(m) and the tþ, are defined as above. The þ¿¡ need not be cocycles and indeed their failure to be is measured exactly by the 2-cocycle gt¡n lor a calculation similar to that in the proof of Lemma 4.1 above shows that g¡"ïot"gn¡: i(g¿¡x)' where ,i: C." -, EC is the homomorphism induced by the inclusion of the subgroup C*. In other words, the lifting bundle gerbe Xp has Dixmier-Douady class equal to lgn¡rl' ! 4.2 The singular theory of CX bundles In this section we will show that there is an analogue of the curvature two form F associatedtoaconnectiononeformAonaprincipalC*bundler:P---+Xinsingular cohomology. To be more precise, we wiil construct a singular 1-cochain A e SI(P;Z) and a singular 2-cocycle F e S2(X;Z) such that zr*F : d,A. We first digress for a moment. Recall from Chapter 3 that we can replace the structure group C" of a bundle gerbe P + Xlzl with any abelian Lie group A and still get a weli defined notion of bundle gerbe. In particular 'ù/e can repiace C" by Z and get the notion of. a'Z bundle gerbe'. Zbundle gerbes arise whenever rù/e have a Ct bundle pz : P - X, since we have the canonical map r : Pl2l -- C" defined by Pf (h,Pz) for p1 and p2 points in the same fibre of P. Then v¡e can pullback the universal Z bundie C --- C' with the map î to get aZbwdle F - Pf2l. Because the map z satisfies r(pz,ps)rlptpz) : r(pr,ps) we clearly get an associative product on P and hence the triple (P, P,X) defines aZbundle gerbe. Of course this is a rather artificial construction, but it does have the useful_property that if one calculates the class in H'(X;Z) defrned by the Zbundle gerbe P then one finds that it is equal to the Chern class of the C* bundle P (as one would hope). To see this we first choose an open cover {U¡} ofX such that there exist local sections s¡ | U¿ ---+ P of P - X. We then form the : : (J¿¡ is like puilback Z bundles (s¿, s)-1P h¡ and choose sections o¡¡ -, Þti - this choosing lifts z¡¡ i Ui¡ --+ C of the transition functions gt¡ : U" -+ C" of P. Finally we can define an integer valued Cech 2-cocycle n¿¡¡, : U¿¡¡" --+ Zby m(o¡nØzo¿¡) : oip*Ujn : FTom this it is clear that - this translates into the condition z¡u*2.;¡ z¿¡lrù¿¡¡. [n¿¡¡] is the image of lg¿¡l under the isomorphism H'(U;Ain) = H'(U;Z¡) induced from the long exact sequence in sheaf cohomology arising from the short exact sequence of sheaves of abelian groups Zw-C¡rr-Çir. Therefore the class [n¿¡¿] in H'(M;Z) is the Chern class of the C' bundle P. We have the following iemma. Lemma 4.3. There erists a singular cocycle c e ZL(C";Z) and a singular cochain b € So(C" x C*; Z) such that dö" - dic + dic : d,b (4 2) and d,åb - dib + d,;b - d,ib: 0. (4 3) 37 Here the d¿ denote the face operators in the simplicial manifold ¡/C'. Proof. First of all note that a singular O-chain in S¡(Z;Z) is a linear combination of O-simplices o : A0 --+ Z with integer coefrcients. Since A0 is just a point, it follows that we may identify a singular 0-chain of S¡(Z;Z) w\rh a formal linear combination nflr\ I . . . I n¡Tnu. A singular O-cochain is a linear map SI(Z;Z) + Z. There is a canonical choice of a singular 0-cochainin So(Z;Z), namety the linear map which sends a formal linear combinatiorytlllrlll...ln¡m¿ to the integer n1n\1..'ln¡"m¡. Let us call this singuiar O-cochain a. It has the property that prfa+prjø : rn*&, where pr1 and pr2 are the projections on the first and second factors of. Z x Z respectively, and where m : ZxZ --+ Z is addition. This is because a singular O-chain of ZxZ may be identified in the same way as above with a formal linear combinationnl(m1,*lr)t.- .+nk(rnk,rn'k) and it is easy to see that the value of.m*a on such a 0-chain is equal to the value of priø*prio on such a O-chain. We also have da: 0, where d is the singular coboundar¡ since a singular l-chain of SíZ;Z) must be a linear combination of constant maps A1 -* Z with integer coefficients from which it follows that da evaluated at such a l-chain must be zero. Next we have the short exact sequence of groups Z -. C -t Ct, where V' --+ C' is the inclusion and C -- C* is the map sending z € C to exp(2rJa). We will think of C as a'principalZbundle'on Ct. Form the fibre product Clzl. We then have a canonical map 7 : CL2l -- Z. Form a O-cochain r*o, of So(CtzJ;Z) by pullback with r. The property pria * pr$a: rn*a of ø shows that ô(r*a) : 0 in ,9o(CteJ;V'), where ô:.9o(Ctzl;Z) - 501çtel'Z) isthe coboundary defined in Proposition 3.6 by adding the pullback maps zri with alternating signs. It follows from Proposition 3.6 that there or¡isfc î c .qo(C.V,\ c'r,eh f.hat r*n : Ã(î\ Sinne d.n. O we hnwe 8(d.î\ :0 so l'¡v \-)-,/-\--l.--\-.-/.J - Proposition 3.6 again we get dô: p*c for some c € ,S1(C";Z), where p: C ---+ C" is the projection. Denote muitiplication in C' by rn and denote multipiication (addition) in C by rîr. We have a commutative diagram xZ I + T fv //) 6tzl 1 where now pr1 : C x C -+ C and pr2 : C x C -t C denote the projectiorrs orl the first and second factors in C x C respectively and Z x Z + Z is addition in Z. fuom this diagram we ger (ñl2l)*r*a: (r åptl'l)-; * (r o prlr'l).o and since r*0,: ô(ô), this becomes 6(ñ.ô): d(priô+pr;¿). Using Proposition 3.6 again, v/e see that there exists b e SO(C' x C"; Z) such that (p xp).b: prlô- ñ}ô+pric, where p xp is the projection C x C -- C' x C'. Fþom this last equation, it is easy to see that ä satisfies equation 4.3 above. Also from the last equation rù/e see that (p xp).db: (p xp).(pric -m*c*prlc), in other words equation 4.2 is satisfied. ! Note l.-1. We will briefly check here that the singular cochains ô and c defined above are not identic ally zero and that c is not the coboundary of a O-cochain. To solve r. a : 6(ô) for ô, recall from the proof of Proposition 3.6 that we put ô equai to the 0-cochain that sends a 0-simplex o : A0 -' C to r*a(o,s(p(ø))), where s : .9s(C";Z) -- So(C;Z) is a section of the set map p* : ,Ss(C;Z) - SI(C";Z). Since we may identify So(C,;Z) 38 and ,Sg(CX ;Z) with the free abelian groups on the points of C and C" respectively, choosing such a section s amounts to choosing a section of. p : C -t C* where we regard C and C* as sets so s need not be continuous. The obvious candidate in this case is the map s : C' ---+ C send.ing rexp(fid) (where 0 < 0 < 2n)to ft1(togr* t/=0) Now consider the O-simplex 1 sending the point 1 of A0 to 1 € C. We have ¿(1) : 1))) : a(r(0,1)) : a(t) : 1. In a similar'ù/ay one can show that dô is not"(r(#log(1), identically zero, and so c is not identically zero. We will now check that we do not : have c - de, for some e € ^90(C'; Z). Suppose we did, then we would have d(ô-p*e) 0 This means that ô must be constant on the O-simplexes A0 --+ Ç. in ^91(C; Z). -p*e Let this constant value be n say. Then e*n defines a new O-cochain of ,So(C";Z) and we have d(e+n): c. AIso we have ô : p*(e*n): p*e* n and so ô(ô) :0, which contradicts the fact that ô(ô) : r*o,10. In fact, one can give an expiicit formula for c. Pirst of all suppose we have chosen a set map s : C' -t C which is also a section of.p: C -- C'. We need to find a section of p*:,St(C;Z) - SíC";Z). There is a canonical way of doing this. Let o : A1 -t C" be a 1-simplex. Then there is a unique way of lifting o to a l-simplex ã: L,L + C with poõ: ø and o(0) : s(ø(0))' Nowwe can write down a formula for c: c": ã(L) - s(o(1)), where ø : Al -r Ct is a l-simplex. It is not hard to check that c defined in this manner is closed and it value on the canonical generator of. Ht(C";Z), a triangle in C" around 0, is the winding number of the triangle ie 1. Hence c represents the generator or fundamental class of Ir'l(C*;Z). Similarly one can write down a formula for b, bor,or: s(op2) - s(ør) - s(oz), where ø1 : A0 - Ct and o2: A0 * C' are 0-simplexes. Now suppose v/e are given a principal Ct bundle P + X. Let rp denote the projection P -+ X. Let, as usual, P[2] denote the fibre product of P with itself over X and let Pþl denote the p-fold such fibre product. We have the natural map r : PÍ21 - C' defined by pz : pr . r(h,pz) for p1 and p2 in the same fibre of P. r satisfies r(p2,Æ)r(h,Pz) : r(h,ps) Ø.4) for (p1, pz,ps) € Pt3l. We have the following commutative diagram: Þ ' =c 1fÞ tl P Ptzl 3çx. Let a : T*c and let ã: i*ê, so dã,: Tþa. Notice that even if P has zero Chern ciass in H2(X;Z), in which case the structure group of P will reduce to Zn for some n, a will be non zero since the map 7 : Pl2l - C* is onto Cx. Then equations 4.2 and 4,4 imply that nicl - ria t nia: dP, (4 5) 39 o where B € ^90(PI3I;Z) is defined by þ : (, o nr, r rs)*b. Another way of seeing this is to note that we have the commutative diagram , (7rr-1 '7opr1- ,Í3-1- ' r-oprz) nr'Þ x ppl nsr Þ (- xC *el_t l^ v Í2'T-1 - + n;t P C, where ñ.p : rrr Þ x pp1 rsr Þ -- r;r P is induced by mp. Flom the commutativity of this diagram and the equation priô+ priô : ñ.*ô+ (p r p).b in x C;Z), we get tÞ ^90(C the following equation in ^90(zr, xp¡r¡nrtP;Z): prlzr¡lô + pritr;tã: ñ1.Þr;rã + (trp x rF). P. (4.6) Taking d of this equation gives equation 4.5. Aiso it follows from equation 4.6 o¡ equation 4.3 that õ(P) : 0 in ,S0(Pl4);2,). Hence Proposition 3.6 implies that there p. exists 7 € ^90(Pt3);Z) suchthat d(7) : Then, from equation 4.5, we get ô(o) : 6(dl) and hence from Proposition 3.6 again, we get a: d'y + 6(A) for some A e Sr@;Z). Since do : 0 we have that 6(dA): 0 and hence there exists F e S2(X;Z) srch that rþF : d,A. Lemma 4.4. F is a representatiue of tlte Chern class of P ---+ X ln H2(X;Z). Proof . The problem is to show that F is a representative in singular cohomology of the Chern class of P. singular representative for the Chern class of P can be constructed ^ gives gerbe by observing as above that P rise to a Z bundle (Þ, P,X) on X - the Iifting Zb:;ndle gerbe associated to the short exact sequence of groups Z -- C -- C'. As we saw above, the Õech 2-cocycle n¿jn : U¡¡¡ --+ _Z relative to some open cover U : {U¿}¿ç1 of. X associated to the Z bundle gerbe P maps to the Chern class of P under the isomorphism n'(X;Zñ = n2(X;Z). To construct the singular cocycle representing the Chern class of P we use the fact (see [a] and the note following Proposition 3.6) that the singular Mayer-Vietoris se- quence for the open covering {tJ¿}¿ç¡ is exact. Thus we view n¿¡¡" as living in S0(U¡¡¡;Z) and use the exactness of the singular Mayer-Vietoris sequence tc¡ solve 'iL¿¡¡ : 6('un¡) for some unj e S0(flnt;Z). Since n¿¡¡" ís locally constant we have dnnjn : 0 and so 6(du¡¡): 0. Applying the exactness of the singular Mayer-Vietoris sequence again im- plies that there is an A¿ e S|((J¿;Z) so that d,u¿¡ : 6(A¿). Applying d to this equation gives us 6(dAc):0 and so there exists F' in S2(X;Z) with d,F' :0 and 6(F'): d,A;. We want to show that F and F' represent the same class in cohomology. We use the same open cover {t/,}¿e.r âs above and form the principalZbundles fl¡: (s¿, s¡)-tP over Ui¡. We have the following commutative diagram: ("r'"j), , P¡j Þ ,C (r¿, ) "r tl ¡Ju 40 which yields by composition the commutative diagram P¿;3C- o;; Y^Vtl u¿¡ ! c'. We let ãu¡ e S|(k¡;Z) denote the pullback pi,ô and we let a¡¡ e SI(\J¿¡;Z) denote the pullback gî¡c. By pulling back equation 4.6 to S0(P¡r xun,u Pni;Z) we get priã¡r * priã¿¡ : ñzþã¿n + (Tp¡o xun¡n Th¡)" þo¡r, where /¿jx: (s¿,s¡, su)*0.Since 0: õ(l), rù/e may rewrite 0;¡x aslik-1a"*1;¡ where .y¿j : (s¿, s¡)*7. Now the lifts z¿¡ : U¿¡ + C of the g;¡ furnish us with sections o¡¡ of. P¿¡. Also it follows that we have ñt p(o¡r, o¿j) : oik I n¿jr. Hence from the equation above we get ojr,ãiu * oirã¿¡ : (o¿t" * n¿¡¡")* ã'¿¡" +'Yik - 7u, * 1¡¡' It follows from the definition of. ã.¡¡, that (o¿¡,in;¡¡")"ã,¿¡ : a\rãu,tn¿jn. This is because (o¿n + n¿¡*)*ã¿x : (o¿t" + n¡¡x)*7it"ô : (z¿x * n¡¡¡)*ô, so if we can show that ô(o * n) : ê(o) + n f.or a O-simplex o : A0 --+ C and n € Z then we will be done. By definition ô(o + n) -- (r.a)(o t n,s(p(o + n))) : (r*a)(o t n,s(p("))) : (r*a)(o,s(p(ø))) * n: c(o) + n. Hence we have olnãi* -'Yik * oitã¿¡ - 7¿i : oirã¡x -'lu" * n¿iu' Therefore oi¡a¿¡ - 7¿¡ plays the role of. u¿¡. Note that d(oita¡¡ - l1i) equals ol¡P*a¿i - d,"y¿j : a¡j - d.y¿¡ : sîA - sîA. It follows from this that F and F' represent the same class in cohomology. ! 4.3 The singular theory of bundle gerbes In this section we utilise the resuits of Section 4.2 to show that there are singular analogues of the curving and three curvature of a bundle gerbe. So let (Q,Y,M).be a bundiã gerbe. We wili assume that we have constructed ã e S}(Q;Z), o e Sr(Qlzl;Z¡ sarisfying equations 4.5 and 4.6 leading to A St(Q;Z) and and, B € ^90(Qt31;Z) e r e S2(Yt2l;Z). Formthe pullback Cx bundles TrorQ over Y[3] for i:I,2,3 so that there exist Cx LQ bundle morphisms iti : Ti + Q covering n¿ as pictured in the foilowing diagram. nn nurQ 'Q *ttl ylsl "t . ylzl The natural map r : Qlz) -- C' induces maps r' : þr;LQ¡tzl * C' such lhal r' : ,onl'1 . Form the fibre product nl'Q xyp¡ rrLQ. Let 11 denote the map (r o nrl2l o p?l,r ttfl pl:l) , (nr'Q xy¡sl rrrQ)t'l -- c' x c* ' " " 4L where p1 and p2 denote the projections zrtlQ xypl rslQ --. TtrQ and zr'f1Q xy¡.1 nlQ --- r3LQ respectively. Also, let 12 denote the map î "nfl , (T;rQ)l2l -, C'. The bundle gerbe product rne gives rise to a map ñq : rrtQ xypl rrtQ -- T;rQ and we have a commutative diagram (nl' Q xyls1 trrt Q¡lzl ^8t, (n;L q¡tzl T2 "l m rf' "x c X CX Hence we have r;(n$l).c - rfm*c. Using equation 4.2 we get ** Ttrn c : (¡?\. o + (n?\. d(, nlt x r rfi)"b @?t). @?t). " - " " : Ø'18\.(î,\"a. We can rewrite this as 6(eiriA + pin[A) d(, o nf) o p?] r o nfi pt:t).b d@,råt). (r|\-t - " " - : õ(rnbr;A) - a@?\.(4fl)-t - [email protected]).(¡?\.t. Ø.7) Let p e S0((zr¡1Q xy¡zlr3tQ\lzl;V,) be defined by p: (r oñfi,r "rf) opt:]).b+ ?ntå\-(r'fi)^t - þi'r).(rl'l)-t - @ït).(rl'l).r. Then equation 4.7 above becomes õ(eiriA + piräA) : õ(rnbriA) + dp. (4.8) Lemma a.5. 6(p) :0 in So(("ftQ xypl r{rQ¡tsl;Z). \Me will omit the straightforward but tedious proof of this lemma. Thus there exists B e Sj(nrtQ ¡rlrQ;/,) suchlhat p: d(B). Hence from equation 4.8 we get 6(pitiA + p;räA)""pl : õ(^bfiA + dB). Therefore there exists C e SI(Vl3l;Z) such that piÌiA + piniA: mbriA + dB * rþC, (4 9) where we abuse notation and denoteby rq the projeclionnrLQ x"p1 rrrQ -- Y[3]. Taking d of equation 4.9 implies r.(triP +n[F) : f (triF +dC) and hence 6(F) : dC. We would like to show that ô(C) : 6. A singular 1-simplex in S1(Ytzì;Z) is a pair (ot,oz) of l-simplex€s ø¿ : A1 - Y such that zr(ø1) :r(oz).Given such a pair (o¿,o¡) there exists a lift o¿¡: L,r - Q of. the l-simplex (o¿,ø¡). Notice that (o¡n,o¿) is then a l-simplex in 51("r'Q xy¡tlrsrQ Z) which satisfies nq(o¡x,on¡): (o¿,o¡,o¡). Fbom equation 4.9 and from the associativily of. mq we see that d'B Arírq lorn,o"r) * Ao r, - @q @u,ozs),on) - C ço1,o2,oa) Ao Añrq d' B C rn * ço rr,o r", - ç rn,rhq (o2s,o p)) - @ t,o g,o ¿) 42 which, using equation 4.9 again, gives Aorn*Ao*lAor'-dB(orn,orr)-dBçnq1os¿,ozs),on)-C7o",o",on¡-Cço.,oz,oa) : Aorn * Ao* I Ao,, - d,Bç2s,op) - dBçorn,n eþzs,oþ)) - C1or,or,o"7 - Cçr,os,oa)' and hence 6(C)(or,or,or,on) : d,B@zg,op) - d,BØrqlos¿,ozs),op)* dBçrn,arq@zs,on)) - dBçr^,[*rl}) Notice that if lhe o¿¡ are now ail zero simplices and ',ve can show : B(or",orr) - Blnqlou,ozs),otz) t Bçosa,nq(ozs,op)) - B(6sa,o2t) 0, (4'11) then we will have ô(C) : 0 as we want. To establish equation 4.11 we first need a more explicit formulalor B. To achieve this we need to digress for a moment. Notice ylz) that if we regard Q and Y[2] as sets then because rq : Q - is onto we can find a section s say, of. rq (obviously s will not be continuous unless Q is trivial). Then we can define a set map c: Ylsl t C* satisfying 1 c(a y : L, c(a z, a s, u ù c(a t, as, a ò-r c(u r, u z, a ò t, az, s)- f.or g¿ alt lying in the same fibre of Y over M,by defining mq(s(Az,Us),s(gr,Az)) : ytsJ. s(AL,A¡)c(gr1g2,gs) for (gr1, Az,as) ç Since r : Y - M is onto, 'ü/e can find a set map o : M -+ Y which is a section of r'. Therefore we can define a function ¿' 1yl2) -- C' by c(at,az,o(m)), where rn: T(uL): n(az). Then it is "'(ar,az): easy to check that c'(Uz,as)c'(Au7s)-rc'(Ar,Ar) : c(UuAz,4s). Hence we can rescale our section s so that it satisfies mq(s(az,us), s(at,az)) : s(at,az). Ø'12) Next, observe from the proof of Proposition 3.6 that a formula f.or Bç2s,o,r¡ is then given by : b B (o r",o rr) ç, ç 1p1or3 )),ø23 ),r (s (p (o p)),o p))' Flom this and equation 4.12 we see that B satisfies equation 4.11. Therefore C satisfres 6(C) :0 and so l¡/e may find D e Sr(Yl2l;Z) such that C : 6(D). Thus if we rename F to F - dD then we have ó(F) : 0 and F is still a representative for the Chern class y[2]. of. Q - Also we can solve F: ô(/) for some f < S'(Y;Z) which must therefore satisfy õ@,f) :0. Hence there exists a e S\(M;V') salisfying n*u : df and da : 0. We have the following proposition. Proposition 4.1. The class in H3(M;Z) defi,ned by the s'ingular ?-cocycle a i,s a rep- resentatiue for the Dirmier-DouadE class of the bundle gerbe (Q,Y, M). We will assume this result for the moment and delay a proof until later. We use Proposition 4.1 to give an alternative proof of the following theorem from [11]. Theorem 4.1 ([11]) . Let u e S2(G;Z) be a representatiue for the Chern class of G. Then the transgressi,on of lul in the pri,ncipøl bundle P -+ M 'is the Dirn¿i,er-Douady class of the lifii,ng bundle gerbe (Q, P, M) assoc'iated to P -- M uia the central extension Ct+G-G. 43 The fact that the transgression of [z] is the Dixmier-Douady class DD(Q) of Q and not minus DD(Q), as is the case in [1i], is due to differing sign conventions. Proof. Suppose in the above construction of the singular 2-cocycle F e 52(Vl2);Z) and the singular 2-cochain C e Stlytel;Z) thar we had been unable to define a set map s:Y[2) Q which was a section of.rq: Q Y[2] and which satisfied equation 4.12. 'We -+ - rewrite equation 4.10 as (trq xylnl nq xyta) nq).6(C)@sa,ozs,orz) : dB'lorn,orr,orr7 (4.13) where rq xy¡a1 ltq xy¡a1zrç denotes the projection from nfl("rtQ xyplr3LQ) *"rnt ni'nstQ : r;rrtlQ xy¡a1 nf,r("1'Q xypl rsrQ) I Qrrrn n Ylal (compare with Defi- nition 3.1) and where B' e So(Qps¿,;Z) is defined by B' : I x B -t (r x mq).8 - B x r - (mex 1).8. If we can show that 6(B'): 0 in So(Q?)rn;Z) then we will have B' : 6(E) for some E e So(Vla);Z) and hence 6(C) : dE. Also it is easy to see that we must have 6(E) :0 in ,S0(Yl'l;Z). We see that ð(B') : 0 as follows. Recall that d(B) : p € ,So(("itQ xyplrsrQ);Z). Hence we get ô(B') : p'where p' € So(Qlr')salZ) is defined by 1 x p+ (1 x ñl2l)*p - p x 7 - (nlzl x 1)*p. It is a tedious but straightforward calculation to verify, using equation 4.3, that p' : 0 and we will omit it. Notice that we could have considered a simplicial line bundle (Definition 3.2) on an arbitrary simpliciai manifoicÌ, ihe constructions above wiii stiii work but we will replace lhe n¡ by the face operators d¿ for whatever simplicial manifold \¡/e are working with. Of course we will not be able to use Proposition 3.5, as that only applies for the special simplicial manifold Y : {Yo} with Y, : yln+t). So it follows that we could consider the central extension C" -- G -- G as defining a simpiicial line bundle (or simplicial C* bundle rather) on the simplicial manifold l/G as^per Proposition 3.2 and constructed u € S'(G;Z) rcpresenting the Chern class of G in H'(C;Z), tt e 51(G2;Z) such thar dfiu * di, + diu : dp, and À e S0(G3;Z) such that dåp- dip+ditt- däp: dÀ and dðÀ - diÀ+ di^- dä,\ + di,\:0. (Thus the triple Q, p,À) represents a class in I/3(,S'''(¡fG); Z) - the cohomology of the total complex of the double compiex .9'''(¡/G) of Theorem 2.7 - and hence a class in H3(BG;Z)). We have the natural map r : Plz) --+ G defined by p, : pf(pt,p2) for p1 and p2 in the same fibre of P. In fact r gives rise to a simplicial map P -' l/G where P : {P^} is the simpiicial manifold wittr P, - pln+lÌ, a portion of which is pictured in the following diagram: ltllltplrl _-GxG lillPlzl --1- ç P.______-______* {1}. By pulling back u, ¡.r and À to .92(ft'l;Z),5t1ptsì ;Z) and 5olptal ;Z) respectively by this simplicial map and using the conditions df,u-d\u+diu : dþ, d[¡r-di¡-t+dip,-di¡t": ¡i) 44 and dðl - diÀ + d;^ - däÀ + di^: 0 it follows from Proposition 3.6 that we can adjust r"u by u .óbo,rtráary d,a so that F : r*u * d,a satisfies ô(F) : 0 in S2(Pþ|;V'). It follows that there exists f e S2(P;Z) such that ô(/) : F and hence there exists u € S3(M;Z) wíth d,a :0 and ir*u) : df . We need to show that [ø] is the transgression of ful. Recall from [2] that the transgression [t(z)] of [z] is characterised by the property that there exists ee S2(P;Z) sruchthat de:'r*c, where c is arepresentative of lt(r)], and [i.*ele^] : lrl, where i, : G + P^ rs the injection of the fibre G into P^ and where elp- denotes the restriction of e to the fibre P-. So we need to show lhat li,. f lp^] : [r]. The injection i is defined by choosing a point p of P^ and mapping g to pg (this needs G connected). If we consider {p} t P^ C P[2] then ô(/) : F : r*u f do gives us : transgression ¿2. Hence from Proposition 4.1 we have l¿- llr^l lrl. Thus ø is the of that the transgression and the Dixmier-Douady class are equal. ! In general, given a central extension of groups A -, G -- G, if one gives each of the groups the discrete topology then there wiil not exist a homomorphism s : G -- G which is aiso a section of G + G. In certain circumstances however, such a section can be shown to exist. An example of this occurs if A is an abelian topoiogical group and we consider the central extension of groups A+ EA---+ BA. Let A6, EAõ and BA6 denote the groups A, EA and BA equipped with the discrete topology. Then we can define a section s: BAõ -, EA6 of EA6 -- BA6 which is also a group homomorphism as follows. A point of BA is of the form l(r0,. .. ,rp),(ot,... ,ap)|, or, in the non-homogenous coordinates, . . - l(¿t, . . .,tr),[rtl loo]1, where t¿: to +'" + r¿-t. A point of EA is of the form l("0,. .. ,rp),(o10,... ,ol)1, or, expressed in the non-homogenous coordinates, . . . lú,, . . . ,te, a:ol1.c,:o)-'oirl l(ar-r)-'aloll. The projection p : EA -> BA is given by p(l@o,... ,rp),(o1o,... ,oo)l) : l("0, . . . ,rp),((o;)-tor, ... ,(ao-)-'ar)l or, alternatively, p(!tt,... ,tp,a'ol(a'o)-ratl l@r-)-'oo]l) : lúr, ... ,tp,l(o'o)-to'rl "'l(o"-t)-'d'l|. Sowhatweseekisahomomorphism s: 8A6 -- EA6 suchthatpos:1. Let l("0,. ..,rp),(or,...,ap)l beapoint of BA6 andassumethatthereareno0'sappearing amongst the r¿. Define s(l(øg, ... ,rp),(or,.. . ,ap)l) : l("0, ... ,rp), (1,4r, ara2,... ,aL"'a)1. We just need to check that s is well defrned with respect to degeneracies. 'We have s(("0, ...,ti * r¿+t,'..,rp),(ot,..',or-t)) : (("0, . . . ,ri* r¿+r,... ,rp), (1, or, ara2,.. . ,Qr ' ''oo-t)) : (("0,. .. ,rp), (1,ot, ... ,at"'ai-r,a1 "'aira!"'airQr"'ai+r,... ,at"'ap-t. 45 On the other hand we have s((zo, ...,rp),(ot,...,ai,I,ai+r,...,oo-r)) : (("0,. .. ,Ip), (1,ot, ... ,at"'CLi-!,a1 "'ai,at"'di,ar"'di+Ir... ,dt"'oo-)) : s((r0, . . . ,ti* r¿+t,.. . ,rp),(ot,. . . ,ae-L)). Hence s is well defined. To check that s is a homomorphism of discrete groups we will use the non-homogenous coordinates. The condition that there axe no ø¿'s equal to zeroin (r0,...,rp) translates into no ú¿'s repeated in (ú1,...,tò. s expressed in the non-homogenous coordinates is s(lút,. .. ,tr, [orl.'.1ø]l) : lúr, ... ,tr,llotl. ..ltr]l Let lto¡1,..- ,tp+q,lap+|"'lor*nll be another point of BA6. We have s(lút,. .. ,to, lotl...loo]l)s(l¿o*1,... ,tp+q, [øp+rl "'loo*n]l) : lút,. . .,to,Ilarl''' lrr]l' lto*r,...,tp+.qtrloo*tl "'lor*n]l : lt"g),...,to(p+q),1[o"(r)l .' . lo"fo*nl]1, where o is a permutation of {1, ...,p*q} such that ú,11¡ <'.. < . . . Itoy), to(n), . . ., to(p*q¡, Lla,O)lo' 1 lo"ll for some a' , a" upon removing all repeated ú's. This is exactly the result of applying s to the point . . . . lúr, . . .,to,fa1l. ltrll. ltp+t, . . .,tp+q,lap+tl. lor*n]|. Therefore s: BAõ -- EAõ is a homomorphism and it is straightforward to check that s is also a section of p. Note however that s considered as a map s : BA -+ EA is not continuous. It now follows from the remarks made during the course of the proof of Theorem 4.1 that if we regard the central extension C" + EC.' + BCt as a simplicial C' bundle on the simplicial manifold /üCX then'\¡/e can choose arepresentative r' €. S2(BC";Z) of the Chern class of EC' ---+ BC* so that there exists C e SI((BC')';Z) such that the following two equations are satisfied: dåF - diF + d,\F : d,C, (4.14) döC - dic + d;C - däC :0. (4.15) It also follows that we can find Az € Sl(EC';Z) such lhat p*F - dAz, 6(A2) : r*c-de,, and such that there also exists B2 e S0(EC' x.ÐC';Z) so that the following equation in 51(,EC* x ,EC*;Z) is valid: piAz - m* Az + piAz : (p x p).C + dBz. Let us rename F to q and C to n2. If one examines the construction of. 4 and n2 then one notices the construction relies upon the fact proved in Lemma 4.3 that there exist c € ^91(C";Z) and ö e SO((C")';Z) satisfying the analogues of equations 4.14 and 4.15. 46 Let us see what happens if we try to iterate this procedure using a2 in place of c and n2 in place of b. So we start with the principal BC" bundle EBC" --+ BBC,X. We form the canonical map r: (EBC")t2l -' BCt and pullback r'2t'o a class r*r2 on (EBC")I,1. Th.n as above we get õ(r.4): d(r o'It1,r or3)*n2. Because rc2 satisfies the analogue of equation 4,3 we get ô((r oTrL¡r ors)*n2): 0 and hence there exists as € 51((EBCX)t2l;Z) srchthat (ron'1 ,rors)*K2: ð(os). Therefore r*o-^das satisfres 6(r*q-'¿or): 0 and so we can solve r*12- d,c.3: ô(Ar) for some Az e S2(EBC";Z). Hencethereexists4e13(BBC';z)withP*t's:dA3,'Followingthisprocedure through leads in a straightforward way to the following analogue of equation 4.9: piAs + piAs: m*As + dBs + (p x p)* ns, where ns e s2(BBC' x BBC";Z) and where fu e x EBC";Z) satisfies 'L(EBC' ô(Bs) - (, op?t ", opf;l¡"or- þl'l).o, - @91)*o, + çml2)¡.ar. This time however there will not exist a section s: S1(BBC";Z) -- S{EBC";Z) compatible with the group structures on ^91(BBC,";Z) and S{EBC";Z) induced by the group structures on BBC' and EBC" and so we can only conclude, following the argument of Theorem 4.1, with 12 in place of c and rc2 in place of b, that there exists Às € ,S1((EBC,")t;Z) such that dôot - diot + di*t - d'ins: fl¡t (4'16) g. dðÀs - diÀ¡ + döÀt - dä)s + dll3 : (4.17) Therefore it follows that we have defined 13 € S3(BBC";Z), Az € S2(EBC";Z), and Àg Sr((EBC')t;Z) such that rc3 e ^92(( BBC")';Z), Bt €,S1((.EBC*)';Z) e d,4 : 0, P* ts : d,As and piAz piAs: m* As dBr (p p)* nz + + + " ditt - ditt + di4: ¿¡at döot - di^t + di^t - d'lns: fl¡t, and À3 and rca satisfy equations 4.I7 and 4.17 above. It is clear that 13 is the trans- gression of.4 in the principal BC' bundle EBC," ---+ BBC". It is also clear that 12 is the transgression of c in the Ct bundle EC,' '- BC". The fundamental class o¡ ¡¡n+r(BpC,";Z) is mapped to the fundamental class o¡ ge+2(Bp+rCx;Z) under the transgression in the principal BpC" bundle EBPC" -+ BP+rçx. Indeed if we consider the long exact sequence in homotopy for the fibering BnC,x -+ EBPC -- Bp+lCt then we get r,pa(BPC") = ro¡2(Be+lCx). Moreover, since these are Eilenberg-Maclane ,pu"". *å hulr" the sequence of isomorphisms rn¡1(Bqc") = Hs+r(BsC,x;72) = Z and with respect to these isomorphisms the isomorphistÍL 'ttp*r(BpC'x) = tr¡2(B'+lC" ) can easily be seen to be the transgression (in the case we are considering the transgres- sion //p+1(BpCr;Z) - HI+2(Bv+LC";Z) is an isomorphism). It follows that i2 is a representative of the fundamental class of. H2(BC";Z) and that r.3 represents the fun- d.amental class of H\(BBC";Z). h would not be too hard to go further and prove a similar result by induction for the fundamental class 6 e S*(Bn-1çx;Z) allhough we will not do so here. the recipe for constructing from a Õech cocycLe g¿¡¡: Recall from Section 4.1 above : (J¿jt" - C' maps g;¡ : [J¿, -t BCX satisfying the cocycle conditiorL giigik !¿¡ aîd 47 maps g¿¡: [1" --, EC" Iifting g;¡ such thaf g¡ng¿trþ¿¡: gt¡r". It is not difficult to go further and construct maps g¿ : U¿ + BBC" such that g¿ : gj on U¿¡ and iifts 0¿:U¿ - EBC" of g¿ which satisfy g¡gn' : g¿j.The map g: M - BBC is obtained from the g¿. Suppose v,/e are given such a cocycle g¿¡t . We wili use the Õech-singular double complex (see [a]) to show that the class in H7(M;Z) represented by 9;¡x and the class represented by g*ts are equal. Let,iu,: U¿ + M denote the inclusion map. We have 'if¡ng*ts: glts: (po gn)*tt: dgiAs. Next, gîAt- gîAz: (0n,0).6(As) : (0¿,þ)*("*tr- do.ù. Since g¡gu' : go¡ we get thar þjAs- giAs: 9I¡q- d(gn,0¡)*os. Using p(g¿¡): 9;¡ we get girt2: d7i¡Az ,where Az € St (EC";Z) is constructed as in the paragraph preceding Lemma 4.4. Let a¿: þiA2 and let 0o¡: gi¡ - Øn,}¡)"ot.Since ô("r) - (, o7T1,r o¡rz)"n2, we get 0¡n - 0¿¡ * þni : gjuAz - gi,"A, + /i¡A, - (g¡*,9;¡)" rc2' Using the equation piA2 + piAz : nL" A2 + dBz + (p x p)" nz, we get 6(0n¡) : (g¿r ' g¿¡*)* Az - iixAz + d(g¡u,9¡). Br. Using the same equation again we get (i(go¡r))" Az - d(gnt",i(sn¡*)). Bz * d(þ¡*,9;¡). B, - (gnr,L)* or. It is straightforward to check that (g¿¡,,1)"n2:0. It is also not difficult to check that 'i* Az - c. Let gqx : U4* -' C be a lift of g¿¡*. We have -t 0¡r - 0¿u * þri : d(|i¡rê - (gnn,i(gn¡r)). Bz (g¡n, sq). Br¡, where ô e SO(C;Z) is constructed in Lemma 4.4. Pú'yt¡r": 9i¡¡ô - (/ut",i(gu¡n)).8, + (0¡¡,g¿¡).82. Define zijj zjk and z¿¡ in Ct by z¿¡: r(s(g;¡),g¿¡), z¡t": r(s(g¡n),g¡n) and z¡¡,: r(s(g¿¡),gr*) respectively. Then one can check thal l¡¡ is given by ei¡þ - (o(z¡r) - o(z¿r) i o(z¿¡)), where ø: (C')'* (C)'is the section constructed in Lemma4.4. It now follows that 'yjkr - ^l¡w *'k¡t - yt¡n : n¡¡¡¡ where r,.¿jn¿ : (l¿¡¡"¿ --+ V' is t,he image of the cocyc,le g¿¡¡ under the coboundary map H'(U;Ain) * Ht(tW;Z¡). Hence g¡jt and g*a3 represent the same class in Ht(M;Z). This argument constitutes a proof of Proposition 4.1 As an application of this, if we are given a bundle gerbe (P,X,M) on M then we know that there is a map g : M n BBC", unique up to homotopy, such that the Dixmier-Douady class DD(P) of P is equal tro g"ts. It follows that the pullback Tkg*ts of the Dixmier-Douady class to X is cohomologous to zeto. Hence there is a lift g : X ---> EBC' of the map g o'îx : X ---+ BBC", ie amap g t X + EBC" making the following diagram commute: X a,EBC" 1tx p M --¡* BBC" Notice that we can therefore define a map f : Xlzl -* BC' satisfying Í (rr, rù Í (rt, rz) : f (rt, rz) 48 f.or n1, nz aîd. ø3 in the same fibre of. X,by f @¡rr): ¿-tþ@ùg(rt)-t)' It is easy ylz) to see that the C" bundle f-tEC" - has a bundle gerbe product and that the resulting bundle gerbe (f-'EC",X,M) has Dixmier-Douady class equal to DD(P)' Therefore the bundle gerbes f -'EC} and P are stably isomorphic and so there exists a C* bundle L+ X suchthat P- |-LEC,E)ô(I). Let ft:X -BC" beaclassifying map for.L and redefrne O fo g./¿. Denote the rescaled / by the same letter as well. It follows that the bundle gerbes P and Í-t EC" are isomorphic. In effect we have proved the following Proposition. Proposition4.2. Let (P,X,W" a bundle gerbe. Then there zs a bundle gerbe morph'ismg, (P,X,M) -+ (EBC,",EBC",BBC) withg: (0,g,g)' Themap g: M -> BBC" is un'ique up to homotoPY. 49 50 Chapter 5 Open covers 5.1 Bundle gerbes via open covers We now turn to some calculations involving bundle gerbe connections and curvings on open covers. Similar results for gerbes appear in the lecture notes [22]. suppose_(P,,YrM) and (Q,X,M) are bundle gerbes. and we have a bundle gerbe morphism Í-: (Í,/,id) which maps (P,Y,M)ro (Q,X,M). It is easily seen that P and Q must have the same Dixmier-Douady class. We will exploit this fact to give a method of calculating the Dixmier-Douady class of a bundle gerbe P on M lrom an open cover of M. LeI {U.),.> be an open cover of. M, and suppose rû/e are given line bundles LoB --+ U,B. Suppose further that we have sections ooB, of. the line bundles Lp., Ø L\., Ø L"B over (Jop, These sections are required to satisfy oBrdoo]6ooBaooþ., :7, where 1 is the non-vanishing section induced by the canonical trivialisation of õ(L"pr) : Lptt Ø LLr¿ Ø Loþõ ø LLB., over (loB.r5 and where we have ptú Lop, : Lpt 6 LLr g Log.Assume that we are also given connections Vod on each LoB. Denote the induced connection on Lop, bV Y oh. Hence we can find l-forms Aoh on Uop., such that V..¡."(o.,pt) : Aop.vooBt. If we denote the induced connection on õ(L"B) by ô(V"pr) then ô(V"Bt)(f) : O to clearly we will have Ap.,o - Aoú I Ao9õ - Aop, :0. (5'1) Suppose we have a partition of unity {rþ.},r> subordinate to the covel {U"}".> of M. Define a l-form Bop on (JoB as follows. Lel B.p : Dr." 1þtAop.t. Then it is easily checked that we have Bp,, - Bot I Bop: Aogt (5.2) 51 Define new connections on the L"B by V.,þ - BoB. If. Fop denotes the curvature of the connection Vop, then the new connection will have curvature FoB eelualto Fop-dBap. Flom the equations above, we see that Fp., - Fot * Fog: d,Aoh. (5 3) Hence the new curvature will satisfy îB-, - Fo, I fop:Q. (5 4) Using the same trick as before, define 2-forms þa on [/" satisfying lrp - þa: ToB by þo: -Df"prþ,. (5 5) 9e> One can check that on UoB wehave d¡to: dltp, and so lhe dp,. are the restrictions to Uo of. a global 3-form a on M. It will follow from the discussion in the next section on Deligne cohomoiogy that ø lies in the image of H3(M;Z) insíde ø3(lø;m.). 5.2 Bundle gerbes and Deligne cohomology We now briefly review the notion of. sheaf hypercohomologE. We refer to [7] or [21] for more details. Suppose 'üue are given a complex of sheaves K' on our manifold M. Thus we have sheaves of abelian groups Kq on M together with morphisms of sheaves dq : K - Kn+I which satisfy dq¡tods:0. Suppose r,¡¡e are given an open cover U: {U"}..ç2 of. M. Then rü/e can form a double complex Ki¡'with Kl¡n : Cr(tl,Kq), \e the Õech p cochains with values in the sheaf Kq . The differentials of Ki¡' are d, t Koün + Kftø+r and ð , KJin ---+ ftPit'o and are induced by the morphism of sheaves d, : Kq ---+ Ks+r arrd tlre Cech coboundary operator 6: Ce(Ll,Ko) - C'+1(U,Kq) respectively. d and ô clearly satisfy d2 :0,62 :0 and d6 l6d,:0, thus Ki¡' is a double complex. LerTot(Kir) denote the total complex of Ki;'. A refinement V < U wlllinduce map- pings fo{Ki) -,Tot(Ki,) and mappings H'(Tot(Ki,)) --+ H'(Tot(K$) of the corre- sponding cohomoiogy groups. We define lhe sheaf hgpercohomology groups H'(M; K') of M with vaiues in the complex of sheaves K' to be H'(M; K') :ryY H'(Tot(Ki,D As an example consider the complex of sheaves çi, 4ry QI/ otr a manifold M. Then, relative to an open cover U : {U.)oeE of M, a class in the hypercohomology group qE H'(M;çi, CIfu) consists of smooth maps gap : UoB --* C" and one forms Ao on (Jo which together satisfy the conditions gB-,silg,B 1 -1 AB-A* dg"Bg ap Then it is not hard to see that classes in I/1 @;Aî,t '9 g:*) are in a one to one correspondence with isomorphism classes of line 'oundles with corrnection on /Vl 52 Consider the complex of sheaves aioogQr*1Q2* on a manifold M. It is shown in [33] (see also [34]) that a bundle gerbe (L,Y, M) with bundle gerbe connection V and curving / gives rise to a class D(L,V, /) in the Deligne hypercohomology group Ht(tW;Ci, - Qt¡r,t - Q2*). Let us briefly recall how this ciass is constructed. Choose as usual an open cover U : {U"}oeD of M such that there exist local sections of. r : Y + M above each Uo and ail of whose non-empty finite intersections (JooÀ...n(Jan are contractible. Form the maps (so,sp) 1(Jog - ylzl and choose sections ooB of the pullback line bundle Loþ: (so,sp)-rL ovet UoB. The connection V induces a connection VoB on Lop and we get V.B(o,B)- Aoþ6oaÞ for some one form Aop onU..p. Loþ has curvature Fop: (so, sB)*Fv with respect to Y ag. Since Fv : ô(/) we get FoB : f B - Í,. On the other hand, we clearly have Fog : dAop, and so we get dAoþ : rB - r" (5 6) Finally, the isomorphism of line bundles m : r¡r L Ø r¡L L --> r;L L induces an isomor- phism of line bundies m: LBr6Log -- Lat and we have m(op-rØo..p): aa-,tgaþt where g,h : [Joh + C' is a representative of the Dixmier-Douady class of L and satisfies the cocycle condition 9ø¿9i]¿9.,P¿9,Åt: L' (5 7) Therefore we have on the one hand (VB, * V,ò(oø Ø o.e) : (Ap-, + A"ò Ø (oB, Ø o.,ø), but on the other, from equation 3.5, (Vp, + V.p)(oø Ø o.g) : rr¿-L oVor o m(oB-, Ø o,p) : a;ri-;;ii rt;sî,nu;u" and hence we get Ap, - Aot * Aoþ: g,àrdg.B^,. (5 S) Fbom equations 5.6,5.7 and 5.8, I¡/e see that the triple (Ío, Aop,9oB1) represents a class D(L,V, Í) in the Deligne hypercohomology group H'(M;C'í,0 - A'* - C)fr.) and one can check that this is independent of the choices made. If we consider bundle gerbes (L,Y,M) equipped only with a bundle gerbe connection V then the discussion above shows that every such bundle gerbe ,L with bundle gerbe connection V gives rise to a class in the truncated Deligne hypercohomology group H'(M;A,fu -- ALM) 53 Suppose we are given a bundle gerbe (L,Y, M) wifh bundle gerbe connection V¿ and curving / and suppose moreover that the bundle gerbe Z is trivial. Hence there exists a line bundle J onY such that there is an isomorphism of bundle gerbes þ: L + 6(J) over Y[2]. Suppose that the line bundle .I has a connection V.¡. Then ô(J) inherits the tensor product connection ô(V"r). Therefore there exists a complex valued one form 4 on Yl2l such that V r : ó-L oô(V",) o ó +4. Since / commutes with the bundle gerbe products, we must have ô(4) : 0 and hence there exists a complex valued one form ¡.1 on Y such that ô(¡l) : 4. Then if we give J the nerù/ connection V.¡ - p, and rename this new connection V.¡, it is not hard to see that we have an isomorphism of bundle gerbes with bundle gerbe connection Ó : L - 6(J) over Y[2]. This result is not surprising in light of the fact that there is an isomorphism H'(M;A[ò = H'(M;C\, - CIla) of cohomology groups. Suppose that the class D(.L,Y ¡,, f) of .L in the Deligne hypercohomology group H'(M;Cin - Q'u - C¿?z) ir zero, so that (using the notation above) we have goh : hplh;+h.p Á^ naþ -kB i k, + dhrBh.A Í" : dko' F}om fo : dko we see that the three curvature ø of the bundle gerbe ,L equipped with the bundle gerbe connection V¿ and curving f is zerc. In other words the map H'(¡W;Cio - At, - Q'r) - Ot(M) applied to D(L,Y t, f) is zero. So we must have dÍ:¡r*(u) :0. The bundle gerbes L and ô(.I) may not have the same curvings, in general the curvings / and Fs, of..L and 6(J) respectively will differ by the pullback of a two form K on lui (here .F'y, is i;he curvature of .¡'wiih respect i;o i;he connection V.¡ defined above). Since Fv, is the curvature two form of a connection on J it is closed and we have just seen that / is closed, so it follows that K must also be closed. Since K is also an integral two form, there exists a line bundle I on M with Chern class equal to the class [K] in H2(M;Z). We may endow / with a connection V¡ whose curvature is equal to K.It foliows that the line bundles.L and ô(JS ¡r-rI*) are isomorphic by an isomorphism which preserves the bundle gerbe products and the connections V¿ and V; - r-rV t. What is more, the bundle gerbes L and ô(J I r-rl") have the same curvings. In effect, we have proved the following proposition. Proposition 5.1 ([34]). A bundle gerbe (L,Y,M) wi.th bundle gerbe connecti.onY¡ and cura'i,ng f has triuial class D(L,V t , Í) in Hz(M;Cio - Q!, - 9'ò tf and onlg if Vt : ô(V"¡), whereVt is a connect'ion on a line bundle J -- Y which triuial'ises L and Í : Fv, , where Fe, is the curuature of the l'ine bundle J with respect to the connecti,on V ¡. Note 5.1. One can define a notion of stable isomorphism of bundle gerbes with bundie gerbe connection and curving - see [34]. One says that two bundle gerbes (LuY, M) and (L2,Yz, M) with bundle gerbe connections V1 and V2 and curvings fi and f2 respectively are stably isomorphic if there exist line bundles ,S and T on Y7 x ¡a Y2 with connections Vs and V7 respectively and there is a line bundle isomorphism @?l)-'l,rsd(s) -, @l:l)-'LzØõ(T) which commutes with the bundle gerbe products and preserves the connectionr þl'l)-tvr +ô(Vs) a"a þl2l)-tV, +ð(Vr) and such that the curvings piå * Fvs and pif2¡ Fv, are equal. One can then prove ([¡¿]) a version of the above proposition for stable isomorphism classes of bundle gerbes with bundle 54 gerbe connections and curvings. One can also show ([34]) that there is a bijection between stable isomorphism classes of bundle gerbes with bundie gerbe connections and curvings and the Deligne hypercohomology group H'(M;Cir - QL* ---' Q2*). Given a bundle gerbe .L with bundle gerbe connection V¿ and curving / with trivial Deligne class D(,L,Y ¡,, f) we will explicitly construct a line bundie J + Y which trivialises L and a connection V7 on J such that V¿ : ô(V¡) under the isomorphism L -+ 6(J) and such that the curvature Fv, of V.¡ is equal to /. Since -L has trivial class in H'(M;C'-- Qtu - Olo) tfrenthere exists hoB:(JoB -- C' and ko € f-¿1(%) such that Toh : hyih,;hap (5.e) dhaþ Aog : + le" * (5.10) -leP hoþ f* : dko (5.11) where we have continued to use the notation from above. First we construct the bundle J --+ Y which trivialises .L. This is done in [33] but we need to review the construction. We define line bundles ,/* - Yo by J' : S;IL where 3o : Yo -- ylz) : use is the map g - (U,s"("(E))). V¿ induces a connection Vo ,6;1V on Jo. We the sections cag €. l(UoB, L.B) to define line bundle isomorphisms 4lop : Jo ---+ JB by ó,8@) : m(ooph;àø") f.or u € Jo. The condition 5.9 above shows that SB.,oþoF : Óot on Yop.r. Hence we can giue the ../o together using the þop to get a line bundle J on ,-Io satisfled Vo : Y Y . If is easy to see that if the connections Vo on the Ó.) " B " 6"U then we could glue them together to form a new connection Vy on J. We calculate o for an arbitrary section t, € l(Y.,/o) restricted to YoB. we have Ó;à " Y B ÓoB Ô;à"YBoþ'B(t*) : o m(ooph;Å Øt") ó;à "V B : ó;à. (m(tr-LY.p(o.ph.þ) I ¿.) * m(o.Bh:¿ I V,(ú"))) (m(((tr.A.u n.fu)o,ph;À)ø ¿.) -t m(o,Bh-l ø v.1t.¡¡ : ó;å " - -*Ár1¿¡1 ,, , , qv ^ s-n*4!.Ê-+V"(ú,)hog : -r"ko * r*lco + V.(¿,). Hence if we let Vl : Vo -l- n*ko then w and so the V'" patch together to define a connection V7 on ow that V¿ : ô(V"') and that f : Fvr. The map m : r[r rise to isomorphisms of bundle gerbes q, : õ(J,) -> -L over have a commutative diagram 6(J") 4a 6(ó.B) L ,4 6(Jp) 55 over Y:2¿. Thus rü/e can define an isomorphism of bundle gerbes rl : 6(J) --+ .L over ylz). We have ð(V, + ¡*lc.'): ô(V,) which equals n;r oVzorlo and thus we get V.¡: rl-t oYroT.Finally Vo * r*kohas curvature F,: fV_*f (dko- /. which equals fþ.by equation 5.11. Thus J has curvature Fv, equal to ¡. 56 Chapter 6 Gerbes and bundle gerbes 6.1 Torsors In this section we briefly review the defrnition and basic properties of. torsors. Definition 6.1 (t7]). A G-torsor on a space X (here G is a sheaf of groups on X) is a sheaf P on X plus an action of G on the right of. P, ie a morphism of sheaves P x G - P, with the property that for any point of X there is an open neighbourhood [/ of that point such that for any open set V c U the set P(V) is a right principal homogenous space under the group G(V). Similarly one can defrne the notion of a morphism P -+ Q between two Ç-torsors on X. One can show that any such morphism is automatically an isomorphism. A good example of a torsor is the torsor l¡ç associated to a principal G bundle P on a space X. Here Px is the sheaf of sections of P. It is then easy to see that the sheaf of groups G¡ acts on P I on the right so as to make l¡ into a G¡¡-torsor. Suppose we have a morphism of sheaves of groups G -- .F/ over a space X and a G-torsor P on X . Then one can construct (see [7]lpugu 191]) an I/-torsor on X denoted by PxaH. PxaH isthesheaf associatedtothepresheaf U + (P(U)xn(U))lG(U), where G(U) acts on H(U) via the morphism of sheaves G - H. So a section of this presheaf would be an equivalence class lp,hl, where p is a section of P(U) and h is a section of H(U), under the equivalence relation (p1, ht) -,(pr,hr) if and only if there is a section gn of G(U) such that Pz: Ptgtz and, h2: gtzL' h1. Clearly Il witl act on the right of. P xA H so as to make P xA H into a f/-torsor. From norÃ/ on we wili only be interested in torsors under the sheaf of groups Ax of smooth A vaiued functions on X associated to an abelian group -4'. Given another space Y and a continuous mapping f : Y --+ X we have the notion of the pullback of. a Ay-torsor P on X (see [7][pages 190-191]). We can pullback the sheaf P to a new sheaf f-tp on Y. This is noìM a torsor under the pullback sheaf of groups f-rgv onY. There is a canonical morphism of sheaves of groups Í-'Ax - Av over Y. Hence we can construct, using the method outlined above, an Ay-torsor f-tp xl-'Ax Av oî Y. We will denote this Ay-torsor by f.P. Given another space Z and a continuous mapping g: Z rvl/e can form two A2-torsors on Z froman -Y, * ¡!¡-torsor P on X; namely U ù. P and g" f P. As mentioned in [7][page 187] these two torsors are not identical but there is an isomorphism g* T* P - ff g)- P between them which satisfies a commutativity condition similar to that of diagram (5-4) on page 187 of [7]. 57 Given two ,,4.¡ torsors Pt and P2 on X we have the notion of the contracted product ¿!¡-torsor Pr I P, on X (see [7][page 191]). To construct Pr SZz we first take the Ax x d¡¡-torsor Ptx P, on X and then using the canonical morphism of sheaves Ax x Ax - d¡¡ induced by the product map A x A -+ A, we form the A¡- torsor (4, x P2) xAx"A, Ax,: ZrØ Pz. 6.2 Sheaves of categories and gerbes Definition 6.2 ([5],[7]). A presheaf of categories C on X consists of the following data. 1. for each open set [/ C X a category C(U) 2. given open sets V c U C X there is a functor pv,u : C(U) -+ C(V) - t]ne 'restriction' functor. We will sometimes denote the image pv,u(C) of an object C of C(U) by Clv and the image pvp(ó) of an arrow Q in C(U) bv ólv. 3. given open sets W CV CU C X we have pw,v o pv,u: pw,u. Strictly speaking we should require that rather than the functors gw,vopv,u and pwp coinciding, they should differ by an invertible natural transformation which satisfies a certain coherency condition given a fourth open subset of X. In most examples that we meet however we will not need this generality. A morphi,sm or functor þ: C -+ D of presheaves of categories C and D on X consists of the assignment to every open set U C X of a functor óu : Cu -+ Du in such a way that if V is another open set of X with I/ C U CX then the following diagram commutes: cu LDu óv Cv + Dv where the two vertical functors are the restriction functors in the presheaves of cate gories C and D. A natural transformat'ion r : ó +Ty' between morphisms of presheaves ofcategoriesó,1þ:C+2consistsofanaturaltransformationru:óu+tþuforevery open set U c X which is compatible with restrictions to smaller open sets. Definition 6.3. A staclc or sheaf of categories on X is a presheaf of categories C on X which satisfies the following'gluing' laws or'effective descent' conditions on objects and arrows. The effective descent conditions on objects are as follows. Given an open set U C X and an open cover {Un}nq of [/ together with objects C¿ e Oñ(U¿) and arror¡¡s ó¿j : prJ;¡,uo(Cn) - pu¿¡,tJ¡(C¡) satisfying pu,u,urr(ó¡r)pvnrn,ur¡(ó¿¡) : pun¡u,uo^(ó¡*) then there is an object C of. C(U) and isomorphisms tþ¡ : pu,,u(C) -- Cr compatible with the /¿¡. The effective descent conditions for arrows are that given an open set U C X, objects C1 and Cz of. C(U) and an open cover {U¿}ø of U together with arror,¡/s ó¿ : pur,u(Cù - pur,v(Cz) satisfying pvr¡,vr(ô) : ptJ;¡,u¡(@¡) then there is a unique arrow ó , Ct --+ Cz in C(U) such that pu,,u(ó) : þ¿ for all ¿ e ,I. In other words the set of arrows from C1 to Cz in C(U) forms a sheaf Hom(C1,C2). 58 A. morphi,sm or functor of sheaves of categories is just a morphism of the underlying presheaves of categories. A natural transformati,on between morphisms of sheaves of categories is defined in a similar way. An equivalence of stacks Ó : C -+ D is a morphism of stacks / such that there is a morphism of stacks tþ , D + C together with natural ry' o idc and rþ idp. If such an equivalence exists then the transformations ó + ô " =+ sheaves of categories C and D are said to be equi,ualent. The objects and arrows in an arbitrary presheaf of categories may not satisfy the ef- fective descent conditions listed above however, in analogy with the case for presheaves and sheaves of abelian groups, there is a functor, the 'sheafification' functor or 'associ- atedsheaf'functorC-rCwhichtransformsapresheafofcategoriesConXintoasheaf of categories Õ. This is subject to the usual universal condition, namely that given a sheaf of categories 2 together with a morphism of presheaves of categories C + D, there is a unique morphism of sheaves of categories C --+ D making the following diagram commute: c a Y D. We will briefly discuss the construction of Õ and refer to [7][pages 192-194] for a more complete discussion. Given an open set [/ C X the category Cu has objects consisting of triples ({Un}orr,{P¿}¡a,{fo¡}0,¡r¡) where {Un}nq is anopencover of.U, P¿ is anobject of. Cuu for each i e I , and f¿¡ , P¡lur¡ --, P¡lun¡ are arrows in Cu,, which satisfy the descent ({U,}' and condition Í¡t"lun,uÍn¡luniu 7 fn*lun,n inC¡¡,,u. Given such objects {Pr},{fu¡}) ({%}, {Qù,{g.BÐ of Cu, an arrow ({Uo),{k},{ln¡}) * ({%},{Q"},{g"B}) it uo equivalence class [{W¡.},{h0,,}] where {Wn,o}'.4'*¡ is an open cover of [/ such that W¿,o C (I¿ ÀWfor all i e I, d € t and h¿,o : P¿lw,. - Qolwn,. is an arrow in C¡ry,. which is compatible with the descent data f¿¡ and goB in the sense that the following diagram of arrows in C1ryn,-¡,nyr,, commutes: ht,olwn,onwr,p Ptlwn,,nwr,B Qolwn'nwr,u l¿ilw¿,o¡wj,p In-ul*r,.n*,,u Pjlwo*nw, Qplwn,.nw,,u, 'B hi,gl*n,on*r,u and where two such pairs ({tr4l,,"}, {h,,,}) and ({w',.}, {h;,"}) are declared to be equiv- alent if there is an open cover {Wi,,}a,.€D of U such thatWi,o CW¿,olW'n* for ali i e I and all o € E and such lhat lt'a,ols,,,- : h|;,olw;:. fot all z e f and all a e X. If the presheaf of categories C has the property that given any two objects P and Q of. Cu for some_open set U C X, the set of arrows Hom(P,Q) is a sheaf then one can show lhaf C is a sheaf of categories. Otherwise it_is not too hard to see that the presheaf of categories C has this property and hence e, is a sheaf of categories (this is analogous to the case for the associated sheaf of a presheaf, in general one has to apply the sheafification procedure twice to yield a sheaf - see [2S] for more details on_this)' It can also be shown that there is a morphism of presheaves of categories C -+ C with the universal property referred to above. 59 Definition 6.4 ([5],[7]). A gerbe Ç on X is a sheaf of categories Ç on X such that the foilowing three conditions are satisfied: I. Ç is a sheaf of groupoids on X, so for each open set U c X the category Ç(U) is a groupoid. 2. Ç is locally non-empty. This means that there is an open cover {Un}n¿ of X so that the groupoid ç(Uù is non-empty for each i € 1. 3. 9 is locally connected. This means that for every open set U c X and all pairs of objects P1 and P2 in ç(U),there is an open covering {Un}na of U such that H om(p¡¡o,u (Pt), pu,,u (pr) ) it non-empty. Erample 6.1. LeI G be a sheaf of groups on X. For each open subset U C X, Iet Tors(G)r be the groupoid with objects the G torsors P on U and morphisms the isomorphisms between G torsors on [/. Given another open set V c U C X there is a natural puliback functor pv,u : C(U) --+ C(7) induced by the restriction functors of the G torsors. By definition of a sheaf these satisfy pw,vpv,u : pwp given a third open set W c V C U C X. Ir is straightforward to show that the assignment U È-+ Tors(G)u defines a sheaf of groupoids on X. Tors(G) is locally non-empty since to any open covering {Un}na of X there is an object of each groupoid Tors(G)uu - nameiy the trivial G torsor on U¿. Tors(G) is also locally connected: given an open set U c X and G torsors P, and P, then locally P, and P, arc isomorphic to the trivial G torsor and hence to each other. Therefore Tors(G) is a gerbe on X. Note 6.-f . In several examples that we will encounter we will be given a presheaf of groupoids Ç on X which is locally non-empty and locally connected. One can show that the associated sheaf of groupoids I on X will be locally non-empty and locally connected as well and hence will define a gerbe on X. Definition 6.5. We say that a gerbe Ç on X is neutral if there is a global object of Ç; ie 1f Çx is non-empty. Erample 6.2. The gerbe Tors(G) is neutral since it has a global object namely the trivial G torsor on X. - Definition 6.6. Let Ç be a gerbe on X and let Abe a sheaf of abelian groups on X. We say that Ç is bound by A, or has band equal to A if for any object P of Ç(U), where U c X is an open set, there is an isomorphism of sheaves of groups Aut@) t3 A over U which is compatible with restriction to smalier open sets. If P1 and P2 are two objects of ÇQ) and / : Pr -r Pz is an isomorphism between them then we also require lhat qprAut(/) : 4p,. Here Aut(f) : Aut(P) -, Aut(Pz) is the isomorphism of sheaves of groups induced bV f. V/e will only be interested in gerbes with band equai to Çi. A morphism of gerbes Ç to 11is a morphism @ : Ç --+ 11 of the underlying sheaves of categories which satisfies 60 the extra condition that given an open set [/ C X and an object P of. Çu, then the following diagram commutes: Autçu(P) Ó", AutTr@u@)) 4ou(p) rlP \L¡¡Í\x A natural transformation between morphisms of gerbes is deflned similarly (notice that any natural transformation between morphisms of gerbes must be invertible). There is also the notion of an equi,ualence of gerbes and hence the notion of. equiualenú gerbes. Ðquivalence of gerbes is clearly an equivalence relation. Recall from Definition 6.3 above that the set of arrows between two objects Pr and P2 in a sheaf of categories is a sheaf H om(P1, Pz) . If Pr and P2 are objects of a gerbe then more is true. The sheaf of groups Aut(P2) acts on Hom(P1,P2) on the right and in fact makes Hom(P1,P2) into a Çi torsor. We have the following Proposition from l7l Proposition 6.1 (t7]). LetÇ be a gerbe onX withband equalto A for sorne sheaf of commutatiue groups A on X. Then we haue 1. Let U C X be an open subset and let P1 and P2 be two objects of Çu . Then the sheaf of arrows Hom(Py,P2) i's an A-torsor. 2. Let P be an object of 9u for some open set U C X and let I be an A-torsor on U . Then there is an object Q of Çu such that there is an i,sornorphi,sm H om(P, Q) = I of A-torsors on U. The object Q is uni,que up to isomorphism and is denoted by PxaL 6.3 Properties of gerbes We wiil briefly review some ways of constructing new gerbes from existing gerbes from [7][pages 193-200]. First of ail suppose we have a gerbe Ç on X with band equal to the sheaf of abelian groups A on X. Suppose that there is a morphism of sheaves of abelian groups A + B on X. Then rù/e can construct a new gerbe on X, bound by B and denoted Ç xA B as follows. First we construct a presheaf of groupoids on X by letting (Ç *¿ B)u have objects the objects of. Çu but given two objects Pr and Pz of (Ç xL B)u *. Iet HomçÇ xL B)u(P1, Pz) : Homçu(Pt, Pz) xa B. This presheaf of groupoids is cleariy locally non-empty and locally connected. We define Ç vL B to be the sheaf of groupoids associated to this presheaf. Given two gerbes 9t and gz onX bound bV Ai we have the notion of the contracted product gerbe h Ø Çz on X. This is also a gerbe bound bV Ai It is constructed as follows. We first form the gerbe Çt x Çz on X, ie the sheaf of groupoids [/ -¡' Çr(U) x Çz(U). This is gerbe bound bV Ai x Ci. Ordinary multiplication in C' induces a morphism of sheaves of abelian groups Ci , C} * C! We defrne ÇtØÇz: (h x Çù xÇi."ai C; - see [7][page 200] for a direct description of this gerbe. Suppose Ç ls a gerbe on X. Then we have the notion of the dual gerbe Ç* on X associated to Ç. Ç. is defined as follows. We let Ç" have frbre Ç*(U) at open set U c X 61 equal to the opposite groupoid ç(U)" of. Ç(U). Given open sets V c U C X , ç* inherits restriction functors Ç.(U) - Ç.(V) from the pu,v: ç(U) - ç(V) These restriction functors satisfy the compatibility condition given a third open set W C V C U C X. It is not hard to see that the assignment U ,- Ç.(U) defines a sheaf of categories and it is also easy to see that this stack is locally non-empty and locally connected and hence defines a gerbe. Given asmooth map,f :Y + X between smooth manifolds X and Y and agerbe C on X bound by Çi, we have the notion of the pullback gerbe f -rC on Y - see [7][page 19S]. This is a gerbe bound by the sheaf of groups Í-tç;, on X. f-rC is obtained by sheafifying the presheaf of categories [/ ,- f-rCu, where Í-'Cu is the groupoid whose objects consist of pairs (V,P), where V c X is an open set of X such that f (U) cV and P is an object of.Cv. An arrow (Vt., Pt) - (Vz,pr) is an equivalence class lWn,anl,, where WnC X is an open subset of X such that f (U) CWp aîdW;¿ CVIVz and a,p : P1lwr, - Pzlwu is an arrow in Csr", under the equivalence relation (Wp, a1.¿) - (W'rr,a'rr) lf. there exists an open set Wi.'2 C X with Wi, c WnìWrr, f (Ø c W'rl" and o12l*l,r: a'nlw,rr. If we have a third object (ys, P3) and an arrow lW2s, a2s] from (Vz, Pz) to (yr, P¡), then the composite arrow lWzs,azsllWp,anl , (V, Pt) ---+ (V2, P2) is defined by lWrt, azsllWn, ot ] : lWzsaWp, a2slwzsnwrzd1zlwrrnwrrl. One can check that this respects the equivalence relation - defined above. One can check (see [7][Proposition 5.2.6]) that the presheaf of groupoids f-rC is locally non-empty and locally connected and hence the same will be true of the sheaf of groupoids associated to f-rC. We will usually want f-rC to be a gerbe bound bV Ai. We can arrange this as foiiows (see [7][page 203]). Observe ihat we have a morphism of sheaves of groups f -'A;( - Ci and hence we can form a new gerbe bound bv Ai in the manner described above. We will use the notation of [7] and denote this new gerbe on Y bound by C] bv f.c. We also have the following Lemma. Lemma 6.1 ([6]) . Suppose we haue smooth manifolds X, Y and Z and srnooth maps f :Y + X andg:Y ---+ Z. Supposewez,realso giuenagerbeC onX. Thenthere'is an equiualence of gerbes ót,n, g-'f-LC 3 ffù-'C on Z such that, giuen a fourth smooth mani,foldW and a smooth map h: W --+ Z then there i,s a natural transformati,on between the equiualences of gerbes as pi,ctured in the followi,ng di,agram: ¡-t n-r ¡-tgn-'öt:¡-tU g)-tc t, ^J f,s,h |lr,,,^ (sh)-t (Ígh)-'c ¡-t¿ r.--vt '9h The natural transforrnat'ion ?¡,n,n satisfies the followi,ng coherency condi,ti,on: gi,uen a fi"fth, smooth'manifold,V and a smooth møp k : V -+ W we requi,re the following di,agram 62 of natural transfortnati,ons to commute. ¡r-t¡-t n-t f -rC , lr-Lh-|(l g)-rc (htc)-t n-t ¡-r, k-'(gh)-'f-rc ¡t-rU gh)-'c l,sh,k Ghk)-t ¡-tg (fshk)-tC le-rh-I g-I l-LC tc-Lh-IU g)-rc (hk)-'g-'f-LC , (hk)-r(fs)-rc k-r (f sh)-rC o J n,n,* (shk)-tÍ-rc (fshk)-tc. There i,s also a uers'ion of thi's Lemma with f -r replaced bA Í.. We now turn to discuss the characteristic class associated to gerbes, lhe D'i,rmi'er- Douad,g class. This is a class in the sheaf cohomology group H'(X;C|) and hence we can regard it as a ciass in Ht(X;Z) via the long exact sequence in sheaf cohomology induced by the short exact sequence of sheaves of abelian groups L-Ø.x*Cx+C|+1. We briefly recall how to construct this class and refer to [7] and [S] for more detaiis - sheaves particularly [5] where there is no assumption made about gerbes being bound by of abelian groups. We also assume throughout the remainder of the chapter that all the gerbes we encounter are gerbes over a smooth paracompact manifold X. Let C be such a gerbe and choose an open covering {Un)nq of X such that each non-empty intersection U¿oì. - .n¡J;,n is contractible. Since I is contractible we can choose objects C¿ of C(U¡). Choose arrorù/s f¿j : puo¡,ur(Cr) - pui¡,(r¡(C¡) in C(Uo¡). It is not immediately obvious that such arror'¡/s exist - indeed the axioms for a gerbe only imply that given a choice of the objects C¿ there is an open cover of U¿¡ such that there exist arrowt f , {U$} "ezri ü pu¡,un(Cn) - puy,,u,(C¡) for each c , Euj. This is the viewpoint taken in [5]. To see that iniact the arrowi fi¡ exist in the present case we use Proposition 6.1 to conclude that the sheaf -Isorn (pur,,un(C¿), pur¡,u¡(C¡)) is torsor. Such objects are classifi.ed by the ^ÇÜn, sheaf cohomology group Ht(Un¡,Ci,,-) which is zero, since U¿¡ is assumed contractibie. This means that there is an isomorphism Íu¡ i P(t.;¡,u¡(Cn) - Pun,,u¡(C¡). Now' in p(J¿¡¡,(.r¡¿(fo)Punrr,u,n(f¡r)puo,r,uoi(f¿) Cu,n,we form the automorphism .o! ,Pu,,ryr.(Cn) aná identify it with a map e;¡¡: (I¿¡¡ - C" under the isomorphism Aut(C¿) -+ Ci',. It is easy to see that in fact e¿¡¡defines a Cx valued Õech two cocycle which is independent of all choices involved. The class e¿¡¡ defines in H2(X;qi) it called the Dixmier-Douady class of the gerbe C, and denoted by DD(C). We summarise the main properties of the Dixmier-Douady class in the foliowing proposition. Proposition 6.2 (t7]). The Di.rrnier-Douady class has the followi.ng proper-ti'es: 63 1. The Dirmier-Douady class of the triuial gerbeTors(Qi) 'is zero. 2. The contracted product C ØD of two gerbes C andD on X has DD(C ØD) : DD(c) + DD(D). 3. The dualC* of a gerbeC has DD(C.): -DD(C). 1 ï f :Y ---+ X is asmoothmap andC is agerbe onX then f.C has Di,rmi,er- Douady class DD(f*C): l.DD(C). Equzualence between gerbes defi,nes an equiualence relation on the class of all gerbes on X and the operations of contracted product and duals malce the set of equi,ualence classes of gerbes on X into an abelian group with 'identitg element the triuial gerbe of torsors on X. The Dirmier- Douady class suppl'ies an homomorphi,sm from the group of att equi,ualence classes of gerbes to H2(X;Çj). Th'is homomorphism'is an isomorphism. It is worth noting that given gerbes C andD on X with a morphism Ó: C -+ D between them which is not necessarily an equivalence, then C and D have the same Dixmier-Douady class. Indeed, given an open cover {Un}nq of X and objects C¿ of C(U¿) and arrows /¿¡ as above, then we get objects óur(Co) of D(U¿) and arrows Óun¡U¿¡) of D(Ur¡) which give rise to the same Õech two cocyclê e¡¡¡ ã,sf.or C. 6.4 The gerbe HOM(P,Q) Proposition 6.3 ([19]). Gi,uen two gerbes P and Q on M, both bound ba Çít, we can fomn a new gerbe H O M (P, Q) on M also bound by Çit. The fi,bre of H O M (P, Q) ouer 0,nopensetU of M is HOM(P,A)(U): Hom(P(U),a(U)) -that'isto saythe cate- gory wi,th objects equal to the morphi,sms between the gerbes P(U) and Q(U) and whose o,r'rous are the natural isornorphisms between these morphisms. Furthermore there i,s an equiualence of the gerbes HOM(P, Q) andP. ØQ on M. There is also a 'subgerbe' of HOM(P,Q), namely the gerbe EQ(P,Q) whose fibre at 0,n open setU C M i,s EQ(P,O)(U) : Eq(P(U),Q(U)), i,e the category whose objects are the equiualences of gerbes Pu - Qu and, whose arrows are thc natural transfortnations between these equiualences. Clearly there i,s a morphism of gerbes EQ(P,Q) -- HOM(P,Q) and hence the gerbes EQ(P,Q), HOM(P,Q) and QØP* are all equiualent. Note that this does not require that P and Q are equivalent gerbes. Proof. To show that this does indeed define a gerbe we first of all need to show that the assignment U -- HOM(P,Q)(U) defines a stack in groupoids on M - that is the gluing laws for objects and arrows are satisfied - and that HOM(P, Q) is locally non-empty and localiy connected. Note that by definition HOM(P,, Q)u is a groupoid. Also H O M (P , Q) defines a presheaf of categories by definition of a morphism of gerbes. Explicitly 1f V C U are open subsets of. M then there are natural restriction functors HOM(P,Q), -, HOM(P, O)v obtained by restricting a morphism Óu , Pu + Qu lo V and similarly for natural transformations. We have to show that in facl HOM(P, Q) defines a sheaf of categories. Suppose then that we are given an open set [/ C M together with an open cover {Un}or, of. U by open subsets U, of M such that there exist objects (ie morphisms of 64 gerbes) þ¿ of HOM(P, Q)u, for every i, e I and arror¡/s (ie natural transformations) á¿¡ : ó1u,, + ó¡lun, in HOM(P,Q)uo¡ which satisfy the gluing condition ?¡nluo,u14luu¡r: g¿ulun,^ in HOM(P,Q)uu¡o. We need to construct a morphism of gerbes Ó:Pu + Qu which is isomorphic to /¿ when restricted to U¿. Let V C U and suppose P is an object of. Pv. Consider the objects ó¿lvnuo(Plv"ur) of Qv¡un The natural transformations á¿¡ provide arrows 0n¡lvnvo,(Plv¡vo¡) , ónlvnun(Plvnu)lvnuo¡ - ó¡lvnu¡(Plv"u,)lvnu,¡ which are descent data in the gerbe I because of the condition 0¡nlun,u?;¡lur¡^:înnlun,u. Let the descended object of Qv be Q say. We put Óv(P): Q. In a similar way one can define the action of Óv on arro'vr/s of. Pv and it follows from the uniqueness of gluing in sheaves that this defines a functor. It is also not hard to see that that this functor is compatible with the restriction functors in P and Q and induces the identity morphism on the bands of the gerbes P and Q. Finally, note that since Q is the descended object for the descent data þ¡ly¡uo(Plvau) and 0¿¡lvnun,(Plv¡un¡) there are isomorphisms X¿ : Qlvnu, - ó¿lvnun(Plv"u) compatible with the 0¿¡lvnun,(Plvnv,,). These isomorphisms induce natural transformations, also denoted X¿ from Óuu to Ô¿. We now have to show that arrov¡s glue properly. Suppose v/e are given morphisms of gerbes ó,tþ,Pu -+ Qu,ie objects of. HOM(P,Q)u, and suppose also that we are given an open cover {U¿}ra of. U by open subsets [/, of M such that there exist natural transformations r¿ : Óuo) tÞurfor eachi € / such thatr¿l¡¡0,: rjlun¡. Let P be an object of. Pu and let P¿ : Pluo. The r¿ provide arro\Ms r¿(P¿) , Ôuo(P¿) - tþuo(P¿) in Quo. Since $ and þ are morphisms of gerbes v¡e can think of the r¡(P¿) as arro'ù/s óu@)lr. - ,þu(P)Iu,. We have r¡(P,)lun, : r¿lu,¡(Pluo,) : ,¡1u,,(Plu,) : r¡(P¡)luu¡. Since g is a gerbe and hence a sheaf of groupoids, there is a unique arrorù¡ r¡¡(P) : óu@) - tþu(P) such that tu(P)lu,: r¡(Plu,). The assignment P r-> ry(P) defrnes a natural transformation. Hence HOM(P,8) is a sheaf of groupoids. We wili omit the proof lhaf H O M (P , Q) is locally non-empty and locally connected and refer instead to [19] where it is also shown that HOM(P,Q) has band equal to w¡¡.^X To show that there is an equivalence of gerbes between H O M (P , 8) and P* Ø Q, we calculate the Dixmier-Douady class of HOM(P,8). Choose an open cover U : {Un)nrt of M such that there exist morphisms Ó¿ : P¡ + Q¿ where P¿ and Q¿ denote the restriction of P and Q to U¿ respectiveiy. We can choose the open cover U so lhat there exist natural isomo¡phisms a¿¡: Q.ilun, + Ô¡lø, in HOM(P,Ø¡(U1). Then the Dixmier-Douady class of HOM(P,Q) will be given by choosing an object P e Ob(P(Urjk)) and forming the cocycle an¿lu¡¡x(P)a¡xlur¡o(P)a¿¡l¡¡,,r(P). We want to relate this cocycle to the cocycles representing the Dixmier-Douady classes of. P and a. To do this we first need to choose objects P¿ e OU (P(Uù) for each i € I. Again rvl/e can assume that there exist arrows f¿¡: |luo¡ - P¡lun, inP(U¡¡)' Since o¿¡ is a natural transformation we get the commutative diagram óulu,¡(Polun, .or(¿lror)l ó¡1u,,(Pulu,, This induces the foliowing diagram of arrows ín Q(U¿¡n) (We have omitted some re- 65 striction functors for convenience of notation) r ( P "," ( ôo(Pn)lu:'Wb,e)l ,,! 6 n r) *' þ n k) I u,, r ;(W tlv,! "' " (r¿jt" + Ó¡@)l (r¿j t" + Ó¡ (P*)lu¿¡ t" Ó ¡ (Pn)\u,, * - I ón(P,) ,orr) + ôt (P) u¿j t" + Ón(Pn)lu¿¡ t" ÓúPn)lun, r on¿(P¿)lv¿¡r - I I ô¿(P,)lu¿¡ t" ô n(P¡)lu¿i t" óo(P*)lu ¿¡ n ó n(Pn)\u,, * - - The diagonal arrow is a cocycle- representing the Dixmier-Douady class of the gerbe Q, while the bottom horizontal aüol¡/ is a cocycle representing the class of P. Since the left vertical arrow represents the Dixmier-Douady class of HOM(P,Q),we must have DD(HOM(P,Q)): DD(Q) - DD(P). ! For us the main point of this Proposition is that one can glue a family of locally defined morphisms between two gerbes together to form a global morphism between the two gerbes. This is also the case for bundle gerbes and stable morphisms between bundle gerbes. More precisely, suppose we have bundle gerbes (P, X, M) and (Q,Y, M) together with stable morphisms S; : Plrn --+ Quo where U : {U¡}¿.¡ is an open cover of M. Suppose we also have transformations 0¿¡ : ó¿ + ó¡ between the restrictions of the stable morphisms þa and ô¡ to U¿¡ which satisfy the giuing condition 0¡u0;¡ : 0¿k on U¿¡¡,. Then there is a stable morphism ó , P + Q together with transformations €¿: ólu, + /¿ which are compatible with 9¿¡. To see this suppose the stable morphisms @¿ correspond to trivialisations L6n: L¿ of P. ØQ on X xsuY. Then the transformations 0¿¡ provide isomorphis*" ãn¡ :-L¿ 7 L¡ gver X xrJr¡Y. The gluing condition 0¡*0¿¡: d¿* translates into the condition 0¡*o0;,i : d¿¡. Hence \¡/e can use the standard clutching construction to glue the various principal C" bundles.L¿ together to form a trivialisation -L defined on the whole ol. X x¡aY. Unfortunately there is no general method for gluing together bundle gerbe mor- phisms; ie given bundle gerbes (P,X,M) and (Q,Y,M) together with locally defined bundle gerbe morphisms f¿ : Plvn - Qlun and transformations of bundle gerbe mor- phisms 0¿¡ : Í¿lvn¡ + f ¡fu'¡ satisfying the gluing condition 0¡x04 - 0¿k over [/¿¡¡ then there is no general way to construct a bundle gerbe morphism f , P + Q which restricts Io Í0. 6.5 The gerbe associated to a bundle gerbe We will briefly review a construction given in [34] which associates a gerbe Ç(P) to a bundle gerbe (P,X,M) in such a way that DD(P): DD(Ç(P)). Given an open set [/ C M we define a groupoid Ç(P)u as fol]ows. The objecls of Ç(P)y consist of line bundles L -+ Xu : r-r(U) together with an isomorphism q : õ(L) -- P*wt of line bundles covering the identity on Xjl : (7rt2l)-1(t/) and commuting with the bundle gerbe products. (Here we regard ô(I) as a trivial bundle gerbe). An arrow in 66 from an object (Lt,nt) to an object (Lz,qz) is an isomorphism of line bundles Ç(P)u : a : L1 - Lz which is compatible with rl1 and 42 in the sense that q2õ(d) rÌt. Given another open set 7 with V c U C M then are canonical pullback functors pv,u : : assignment Ç(P)u - Ç@)v satisfying pw,vpv,u Pwp. In [3a] it is shown that the U -, Ç(P)y defines a gerbe. To see that the band of this gerbe is equal to Cfa, note that if (L,ù is an object of. Ç(P)y then, since we may identify an automorphism o of. (L,a) with a function Í , Xu -- C], the condition 4ô(a) : ,l forces ô(/) : t and so / must descend to a function / : U - C'. In this way v/e can construct an isomorphism Auú(L,q) = Ci, which is compatible with restriction to smaller open sets and with isomorphisms Aut(L1,qù - Aut(L2,42) induced by an isomorphism (Lt,q) - (Lz,qz).The advantage of this construction is that one does not need to apply a 'sheafification' procedure to force objects and arrows to satisfy the effective descent conditions. To see that the objects o1Ç(P)y satisfy effective descent note that (L¡,T)lun¡ tf {U¿}¿et is an open cover of.U, (L¿,rl) are objects of.Ç(P)u, and o¿¡ : - (L¡,n)lun, ar" arroYm¡s of Ç(P)u, satisfying a¡na1 : di¡ then we can glue the line bundies .L¿ together with lhe a¿¡ using the standard clutching construction to form a Iine bundle .L on Xu. The 4¿ glue together to form an isomorphism 4 : õQ) -- P*IJI which trivialises Pxl]t. Similarly one can show that the arrows of. Ç(P)u satisfy the effective descent condition. To see that 9(P) had Dixmier-Douady class equal to that of the bundle gerbe P, choose an open cover {U¡}ø of M, all of whose non-empty intersections U¿o À' ' 'ìU¡o are contractible, such that there exist local sections s¿ : U¡ -, X of zr. We can defrne objects (L¡,rt¿) of Ç(P)u, by setting L¡ : êotP*g), where 36 : Xu. -- XI]) is the map which sends r € Xu. to (r,s¿(zr(ø))) e X!j). r¡¿ : 6(L¿) -- Px[]) is induced by the bundle gerbe product on P. We can define arro\Ms a¿¡ : (La,q¿)lun, - (L¡,rl¡)luo, by first choosing sections o;¡ : IJ¿¡ * (so, s¡)-lP and then defining a¡¡(u¿) : mp(o(trLr(ui))gu¿) '. proj we now calcuiate the Cech f.or u¿ € tr¿ , wher e, 7r ¡, L¿ - Xy, denotes the ection. If cocycle representative for the Dixmier-Douady class of Ç(P) from this data, then we get it exactly equal to the cocycle for the bundle gerbe P. We summarise this discussion in the following proposition. Proposition 6.4 (t34]) . Gi,aen a bundle gerbe (P,X, M) the assignmentU -> Ç(P)u defi,ned aboue is a gerbe with Di,rmi,er-Douady class equal to the Di,rmier-Douadg class of the bundle gerbe P. Note 6.2. One can show that a bundle gerbe morphism Í, P + Q induces a mor- phism of gerbes ÇU), ç(P) - Ç(Q) and that a transformatíon 0 : f + 9 induces a transformation Ç(0) , ÇU) + Ç(g) between the induced morphisms of gerbes. Thus rù¡e can think of. Ç as furnishing us with a 2-functor from the 2-category of bundle gerbes and bundle gerbe morphisms to the 2-category of gerbes. Given two bundle gerbes P1 and P2 on M one can show that there is an equivalence of gerbes 9(P, ø Pz) = Ç(Pr) ø Ç(Pz) and one can show as well that given a bundle gerbe P on : lhere is an equivalence of gerbes M and,a smooth map "f -fü -> M ç(Í-LP) = f.ge). Unfortunately I have been unable to show that these equivalences behave as one would Iike; namely that the equivalence Çff-tP) = f.çe) commutes with the equivalences óÍ,n of Lemma 6.1 and the equivalence Ç(PtS Pr) = 9(Pù øÇ(Pz) commutes with the equivalence Çt Ø (Çz Ø Çs) = (h Ø Çr) Ø Çs of gerbes Çt, Çz and Çs on M - 67 We now wish to show how a bundle gerbe (P,X,M) with bundie gerbe connection V and curving / gives rise to a connective structure and curving on the associated gerbe ç(P). We first need to recall the definition of connective structure and curving from [7] and [8]. Definition 6.7 (171, [8]). Let Ç be a gerbe on a smooth manifold M. A. connect'iue stracture on Ç s given by the following data. 1. For every open set IJ C M, an assignment to each object P of 9(Ø of an Ola torsor Co(P) on U which is compatible with the pullback functors pv,u. Sections of Co(P) are usually denoted by V. 2. Given objects Pr and Pz of. Çu and an isomorphism / : P1 + P2, there is an isomorphism .f* : Co(P) --, Co(P2) of f¿la torsors on (J, which is compatible with composition of morphisms in Ç(U) and with the pullback functors p4u. 3. Given an object p of Ç(U) and an automorphism / : P - P then we require that /. : Co(P) -, Co(P) is the automorphism defined by V r- V +df lf , where we regard / as an element of Cf . A curuing for a connective structure on a gerbe Ç on M consists of an assignment to each open set U c M, each object P of Ç(U) and each section V of Co(P) of a two form /(P, V) on U satisfying the foliowing conditions: 1. For objects Pr and Pzof.Ç(U) and isomorphisms þ: Pt- P2ínÇ(U)wehave /(Pt, V¡ : f (Pz,ó.(V)) for ail sections Y of. Co(P). 2. If A is a one form on [/ then for any object p of Ç(U) and any section V of Co(P) we have l(P,V + A) : f (P,V) + dA. 3. Given open sets V cU C M, an object p of Ç(U) and a section V of Co(P), the assignment (P,V) - f e,V) is required to be compatible with restrictions to smaller open sets. We now show how a bundle gerbe connection V with a curving / on a bundle gerbe (P,X,M) gives rise to a connective structure and a curving on the associated gerbe Ç(P) on M. LetU c M be an open subset of M. Let (L,ri) be an object of Ç(P)y. LetVt be any connection on L. V¿ induces a connection ô(V¿) on ô(I). We also have the connection T-L o V o n on ô(I) induced by V. These two connections must differ by the pullback of a one form A o Xljl. Since V is a bundle gerbe connection, we must have ô(A) : 0. Hence r¡/e can solve A : 6(B) for some one form B on Xu. Therefore if we had originally defined V¿ to be equal to V¿ - B then we would have ô(Vz) : n-r o V o 4. Therefore r¡/e define Co(L,4) to be the Ol7 torsor consisting of the connections V¿ on -L satisfying ô(V¿) : ,t-'o V o 4. It is clear from the previous discussion that this does indeed define an Of, torsor. It is also ciear that the conditions (1) and (2) above are satisfied. To see that condition (3) is satisfied, let f : U - C* be an automorphism of (L,q) under the isomorphism Aut(L,ù = Air. Then if V¿ is a section of Co(L,4) then we have : V * as required. To define /-(V) ".(d,f lÍ), a curving for this connective structure, simply Í(L,rl,V¿) be the two form Fy, - / where Fv" is the curvature of the connection V¿ on L. Since 6(Fr" - /) : 0 we may identily Í(L,n,Vr) with a two form onU. This clearly defines a curving for the connective structure on Ç(P). 68 6.6 Higher gluing laws In this section we discuss how to glue together a family of gerbes Ø defined on an open cover {Uu}n.t of. M rc get a gerbe defined on the whole of M. This has an extra level of gluing : need not satisfy : complication in that the morphisms Ö¿¡ Ç¡ - Ç¿ Ó¡*Ó¿¡ Ó¿t" but rather there is a natural transformation 4s4x , Ó¡nÓ¿¡ + Ó¿u satisfying a certain cocycle condition over U¡¡¡¿. The triple (Ç¿,ó¿¡,tÞ4u) is called Z-descent døú¿. We will need the results of this section in Chapter 13. Proposition 6.5 (t5]). Suppose we are g'iuen an open couer {Un}nq of X and gerbes Ç¡ on IJ¿ for each i, € I as well as morphr,sms of gerbes gt¡ : Çn - Çi between the restri,cti.ons of the gerbes Ç¿ and, Ç¡ to U¿¡, whi,ch sati,sfg g¿¡ : i,d,gn. Suppose we are also giuen natural transformati,ons tþ¿¡¡ , gjxlun,ng¡jlU¿¡t" è g¿xlvniu ouer U¿¡¡ which sati,sfg the foltowi,ng condition (the non-abelian 2-cocgcle condi,tion): the di,agram of natural transforrnati,ons below commutes : 9it 9¡ 9;t lc Çt What thi,s means is that after pasting together the natural transformati,ons in each di,agram, the two natural transforrnations between the morphisms boundi,ng the øboue di,agrams are equal; so 'in other words we haue tÞ*t(Lr*, o tþ¿¡ *) : tþ¿¡ítþ ¡nL o Lrn,). Then there eri,sts a gerbe Ç on X and an equ'iualence of gerbes X¡ : Çluo ---+ Ç¡ ouer IJ¿ together with no,tural transforrnations {¿¡ : g¿¡X¿lu¿¡ } X¡lvo¡ whi,ch are compati,ble with the natural transfonnati,ons tþ¿¡¡ in the sense that the followi,ng d'iøgram of natural tran sf o rm ati o ns co mmut e s : Xi Çluo,r +9¡ Çlu,,r xk ,/ Xk 9ij 9ij Çi Çn 9jk Ç¡ Çn' ,* In other words we haue €¿u(tþo¡r o 1"n) : Ç¡n(Le¡r o €¿¡). We have an analogue of the above proposition for bundle gerbes and stable mor- phisms. We have to exercise a bit more care here though because composition of stable morphisms is not strictly associative but rather associative up to a coherent transformation (see Proposition 3.9). 69 Proposition 6.6. Let {U¿);ç¡ be an open couer of M by open subsets U¿ of M . Suppose for each i € I we haue a bundle gerbe (Q1,Y,U¡) ouer U¿ such that for each pai,r i, j e I with the intersectionU¡¡ non-empty there'is a stable morphism ft¡: qo -- Q¡ correspondi,ng to a triu'ialisation (L¡0,,,ó¡.,,) of 8i ø Q¡. We requ'ire that the stable morphisms Ír¡ sati,sfy f ¡n: Ín¡ i,n the notation of Secti,on 3./¡ so that i¡¡ correspond,s to the tri,uialisation of Qi øQ¡ giuen bA (Lï,,, ói,). Suppose also there 'is a transfortnat'ion 0¡¡¿ : f ¡t o f¡¡ è f¿t" of stable morphisms ouer U¡¡¡ which satisfi"es the cond'ition 0¡¡¿(0¡m o 7 Ír¡)0 ln¡,f¡t",ft"¿ : ?¿m(l ¡u, o 0¡¡¡) ouer U¿¡¡,¿. Then there i,s a bundle gerbe (Q,Y, M) on M together wi,th stable morph'isms g¿ : Qlu, - Q¿ ouer U¿ such that there erist transform,at'ions of stable morphisms €t¡ : ¡n, o g¿ + !¡ oaer U¡¡ which satisfy the compati'bili'ty condi,tion €¡u(7¡ : €tn(04¡oLn) ^o€¿¡) with 0¿¡¡, ouer U¿¡¡,. Proof. Recail that the stable morphism f ¡x o f;¡ corresponds to the trivialisation (L ùroÍo¡ ' Ó¡5uo¡.r) of Qi 8 8¡, which is obtained by descending the CX bundle L¡,n Ø Lto, Ø Qj on Y x u Yt"xuY]'f roYxpt Yr". In particular, this implies that the following isomorphism is true on YxuY¡,x¡aY]2): LÍ¡uorn¡ - Lr,u Ø Lto¡ Ø A; (6 1) The two trivialisations -L¿* and L¡,no¡r, of Qi A Qr differ by the pullback to Y¡, x ¡a Y¡ of a C' bundle J¡¡¡ on U¿¡¡". So we have Ll¡ooÍr¡ = Llor Ø ryurr*ynJü*. (6 2) 0¿¡¡, is then a section of the bundle J¿¡¡a oî U¿¡¡. fuom equations 6.1 and 6.2 we have L(Írol¡t")o1n¡ = LÍu,of¡u Ø L¡r, Ø Qi æ L¡,, Ø Ltr, Ø QiØ r-rJ¡nr - LÍle1n¡ S 8; S Q¡ Ø r-L J¡ut æ L¡, Ø r-1(J¡*t Ø J¡¡ù, where we have denoted the various projections lo M by ur. Using equations 6.1 and 6.2 again we get Lft"p¡¡xof¡¡) = Lio,Ø L¡r^"¡rrØQi 70 where again we have denoted the projections to M by zr'. The transformalion 0¡,,,¡,n,¡r, provides an isomorphism -L¡u,o ç¡xoir) = L(Ínpf¡òo¿, which is compatible with the triv- ialisations óJ¡¡o(J¡¡"o¡¿¡; and ó6n,o¡,u¡o¡n,. It follows from the series of equations above and the property of 0lo¡,Í¡u,Ír, just mentioned that there is an isomorphism J¡mØJ¿it=JuaØJ¿¡u over U¿¡¡"¡. The cocycle condition on 0¿¡¡, is that the section 9im Ø 0¿¡¿ of. J¡u Ø J¡¡¡ is mapped to the section 0¿n Ø 0¡¡¡ of. J¿m Ø J¿¡¡ under the this isomorphism. We now define the bundle gerbe (Q,Y,M) as follows. We put Y equal to the disjoint union IJ.nrrY with the obvious projection Y -+ M. The fibre product Y[2] is equal to theãiÀjoint union If.n,j.rY xu,, Y¡. Define a C' bundle Q on Yl2l by putting Qlurrv¡ : Ltn,. \Me now need to define a bundie gerbe product on Q. This will be a C" bundle isomorphism L¡,Ø Llrr - Llnn covering the identity onVxuY¡ xuY* which satisfies the associativity condition. Note that from equations 6.1 and 6.2 we have the isomorphism L¡,u Ø Lin, = Llnu Ø Q¡ Ø r-L Jnr¡ of C" bundles onY¡x¡aYrl'l ,*Yk. Ilwe embed YxuYi xuY¿ inside VxuYj'l r, Yrby sending (An,yj,An) to (An,A¡,U¡,Ux), then using the section 0¿¡¡ of.4ir and the identity section of Q¡ over the diagonaì., we can construct an isomorphism Ll,uØLln¡ - L¡nr. The coherency condition on 0¿¡¡ shows that this product is associative and hence (Q,Y, M) is a bundle gerbe. We now need to show that there is a stable morphism g¿: Qluu - Q;, ie a trivialisa- tion (.Lro , ón) of Q. ØQ¿ over U¿. The C' bundle .Lr, Iives over Y , *Y: L[r., Y¡ x mY' Let Ln, be the disjoint union of the C* bundles LÍ,r. Since f¡¿: l¿¡ the isomorphism Lfn,ØQ¿= LhoØLln¡ becomes L],oØQ¿= LlonØL.¡,, which shows lhat Lnn trivialises Q. Ø Q¡. Finally we have to define transformations $¡ : f¿j o g¿ è gj, compatible with 0¡¡¡ in the appropriate sense. \Me have two trivialisations Ltn,onn and Ln, of Q* I Q¡ and hence there is a C* bundle D¿¡ on U;¡ such that, L¡n,osn : Lgj Ør-t D¿j where n' denotes the projection Y xtrY¡ '--, Ui.Therefore we have Lfn,on,ØQ¿ = Ln,ØQtØr-rDrr,, an isomorphism of C" bundles on l[*., Yt x u Y¡ x t,t Yottl. Ot each constituent Yt" x ¡,t Y¡ xuYt'l ofthe above d.isjoint union this isomorphism takes the form Lfn, Ø Llo., = Lh, Ø Q,i Ø n-L Dnr. Fbom equations 6.1 and 6.2 we have LÍro Ø Lh, ñ L¡uno¡n, Ø Qt N Ltu,Øn-rJuûØQ¿. This shows that there is an isomorphism r-rD¿¡ - r-rJt"¿j onY¡ xvY¡ xMYl2l. Therefore n-Lït;.i induces a section €n¡ of n-t Dü which descends to a section {,¡ of pfi. Because of the coherency condition satisfied by 0*o¡, it follows that {¿¡ will be compatible wílh 0¿¡¡. n We have the following generalisation of the above proposition which we will need in Chapter 13. 7I Proposition 6.7. Suppose¡r : X -+ M 'is a surjection admitti,ng local sections andthat (P,Y, X) is a bund,le gerbe on X such that there is a stable morphi,sm f : n¡r P -, r;1P correspondi,ng to a tri,u'ialisation (L¡,ó¡) oÍ nl'P* Øz.;1P. Suppose also that there is a transforrnati,on þ t rlr Í o 7T;1Í + n;'Í as p'ictured i'n the foltowing d'iagram: --r ¡ r;rr;r P "' ', nl'n;r P : r;rn¡r P : r;Lr¡L P ffi rlrr;r P n;rr;l P. þ is required to sati,sfy the coherency condition that the following diagram of transfor- mat'ions commutes: nl'nll f o r;rr;r f o t;Ln;L f 1 v;Lvlt Í " n¡'nr' f o r;rn¡rþ 1,r-r";trotitú.lf [n;',tot*-,,-r, rrllnst f o rrrn;'r n¿'n;'f o r;rr¡t f ll nlltrtr f o rlr ir r rl'¡rt'f o n;rntt f "r-trll n*r'r c /13--l*-l tt2 J t 'n;t f- We will wri,te this cond,i,tion f,guratiuelE as n¡rþ Ø r;r1þ* Ø nl'rþ Ø n;rtþ* : 1. Then there etists a bundle gerbe (Q,Z,M) on M and a stable morphismn : P ---, ,r-rQ together wi,th transforrnattons € , nltrt o ó + rltr¡ whi,ch are compatible with the transformations þ zn the approprtate sense. The conuerse is also true. Proof. Choose an open cover {Un},r, of M such that there exist local sectioûs s¿ : IJ¿ - X of ¡r : X + M. Form the pullback bundle gerbes P¿: s¿rP. The stable morphism f : r;rP ---+ r;LP induces a stable morphism f4 : P¡ '--+ fl by pullback: f4:(s¡,t¡)-Lf,andthetransformationT/inducesatransformationtþ¿¡*:f¡¡of¡n+ fu* by pullback as well: ,þ¿jx : (s¿, s¡, su)-Lrþ. The transformation ú¿¡¡ satisfies the condition tþ¿¡ÁlO . tþ¡xù|¡ tþ¿ta(tþ¿¡n o 1d*,). " ,¡ ¡,Ít t: Therefore we can apply Proposition 6.6 to find a bundle gerbe (Q,Z,M) on M and a stable morphism g¿ : Qlu, + fl and transformations €ø : 6ut o gj è 9¿ which are compatible with the transformations tþ¿¡¡ (the fact that the stable morphisms ,f¿r 8o from P¡ to 4 rather than from P¿ to P¡ is of no consequence). We need to construct a stable morphism q , ¡r-LQ - P. Let ,9¿ : X¿ + X[2] denote the map x ¡+ (r,s¿(zr(z))). / induces a map (s¿on)-L P - Plxrby .ît1f. Therefore we can define a stable morphism qi : (raLQ)I", * Plx, by composition: uÜ (n-'e)lxn : ,r-1(elu) "30 n-t P P1*,. "ot 72 Next we define transformations á¿¡ : rl¿ è q¡ by composition as indicated in the follow- ing diagram: 7t 1 si- P It' -1 s:1 f I <,¿jc- _1 r-L(Qlu,¡) T- Plx,i ^- 1 , þ "¿j 7f -1 'Q; ê;'f 7--rs;r p --+ r+ ie 0¿¡ : GåjL1þ o 1,-,er)(1st,J o n-'€oj\. Here .î¿¡ : X¿¡ Xt3l is the map z (r, s¿þr(r)), s¡(n'(ø))). Because of the compatibility of €¿¡ with rÞ;¡nwehave 0¡n0,i¡ : 0¡k. Hence we can glue the stable morphismsr¡; : (tr-LQ)lx, * Pl¡" together to form a stable / morpnrsm n:T -1 'Q -+ P. We nowneedto definethetransformations X: I ontLn+ rrLr¡ and prove they are compatible withT/. It is sufficient to define transformations X¿: Í orlrr¿ è nltrt, which are compatible with the gluing transformations 0¿¡. X¿ is ind.uced fromþ by putlback with the X[4 -- ¡tsJ which sends (rr,rz) e X:21 ^ap to (r1, 12,s¿(m)) whe¡e n(rù: r(r2): rn. We try to indicate this in the following diagram: nlrn-'g¿ r.¡ 1.30- 1d nr'n-t (Qlu,) r¡Lr-r s;r P NIL P Xí f -t nl'rr-t(Qlu,) ;Lr-r s;L P 2 P One can check that the l¿ àÌe compatible with the 0¡¡ and also that the glued together transformation X is compatible with'r/. ¡ We conjecture that there is aiso a version of this proposition for gerbes: Conjecture 6.1. Let r : X ---+ M be a surjection adm'itt'ing local sections. Suppose therài,s a gerbeÇ onX togetherwith amorphi,sm of gerbes þ:riÇ +riÇ ouerXl2l whi,ch malces the following di,agram commute: L.riÇ a"Ó, A*rlÇ =l ç ç, where the two aerti,cal equiualences o,re i.nduced by the equ'iualence of Lemmø 6.1 and A, : X - ylz) i,s the inclus'ion of the di,agonal. Suppose ølso there 'is a natural transfor- -tf() n'¿o,t'ion þ between morphisms of gerbes ouer Xl3l as pi,ctured in the fottowing diagrarn * *n niÓ * *n / \-lr*a / \*a 7t17t19 ------+ 7r17r29 + = l7t2o T1) ')-9: \tr1 o 13) ! (rr o rr)*ç thlti9 I *;o 1þ v (Tr o r2)"9 7t37ttg = o ririg -ñ rir|Ç ' (tr2 o r2)*Ç: (tr2 rs)*Ç. We also require the natural transfomnatton þ to satisfy 0, coherenc7 conditinn ouer Xl4l . Thi,s is bas'ically the non-abelian 2-cocycle condi,t'ion refemed to in the last propositi,on but with the ertra complicati,on of hauing to insert equiaa,lences of gerbes between dou- ble pullbaclcs and natural transforrnat'ions between diagrarns of such equiaalences as i,n Lemma 6.1. Obserue that we rnl,A pullback the natural transforrnat'ions tþ to forn new natural transforrnati,ons rfþ, r$tþ, and so on between the appropriate morphisms of gerbes. We can then fonn the di,agrams rin!ö * * *n ririr$Ç = . rinitriÇ 7t 17t g7t2g ,þ rþr$tr$Ç +rþrþriÇ nir)ó niriö =l ¡rir2tt I ç:r$ririÇ 11à îj 1þ : I ó riniô ,l 7t iît1'Ï i ç 3/l tTiç -l_1 rþrþriÇ riniö rþrþriÇ -*rirþrgÇ and rþrië * * *n rgnþr$Ç 3 ri¡ri¡rig 7t alt i7t29 t_ lt ,t- nþrió niri¡riÇ -z- nþrþriÇ rþrIriÇ 7fÄ"âó 1t à1tà 1tÀ1p TiTl1ti9* * *ô nirþiÇ = , rir$r$Ç ririö =1 I ll * * *n 7t27tò1t rþrlr$Ç iv nitrió riririÇ -- Paste together the natural transfomnations in the first of the aboue two d'iagrarns to yield a si,ngle nl,tural transformat'ion between the morphisms of gerbes bounding the 74 di,agram. Fi,nallg we co,n state the coherency condtt'ion on the natural transfortnati,on d: the followi,ng di,agra'm of natural transformations, after bei,ng pasted together, i,s the same natural transformat'ion as the pasted-together natural transforrnation in the second di,agram aboue: nþriô rirþriÇ 7f4 TíTt ç rþringÇ =l rþr[rfÇ firiö ri ^ ltl öç -:--> rþr[riÇ A - I r$rþriÇ * riç < rþrþriÇ -- rillir 1 ç _ I ^ I I 'l rirþriÇ --rgtririç - =--- rir[ngÇ rririó * rlrþrgÇ 4 tts 2 ç rirlriÇ Íirtq ^ ^ Then there etists a gerbe D on M together wi,th an equ'iualence of gerbes y : r*D '-+ Ç and natural transformations ( between the morphisms of gerbes as pictured i'n the diagram: following ó irig =rîÇ ,t "î" "r*l I ritr*D + rir*D =1 (tr o r1)*D: (n o r2)*D. We then require that the natural transformati,ons riÇ, ri( and r!( are compati,ble with þ i,n the sense that using the natural transformati,ons ri€, ri€ and rl( and the natural transformations 0¡,n,n of Lemma 6.1 we can construct a natural transformat'ion between the morphi,sms of gerbes bounding di,agram (1) aboue. The compati,bi,Iitg conditi'on is that this natural transformati,on and þ coinc'ide. The conaerse 'is also true. Note 6.3. 1. One should be able to prove this conjecture in a similar manner to the proof of Proposition 6.7 but with the additional complication of having to insert the equivalences Ó¡,s and natural transformations d¡,e,r" of Lemma 6.1' 2. If the equivalences ót,s : g*Í*g -- Ug).Ç of Lemma 6.1 were the identity, we could write the coherency condition on the natural transformation ty' more suc- cinctly as : o Ln,nió). r7þ(I*â*äö " rirþ) ritþ (tritþ Unfortunately we have to cope with having the equivalences þ¡,s and natural transformations 0¡,s,n and so, in order to introduce a nevv notation to avoid the diagrams above, we will abbreviate the coherency condition on tÞ to 6(tp¡ : 1. We will encounter such diagrams again in Chapter 13. 75 76 Chapter 7 Definition of a bundle 2-gerbe 7.L Simplicial bundle gerbes and bundle 2-gerbes Recall in Chapter 3, Section3.2, that we saw how bundie gerbes (P,X,M) could be regarded as simplicial line bundles on the simplicial manifold Xþ|. We will use this observation to motivate our definition of a bundle two gerbe. Our first step is to see if there is a workable notion of. a s'implicial bundle gerbe. Note that Brylinski and Mclaughlin define a notion of. si,mpl'icial gerbe (see [8], [9] and also Definition 13.1) so this wiil also serve to guide us. We will work on the simplicial manifold X : {Xp} with Xo : yln+tl associated to a surjective map r : X --+ M admitting local sections. Recall, see Definition 3.2 and also [8], that a simplicial line bundle on a simplicial manifold X : {Xo} consists of a line bundle L on Xy together with a non-vanishing section s of ô(I) over X2 which satisfies ô(s) : 1 on X3. We attempt to generalise this to the case of bundle gerbes as follows. Let (Q,y, ylzl) be a bundle gerbe on X[2]. We suppose there is a bundle gerbe morphism m : r;1Q Ø nl'Q -- r;'Q over X[3]. We think of rn as defining a 'product' on the bundie gerbe Q. Next observe that we have nl' (lírQ rrtr;L : r;trlLQ Ø r;t (tr¡LQ ø r;Lq¡, I "ttQ) Ø Q by virtue of the simplicial identities. The crucial observation is that rh" gives rise to two bundle gerbe morphisms over Xlal as indicated in the following diagram. (j\tQ n;tQ) rlt (nL'Q Ø "lQ) Ø rrLr;rq nlrn-tQ I úL I rl - 'it,aørJ ]tø"n-lr, nl'n;'Q Ørrtr;rQ nl'nr'Q Ørnrr;IQ llil il ll nl'(nt'Q Ø r*rQ) r;L(rtrQ I ît18) 'r'-l !"r'- nl'n;tQ nltn;'Q. 77 'We define bundle gerbe morphisms rn1 and rnz as follows. m1 : nlln'Lo(zr¡1ø41) m2 : T;rñL o (14 T LriL). At this point it is convenient to introduce a different notation which avoid cumbersome diagrams such as the one above. We will denote nl'Y by Yzs, T;rY by Yrs, r;rY by Yz, rtlrl'Y by Y2a aîd so on. Similarly we will denote the CX bundle rtrQ onY,!l by Qzt and so forth. More generally we shall denote nr'Y xy¡slrrrY - Yzzxy¡slYpby Ytzs, ntLQØnltQ by Qrzs and similarly for other combinations. Hence m , Qns - Qn and ml,rlrz:. Qns¿ - Qt¿. Therefore the diagram above becomes in the new notation Qrz -*, I { QzaØ ^l+ Q, Rather than demand the two morphisms rn1 and m2 are equal, we settle for the weaker condition that there is a transformation a : r7t4 è rnz of bundle gerbe morphisms. Recall, see Definition 3.8, that this means that ¿ is a section of the descended C' bundle D^r,^, on X[a] of Lemma 3.3. The transformation a gives, in some sense, a measure of the 'non-associativity' of the 'product' rn. Accordingly, we wili call a the 'associator' transformation or 'associator' section and we will call the (trivial) C* bundle A: D^r,^, on Xltl the 'associator' bundle. The final condition is a coherency condition on ¿. To write it down first observe that Q induces a bundle gerbe (Qrrrnu,Yzzas,¡[si) over Xt5]. The bundle gerbe morphism rn induces five bundle gerbe morphisms M¿ : Qntnu - Qs given as follows. Mt:no(na1) o(nø181) Mz:no(na1) o(1 8?it81) Mz: rho (L I r7r,) o (1 I ?7¿ I 1) M¿ : rh o (78 r7r) o (t ø t ø ra) Ms:no(na1) o(tøtøø) Notice lhat Ms can also be written as Ms : ñ, o (I8 rn) o (rn I 1 I 1). It is not too hard to see that we have the following isomorphisms of C" bundles on ¡tsJ. We have DMr,ñr, - rtrA, D,rrr,,û, - ¡r1rA, Dñrr,,rrn - rlrA, Drurn,rlo, - T;rA* and D¡au,¡a, - rq|A*. Since Dîor,frr, is canonically trivialised, and the bundle gerbe product on Q gives an isomorphism D vr,t t, Ø D vr,¡ut, Ø D Mr,tutn Ø D vn,¡it, Ø D Mu,¡it, = D Il4r,t rr, we have that ð(A) otr ¡[sJ is canonically trivialised. The final condition that we require is that the section d(o) of ô(A) matches this canonical trivialisation. 78 Notice that the thing that makes all this possible is the fact that the r; are the face operators for the simplicial manifold Xp: XIP+L1. Therefore all that we have said above applies equaily well to an arbitrary simplicial manifold X : {Xr} Thus we make the following definition. Definition 7.L. A. si,mpli,cial bundle gerbe on a simplicial manifold X : {Xo} consists of the following data. 1. A bundle gerbe (Q,Y,X1) on X1. 2. A bundle gerbe morphism n: d;rQ Ø d;'Q - dr'Q over X2. 3. A transformation a'. rU è ñLz between the two induced bundle gerbe morphisms ñt4 a,nd rn2 ovet Xs. mt and m2 are defined in the following diagram. d'o' @6t Q Ø d';' Q) Ø d;r d;r Q : ¿;t ¿ttq ø dlt @lt Q Ø d;' Q) a;l'aøtj I tØd;r^_ ' * d,;td,r'Q Ø d;rd;rq dr'd,;'Q Ø dltdrrQ a d,;'(d,ltQ Ø d;tQ) dr'(dotQ Ø d;'Q) d;'*ltl |or'- d,;'d,rt Q d,rt d,r'Q. So rn1 : d;rñ o (d;LnE¡ 1) and 'ñtr2 : dr'n o (1 El d,l'r1r). Thus ø is a section of the CX bundle A: D^r,^, ovet Xs. 4. The transformation ø satisfies the coherency condition d;t a Ø drto* Ø d2ra Ø d3ra* Ø dara : r, where 1 is the canonical section of the C' bundle 6(A) over Xa. This definition should be compared with the definition of simplicial gerbe given in [8] and [9]. With this definition in hand, lve now turn to the problem of defining a bundle 2-gerbe. We make two definitions, the first of which includes the following one as a special case. Definition 7.2. A, bundle 2-gerbe is a quadruple of manifolds (Q,Y,X, M) where zr : X + M is a smooth surjection admitting local sections, and (Q, Y, Xt2l) is a simplicial bundie gerbe on the simplicial manifold X : {X} associated t'o r '. X + M wilh Xp: XIP+LI. Definition 7.3. A strict bundle 2-gerbe is a quadruple of manifolds (Q,Y, X, M) where r : X - M is a smooth surjection admitting local sections and where (Q,Y,¡tzl) is a bundle gerbe on X[2] such that there is a bundle gerbe morphism n : rrLQ ØrrrQ -+ TrrQ such that the two induced bundle gerbe morphisms r7l1 and i7L2 over Xlal are equal. 79 The following diagram shows what a bundle 2-gerbe looks like a J ylz) ---+Y J ytzt X = J M For future reference, we will record here some properties of the associator transforma- tion a and its lift â. So â is a section of the C' bundle A : (mt,mz)-LQ onYpsa. Since â descends to the section a of A, it must commute with the descent isomorphism of ,4.. property: If. usa 'u23 und This implies that â enjoys the following € Q tu"n,oLù, e Q @rr,oLr) up € Q(srr,y,r") where Uzq,ULs, e Yç"s,,a), Azs,Aizz €Y62,*t) and gp,y'r, €.Y6r,rz), then mq(ñ'2(us¿, Ø uzs Ø un) Ø ã(asn, Ezs, an)) : Ø ña(usa Ø uzs ø un))' ^q(ã(y'r¿,g'2s,g'r2) The coherency condition on a is a consequence of the following property of â: suppose E45 e Ypa,ts), Usa €. Yps,"a), Uzs € Y@r,,r) and ye e Y@r,,2)' Then we have the following equation (we have denoted the bundle gerbe product mqby. - since the bundle gerbe product is associative there is no danger of confusion) ã(*(anu,as¿),uzs,up)-r ' ã(anu,usa,m(uzz,a,,z))-r 'ñ("(unò Ø ã(az¿,azs,gp)) ' ã (a nt, m (a za, a zs) ., a n)' ñ(ã(u nt, u s¿,, u zs) Ø e(a tù) : e(m(m(rn(A qs, A sq), Azs), A n)) 7.2 Stable bundle 2-gerbes We also define a notion of bundle 2-gerbe with bundle gerbe morphisms replaced by stable morphisms. It will turn out that this definition of a 'stable bundle 2-gerbe' will turn out to be the most versatile, for example in Chapter 13 we will show that a stable bundle 2-gerbe on M gives rise to a 2-gerbe on M. Definition 7.4. A stable bundle Z-gerbe consists of a quadruple of smooth manifolds (Q,Y,X,M) where r : X + M is a surjection admitting local sections and where (Q,Y,¡¡tzl) is a bunclle gerbe on XÍ21. We suppose there is a stable morphism rn : ¡rtrQ øzrs'Q T;L Q corresponding to a trivialisat ion 6(L, ) of r;tQØrlIQ. øn;Lq. 'We - also suppose there is a transformation 0" : nlo (rn 8 I) + mo (1 A rn) as pictured in the following diagram: Qrrrn Æ Qnn 18rr¿ r/r Ìn Qßa, ^- Qv, 80 ø is required to satisfy the coherency condition that the following two diagrams of transformations are equal: Qns¿s Qtzzu Q:n¿s Qnns Qns Qus Qru Q.¿znu Q.r,ss Q:nEs "yQßs 47235 Qrzu Qus Qs Note 7.1. 1. Notice that by (1) of Note 3.2, every bundle 2-gerbe on M gives rise to a stable bundle 2-gerbe on M. 2. As in Section 7.1 let n'Lr: Qnsa - Qube the stable morphism mo(mØl) and let mz: Qrzs¿ - Qu be the stable morphism mo (LØm). Then the transformation a i Trrl è rnz is a section of the C' bundle A on Xlal such that L*, ? L^rØrtluA onYtzs¿. As before, the C" bundle ô(A) on ¡[al ir canonically trivialised and the coherency condition on a is that the induced section ð(o) of ô(,4) matches this canonical trivialisation. 81 82 Chapter 8 Relationship of bundle 2-gerbes with bicategories 8.1 Bicategories \Me frrst recall the definition of a bicategory (see for instanc" [t]) Definition 8.1 (11]). A bicategory ß consists of the following data: 1. A set Ob(B) whose elements are called the objeús of. B. 2. Given two objects A and B of B there is a category Hom(A,B). The objects of Hom(Á, B) are called l-arrows of B whiie the arrows of Hom(A, B) are called 2-arcows of. B. If o and B are l-arrows of Hom(Á, B) and / is a 2-arrow with source o and target B then this is denoted by ó, a.+ P. The following diagram gives a useful way to picture 2-arrows: A B 3. Given three objects A, B and C of. ß, there is a functor, called the composition functor, m(A, B, C) : Hom(B, C) x Hom (A, B) ---+ Hom(Á, C). If o and B are l-arrows in Hom(B, C) and Hom(,,4, B) respectively then we usually write rn(A,B,C)(o,þ) as oo p and',n(A,B,C)(Ó,rþ) is usually written as þorþ where ót al standþ: P + B'are 2-arrows in Hom(B,C) and Hom(,A,.B) respectively. 4. For each object A of 6 there is a l-arrow Ide in Hom(Á, A) called the identi,tg 1- o,rrow of A. The identity map of Id¿ in Hom(Á, -4) is denoted by ida : Ida =+ Id¿ and is called the i,ilenti,tg Z-arrow of. A. 5. Let A, B ,C and D be objects of 6 and let Hom(A , B, C, D) denote the category Hom(C, D)xHom(B,C) xHom(Á, B). There is a naturai isomorphism, called the 83 o,ssocio,tiai,tg i,somorphism and denoted a(A, B , C , D) , as pictured in the following diagram: Hom(.A, a,C, o¡M2\ouom(B, D) x Hom (A, B) idxm(A,B,C) rn(A,B,D) I ,B,C,D) Hom (C D X Hom( A, C) Hom(Á, D) ) m(A,C,D) So a consists of an arrow a(''1, þ, a) : a(A, B , C, D)(1, 0., *) in Hom(/, D) from Oo þ) oo to .y"(P oa) for all 1-arrows 7 in Hom(C, D), P in Hom(B,C) and a in Hom( A, B) such that given 1-arrows ^l' , 0' and o' together with 2-arrows ó: I +'y' , rþ , 0 + þ' and ¡ : a è o' then the following diagram commutes: 0.Øoo--@@- (1'op')oc"' . ^ .11 il , , (t' ,p' ,o') "(r'É'").ll ll" .y o (p o o)------]%. 1' o (p' o o'). 6. Suppose A and B are objects of 6 and that a is a l-arrow of Hom(A, B). Then there are natural isomorphisms id¿(Á, B)("): aè oolda and id¿(.A,.B)(a) : a =à Idsoa in Hom(Á, B) called left and right i,denti,ty 'isomorphisms respectively. The isomorphisms are natural in the sense that if $ : a. + P is a 2-arrow in Hom(Á, B) then the following diagrams of 2-arrows commute: u id/A,B)(a)+oold.A 'll idLØ,8)(p) lf'".' p p o Id¡, and idp(,a,8)(a). a. Id6oo 'll idp(A,B l|,0""' p Ids o B. 7. This axiom forces a coherency condition on the associativity isomorphism ¿. So suppose we have objects A,B,C,D and E of B. Suppose we have l-arrows a of Hom(Á, B), P of Hom(B,C), ^t of Hom(C, D) and ô of Hom(D, E). Then we require that the following pentagonal diagram of 2-arrows in Hom(A, E) com- mutes: (ðo(ryoB))oa a( a(6,1oP,a) ((ôory) oB)oa ôo((ryoB)oa) \o(óo7,B,a) \ Á0"^',,u,., (ô o (þ o (r o (P o a)) ry) " " ") ffid This condition is called associatiui,tg coherence. 84 8. Suppose A, B and C are objects of. B and o and B are l-arrows in Hom(A, B) and Hom( B, C) respectively. \Me require that the foliowing diagram commutes: po(IdBoa) o (BoId6) oa lpoidp(A,B) id¡(B,C)or. 0oa One can also define a notion of morph'isrn of bicategories, transformat'ions between morphisms of bicategories and finally a notion of. modi,fi,cationbelween transformations of morphisms of bicategories, however we will not spell out these definitions and refer instead to [39]. We will also need the notion of a bigroupoid. Definition 8.2 (t5]). A bigroupoid consists of a bicategory B which satisfies the fol- lowing two additional axioms: 1. 1-arrows are coherentiy invertibie. This means that if o is a l-arrow of Hom(Á, B) there is a 1-arrow B of. Hom(B,A) together with 2-arrows ó, þ o d + Ide in Hom(,4,.4) and tþ : ao P + Idp in Hom(B,B), which satisfy the following compatibility condition. We require that the following diagram of 2-arrows in Hom(Á, B) commutes: ao(Boa) o(o,þ o$ (aoB)oa oold¡ tþoIo IdBoaffia 2. We also require that all 2-arrows are invertible. One can show as a consequence of the definition of a bigroupoid that the following diagram of 2-arrows in Hom(B, A) commutes: Bo(aoB) a(p otþ @ocr)oB B olda óoLp Idao p B idR(B,A)(B) Enample 8.1. Let X be a topological space. Define a bicategory II2(X) as follows. Take as the objects of II2(X) the points of. X. Given two points ø1 and ø2 define a category Hom(ø1, 12) whose objects are the paths 'y : I '-+ X with 'y(0) : u1 and 'y(I) : rz, where.I is the unit interval [0,1]. Given two such paths 11 and'fzwe define the set of arro'ù/s from 71 to 72 in Hom(ø1, 12) to be the homotopy classes [¡r] of maps ¡.t, : I x I + X such that p(0,t):'yr(t), t"(I,t): lz(t), p(s,0) : rr a,nd f¿(s,1) : Øz where it is 85 understood that two such maps p, and, p' belong to the same homotopy class if and only if there is a map H: I x.I x I --+ X such that.F/(0,s, t): ttþ,t), H(I,s,¿) - tt'(s,t), H(r,l,ú) : H(r,s,0) : and,Í/(r, s,l): 12. Notice that H(r,O,t):'yt(t), lz(t), "t every arror¡,/ in Hom(21,r2) is invertible. If we have a third path 73 from 11 to z2 and a homotopy class [)] of maps ) : ,I x I ---+ X from 12 to 73 leaving the endpoints fixed, then we define the composite arrow [À¡,r] from 1t lo 'ys to be the homotopy class of the mapÀp:IxI--+Xgivenby s € ¿ € 1]' (1,')l's' Itt7s't)' [o' å]' [o' '^r¿'l\ùr'i - '\: \,r1zr - r,t), r € [å, 1], t e [0, 1]. whe¡e p, and À are representatives of the homotopy classes [¡;] and [À] under the ho- motopies .[/ above. It is straightforward to check that [À¡r] is well defined and that this law of composition makes Hom(r1, r'2) into a category. To recap, the objects of n2LX) are the points of. X, the l-arrows of II2(X) are the paths in X and the 2-arrows of II2(X) are the homotopy classes of homotopies with endpoints fixed between such paths. We need to define a composition functor m(rt, rz, rs) : Hom(ø2, tr) x Hom(21 , rz) - Hom(r1, r3). If 723 is a 1-arrow of Hom(ø2, 13) and ?rz is a 1-arrow of Hom(r1, r2), then we define m(rt, rz, rz)('yzs,'yn) : 1zs a ^ltz, where 1zs o ^ltz : I ---+ X is the path tn(2t), ¿€ [0,å], ?Ytt o tn)(t) : tzs(2t - r), t € [å, 1] If we are also given l-arrows lzz of Hom(ø2,ø3) and ltz of Hom(ø1,ø2) together with 2-arrows lprtl t 'yzs è y3 and Îpt] , 1n è lrz inHom(r2,ø3) and Hom(ø1,z2) respectively, then we define m(rt,rz,rs)(luzt],lt tl):lltzso þnl to be the homotopy class of the map s€ [o' 1]' ú € [o' å]' o- rr'/\"1"/tttz)(s,ú) : {u"G'2t)' [øæ(t,2t-7), s € [0,i], ú € [+,1]. where pzs is a representative of [¡.1¡] and þp is a representative of lprrl. Aguin, it is straightforward to check that this defines a functor. Next, we need to define identity l-arrows and identity 2-arrows. Given an object r of tI2(X), we define Id, to be the constant path at r and we define the identity 2-arrow id, to be the homotopy class of the constant homotopy from the constant path to itself. Now we need to define the associator isomorphism. Given 1-arrows 7s¿ in Hom(23, rq), 1zs in Hom(22,ø3) and 'yp in Hom(ø1, r2), we need to define a 2-arrow a(1s¿,12s,'Y9) , (7s¿ o 7zs) o 'Yn ) .yzao (.yzso712) in such a way that given l-arrows Zåa in Hom(r3, ro), l'zz in Hom(r2, 13) anð,l* in Hom(r1 ,r2),logether with 2-arrows ltrtnl t'ysa è'y'sa, lpzzl: ^lzs ) y, and [ttrr) , 'ln ] 7i, then the following diagram commutes: Q4¡op2s)opn,t/\t ('ytnol'lzs)o1r, -----ì (7s¿ o ^lzs) o.yn o('ys¿,zzg,T2)l 1l"0!"n,irr,t!rr) 'Ysqo('Yzsol:¿) o t.Ì?3 o.ln), t sqo(t"zsol4z) 'ó+ \ /2 86 -?o, and that the pentagonal diagram of axiom (7) above commutes. There is a standard choice lor a(13a,^tzz,'yn) - see for example [38]. We set a(^lsq,^tzz,'Yn) to be the homotopy class of the map a('Ys¿,'lzs,^ln) : I x I --+ X defined by nz(*), se [0, 1], ú€[0,?], o('ytn,'Yzt,'Y:¿) (s, t) : lzz( t -2+s), s € [0,1], ú € [?, ?], r*(ff), t6lo, 11, ¿€[?, 1]' Next suppose we are given 2-arrows ltttnl t 'Ysa è lrn, lttrtl : lzs è y, and lpt ] , 'yn è 'y'tz. We need to show that ã'(1sa,"tzs,nz)((þn o þzs) o ttz¿) = }tn o (Pzt o trs¿))a(ls+,'yzs,'yy-), where F4 is a representative of the homotopy class [pn¡]. Ott. catt check that a homotopy with endpoints fixed between these two maps is pn(2-2rs,A6kÐ+ir, r € [0, 1], s € [0,å], ¿ € [0,k#tÐtl], pzs(2-2rs,4t-2(I-r s*rs) 1), € s e - - " [0,1], [0,]], 2(1-r-s*rs)*2 1 ,-=t----T-,--T'' - ¡2(L-r-s+rs)+1 ps¿(2 - 2rs,æP),, e[0, 1], s € [0, å], ú € [k;ÍÐË, 1], ¡.r,p(2rs-2s*1,;6-¡-g¡;i), , e [0,1], s € [å,1], ú€ [0,4Liitl], Fzz(2rs-2stir,4t-r(1 - 2t)- 1), r € [0,1], s € [å, 1], r(L-2s)+21 rüE - [--A-t-f¡r(1-2s)*7 lt p,ga(2rs-2stL,ffi),"E[0,1],s€[å,1],telryl2,\. This proves that ø is a natural transformation. We need to check that the associativity coherence condition is satisfied. So we need to check that the following diagram of 2-arrows is the identity 2-arrow from ((7a5 o ?s¿) o'Yzs) o Jp to itself. (.ynu o (.ytn o 1zs)) o 7r, o('y¿s,''ts¿,^rzs)o ,1s¿o1zs,1:u) ((,ynu o .ys¿) o'yzs) o 1t l¿s o ((ls¿ o'Yzz) o'Yn) o(.'t¿so^ts¿,'tzg,rrN , -- \-1 z4oahs¿,'rzs,'nz) (7ru o ls4) o (tn ""liÐ'Y+'luå (t n o ('yzzo rrr)). 87 We have the following expressions for the 2-arrows (here s € [0,1]): ln(2t),¿€[0,å], ns(#), ú€ [å,?], (d(lnu, ^lsa,,'yzt) o idrr, (s, ú) ) ts¿(8t*s-6), ¿€ [?,?], ryrs(e#), ¿€[?,i], nz(*),ú € [0, ?], 112s(8tt-2s-4), te [?,?], a('y u,'fz¿ o'Yzs, l1.¿) (s, t) ts¿(8t * 2s -5), t e [?, ?], rø(ff), ú€[?,1], xz(*),¿€[0,?], (id"r, o -a('yzq,, (s, tzs(8t*s-2),te[?,?], ^ln, "y n)) t) r*(H), ú€[?,å], us(2t-1), ¿€[å,1], zrz(ft),te[o,f], 'Yr'(H), ú€[+,+], a('yas,"Yst,'Yzz o 712)-1(s, ú) tzs( t-s-1), re [+,?], uu(E), t el\t,tl, nz(fr),te[o,f], ns(4t-s-1), ú€[+,?], d('yqu o 'lst,'f2s,"yrz)-1(s, t) x¿,(u#Í), ¿ e [?,+], r*(H), te [S,t]. Composing these 2-arrows gives the 2-arrow from ((7a5 o'ysa) o'yzs) o ^'/12 to itself. We want to show that this 2-arrow is the identity 2-arrow at (('y¿u o'ys+)o'yzs)o712. The plan is to find homotopies from each of the individual 2-arrows above to the identity 2-arrow at ((t u o.,tsa,)o.yzz)o1p. Of. course such homotopies will not fix endpoints but we will see that the homotopy obtained by composing each of the individual homotopies will. 88 The homotopies are given as follows (here r € [0,1] and s € [0,1]) nz(2t),ú€[0,å], ¿ e [å,I=*te], F1(r, s,t) n'(ffi¡l), n+(8t+s(1 -r)-6), te [o+4,44], r*(ffi),¿e [L(P,r], tn(6#), te [0,#], 8t-2(r-I)s-4\ ¿ ¡(r-1)s*2 2(r-l)s*r*51 nt( )t t' - L----l- t --ã--l' F2(r, s,t) 1+, = k:1à41, lsa,( 8t - 2(r- 1)s - r -5), ¿ € t?E#, 8t-2(r-l)s-r-6¡, - ¡2(r-L)s*r*6 11 us( 2+l!ry )t u = [-----ã-' 'J' 'yn( 1 s-2-2r lzs( 1+r F3(r, s,t) 8ú- (r-1)s-3-3r ls+( 1- (r-1)s znu(W), ¿e [#,1], tn( nt( Fa(r, s,t) 8¿- 1 -4r-2 w(ffi¡,tm¡ff,t1,lsa( 2-r r,z(ffia), ¿e [0,l$1], ns( t-s(1 -r)-r-!), ú€ t{l+ttl ,Wl, F7(r, s,t) 8t-2s(l-r)-2r-4 s _T ls¿,( 2-r-s(1-r) 8t-s(1-r) -r-6 us( 2-r-s(l-r) t It is not too hard to check that if one composes these homotopies then one has a homotopy with endpoints fixed between a representative of the 2-arrow above and the identity 2-arrow at ((lnsoTs¿) o.yzs) o712. This proves the associativity coherence of a. We now need to define left and right identity transformations id¿ and id¡ respec- tively. If. 1e Hom(ø1,ø2) is a l-arrowthen id¿(øunz)(l) is a 2-arrow'y + ^r oId",, \Me define id1(nyrz)(l) to be the homotopy class of the map ("''€ (s,ú) + [o';]' lz(H),'t å ¡;, 11. We need to prove that the assignment 1 e íd1(rt rz)(l) defines a natural transforma- tion, so we need to show that if we are given l-arrows 7 and 7' of Hom(rt,rz) together with a 2-anow lttl t l + "y' in Hom(ø1,r2) then the foilowing diagram of 2-arrows commutes. ^l .Y id,¡(q,r2)(1) id¿(qp2)(1') ll lt 7 o Id,1 ffi,r^l' o Id", 89 We show that the homotopy classes ([¡r] oId",id¿("r, rz)(ù and id¿(21 ,rz)(l')[p] are equal. Firstly, id¿(r1, rù(l')[¡r] is represented by the map o,Il,t € [0,1], t e [0, T], € [å, 1], t e l\,t1, where p : 12 - X is a representative of the homotopy class [¡.e], and the homotopy class ([p] o Id",)id¿(r1, rz)(l) is represented by the map rr, s€[0,å],úe [0,s], se [0,]1, te [s,r], (s, ú) r* 7(Í=), rrj s€[å, 1], ú€[0,T1, p,(2s-7,%#), s€[å,1], te [ç,1] Then rr, r € [0,1], s e [0, L), t, [0,rs], pQG-r)s,l¡fi), r € [0,1], s € [0,å], ¿ € [rs, 1], (r, s, ú) r-> 11, r e [0, 1], s € [å,1], ú € [0, Tl, p((2s-2)r+t,#),re[0, 1], s€[å, 1], te l2s],t1, is a homotopy with endpoints fixeci between (pold',)id¿(ø1, rz)(ù and id¿(ø1 ,rz)(l')p. Similarly, if 7 is a 1-arrow of Hom(r1 ,r.2) then we define id¡(ø1 ,rz)(l) : 'y + Id,, o 1 by setting id¿(ø1, rz)(l) equal to the homotopy class of the map (s,ú) ,.+ {ri:i:?J,",. '0,#,, Again one can show that given another l-arrow 7' in Hom(rr,rr) and a 2-anow lp,]: 'y +'y' in Hom(u1 ,z.2) Ihen the diagram of 2-arrows below commutes. 1 7 idn(zr,øz)('y) idp(ø1,ø2)("y') Id,, o 1ffirrl{,"o1' Finally, we need to show that axiom (8) of Definition 8.1 is satisfied. So we need to show that, given objects r¿) r j and n¡" of II2(X) and 1-arrows '/¿3 and 'y¡n inHom(r¿, r¡) and Hom(r¡,ru) respectively, that the foilowing diagram of 2-arrows in Hom(ø¿,ø¡) commutes. old,r)o^yn¡ \uo(Id,,o1¿¡) a(1¡¡,1d., (1;¡ id.t, n oid p(r ¿,a ¡ ) )(1r)oidr¿¡ .yjx o.hj 90 The 2-arrow a(1¡x,Id,,,'yo¡)(id¿(z¿, r¡)(l¡x) o idr,¡) is represented by the map t,i¡(Zt), s e [0, l], t. [0, å], r¡, s€ [0,å], tel],,+1, s € [0, t, (s, ú) r-> nr(z#), l), [+,1], tn¡(&), s € [å,1], ú € [0,+], r¡, s€ [å,1], tel+,+1, w(Y), se [å,1], ú€[+,1), while the 2-arrow id"ru o idp(z¿, ,¡)Øn¡) is represented by the map s e [0, à), t , [0, æb], l' 1 rlrJ' L2(2s+1) ' [å, 1], t€[ï,1]. We want to show that these two maps are homotopic by a homotopy fixing endpoints. One can check that such a homotopy is given by Lr), 'y¿¡(2(2rs+ 1)), s e [0, t, [0, ffi¡], r¡, s€[0,å], tel#çry ltPl, z¡*(ãì*#), 6 [0, å], ú € 1], (r, s, ú) r- " lryi, 'yni(##),' ç [å, t], t e [0, ffi), újr.r: o:L2j-J)c c 11 1l ¡. ¡t-2(L-r)s @], "-12(2rs*2-r), 2 w(W),'e [å,1],telry,\, Thus we have shown that II2(X) is an example of a bicategory. In fact, II2(X) is a bigroupoid as we will now show. We wili call II2(X) the homotopy bigroupoi,d of X. Firstly we have to show that the l-arrows of II2(X) are coherently invertible. So let u1 and 12 be points of X and let 7 be a l-arrow of Hom(ø1,r2). There is an obvious candidate for the 1-a¡row 7-1 which is to be a weak coherent inverse of 7, namely we set 7-1 to be the path opposite to 7, so 'y-L(t) : 7(1 - ú). A homotopy p joining 'y-L o 7 to the constant path at z1 is given by o ( s s 1' o < t < l' tt(s,t)')"t : {'l','-('-,:])' lr((t - s)(2-2t)), oS s < 1, å 91 Definition S.3 ([25],). A 2-categorg C consists of the following data: 1. A set Ob(C) whose elements are called objects of C. 2. Given two objects ø1 and rz of. C there is a category Hom(r1, ø2) whose objects are called l-arrows of C and whose arrol,¡¡s are called 2-arrows of C. As before if c and p are l-arrows of Hom(ø1, ø2) and / is a 2-arrow of Hom(ø1, 12) with source a and target É then we indicate this as þ: a + P. 3. Given three objects tt, 12 and 13 of C there is a composition functor m(rt,rz,rs): Hom(r2, tr) x Hom(ø1, ,r) - Hom(r1, 13) such that, if za is another object of C, then the two functors m(q, rs, r¿) (l x m(q, rz, rs)) m(rt, xz, r+) (*(*r, rs, na) x L) from Hom(rr,rr,xs,rt) to Hom(21, øa) coincide (here Hom(ø1, 12,rs,øa) denotes xHom(r2,"r) Hom(r1,ø2)). As before, if o and the category Hom(øt,ra) " þ are l-arrows of Hom(ø2,ø3) and Hom(ø1,r2), then we usually write ao P f.or m(r1,r2,4)(a, B). AIso if ó, a I q" and, þ : P + B' are 2-arrows in Hom(ø2,13) and Hom(21,ø2) respectivel¡ then we write / otþ f.or the 2-arrow m(rurz,rs)(ô,rþ) in Hom(r1, ø3). 4. For each object ø of C there is a l-arrow Id, of Hom(2, r) called the identity l-arrow of r. The identity map of Id, is denoted by id, and is called the identity 2-arrow of ø. If g is another object of C and a is a 1-arrow of Hom(ø, E) then we have Id, o e: e,: d o Id,. If 0 is another l-arrow of Hom(z,E) and þ: a + p is a 2-arrow of Hom(r,g) then ido o þ : Ó: þ o id,. 8.2 Bundle 2-gerbes over a point and bicategories We first review a construction of [33] which shows how to associate a C* groupoid to a bundle gerbe (P,X,M) restricted to a single point rns of. M. Let X,,o: n-t(mo). Take the objects of the Ct groupoid to be the points of X,nr. Given two such points ø1 and 12,let the set of arror¡/s from ø1 to r,2be the points of the fibre P1r,,"r). W. have to be able to compose arrows and we also have to show that there exist identity arrows. Given points rt, 12 and ø3 of X^o as well as a,rrov¡s u12 : 11 , 12 and u2s : 12 -'> ts (ie points un € P@r,r") and u2s e Pçrr,,r)) define the composed arrow uzs'.ttz: 11 '+ fi3 to be the point mp(uzzSurz) €. P67,,s). Here mp denotes the bundle gerbe product in P. Since the bundle gerbe product is associative, this process of composing arrows is associative. The identity arrow 1'. x ---+ r is given by the identity section e(A) e P6,,¡. Note that the category so defined is a groupoid (ie every arrol¡/ is invertible) and that the set of automorphisms of an object is isomorphic to CX. Now suppose that we have two bundle gerbes (P,X,M) and (Q,Y,M). Denote by P,no and Q,no the C" groupoids constructed from P and Q respectively. Suppose we have a bundle gerbe morphism f : ff,f,idu): P - Q. Then it is easy to see that becaur. / .o**utes with the respective bundle gerbe products, we can use / 92 to define a functor Í , P^o - Q^o. If there is a second bundle gerbe morphism g : þ,g,id¡rt) : P ---+ I then we can form the C' bundle (f ,g)-'Q + X-o âs in Lemma 3.3 and, by the resuits of that Lemma, we know that (/, g)-LQ 'descends to rn6'. Henceitispossibletochooseasection of U,g)-LQ whichiscompatiblewiththe descent isomorphism. Such a section then gives rise to a natural transformation f + g. Bicategories are related to bundle 2-gerbes over a point in the following way. Let (Q,Y,X, M) be a bundle 2-gerbe and let m be a point of. M. We define a bicategory Q,n as follows. The objecls of. Q^ the points of. X,n: 7T-r (rn). Given two such objects r¿ and rr. , define a category Hom(r¿, r¡) as above. So we let the objects of Hom(z¿, z¡ ) (the l-arrows of Q^)be the points gr lying in the fibre Y1"o,"r¡. Given two such points gr and A' wedefine the set of arrows ftom g to g' tobe points u € Qfu,v,). Composition of arrows is via the bundle gerbe product in Q as explained above. The bundle gerbe morphism m : r;rQ Ø nltQ -. T;rQ can then be used to define the composition functor rn: Hom(ø¡,nx) x Hom(ø¿, r¡) - Hom(ø¿,ø¡) by mapping an object (g¡n,U¿) of. Hom(ør', ø¿) x Hom (r¿, r¡) to the object m(g¡r,U;¡) of Hom(r¿, z¿) and mapping an arroru (u¡*,u;¡) in Hom(ø¡, ø¡)xHom (r¿, x¡) from (y¡¡,y¡¡) to (í¡n,A';¡) to the arrow ñt(u¡¡Øur¡) in Hom(r¿,rn) fromm(y¡n,A¿) to m(g'r*,U'r). The fact that m is a bundle gerbe morphism and hence commutes with the bundle gerbe products ensures that m is a functor. We now need to define the associator natural transformation a(r¿,t¡ttretø¿) between the functors bounding the diagram below: Hom(r¿, r¡tttet"ù .-g*Hom(ø7, ,r) , Hom(ø¿, ø¡) Ixrn m ,ri,, t",rt) Hom(ø¡, r¿) x Hom(rn, r,,) rf¿ Hom(z¿, ø¡) Given an object (Ap,A¡n,A¿) of Hom(ø¿, r¡trretø¿), we set a(r¿,r¡trk,rL)(A*¿,U¡x,U¿) to be the arrorü¡ ã(Arr,A¡r.,A¿¡).The fact that the section ã, of A descends to a section a of A ensures that a(r¿,r¡ttktø¿) is a natural transformation and the coherency condition satisfied by the section o guarantees that the natural transformation a(r¿,n¡,xn,rt) satisfies the associativity coherency condition. To define an identity l-arrow at an object r of. Q*, flrst choose a set map .I : + Y which is a section of. ry : Y + ¡tzl (here A(X) denotes the image of X under^(X) the diagonal map L : X - Xlzl, r -, (r,ø)). So -I associates to each point ø of X an element of the frbre Y1,,"¡. Let Id, : I(r) and let id, : Id" =+ Id" be the identity 'We section of the bundle gerbe Q evaluated at the point I(r) - ie e(/(ø)). now need to define left and right identity transformations. Ignoring such concepts as continuity and smoothness, notice that we can define bundle gerbe morphisms Id¿ : Q '-+ Q and Idp Q+Qasfollows: Id1:Y "+Y a è rn(A, I (n2(ny (y)))), Idp: Y toY s + m(I þr,,(nv (a))), u), 93 and Ið,7: Q - Q y r-+ ñ,(u Ø e(Iþr2(ny(to(ø)))))), Id'p: Q -' Q u + rh(e(I(n{try(trq("))))), "). Therefore we obtain C' bundles (in the set theoretic sense) It : Dre¡a¿ and Ia : Dç1o,la^ on X[2] and hence by restriction set theoretic C* bundles -I¿ and .I¿ on X,n x X^: Choose set maps íd7 : Xl '+ Ir and idn , X?,, ---+ In which are sections of the projections zi'¡, : 11 + Xl and 'ï¡^ : Ip -- Xk respectively. We denote also by id¿ and id.¿ the lifts of the sections to the set theoretic CX bundles (1,Id¿)-18 and (1,Idn)-18 respectively. It is easy to see that these sections provide 2-arrows idy(r¿,"¡)(A) : U + m(A,1,) and idp(r¿,r¡)(A) : A + m(1,¡,g) f.ot a l-arrow y €Yç¿,,¡) which are natural in the sense of (6) of Definition 8.1. We need to check that the left and right identity natural transformations are compatible with the associator natural transformation a(r¿,r¡,rx,rt). It is easy to see that we have an isomorphism of set theoretic C' bundles over Xfl given fibrewise as follows: I n(rt, rr) Ø Ii@", rt) = A(rt, 12, 12, rs). We can defrne a set map Í , XL -- C" such that a(r1, 12, r.2, rs)id7(rz, rs) : ida(ør, rù' f @t, 12, rs). Therefore if. ap e Y(rr,,r) and y2s € Yç,2,,s) then / satisfies ã(Arr,I(rr),g2s)Qd;(rz,rz)(azs) o 1r,r) : la* o id¡(ø1, rz)(An)' Í(rt,rz,rs). Similarly rve can define a function g : Xh + C" by a(r1, n1, 12, rs)id;(tt, rz) : idt (rt, r.z) ' g(ru rz, rz). Again, if. Ep € Y@r,,r) and y2s €. Yça2,,2) then g satisfies a(I (r),an,azz)id;(rb rs)(azs o an) : Ior" o id7(r1, rz)(an) ' g(rr tz, rz)' By considering the diagram of Axiom TA3 of [20] we have the following equation: Í (rt, rr, rr) : f (rt, rz, rq)g(rz, rs, rq)-L . (8 1) We now need to be more specific about our choices of id¿ and idn. First of all, Iet us agree to put idl(r,ø) : ida(r,ø) so that the two 2-arrows id¿(2, r)(1,): I, + I,o I, and idp(r, : I, + I, o I* are equal. Notice also that we can choose id¿ and ")Q) id¿ so that we have õ"(1",1,l,)(ifu(r,n)(1,) o 1¡,) - 7L o id¡(r, r)(1,) Therefore we have made an explicit choice of id¿ and ida over the diagonal A(X) C X3.. For Un € Yþr,,r) with 11 I x2 Iet us agree to choose id¡(21, r.2) so that the 2-arrow idp(r1, rù(Aù : Un è I'z o Utz satisfies ã(Arr,,I,r,I,r)(id;(r2,rz)(I-) o 1r'r) - 7-1,, o id¡(r1, rz)(an). 94 For g21 € Yç,2,,1) with ø1 I r.2let us agree to choose id7(t2,21) so that the 2-arrow id¿(r2, rt)(Azt) : Azt ) Azt o I,z satisfies: a(I,r, I,r,Azt)(id;@2, r)(Uzù o ILr) : 7a* o idp(r2,rz)(I,r). Notice that this has now fixed a choice of ida and id¿. Also notice that we now have f(r,r,r):1 l('r,rz,rz):1 Í ('r, nz, rt) : 1 PuL ra: rz ã,îd rL : t2 in equation 8.1. Then we get f (rr,rr,"r) : f (rr,rr,rz)g(rz,rs,tz)-L. This implies that g(rz,rz,rr) : 1. Now put ra : rz inequation 8.1. We get I (rr, rz, rr) : f (rt, rz, rz)g(xz, rs, r2)-r - I. This shows that id¿ and idp satisfy the compatibility condition with ¿. We record the above discussion in the following Proposition. Proposition 8.1. Giuen a bundle 2-gerbe (Q,Y, X, M) on M ue cz,n associ,ate to each point rn of M a bicategory Q,n where Q^ is defi'ned as aboue. Note 8.1. 1. Recall from [33] that a C* groupoid is a principal C' bundle P over X2 with an associative product. The point of this result was that the existence of an identity and of inverses u¡as a consequence of the product being associative and one did not need to impose further axioms. Let us agree to call a 'C' bicategory' a bicategory Q such that for any pair of objects A and B of. Q, the category Hom(r4., B) is a C" groupoid. It then follows that every CX bicategory arises from a bundle 2-gerbe over a point, or alternatively, as a simplicial bundle gerbe on the simplicial manifold X : {Xo} with Xo : Xp where X is the set of objects of the CX bicategory. In fact one can show that the CX bicategory arising from a bundle 2-gerbe is a C" bigroupoid and hence any C' bigroupoid would arise from a bundle 2-gerbe over a point. 2. Notice that we could weaken the definition of a bundle 2-gerbe by removing the requirement that there exist an 'associator' section and still obtain a C* bicategory (in fact a C' bigroupoid) when one restricts the bundle 2-gerbe to a point of the base manifold. This is because we could choose a (set theoretic) section a of the 'associator bundle' A + XI and define a C" valued function I , XL -- C* satisfying a cocycle condition on Xl by comparing ô(a) with the canonical trivialisation of õ(A).It then follows that we could rescale ¿ to obtain the associativity coherence condition ô(ø) : 1' However, in order to associate a class in Hn(M;V,) to abundle 2-gerbe we need to be able to choose a smoothly. 3. We could require further axioms to hold in the definition of a bundle 2-gerbe so that the construction of the Ct bicategory above was 'smooth' in a certain sense, ie one had a smooth choice of the section I : X --> A-lY and smooth choices of the left and right identity isomorphisms id¿ and ida. These conditions will in fact hold in the main examples of bundle 2-gerbes we consider in the next chapter. 95 96 Chapter I Some examples of bundle 2-gerbes 9.1 The lifting bundle 2-gerbe Suppose we have an extension of groups BCt -, G -- G, with BC' central in ô. Suppose also that P - M is a principal G bundle. Since BC" is an abelian group, we can form the lifting BC' bundle gerbe (P,P,M). So the bundle gerbe product on P is a BC' bundle isomorphismn'L: nr'FØrstP -- r;LF which is associative in the usual sense. (As we have previously remarked, the fact that BC' is an abelian group enables us to form the contracted product r;rÞøzr;1P). Now we can form the lifting bundle gerbe (8,P,plz)). The BC" bundle gerbe product rn then induces a bundle gerbe morphism m : rrrQ Ø nltQ -- T;tQ which is strictly associative in the sense that the two induced bundle gerbe morphisms rn1 and rrl2 are equai. It follows that the quadruple (8, Þ, P, M¡ defines a bundle 2-gerbe in the strict sense (Definition 7.3). An example of this situation occurs with the short exact sequence of abelian groups BC* + EBC" -- B2C'and the universal B2C'bundle EB2C'-t .B3C". We record this discussion in the following proposition. Proposition 9.1. Let BC" -. Ô - G be a short eract sequence of groups with BC" central in G. Let P + M be a principat G bund,Ie. Then the quad,raple (Q, P, P, M) defined aboue'is a strict bundle 2-gerbe. Another situation in which bundle 2-gerbes arise is the following. Suppose we have p y[zl a surjection ¡r : X -+ M admitting local sections and a BC" bundle - together with a BC* bundle isomorphism rn : r;r P I zr;lP + r;L P covering the identity on X[3]. However we do not assume that the two induced BCt bundle isomorphisÍìs rrù1 and m2 are equal. Here rn1 and rn2 are defined as follows m1 : r2tr1L P Ø n;L(nlt PA ?ï-tlP) + r;Ltr;L P TrLl: r;lm o (1 A nlL*) tP rn2 : 'rLL(T, a nlL P) Ø rrrr;L P -, r;Lr;r P îTL2: rlLm o (zrtlrn I 1)' where we have identified r;Lr;tPØr;L(tr;tPA7rtlP) : rtr(trttPør;Lf)Ør;tr;LP and r2rr2r P : r;rr;1P by virtue of the simplicial identities satisfied by the r¿ otL the simplicial manifold X : {X,} with Xp: yln+t). Therefore rn1 and rn2 differ by a map d : Xlal -. BC' which must satisfy 6(Ô) : 1 on X[5]. Suppose finaily rhar þ has a Itfr ô : yln) -- EC" which satisfies 6(d) : 1 on X[5]. Then if we let (Þ, P,¡tzl) denote 97 the lifting bundle gerbe associated to the BC' bundle P --+ X[2] then the quadruple (Þ, P, X, M) is a bundle 2-gerbe. As before the BCX bundie isomorphism rn provides a bundle gerbe morphism rlt,: rrLÞØrr1Þ -- r;rÞ covering the identitY on Xlsl. One can check that the associator bundle A -+ XlLl has classifying map equal to þ and : 1 so / provides a section ø of A - the associator section - and the condition 6(/) is the requirement that ô(ø) matches the canonical trivialisation of ó(A) on X[5]. 9.2 The tautological bundle 2-gerbe Here we verify that the tautological bundie 2-gerbe introduced in [14] is still a bundle 2-gerbe as defined in these notes. To be more precise we only verify this for the special case in which the base manifold is 3-connected or at least has zr3(M) : g. To define this tautological object recall that we start with a manifold M, which rù¡e assume to be 3-connected, together with a closed four form @ on M with integral periods representing a class in Ha(M;Z). Nexf we take the path fibration ¡r : PM '--+ M consisting of piecewise smooth paths ? : [0, 7l -+ M , ry(O) : rns where rns is a basepoint of. M and with zr the map which sends a path 7 to its end point 7(1). The fibre product p ¡4lz) consists of all pairs of piecewise smooth paths (71, 72) with the same end point, that is fr(1) : 1r2G) Denote by 7'the path 7 traversed in the opposite direction and denote by ?r x 72 the path formed from a pair of paths 'Yt,'lz by traversirg ?, at double speed and then 72 at double speed. Of course, this is only a piecewise smooth path if ur(1) : lzQ).Thus if (lt,lù ePMl2l then 71 *1i e QM, where QM is the based piecewise smooth loop space of M. : the loop Therefore we can define a map ev PMl2) x ^91 * M by evaluating ^h*'1i against the angle á. Therefore we may define a ciosed three form ø on P Ml2) by pulling back O toPMl2l with ev and then integrating ev.(O) over the circle. One can check that ø so defined is a closed three form with integral periods. The next step is to perform the tautological bundle gerbe construction onPMlzl using the integral th¡ee form ø. Bundle 2-gerbes enter the picture because we can define a bundle gerbe morphism covering the map p¡4ls\ + pl¡ttlzl define¿ by (ryr, 'Yz,'Yz) * (?r,73). For the details see [14]. To make the calculation simpier assume for the moment that the base manifold M is a point. Then we start with a manifold X and a closed 3-from ø on X with 'We integral periods. \Me construct a bundle gerbe on X2 : X x X in the usual way. define a space Y above X2 with fibre at (rt,rr) €. X2 equal I"o Yçq,,2) : {o : I -+ X, a picccwise smooth : a(0) : 17, a(t) : ø2). Next define a C'-bundle Q + fl2) ylzl whose fibre at (a , 0) ç is all equivalence classe s lt-t, z), where z is a non-zero complex number and p, : a + B is ahomotopy with end points fixed, that is p, : I x I --+ X and pr(O,ú) : o(ú), p(I,t): þ(t), p¿(s,0) : o(0) : P(0) and ¡;(s, 1) : a(1) : p(1). The equivalence relation is defined by (pt, if for any homotopy F t p1.è l-tz "r) - }tr,"t) with endpoints fixed (so F : I x I x,I -> X, F(O,s,ú) : l.tt(s,t), .t.(1,s, t): p,2(s,t) and for each r € [0, 1], F(r,-,-) is a homotopy with endpoints fixed between a and p) we have 22: eKp( F.(a))21 1," If (r1, rz,rs) e X3 : X xX xX note that we can define amap m:Y62,rs)xY1q,,z) + 98 Y(,r,,") bY (o5, a1¿) è Q.23 o Q'12, where (o,, o a,z)(t) : ?Í iir!?'= r, {:::lii'_\"".',1- and where o,zz € Y(rr,r") and a12 e Yp7,,2). This map m cleatly extends to a map m : r;rY xys rrLY -'+ r;LY covering the identity on X3. We can also define a C" bundlã morphism ñ : r;rQ Ø nltQ -- 761Q covering the map ml2l ' (trlLY xv' rlY)l2l --- (r;tY)¡zl by ñ(lttrt, zzsl Ø lpn, znl) : lpzs o þLz, zztzn], where p23 is a homotopy with endpoints fixed between 423 and þzs, lttz is a homotopy with endpoints fixed betweêrL ay2 and Bp and p,2s o ¡ty2 is the homotopy with endpoints fixed between cr2s o d12 and B2s o B12 given bY \, ., ( pr.r(s,2t),") 0 ( s < 1, 0 < t < If2, }trt o rL'/\-7-/l.tn)(s,t) : \' ( [ør.(t,2t-r),0 s <1, Lf2 < ú < 1. One can check, see [14], that ñ, is well defrned and commutes with the bundle gerbe products. Clearly r?¿ is C" equivariant. Therefore rn : (ñ,m,id) is a bundle gerbe morphism. As usual, rn defines two bundle gerbe morphisms ñ\ : (ñr,mt,id) and rn2 : (ñz,mz,id) between the appropriately defined bundle gerbes over Xa : X xX xX xX. The maps rn1 aîd TrL2 àrê defined fibrewise by rn1 :Y@s,tn) XY("r,r") XY(rr,rr) +Y(qp¿) (orn, orr,, orr) - (otn o azs) o aD, nL2 | Y@s,zn) X Y(r",r") X Y(rr,rr) -+ Y(rtp¿) (ou, aze, orr) - o;sa o (ay o cr¡p) for o3a €Yçrs,ra), o.zs €Y@r,rr) and ay2 €Y61,a2). It is a standard result (see [38]) lhat m1 = ff12. This means that the C"-bundle (mt,mz)-LQ - Ynza has a section â, where we write for notational convenience Yrzs¿ for either of the spaces r;tr;rY xx4TAr(T1rY xyth'Y or zrfl (nrtY xvzr{rY) xva r;Lr;rY over Xa. Flom the general theory of simplicial bundle gerbes (see Section 7.1) we know that the C" bundle Â: (mr,,mz)-rQ descends to a C* bundle A on Xa such that ô(A) is canonicaliy trivialised on X5 : X x X x X x X x X. To show that the section â of ,zi descends to a section a of. A, we first need an explicit formula for â. We get this from the homotoq¡ mt = TLz (see Example 8.1 and [38]). Set ã(orn,azs, oy.) : la(o.z¿., o,zs,d\z),1], where d(au, azs,oq¡-) : I x I + X is given by 0<ú<+,Q(s(1, -2),+ 99 We now need to show that the descended section o satisfies the required coherency condition on X5, that is that ô(ø) matches the canonical trivialisation of ð(A). Again this follows from the results of Example 8.1. One can check that the homotopy F : FsFaFsF2Fl joining d(an o o,34) azs, or-)-lo(anu, o,BA, o,2B o o.r-)-L (idonu o a(ar¿, azz, op)) a(a x, Q44 o Q,2s, a 72) (d'(a a5, o,s¿, a zs) o idor, ) to the constant homotopy from ((aa5oorn) oort)oarzto itself has image that is at most two dimensional. Hence F" (r) : 0 which shows lhat a satisfies the required coherency condition. In the general case, where M is not a point, we can apply the construction given above to each fibre of the path fibration r : P M --+ M to obtain a bundle 2-gerbe. 9.3 The bundle 2-gerbe of a principal G bundle Suppose r¡,/e are given a principal G bundle P -+ M, where G is a compact, simply : connected, simple Lie group. Then it is known [15] that "r(G): 0 and H!(G;Z) Z. It is shown in [7] that there is a closed, bi-invariant three form u on G with integral periods which represents the canonical generator of Ií3(G;Z) : Z. If. G :,SU(N), then ¡u is the three form #t (¿gg-t)t - see [8]. Recall from [14] that we can define a bundle gerbe (Q,PG,G) on G with th¡ee curvature equal to z. The fibre of. Q -- pçlzl at a point (a, þ) e PGl2l is all pairs (ó, r), where z Ç Cx and / : a è B is a homotopy ó t I' ---+ G satisfying Ó(0,t) : a(t),þ(l,t) : p(t),d(s,0) : e and d(s,1) : a(1) : þ(I), under the equivaìence relation - defined by (ót, rr) - (ó2, zz) if and only if for all homotopies F : þ1 + óz with endpoints fixed between þ1 and Óz wehave z2 - 21 exp(/¡. F.r). The bundle gerbe product is defrned by zt] Ø ztzzl, lü, ló2, "r) - lótó2, where þ152 denotes the homotopy defined by (óþù(s,t): S1(2s,t) for0( s It is shown in [14] that this is well defined, associative, etc. Propositiong.2. The bundle gerbe (Q,PG,G) is a simpl'icial bundle gerbe on tlte simplicial manifold NG. Proof. We first need to define the bundle gerbe morphism m: (ñ,rn, id) which maps n: d;LQ Ø d;'Q -, d,,,tQ. Define m: d;rPG xc2 d;rPG - drrPc covering the identity on G2 : G x G by sending (a, þ) to the path c o a(I)p given by a(zt), 0 100 Nexb, we need to define a CX equivariant map ñ: d;IQ Ø d,;'Q - dL'Q ml2l , (d.;rPG xç, d,¡Lrc¡Ízl - d¡LPclzl and check that it commutes with the bundle gerbe product. So take pairs (/, z) and (rþ,r) where z,u) € C' and ó t I" -t G and ,þ, 12 -- G are homotopies with endpoints fixed between paths er,d2 and Ér,B2 respectively. Then we put : ñ((ó, z), (1þ,r)) (ó " ô(0,I)tþ, zw) where ÓoÓ(0,I)þ: 12 -+ G is the homotopy with endpoints fixed between o1 oa1(I)87 and a2 o az(I)02 given by þ(s,2t), 0 < ¿ SI/2 (ó o ô(0,1)T/)(s, ú) : þ(0,L)þ(s,2t - I), Il2 < t < r. We need to check firstly that this map is well defined - that is it respects the equiva- Ience relation - - and secondly that ñ, commutes with the bundle gerbe products. So suppose (ô, z) - (ô' , z') and (þ,w) - (rþ' ,w'), where @ and $' are homotopies with endpoints fixed between paths CI1 and a2 and where tþ and rþ' are homotopies with endpoints fixed between paths B1 and 02. We want to show that (ó' o (9.1) (ó " ó(0,I){, zw) - ó'(0,I)$' , z'w'). Therefore we want to show that for all homotopies .FI : 13 - G with endpoints fixed between ó " ó(0,1)T/ and ó' o ó' (0, 1),/' we have z'ur' : zw exp( [ ,.r). JF Note that if Õ : 13 - G is a homotopy with endpoints fixed between S and S' and ù : .I3 -+ G is a homotopy with endpoints fixed between tþ and tþ' , then by integrality of z we have exp( / H*u):svrç (Õ o Õ(0,0, 1)ü).2) Jts t,, Therefore v/e are reduced to showing that 7 u zw exp( (O o O(0,0, 1)ìú).2) (e.2) 'We have rfr exp( / (Õ o Õ(0,0, l)itrr).2) : exp( I Q.r) exp( / (O(0,0, 1)Ü).2). Jts' JF JIg By the biinva¡iantness of L/) we get (Õ(0,0, 1)!ú).2 : ú*1./, hence e*p( / ((Þ *, Õ(0,0, 1)v).2) : exp( [ ,.r)exp( / ú*r), JTs JF JF which impiies the result. Hence rî¿ is well defined. It is a straightforward matter to verify that ñ, respects the bundle gerbe products. It remains to show that there is a transformation of the bundle gerbe morphisms rn1 and m2over G x G x G which satisfies the compatibility criterion over G x G x G x G. This has already been done above for the tautological bundie 2-gerbe and the proof given there carries over to this case. ! 101 Suppose that we have a principal G bundle r : P - M. Pulling back the simplicial bundle gerbe (q, lG, G) ¡6 plzl via the canonical map I : Plzl -, G defines a quadruple of manifolds (Q, P,P,M) which is in fact a bundle 2-gerbe. We have the following proposition. Proposition 9.3. The quød,ruple of manifolds (Q, Þ, P, M¡ is a bundle Z-gerbe- Note 9.1. In the next section we will define a class in Ha(M;Z) associated to a bundle 2-gerbe. It will turn_ out (see Section 11.3) that the class in HL(M;Z) associaled to the bundle 2-gerbe Q is the Pontryagin class of the principal G bundle P. t02 Chapter 10 Bundle 2-gerbe connections and 2-ctrlings 10.1 Definitions and some preliminary lemmas Lef (Q,Y,X, M) be a bundle 2-gerbe. So there is a surjection zr' : X --+ M admitting Iocal sections. Recall from Section 3.3 the extended Mayer-Vietoris sequence Q+ep(M; 5r-ro1r; lerlytzl¡ 4 . \ençytd¡ 3... It is shown in [33] that this sequence is exact (see Proposition 3.5). We will exploit this fact to define the notion of a bundle 2-gerbe connection and a 2-curving for a bundle 2-gerbe connection and construct a closed integral four form on M associated to the bundle 2-gerbe (Q,Y, X, M). Recall from Section 3.3 that a bundle gerbe connection V on the bundle gerbe (Q,Y,¡tzl) has a curving fi which is a two form fi € O2(y). Recall from [33] (see also Section 3.3) that there is a closed three form ø on X[2] with integral periods which satisfres rþu : df . ø is called the Dixmier-Douady three form of the bundle gerbe (Q,Y,¡tzl) or sometimes the three-curvature. Definition 10.L. We use the notation of the previous paragraph. A bundle Z-gerbe connection on a bundle 2-gerbe (Q,Y,X,M) is a bundle gerbe connection V on the bundle gerbe (Q,Y,¡tzl) together with a choice of curving h e Q2(y) for V, such that ô(ø) :0 in C)3(Xtsl¡, where ø e f l3(Xtzl) is the Dixmier-Douady three form with rþa: dfy Proposition 10.1. Let (Q,Y,X,M) be a bundle 2-gerbe. Then (i) bundle Z-gerbe connecti,ons on (Q,Y,X,M) eri,st and (ii.) the three curuature a defi,nes a closed four form O on M with i,ntegral peri,ods. Assuming the existence of bundle 2-gerbe connections for now, suppose we have chosen one on the bundle 2-gerbe (Q,Y,X,M). Let u € Q3(Xt2l) be the three cur- vature, then d(ø) : 0 and rffe can solve ø : õ(Íz) for some fz e As(X¡. We call a choice of. f2 a Z-curai,ng for the bundle 2-gerbe connection (V, /t). Since ø is closed, we have õ@fù:0, and hence there exists Q eQA(U) such lhat df2: r*O and O is closed. We will see later that O is in fact an integral four form and hence is a de Rham representative of a class in Ha(M;Z). O is called the four ciass of the bundle 2-gerbe (Q,Y, X, M). 103 Example 10.1. As an example of this we consider the tautological bundle 2-gerbe on a 3-connected manifold M associated to a closed integral four form O on M (see Section 9.2). Recall that we first pull O back to P M , lhe piecewise smooth path space of. M, using the evaluation map ev : PM x I ---+ M sending (1,t) to 1(t). We then define an integral three form /2 onPM by Ír: .frev*O. It is straightforward to check thaf dfz: ,rT*@ where r : PM + M is the projection. It then follows that the three form ô(/2) onPM[2) defines a class in the image of Hs(PMl21;V.) inside H3(PUtlz);W¡. Alternatively we can think of 6ff2) as defining a ciosed integral three form on QM, the space of based, piecewise smooth loops on M. Recall from Section 9.2 that we then construct the tautological bundle gerbe on each fibre of n : P M -- M . This amounts to constructing the tautological bundle gerbe with bundle gerbe connection V and curving / (see example 3.3) whose three curvature equals d(/2) on p¡4lzl. Therefore (V, /) is an example of a bundle 2-gerbe connection on the tautological bundle 2-gerbe and f2 is a 2-curving of this bundle 2-gerbe connection. We need the following series of Lemmas. Lemma 10.1. Let (P, X, M) and (Q,Y,M^) be bundle gerbes with bundle gerbe connec- tionsVp and,Yq respectiuely. Suppose (f ,f ,i'd): (P,X,M) - (Q,Y,M) is abundle gerbe morphism. i irdu""t an isomorphi,sm of line bund'I"t I , P + fffA)-LQ ouer Xl2l whi,ch commutes wi,th the bundle gerbe products. Then we haue : õ(A), Vp i-, " 1¡lzl¡-ryo " f + forsomeA€ç¿1(X). Proof. Because f commutes with the bundie gerbe products on P and Q, / commutes with the bundle gerbe products on P and (/tzl¡-tq. Si.nce VO Ij a bundle gerbe connection, so is 1¡tzl)-1VO and therefore Vp and /-1o (/tZ)-1V qo f are bundle gerbe connections on P. Since any two connections differ by the pullback of a complex valued one form on the base, we have Vp : Í-'o (/tzì¡-tya o i * a, for some a e O1(Xtzì¡. Because Vp and f-r o ç¡lzl¡-lvç o f are both bundle gerbe connections, we must have ô(a) :0 and so a: á(,4') for some A € 01(x). n Suppose now that (P,X,M) and (Q,Y,M) are the bundle^gerbes of the previous lemma but we now have a pair of bundle gerbe morphisms (fn,Íu,id) : (P, X,M)'--+ (Q,Y, M), i.: I,2. By Lemma 10.1 there exist .41, Az € Ql(X) such that : o1¡tzl¡-1Vo 6(Ao). V p i,' " i¿ * We have the puliback bundle (ft,lz)-tQ -' X of Lemma 3.3. Let Ó , nr'(fr, fù-'Q = r|(h,lù-'Q denote the descent isomorphism of Lemma 3.3 and let (fi, Ír)-'Vq denote the pullback connection on (ft, fz)-tQ + X. Lemma 1O.2. The putlback connection (f1, fù-'Y e satisfies ntr(Í,., fù-tYq: ó-r or;1(h, fù-'Yq o ó + 6(h - A2). Proof. Define a map t h, Íz >, ylz) -- Yl2l by sending the point (rr,rr) of X[2] to the point Ur@t),fz@z)) o¡ylz). The bundle gerbe product onQ gives the following isomorphisms of line bundles with connection over X[2]. U['])-'Q Ø nl'(fr, fù-'Q 3. fr, f, >-' Q (10.1) nrL(h, Ír)-tQ s (¡j'r)-to 3. rr, r, >-t Q (10.2) 104 Here the line bundle ( .fi, fz >-r Q is equipped with the connection ( f1, f2 )-1 Yq, while the line bundles Ul'\-'Q Ø n;'(fr, fr)-'Q and z-¡1(fi , fù-'Q ø (/21)-18 are endowed with the connections (¡j2l;-tVq t rllUr,fù-tYç and nlL(fr,fz)-tVe + (/'f )-tVc respectively. Hence we can combine the isomorphisms 10.1 and 10.2 to get an isomorphism of line bundles n,.r(f,,, rr)-'e s (d'r)-'o 3 ç¡[']¡-te ør;r(rr,, rr)-'e. Note that this is an isomorphism of line bundles with connection. We have the following sequence of isomorphisms of line bundles nr'(ft, Íù-'Q - nrtjr, fù-'Q I P I P- (10.3) nt' (rr, rù-'ea p a p* - rr'(Ír, lù-'e Ø (Í!tt)-'e Ø (r['\-te- (10.4) r,,'(rt, fz)-'e Ø (rl't)-'q Ø (r;'\-'e* 3 n;t(fr, Íù-'e (10.b) (10.6) The descent isomorphism / of Lemma 3.3 is the composition of these line bundle isomorphisms. Here the isomorphism 10.3 is induced by the canonical trivialisation of P 8 P*, the isomorphism 10.4 is induced by the isomorphisms .f1 and fi and the isomorphism 10.5 is induced by the bundle gerbe product mq (in other words by the isomorphisms 10.1 and 10.2). Flom this, and Lemma 10.1 we see that the difference rtL(Ír,fù-'Yq-6-t'or,r(ÍL,fù-tYqolisequaltoô(.41 -Az). ¡ Suppose that we have bundie gerbes (P, X, M), (Q,Y, M) and (.8, Z, M) equipped with bundle gerbe connections Vp, Vq and V¿ respectively. Suppose also that there exist bundìe gerbe morphisms f : P -+ Q and g : Q - -R with f : (Î,/,id) and 9 : (0,9, id). With the notation of Lemma 10.1 we have Vp : 6(A), f-, "1¡tzl¡-1Vo " Í + Vq : ã-r olrtzl¡-1Vn o þ + õ(B), for some Á e f-¿l (X), B € CIl(y). Let $=¡ denote the isomoiphis.T f,Jj. : P = (gl'1"¡lzl¡-tpinducedbythebundle mapþoi: P ---+.Rcovering nlz)"¡lzl ' ylzl -- 7lz). Lemma 10.3. With the notøtion of the aboue paragraph, we haue the followi,ng erpres- s'ion: Vp : (11Ð-t o (þtz) o ¡t2l¡-rv¡) " (1;-Í) + 6(A+ f-B). Proof. The isomorphism g , Q - @lzl¡-rp gives rise to an isomorphism over X[2]: trl)-t( p. Ut2\-L s : fftz)¡-tq - (.f stz)¡-t Also by definition we have an isomorphism of line bundles with connection Ul2l)-L þl4)-tR - çnlzl o ¡lz)¡-L R, where each bundle is equipped with the obvious pullback connection. Next, the diagram canonicar (¡tzl¡-r Glz)¡-tp , Gltl o ¡lzl¡-tp ((/t21)-1ø) 9o P 105 commutes. This is reaily the definition of fflzl¡-tn. Therefore we will have (llÐ-, o ((etzì o /t2l)-lvR) " 6l¡) : ((Ítt)-,9)'/)-'o ((¡tzl¡-t1ntzì¡-1Va) o (((¡tzr¡-'Ð " f) : i-, o (fpr)-rù-1 o ((¡tz1¡-r1ntzl¡-1Va) " ((/t4)-rg) " f : _ 6(B)) o Í-, " 1¡rzl¡-1(Vo f : i-r" 1¡tzl¡-tVo "f -6(Í.8) : Vp- 6(A+f.B) Suppose, as in Lemma 10.3, that we have bundle gerbes (P,!, M), (Q,Y, M) and (R, Z,M), but that we have pairs of bundle gerbe morphisms (fu, Ín,id) : (P, X, M) -' (Q,Y, M) and (0¿, g¿,id) : (Q, Y, M) -' (.R, Z, M) for i : 1, 2. Using Lemma 10.1 again we get vp : in' "(/'l)-tvo " i¿* õ(An), : o 6(Bo), Y e øn' " þ14)-1Vn þ¿ i for some A; e Qr(X), Bo e fll(y) f.oyi: I,2. Lemma LO.A. Wi,th the aboue notation, the pullback connecti,on (h o f t, 9z o fù-l[ n on (gt o f t, gz " fù-r R -+ X satisfi,es nttbto ft,9z" fù-rV n: ó-r orîr(gto Ít,9zo f2)-rV poó+6(Ar- Azi fiBr- fiBr), where ó, nttGro ft,gz" Íz)-rR= rl1(gto ft,gzo fz)-'R is the descent isomorphism of 3.3. Proof. Same as for Lemma 10.2. n t}.z Proof of the main result We now turn to the proof of Proposition 10.1 Proof. Suppose r¡¡e are given abundle 2-gerbe (Q,Y,X,M).Let V be abundle gerbe connection on (8,Y,X12)) with curving.f € C)'(y) such that ó(/) : Fv, where Fv is the curvature of Q with respect to V. The three curvature ø then satisfies ri¡a : df . Let us look at what happens over Xt3l. We have induced connections n'o-lV on the bundie gerbes (nn'Q,r'irY,¡tll¡. Let r;rf denote the curving for z-n-lV induced by f . Let @?l)-'"r'V + @l:l)-rø"tlv denote the tensor product bundle gerbe connection on r¡tQ Ø nl'Q -- r¡1Y12) xy¡z1rsrY[2l. The bundle gerbe morphism m : (n;rQ Ø nl'Q,r;LYlzl xxts) rllYtzl, ¡tsì¡ - (n;'Q,r;|Y, Xtz)1 gives rise to a C'-bundle isomorphism fTL tn;'Q nrLQ Ø nltQ (mÍ21 ) (nrtY x y¡s¡ rrtY)lzl 106 ñt clearly commutes with the respective bundle gerbe products. By Lemma 10.1 there is a one form p e 01(zr'¡1Y xypl rlrY) such that @?l)-r"rtv + þ!'r)-trr;tV : ñ¿-r o (ml2))-LrttV o m + 6(p). (10.7) Hence we get p*r'v (p?l). F*;,o + @91). : lmlzl¡* F*-,o + õ@,p), (10.s) and hence 6(pi"l'Í + pitrll f - **n;'f - dp) :0. (10.9) Therefore, there exists p, e Q2(Xl3l) such that pij\rf +pinltl:m*rlLf +dptnlrrrþ, (10.10) where z16r, denotes the projection Ytzs:7\1Y xyplrsrY * ¡lal. It is easily checked that ria - ria * rla : d"p,. (10.11) Now let us look at the situation over X[a]. We will use the notation introduced in Section 7.1 so that we will denote r;rY by Yzs, nltQ ØnltQ by Qns, r;tr¡LQ by Q3a and so on. As mentioned in Section 7.1, the bundle gerbe morphisms m1 and m2 are defined as in the following diagram: Qnz¿,4 Qrrn ,.r,o,l.tt l^ Qrsa+ QM,. So rn1 : rlt, o (n ø 1) and T7¿2 : r\t o (I Ø rn). Recall from Definilion 7.2 that Á : (mt,mz)-LQr¿ is the line bundle oîYtzs+ which descends to the associator line bundle Aon¡taJ. p¡t¡hermore Á hu. asection â which descends to asection ¿ of A-the associator section. Á inherits the pullback connection V; : (rnt,mz)-LYg (here V1a denotes the induced connection on Q1a) relative to which Á hu. curvature equal to (m1, m2)* 6(n;'n;t Ð : mTtrlrn;' f - mitr;Lt;L f . Now * ¡ ffizT2'Tz--] -] I : (nl'* o (1 x¡¡1n¡ rarm)).r2tn;'Í : (1 x¡1a¡ rnLm).(trrLm)*r;Lr;Lf : (1 x¡1a¡ rnrm).12'@inrt f + pi"l'Í - dp - nhrnnöp) : pinlrrtL Í * (1 x¡¡n¡ rnLm).p\trntn;'f - d,(l x ypl ralm)*nl'p - nhrrnräp : pin;'ntl Í 1- pitrnLr;' f + pinn'r*l f - d,(1 x y¡n¡ rnlm)*rrL p -I x dtrnL p - TI,,"n("itt + riù. r07 Similarly we have mitrt'rr'f* -l -1 pintlntt Í I pitrrrr¡' f + pin,'r¡r f - dtrrr p x r t -d(tr rtm x¡1a¡ l)*zf p - nh"rn(t it-t + näp). Hence we have mltr;Lr;rf -mir;rr;rf : nh,,nõ?ù1-d'(trrtp x 1) + (nl'^xv¡n17)"t3Lp -L x rnl p - (1 x¡¡n¡ nnlm)*rrr p). On the other hand we can write (V;)(a) : ô I â for some one form ã. onYzsa. Thus we must have dô : nîrrrnõ(lt) + dpr, where p1e QL(Yvrrn) is the one form nr' p x L + (tr¡Imx¡1a¡ 1)*r"f'p - L x rll p- (1 x¡1n¡ rnlm)*nrlp' Let þ : n¡IA -- rll Âdenote the descent isomorphism of Lemma 3.3. By Lemma's 10.1 and 10.4 we have ,rrrV,4: ó-r o n;rV ¡o ô + õ(pr). Therefore the connection V¡ - pt on Á descends to a connection V ¡ on A. Let V¿(a) : o8¿ for some a € f-¿1(Xtal). Then we have ã- pr:Th""nd.Hence we get ¡rip - ript + räp - lri.t-r : da. (10.12) If we can show ô(o) : 0 then v/e are done. To do this we need to examine the situation : . over X[5]. 17¿ induces five bundle gerbe morphisms M¿ : Qns¿s - Qu for i 1,. . ,5. We can form five Ct bundle" Dvr,t,tr: (Mt, Mz)-'Qrc, D*r,*, : (Mz, Mr)-'Qru, . . . , ñ*u,*r: (Ms, Mr)-'Qtu onY1234s, all of which descend to CX bundles D,lor,,rrr,Drûr,Ìør, . . . , DñÍr,rú, on X[5]. Since each pair of maps (Mn, M4ù has M¿ factoring through rn1 and, Mn+i factoring through TL2 or vice versa, v¡e can identify DMr,,ø, with zr;14, 1A* D,;t",¡rr"with zr;1.4,, Dtfrr,¡rrnwith zr¡14, DËtn,rrt, with zrn and D¡ør,ñt, with n;1,4* as explained in Section 7.1. Also note that the associator section â induces sections ã,¿¡ of. b*,*,. Note that the bundle gerbe product in Q supplies an isomorphism D*r,*r Ø Dyr,u" A "' A D¡ør,ur + DM.,M, (10'13) and the coherency condition on â is thal ã,p8 âzs I . . . I âsr is mapped to the identity section of. D¡4r,¡4, : (Mr, Mt)-'Qru. Let Vrs denote the bundle gerbe connection on Qrs induced by V. Then Vrs in- duces a connectiott Ûrj on the (line) bundle Ð*0,*, by pullback' y;i (Ma, M¡)-rV1¡.. Since Vrs is^ a bundle gerbe connection, under^ the isomorphism 10.13 above, the connection Vtz+...*Vsr on D¡4r,¡4"4'.'A DM,,M, is mapped to the connection Vr, : (Mt, M)-1V15 on D*r,*r. Hence we have that (Ûtz + "'+ Ûsr)(ãnØ"'8â51) : ¡. (10'14) We now need to calculate i,i¡(ã¿¡). Using Lemmas 10.1- 10.4 one finds V : (tr*ria* 1 x f- (mtx 1)*p (*rx 1)-p) Ø ãp : rz(ãn) ø - Yzslezz) : (tr"r[a+ (1 x m x 1). p1) Ø ãzz Ûrn(ârn) : (tr*r[a+ (1 x mt)* p- (1 t mz). p) Ø ãsa, Vnu(ânu) : (-n*rla- (1 x I x m). p1) Ø ãqs : Vsr(¿sr): (-r*ria-(m x 1x l)-pt) 8âsr. 108 Here we have abused notation and denoted by n'the projection rytzsts:Yy2sas -t X[5]. If one does the calculation then one finds that ,: -a (Vrz + "' + Vsr)(ârz 8 "' 8âsr) : ô(o) Ø (ãrr8 "' 8 âsr) (10.15) and so we conclude that ô(o) :0, completing the proof. ! Let us explicitly write down the bundle 2-gerbe connection (V, /) on (8, Y, X, M) constructed in the above proof. Since ô(c) : 0, we can soive a : õ(þ) for some 0 e Q'(XU¡. Therefore we have õ(p) : 6(d,P) and so r'¡/e can frnd u e CI2(Xt2l) such that ¡r : d,þ+6(u). Now give the bundle gerbe (Q,Y,¡tzl) the bundle gerbe connection V with curving f' : f - Tru.Fbom equation 10.10 we have pînrL f' + pît lt Í' : m*rlL f' I dp * nhrr1t - õ(r)) : m*rîr f' * rir"rd,B. (10'16) This equation implies that the three curvature 4¡' : u - du of the bundle gerbe (Q,Y,¡tzl) with bundle gerbe connection V and curving /' satisfies 6(a'):0. There- fore (V, "f') is a bundle 2-gerbe connection for Q. 109 110 Chapter 11 Comparison of the Cech, Deligne and de Rham classes associated to a bundle 2-gerbe 11.1 A Cech three class Let (Q,V, X, !tI) be a bundle 2-gerbe (Definition 7 .2). We will show how to construct a C"-valued Cech 3-cocycle associated ro Q. Choose an open covering {Uo}oq of. M, all of whose finite intersections are empty or contractible, and such that there exist local sections s¿ : U¿ - Xlun of. ¡r : X - M. We have the usual maps (t,,t¡) i(J¿¡'--+ yl21,m,- (s¿(rn), s¡(m)). Let (Q1,Y¡,U;¡) denote the pullback of the bundle gerbe (Q,Y,¡tzl) to Uo¡ by (s¿, s¡). Thus Y¿¡ is the space over U¿¡ whose fibre at m € U¿¡ is Y¡^ : Yþ¿(m),s¡(rr)). Since LI¡¡ is contractible, the bundle gerbe (Qo¡,Y¡,U,i¡) is trivial. Hence there is a : C'-bundie ¡,q - Y;, such fhar õ(L¿¡) rll L¿j ø r;1Li, = Q¿¡ o"Y!r). Over (J¿¡¡ the bundle gerbe morphism ñ' : (ñ,m,id) induces a map m : : ^Yj* Y¡xxurY¡ -Yn. Let L¡¡¡: Ljt Øm-rL;r*Lø. Notice that we have ô(i¿¡¡) - Q¡o Ø (mÍ2|¡-r}.r Ø Q¿¡, which is trivial. In fact, the C'-bundle L¡¡¡ descends to a bundle Luj* - U¿¡¡. To see this we need the following lemma. Lemma 11.1. Suppose (P, X, M) and (Q,,Y, M) are bundle gerbes w'ith a bundle gerbe morphism(i,f ,¡All. P - Q. If P and,Q arebothtriui,al, sothereeristL - X, J -Y suchthat P - 6(L),8 - õQ), thenthebundle LØ ¡-r¡" ouer X descends to M. Proof . Î , P ---+ Q induces an isomorphism õ(L) 3 6U-'J) as pictured in the following diagram: P = '(Íl2l)-tQ ð(¿) -* 6U-'J) This diagram lives over X[2]. Clearly over X[3] we get the following diagram, induced 111 from the one above by tensor product and pullback, 6(P) j* ô((¡trr;-tç¡ 66(L) = - 66(f-rJ) Let sp and sç denote the sections of ô(P) and d(Q) induced by the bundle gerbe prod- ucts respectively. In the second diagram above, the isomorphism ô(P) -' ô((/tz1¡-tç¡ maps sp to s(/r2t)-re : (/t3l)-lsq, since this isomorphism is induced by a bundle gerbe morphism. Similarly, the vertical isomorphism ô(P) -* ôô(I) maps sp to 1¿ and the vertical isomorphism ô((/lzf ¡-'Q) - 66U-1J) maps i¡tsl)-lsq to (/tsl¡-t17. Here 1¿ and 1.¡ are the sections induced by the canonical trivialisations of õõ(L) and ôô(./) respectively. It follows now that if we define a section t of 6QØ Í-rJ-) using the isomorphism ô(I) - 6(f -r J) from the first diagram above, then ú satisfies ô(t) : 1 and hence LEt ¡-t¡* descends to M. n It is easy to see from this lhat L¿¡¡ descends to some Ct-bundle L¿jn - Ul¡r. Next, over U¿¡¡¿ we have two bundle gerbe morphisms rh and rn2 obtained from the bundle gerbe morphism m: (ñ,rn, id) , TrrQ Ø nrtQ -- T;rQ over X[3]' mt,rTtz: Qn Ø Q¡x Ø Q;¡ + q'* The C" bundle (mt,mz)-|Q¿t onY¡rt:Yu xuY¡n xy Yt¡ descends to a Ct-bundle A¿¡¡,¿ on U¡¡¡¿. It is clear from Definítion 7.2 that A¿¡¡¡ is canonically trivial with a section a¿jkr: a(s¿,s¡,sr,s¿) which clearly satisfies 6(a)4n-: 1, where 1is the section of õ(A¿¡¡¿) on(J¿¡¡¿,,induced by the canonical section of 6(A) -- ¡[sJ. We need to relate (mt,mz)-rQato the principal C' burrdles L¿¡¡. The ùlap rn1 is defined by composition as in the following diagram. Y,¡nt4Y¡tLVt Therefore LntØL¡xØL¿¡Ø*;'Ln : Lnt Ø L¡* Ø L¿¡ Ø (m o (m x r))-l Lir : L*t Ø L¡x Ø L¿¡ Ø (m x t)-rm-r Li, - Lnt Ø L¡t Ø L¡¡ Ø (rn x 1)-1ltr¡t,L¿¡¿ Ø Ljt Ø Li¡l : L*t Ø m-t Llt Ø L¡t * Lu * Lii Ø rynr u,L¿¡¡ N r¡l.r,(Lut ø L¡*ù. The map m2 is induced by composition as indicated in the following diagram t/ lxttl ,. rr¿ Y¡jt"L -t--t+ Y¿m+Yt II2 Then Lnt Ø L¡x Ø L¡¡ Ø rn;l Lit : Ln Ø L¡xØ L¿¡ Ø (m o (r x m))-L Li, : Lu Ø L¡x Ø L¿¡ Ø (t x m)-Lrn-L L[, : Lnt Ø Lit Ø L¡n Ø *-L Lin * Lu Ø ryrr,L¿¡,¡ ni\,,(Ln¡t Ø L¿rù' - 'aJKa' Therefore we get (Lxt Ø L¡* Ø L¿¡ Ø rn;r Li). Ø (L*, Ø L¡x * L;i Ø n'Lll Li) - mlr Lu Ø mrr Lk : (mt,mz)-Lõ(L*) æ (*r,*r)-'Q, N niln'An¡o'' Therefore we must have L¡nØ LintØ L¿¡tØ Li¡¡"= A¿¡¡"¿ over U¿¡¡¡. Now choose sections o¿jn € l(Uoir, L¿¡¡,) and define g¿jm : Uojnt - Ct by o¡m Ø oi*¿ Ø o¿jt Ø ol¡u' 7¿¡t"t : d¿jt"I. We have the following proposition. Proposition 11.1. gq*t sati,sfies the Cech S-cocycle condit'ion g : ¡ w^g ¿xT^g ¿¡ t,,g utn^g ¿¡ kr I, and, hence is a representat'iue of a class tn H3(M;üì: í (U;Z). Proof. AII that one really needs to do is to check that g¿¡¡¿ does indeed define a cocycle. This follows from the coherency condition on ø. One can also check that another choice of the sections s¿, or the line bundles L¿¡, ot the sections o¿jt" changes g¿¡nby a coboundary. n There is another method of calculating the Õech cocycle g¿¡¡,¿ which is similar in spirit to the method. used to calculate the Cech representative of the Dixmier-Douady class of a bundle gerbe. Let (Q,Y,,X,M) be a bundle 2-gerbe. Choose an open cover {Un}oq of M such that there exist local sections s¿: U¡ + X of r: X + M and such that each non-empty finite intersection U¿, l1 . . - ¡ U+ is contractible. As above, form the maps (sn, s¡) : (J¿¡ '-'+ X[2] and let (Q¿¡,Y¡,U.¡) denote the pullback of the bundle gerbe (Q,Y,¡tzì) with the map (s¿, s¡). 5s Y;i has fibre Y¡^ at m e U¿¡ equal to Y1"n1-¡,"¡(rn)) and there is an induced projection ryn, : Y¡ * U¿¡. In certain circumstances, for instance if. ry : Y + ¡[zì it a fibration, then one can choose sections o¿¡ : U¡¡ -, V¡ of. rvu,. We will assume that this is the case (in general one would only be able to choose an open cover {Uü)"r>r, of U¿¡ such 113 that there exist local sections oit of. ry: : Y¡ - U¿¡). Denote, for ease of notation, m(o¡r",o¿¡) by o¡¡ o ot¡. Then we have a map (J¡it" - {[2] which sends m € (J¡¡¡, to (oon(*), (o¡¡, o on¡)(m)). Let Q¿¡¡ : (a¿n, o¡¡ o oq)-rQ¿¡ denote the pullback bundie on (Jo¡u. Choose a non-vanishing section pt¡n of. Q4r - U,i¡n. Define a non-zero section s¿¡¡¡ of the line bundle (o*, (og o o¡n) o o;¡)-rQ¡t ovet U¿¡¡¿ by s¿jkt : mq(fn(p¡nt Ø e(o¡¡)) Ø p¿¡ù. (11.1) Here mq denotes the bundle gerbe multiplication on 8. We can also defrne a non-zero section t¿¡¡¡ of. the line bundle (ont, o,", o (o¡t o o¿¡))-rQ¿¿ by puttiîE t¿iw equal to mq(rn(e(onù Ø pu¡u) Ø p¿*ù. (11.2) Therefore we can use the associator section to define another section of. (o¡,(o¡¿oo¡¡)o on¡)-rQt bY mq(a(o¡r¿, o ¡t", c¡) I s¿y¡¿). Thus we can define a function e¿¡¡¿: U¿¡¡"¿ -t C' which satisfies t¿jt"t: mq(a(o¡¿,o¡¡",o4) I s¿¡¡¿) '€¿ju. (11.3) One can show that e¿r'¿¿ satisfies the cocycle condition e¡nme¿¡"1*e4me1¡L¡" eunt : I. We have the following lemma. Lemma tL.2. The cocyclê e¿jxt defined aboue gi,ues ri,se to the sanl'e class in the Cech cohomologA group Ht(tut;Ci) as the cocycle 94ut of Proposition 11'1 Proof. We have (o¡¡,o;)-rL¿¡¡" = L¿jt. Bú L¿¡¡ - Lit Ø m-LLit * Lu and so (o¡n,o;¡)-rLnr^ : o¡r,1Lin Ø (o¡n o o¿¡)-rLir Ø o;lLnr' Hence we get (o¡x,o4)-rL¿k : L¡¡n Ø o¡*L Lîn Ø o¿jr Lij' Also we have (o¡n o o¿j, o¿t")-LQu" N (o¡¡ o o¿¡, o¿¡,)-L õ(L¿¡) : ,nx'Lot" Ø (o¡r o onr)-L Lir ñ L¡¡t" Ø (o;¡t Lit" Ø o¿kr Lik Ø oijr Lii)* . Therefore u/e can construct a section p1u of. (o¡poo¿¡,ont")-rQ¿¡ by taking the section o¿¡¡ of. L;¡¡ and choosing sections s¿, of ou'Ln¡ and forming the tensor product section o¿ju Ø (sr.r I si¿ I s¿¡)*. Fbom here it is not too hard to show that the cocycle e¡¡¡¿ is cohomologous to g¿¡¿¿. ¡ Erarnple 11.1. Consider the lifting bundle 2-gerbe (Q,P,P,M) of Proposition 9.1 as- sociated to a principal G bundle P - M where G is part of a central exbension of groups BC* -- G - G. We associate a C" valued Õech 3-cocycle g¿¡¡r¡: (J¿iu -r Ct to Q as follows. Choose an open covering {Uu)na of M aIl of whose frnite intersections are contractible or empty and such that P is localiy trivialised over U¿ - say there exist sections s¿ of P_+ M over (J¿. Now let o¡¡ be sections of the pullback BC" bundle h, : (s¿, s¡)-l-P over (J¿¡. We can now define a Cech 2-cocycle 94n : U;¡n --+ BC' 774 representing the class in U'(¡W;BAiò associated to the BC" bundle gerbe (P,P,M) by mp(o¡n Øsa o¿¡) : oit ' g¿jx. It follows that the image of the class [g¿¡¡] defined by gn¡t under the coboundary map H2(M;BAfu) - H3(M;8i¿) i. the class defined by the Cech 3-cocycle g¿¡¡¿ : U¿jm -- C" . g;jt"t would then be defined by choosing lifts gqt,: Ue¡* -+ EC" of the gr¡u and. then settinS g¿¡m: g¡ng¿nLîl¡¡. We can also construct a Õech 3-cocycle g¿¡7,¿ representing a class in H3(M;Cir) : Hn(M;Z) associated to a stable bundle 2-gerbe. We record this in the following propo- sition. Proposition 11.2. Let (Q,Y,X,M) be a stable bundle 2-gerbe. Then there is a Cech 7-cocEcle gijkr representi,ng a class ln Hs(tt[;Cl,¡) associated to the stable bund,le 2-gerbe a. The class g4*t is constructed as follows. Choose an open cover U : {U¿}¿ç¡ of. M all of whose intersections are contractible and such that there exist local sections s¿ : U¿ + X of. r : X - M. Form the maps (sn, s¡) : (J¿¡ --+ X[2] and let (Q¿¡,Y¡,U1¡) denote the pullback bundle gerbe (so,s¡)-'Q. Since U¿¡ is contractible there is atrivialisation.L¿¡ of Q¿¡.AIso the stable morphism m: r;IQØT,LQ - T;IQ induces a stable morphism rn¿ju : Q¡*ØQ¿¡ - Q¿n corresponding to a triviaiisation (L,n,ru, Ó^n¡r) of Q¡nØQi*ØQ¡¡. Therefore there is a C" bundle L¿¡¡ on U¿¡¡ such l,hal L^n,n = L¡nØLixØLn¡Øn-'Lnj*, where n :Y¡r xmYx xtrY¡ -'-+ U¿jn denotes the projection. It is not hard to see that there is an isomorphism L¡xtØ LiutØ L¿¡tØ Li¡t = A¿¡¡¿ of. C' bundles on U¿¡¡¿ (here A¿¡¡¿ denotes the pullback C" bundle (s¿, s¡,sr, j¿)-1,4,). Therefore if we choose sections o¿¡¡ of. the bundles L¿jx, then we can deflne a Cech 3-cocycle g;¡r"t : Uojn¿ - C' by o¡nt Ø oip Ø o¿jt Ø ol¡*: a¿jnt' 9¿jm, where a¡¡¡¿ denotes the pullback (s¿, sj, s¡,s¡)-ra of the associator section a of. A. tL.z A Deligne hypercohomology class Let (Q,,Y,X,M) be a bundie 2-gerbe. Suppose the bundle gerbe (Q,Y,¡[zJ) comes equipped with a bundle 2-gerbe connection (V, /) so that the three curvature form ø of the bundie gerbe (Q,Y,¡tzl) satisfies ô(ø) : 0 in f-I3(Xtsl¡, as in Definition 10.1. Let K € CI3(X) be a 2-curving for ø so that õ(K) : a. We will construct a class D(Q,V,l,K) associated to the bundle 2-gerbe Q with bundle 2-gerbe connection (V, /) and 2-curvíng K in the Deligne hypercohomology group Ht(M;Cio - A'* - Q'u - Air) Choose an open covering {Un}nq of M such that each finite intersection U¿rl. . 'n¡J;,e is either empty or contractible and such that there exist local sections s¡ : U¿ - X of zr for all i e L As in Section 11.1, let (Qn¡,Y¡,t/r¡) denote the bundie gerbe onU¿¡ obtained by pulling back Q with the map (s¿, s¡) : U¿¡ - yÍzl. Therefore V induces a bundle gerbe connection V¿j on Qt¡ by pullback. Let Fyn, denote the curvature of pullback Qq with respect to Y ¿¡. The curving "f on Y induces a curving Ír¡ for Vu¡ by with the map Y¡ + Y. Similarly the three curvature of Q¿¡ with respect to the bundle gerbe connection V¿j and curving fq is uor: (s¿, sj)*, e Qs(U¿¡). Since U¿¡ is contractible, ø¿¡ must be exact and so, using the same notation as in Section 11.1, let L¿¡ -.Y¡ denote the C"-bundle which triviaìises Q¿¡. As noted in Section 5.1, the 115 isomorphism H2(M;Afu) = H'(¡ut;Cñn - C¿þ) of truncated Deligne hypercohomology groups implies that a bundle gerbe with trivial Dixmier-Douady class can be endowed with a trivial bundle gerbe connection - so that if Q is a trivial bundle gerbe with a trivialisation Q - õ(L), then, given a bundle gerbe connection Vq on Q, one may choose a connection V¿ on -L so that the isomorphism Q = ô(I) is an isomorphism of bundle gerbes with connection: (Q,Vò - (ô(¿),ô(V¿)). Therefore'ù¡e may choose a connecti,on V¿3 on L¿j - Y¿¡ such thal (Q¿¡,Vnr') = (6(Ln¡),ô(V,¡)) is an isomorphism of bundle gerbes with connection on Y,fl. Let 4¡ denote the curvature of L¿¡ with respect to V¿¡. Then we must have Fe,, : 6(Íu¡) : 6(Fo¡). Therefore there exists t¡ € Q2(urt) such that F¿j : Ír¡ - r*yr. (11.4) Flom this we deduce 0: dF¿¡ : dÍ¿¡ - n*d1¡j and hence u¿j : d'Y¿j. (11.5) Next we examine the situation over U¿¡¡. Recall equation 10.7 from Section 10.2, t)-r"lrv @ + @tl)-tnttV : fn-r o (mlz\-trttV o m+ 6(p). (11.6) Since (V, /) is a bundle 2-gerbe connection and the associated three curvature ø of the bundle gerbe (Q,Y,¡tzl) satisfies ô(ø) :0, we must have pi:¡:rf +p;rlLf :m*rlrf tdp+rþ,,,0, (11.7) onr;LY xy¡z.rsLY (Compare with equation 10.16). Here B is a one form on X[3] such that V¡(¿) : 6(P) I a. This formula pullsback to Y¡n : Yjr x u Y¡ to give the following equation in Q2(\t¡,): pif ¡* + piÍ¿¡ : m* Í¡t * dp¿¡x 4- riu.rd,B¿¡¡, (11'8) where p¿jt, is the pullback of p to Y¡* via the inclusion map Y¡x * r.,-.Y¡ -- rlrY x xpt n/Y and p¿¡¡: (s¿, s¡,st")*0.Putling equation 11.6 back to (Y¿¡¡)[2ì gives Y¡n*lu ñ,-Lo(n'¿l2l)-\Vitoñ+6(pn¡*). (11.9) Recall also from Section 11.1 that we defined Lrru : L¡r, Ø rn-L Lîk * Lø on Y¿¡¡. L¿¡¡ has the tensor product connection iorr:Vr*-m-LÛu"-lYq' As explained in Section 11.1, i¿¡¡ descends to a bundle L¿¡¡ oî (J¡¡¡ and we may choose a non-vanishing section o¡¡¡a of L¿jn - [4¡t. F]om equation 11.9 we get : o ñ õ(pr¡n), 6(V ¡r,+ V¿¡) ñ,-r " 6(m-rl ¿n) * which implies that the connection Yrr-*-tÛn,"*Vri - P¿jt" on Ln¡,, descends to a : we have connection on L¿¡¡ - call it V;¡¡. Let Ya¡¡(oijk) d¿i*Ø o¡¡¡,. Then d\n¡t ri, rd'a¿¡¡ : F¡t - m* Fu" * Fti - ' 116 Therefore from equations 11.4 and 11.8 we have rþr.ud,a¿¡¡,: nþo,o@þn¡*- õ6n¡) and so dAu¡r:'Yjk -'Y¿tr l'Y¿j, (11.10) where we have put A¿¡¡,: þ¿j* - a¿jn. One can check that the isomorphism L¡*t Ø Li*t Ø L¿¡t Ø Li¡¡ = A¿¡¡"¿ maps the tensor product connection Y¡u-Vu,¿+Y¿¡t- V¿r'¿ on L¡mØLixtØL¿¡tØLi¡n to the pullback connection (s¿,s¡,s¡,s¿)-1V¿ : V¿jnt on A¿¡¡¡. Since V¿(o) : 6(P), Y u¡núan¡nù : õ(þ¿¡u) and so we have ajkt - a¿nt i o¿jt - d¿j* I 9ÇIþS11,L: õ(\n¡n), or 6(A¿¡¡,) : gal,iOr¡¡¿. Therefore we have the set of equations : A¡r,t - A¿xt * An¡t - Ao1 9øItdg¿¡t t .yjk dAo¡r : - l* * 1¿¡ d1¿¡: K¡-Ko- It follows that the quadruple (go¡t"t,A¿jn,'Y¡j,K¿) defines a class in the Deligne coho- mology group Ht(M , Ciú -- 9'* - ç)2u - Qlo). Hence we have the following proposition. Proposition 11.3. Euery bundle 2-gerbe (Q,Y, X, M) equipped w'ith a bundle Z-gerbe connection (V,/) and 2-curaing K giues riseto a class D(Q,V,f ,K) i,nthe Deligne cohomologa group Ht(¡W;Cio - Qt¡r,t - Qzu - AÎr) In particular it follows from this by standard double complex arguments (see eg. [+]) that the class defined by go¡t"t is the same as the class defined UV #O Note 11./. It is also possible to construct singular analogues of the 2-curving andfour class of a bundle 2-gerbe in a similar way to that used in Chapter 4. 11.3 The first Pontryagin class Recall the bundle 2-gerbe (Õ,P,P,M) constructed in Proposition 9.3 associated to a principal G bundle where G is a compact, simple, simply connected Lie group. We have the following proposition. Proposition L7.4 ([8],[10]) . The class in HL(M;Z) associated to the bund,le 2-gerbe Q is the transgressi,on of [u], that is the fi,rst Pontryagi,n class h of P. Proof. We calculate the Õech four class of the bundle 2-gerbe I and get it exactly equal to the Õech cocycle defined by Brylinski and Mclaughtin in [8] and [10]. We can then apply Theorem 6.2 of [8] to conclude that this Cech four class is p1. We set out to calculate the Õech cocycle g¿¡¡¿. First choose an open cover {Uo}oa of M plzl relative to which r : P -> M has local sections s¿. Since_.Þ -- is a fibration, 'û/e can calculate the Cech cocycle representing the four class of Q using the second construction of a Õech co_cycle as detailred in Section 11.1. Recall that we first choose sections o¡¡ : IJ¿¡ -- P¿¡: (s¿, si)-'P. This is equivalent to choosing maps 'y4: U¿t x I ---, G tr7 such that %¡(m,0) : t and y¡(m, 1) : g;¡(m). Next we have to choose sections p4u : (Jqr - (oa,, o¡¡o04)-18¿¡,. This amounts to choosing maps 1;¡n : Uq*x I x I ---+ G such that I¡*(m,O,¿) : %x(m,t),'y¿¡x(m,l,¿) : (lo¡o7¡¡l¡ù(m,t), l¿¡n(m,s,0) : 1 and y¡¡,(m, s, 1) : g¿¡(m)g¡k(rn) (This uses the fact that r{G) : 0 and the cocycle condition g¿¡g¡t - g¿¿ satisfled by the transition functiorLs gij).Define the section f¿¡¿¿ : p¿¡ù pu"ù. we of the bundle- (onr,ot"t o (o¡n " on¡))-r}1 by t¿jt"t mÕ(th(e(o*,) ø Ø If denote ahomotopy with end points fixed /from apath o to apath 0by a 2-arrow þ : a è p then we see Ihaf ñ,(e(ont) Ø pn¡n) amounts to composing horizontally the 2-arrows in the following diagram - ^lih -/.-.\9ih^lht 1 il. 9¿t, tl 9¿t 'l¿jo9íj''ljh 9¿k1kt The bundle gerbe product mq(ñ,(e(orù Ø pn¡n) Ø p¿xt) is obtained by composing the 2- arrows ^l¿t"t: ^l¿¿ è'\¡¿t o9¿t'Yt¿ and 'Y¿jt"o|¿*'YtcL 'l¿*o9¿*^lt"¿ + ?Y¡¡o9¿¡'Y¡¡)og¿t"^Yt"t vertically. In a similar manner, one constructs the section s¿jt"r : rn6(m(OixtØ e(o¿¡)) Ø p¿¡ù of the bundle (o¿¿, o¡¡o(o¡noon¡))-LÕa. We make t¿¡¡¿ into a section of. (o¿¿, (o¡¿oo¡n)oo¿¡)-rÕn by using the associator section: mq(a(out,o¡x,oq)Øt¿¡xt). Finally we define the cocycle g¿¡ntby s¿jkr: m6(a(ort,o¡r",oq) Øt¿¡uù' g¿iw. We can get an explicit formula fot g¿¡¡¡ as follows: we choose a homotopy with endpoints fixed Hnjxt t U¿jw x I x I x I --+ G such that Ho¡nt(m,0, s, ú) : ('',4 o g¿¡'Y¡r¿)'l¿¡¿(s, ú), Ho¡u(*,1, s, ú) : a('Y*t,"Y¡x,'Y;¡)('Y¿¡xo g¿tlrù'yl"t(s,t), H¿¡¡¿(m,r,O,,t) : 'Yu(m,t), H¿¡¡¿(m,r,L,t) : (1u¡ o g¡n'Y¡u) o 9¿t^lt"t, Hn¡n(*,r, s, 0) : 1 and Hn¡u¿(*,r, s, 1) : !¿¿ aîd we set 9¿ju : exPU¡s Hi¡xY)- This is just the integraì of. u over the tetrahedron described in [8] and [10]. Thus our cocycle must agree with that defined by Brylinski and Mclaughlin. n 118 Chapter L2 Trivial bundle 2-gerbes L2.L Trivial bundle 2-gerbes LeI (Q,Y, X, M) be a bundle 2-gerbe. Choose an open cover U : {U¿}¿ç¡ of. M . Form the sets X¿: n-'(Uu). Then the collection {Xrhe¡ forms an open cover of X. Define maps 3¿ : X¡ -- ylzl by sending r € X¿ to (r,s¿(rn)) ç ylz) for s¿ : (J¿ + X a local section of zr and m: r(r). Pullback the bundle gerbe (Q,Y,¡tzl) to X, with the map ,3¿ and define Q¡: 3¿LQ andY: 3¿rY. Thus Y¿ has fibre Y" at r € X¿ equal toY6,s¿(rn)¡ where again m: n(x). We will assume for the time being that we can choose sections o¿¡ : U¿¡ - Y¡ where Y¡ - U¿¡ \s defrned above. Define maps ó4 : 4 - Yj covering the identity map on Xo¡ by sending U e V to m(o¿¡,A) € Y¡. The þ¿¡ extend to bundle gerbe morphisms ór¡ : qo -- Q j, ón¡ : (ón¡, ón¡,id) with ór¡(u¿) : ñ'(e(o¿¡) 6 ,o) for u¿ € Q@,,sr) where (ar,ar) e Yl2l . Let Z : lJ,etY¿ and let n2 : Z + X denote the obvious projection. The fibre product 2lzl : Z xx Z is equal to the disjoint union IJ.,¡YxxY¡. Define maps X;¡:frxxY¡ -Vj2l by Xn¡(Ao,U): (óu¡(a),g),forU¿ €Vand,g¡ €Y¡. Let L¡¡ d.enote the pullback line bundle XijrQj on[xxY¡. Define a line bundle L + Zl2l by setting 7 : lJt,i L¿¡ with projection map -L -- 2lzl induced by the projections L¿j -Y xxY¡. Our aim is to show that the triple (L,Z,X) forms a bundle gerbe. To do this we need to define an associative product m¡: r;LLØr;rL - r;tL on Zlsl. Since 7[zJ : Uu,¡,rY xxY¡ xx Y¡, this amounts to finding a bundle map L¡** Lu -¿ Lrn which satisfies the associativity condition over Y xxY¡ xxYu xxY. Let u¡n € Ljku¡,ort, u¡i € Lii@,o¡) with g¿ € Y,, Ai € Y¡. and g¡ € Y¡". So u¡r € Qtp¡r@),a*),u¿i € Qiron¡ ñ(e(o¡r) Ø u¿¡) € Qk6,*oanjtu¿),öjt @¡))' Therefore (u ñ,(e(o ro¡ mq ¡ x Ø ¡ r) 8 ) ) € Q (ó ¡ *"ót¡ (s¿),sx). (o¿¡,,m(o¡r,ou¡))-LQ and choose as Denote as before by Qr¡t, the line bundie ?nUo¡n before a non-vanishing section o¿¡¡ of. Q¿¡¡. Define a section €,¡r of the line bundle (óu,, ô¡* ón¡)-1Qn % Ot " î €o¡*(a.) : me(ã(o¡k¡ oii ¡ a) Ø ñ(o,¡n8 e(gc)))' 119 where Ar €Yn and where â denotes the section of the pullback line bundle (mt,rnz)-rQu which is mapped to the pullback of the associator section rtls¿a under the isomorphism of line bundles (^r,*r)-'Qu Here 21144 : -+ Xla) denotes the projec- = "t]r+A. Yzsa tion. Finaliy we define f.or u¡t, u¿¡ â,s above, m¡(u¡n 6 uu) : rne(rne@¡x Ø ñ(e(o¡n) Ø u¡,¡)) ø €o¡r(an)). (12.1) We have the following proposition Proposition 12.1. Suppose (Q,Y,X,M) is a bundle Z-gerbe with triui,al four class and suppose L and Z are defined as aboue. Then m7 i,s assoc'iati,ue and the triple (L, Z, X) forrns a bundle gerbe. This uses crucially the fact that the cocycle g¡jt"t is trivial, without this the proof will fail. 'We Proof. need to show lhat m7 is associative, that is m;o (14 *") : mlo (m;@I). \Me have rnt (um @ m;(u¡x Ø u¿¡)) (u (e (o o ((u (e (o o u (a n (a s t"ù ¡ x i t") n ¡)) €n ¡ o n)))) € r, o), where we have denoted composition by bundle gerbe product by juxtaposition and composition via the bundle 2-gerbe product with o. We also have m1(m¡(u¡tØu¡n) Øu,¡) (((u (e (o o u (" u o ¡¿ ¡",¡ ¡ ò) € i u(a )) þ ¡ ù " )) €u ¡ ía n) We have to show these two expressions are the same. For this we will need the following lemma. Lemma L2.1. Under the hypotheses of the aboue propositi,on, we haue ("(o : (o n t (a r,) " €n¡ r(s ù) Êu"t(u n) € ¡ r, ¡ " u ) Ê¿¡ ¿) To prove this one needs to use the associativity isomorphism to regroup brackets and use the coherency condition on â. I20 The first of the above two expressions is equal to o : (u¡"¿((e (o ¡,¿) (o nù) ((u ¡ u(e(o i n) u n¡)) €¿j k(a ù))) €¿kr(a ¿) " " : o ((e(o¡r") un¡)€n¡t (yn)))))€,*,(gu) (unt((e(ont¡ u¡*)("(oo,) " " : (u¡¿(e(o¡¿) o ((e(o¡x) un¡)€o¡n(an)))€,*,(sn)) u¡*))(("(o*,) " " : (unt(e(ot¿) o u¡r))((e(o¡¿)e(o,,¿)) o ((e(o¡o) " un¡)€n¡r(ùD€*t(an) : (u¡"¿(e(o¡¡) o o (e(o¡n) un¡))("(or,) u¡n))(("(or,) " " Ê;¡r(aùЀnr,(an) : o (e(o¡n) u¿))(Çnt(o¿¡ o g)ê¿¡ís¿)) (u¡"¡(e(o¡¿) u¡*))(("(rr¿) " " : (u¡¿(e(o¡¿) o u¡n))ã,(on¿,o¡kta)(G@xt) o e(o¡x)) o u¿¡) ã(o o o;¡ y)-' u, ¡ k t " $ t"t(o¡¡ " A)€¿¡t(A¡) : (u¡¿(e(o¡¡) o u¡*))ã(on¿, c¡ktU)G@nt o o¡*) o u¿¡)(o¡xt o e(o¡¡ o An))€u¡t(Ar) : (ut"t(e(onù o u¡r,))ã(or,¿, o¡kty)(o¡nt " u¿¡)€¿¡t(g¿) : (unt(e(o o u a y o ("(a nù ¡ n)) ã(o ¡¿, ¡ k t ) (o ¡ nte(o ¡)) ì"n¡)) €n¡,(a r) : (u¡"¿(e(o¡¡) o u¡r,))ã(o*¿,o¡kty)@¡*t o e(y¡))(e(o¡t) o u¿¡)€¿¡t(g¿) : (u¡"¡(e(o¡"¿) o u¡r,))€i*,(aùG@ù " u¡¡)Ë¿¡t(uù. This proves that m¡ is associative and hence the triple (L, Z,X) is a bundle gerbe Definition 12.1. Let (Q,Y, X, M) be a bundle 2-gerbe. V/e say that Q is triuial if. there exists a bundle gerbe (L,Z,,X) on X together with a bundle gerbe morphism q : r;LL I I - T;tL over X[21. \Me also require that there is a transformation á between the bundle gerbe morphisms in the diagram below. r;rr;rLØn,,tQØnl'Q 18rn >r;Lr;rLØn;'Q 'i'øølJ'll , r;Lr;L L Ø nlt Q ffi r;Lr;I L Ø n;tQ Tz-1- -\ r;Lr¡r L Ø nl'Q r;rr;t L nl'4 ^-r*-lttg tI2 tJr So d is a section of the Ct-bundle B: Dñr,n, over X[3]. Here 41 :rllrl o(idSra) and, r¡2 : (nrln8 id) and Dñr,ñ, denotes the descended bundle on X[3] whose lift rlrû " to ZaxxptYnz is isomorphic to (rtt,rlz)-tL1 - se€ Section 3.4. There is a canonical isomorphism r;LB Ør;LB* ØrllB Ør;LB* A over ¡lal, where A is the associator id8 - bundle of Q. We require that 7r1 7r;'0* Ør;r0Ør1'0* : ø under this isomorphism. \Me say that the bundie gerbe L tri,ui,alises Q. Note 12.1. The canonical isomorphism lrL'B Øn;t B* Ør;r B Ør4r B* = A is a result of the following. We can define five bundle gerbe morphisms f¿ : La Ø Qns¿ '+ Lr for I2I 1 ,5 by r_ JI 4 o (id Ø rh,) o (id I rn I id), t_ J2 4 o (id I rh.) o (id I id Ørn), r_ J3 n"(n 8id) o (idØterti,dørn), r_ J4 n"(ñaid) o(tøi¿8id), J5:I n"(naid) o(id8?7¿sid). Note that /3 can also be written as fs : fio (id a n) . (n8 id I id). Flom this we can see that we have the following isomorphisms D¡,,¡, æ A, D îr,f, = nll B, D¡r,¡^ - rlr9, ru D ¡^,* rtr B* , D æ rll B" Ìr,¡-, ' In each case, as in the notation used in Section 3.4, D¡,¡ denotes the line bundle on Xlal whose lift to ZaXyla:Ytzza is the iine bundle Uo,f¡)-tLr. The bundle gerbe product on Ly provides a trivialisation of the line bundle (Íu, fù-tL1 I (.f¿, fò-tLr 8 ("fr, Ín)-'Ltø (.fr, fù-tLr 8 (/t, fù-'L, on Za x yvt Ynz+. This provides the canonical isomorphism r;1 B Ø T;18* I 7rt1B I nlL B* - A. The coherency condition on the section 0 of B is as follows. Each bundle (Ìn,Í¡)-tL1 has a trivialising section f¡ induced from g and given as follows: ln(z+, asa, azs, a::) : û("(rn) Ø ã(atn,uzs,a::.)), irrQn, uza, uzs,, un) : â(zo, Usa, m(Azs, Un)), ÎrnQn, as¿, Uzs, an) : AØQn,gsa),azs,,an), J aslz¿, As¿, azt, Utz) : n@Qn,,gsE,uzs) Ø e(atz))-' irr?n, as¿, Uzs, Un) : 0(za,m(Ysa,azs),an)-r' where á denotes the iift of the section 0 of B : Dnr,n, under the isomorphism D - (ry,q2)-rLt. The coherency condition zrlld ØlT;re* Ør;r0 Ønnt?* : a under the isomorphism n¡L B Ø r;r B* Ø r;r B Ø r4t B* ry A translates into ÎttÎnuÎtnirrÎr, : e(nþa, m(*(arn, azz), an))), where e is the identity section of the bundie gerbe (h, fù-rLt. We have the following lemma. Lemma L2.2. Suppose (Q,Y, X, M) i,s a triuial bundle 2-gerbe w'ith tri,ui,alisi,ng bundle gerbe (L, Z, X). Then the bundle 2-gerbe Q has zero four class in ïAQø;Z). 122 Proof. Let, Ll : {U¿}¿q be a 'nice' open covel of M. So each non-empty finite in- tersection Urì--.nUb is contractible and there exist local sections s¡:U¿ -. X of. ry : X -- M. Define maps (tn,s¡) : (J¿¡---, Xt2l by rn è (tu(*),s¡(*)). Pullback the bundle gerbe Q : (Q,Y, Xl'l) with this map to obtain a new bundle gerbe Qn¡ : (Qo¡,Y¡,Ø;¡). So we get a manifold V¿ and an induced projection map 7r¡¡ :Y¡ - U¡¡. If m € U¿¡ then r;\(m) : Yj* : Y(s¿(rn),s¡(m)). Since U¡¡ is contractible the bundle gerbe Qq ís trivial. So let J¡j - Yr¡ be a C' bundle trivialising Q¿¡' Let (L, Z,X) be a bundle gerbe trivialising the bundle 2-gerbe Q. Pullback the bundle gerbe .D with the sections s¿ to get a new bundle gerbe L¡: (L¿, Z¿,U¡) on the contractible open set U¿. Let K¿ -+ Z¿ be a C* bundle trivialising the bundle gerbe L¿. The bundle gerbe morphism ¡:r;LLØQ - n;tL induces abundle gerbe morphism rh¡: LtØQ¿¡ + L¿ over (J¿¡. Consider the C' bundle K¡ Ø Jo¡ ØqørKi on Z¡ xuY¡. Since the bundies K¡, Jq and K¿ trivialise the bundle gerbes L¡, Q¿¡ and L¡ respectively, and because rh¡ is a bundle gerbe morphism and hence commutes with the bundle gerbe products on L¡ Ø Q¡¡ and, L¿, it follows that there is a section $¿¡ of õ(Ki Ø J¿¡ øqijLKî) over (Z¡ x¡rYr¡tzl satisfying the coherency condition ô(i) : 1. Hence the C* bundle Nl ø Jt¡ ø n;'X; descends to a bundle C;¡ on [/¿¡. F]om the construction of the Õech cocycle 9¿j¿¿ associated to the bundle 2-gerbe Q given in Section 11.1 we have that the C' bundle Lojr : J¡x Ø J¿¡ Ø m-L Jix descends to a CX bundle L¡¡¡ on [4¡r and that furthermore there is an isomorphism L¡u Ø Lint Ø L¡n Ø Lit¡" = A¡¡¡,¿ over (J¿¡¡r¿, where Aoj,r, : (s¿, s¡, s¡, s¡)-LA. Also the Õech 3-cocycle g¿¡¡¿ was defined by choosing sections o¿¡¡ of. L¿¡¡ and then comparing the section a¿jkr : (s¿, s¡, s¡, s¿)-14 with the image of the section o¡u Ø oint Ø o¿jt Ø o¡1¿ under the above isomorphism. It is not difficult to show that there is an isomorphism C¡x Ø C¿¡ Ø B¿ju = L¿¡t Ø C¿t" of C" bundles over (J¿¡¡,, where B¿¡¡, denotes the pullback of the bundle B on X[3] via the map (J¿jt - X[3] defined by sending a point m of U¿¡¡ to (tn(-), s¡(m),t¡(-)). Flom here we see that if we choose sections pt¡ of. C¿¡ lhen we can define a section o¿¡¡ of. L¿¡*by o¿jk: P¿¡Ø pjkØ Pit"Ø0¿¡¡,where ?nj*: (s¿,s¡,tr)-'ï. Comparing o¡nt Ø oi*t Ø o¿jt Ø oirr with, o,¿¡¡¿ aíLd using the coherency condition ô(0) : ¿, \Me see that the cocycle 9¿j¡,¿ must be trivial. ! We shall now prove that the converse is also true, justifying the terminology. Let (Q ,Y, X , M) be a bundle 2-gerbe with zero four class in Ha (M ; Z) . We shail prove that the bundle gerbe (L, Z, M) we constructed in Proposition 12.1 trivialises the bundle 2-gerbe (Q,Y,X,M). First of all we need to define the bundle gerbe morphism 4: nl'L88 -' n;'L.Recall Z :Uo.r)Í. Therefore if (ø1, ,r) € ¡[zJ, then nl'Z xyplY has fibre at (ø1, 12) equal to the disjoint union of the spaces Y@2,s¿(,,)) xY(,r,,r), where m: r(rt):T(n2).Therefore the map m: r¡rY xxptzlrY - n;LY induces a map r¡ : r¡t Z XyplY + r;r Z which sends (Ui,ù to m(y¡,3r) for u¿ €_Trr\,and g € Y. .We definã a bundle map i : r;tLØQ - n;'L covering the map rtl2l , (trtrZ xy¡z7Y)Í21 ¿ rrrTlzl as follows. Let u € Qfur,sr) and.u¿¡ € Q1*çn¡,yn¡,rr¡ where (Ar,Aù e Y[2] and Ai e úLYi, U¡ e TrtYj. So (go,g¡) e r;tzlzJ and (y¿,U¡,u.i¡) €. rrLL6n,or¡. Then 723 m((ñ,(u¡¡ ø u) ø ã(on¡,a¡,aù-L) is in Q1-1on i,*(u¿,aù),^(y¡,yz))- Therefore we define n@¿¡ Ø u) : rn((ñ'(uo¡ Ø u) ø ã(oi,,u¿,aù-t)' (I2'2) The problem that confronts us nolv is to show that 4' so defined with f : (n,4, id) is in fact a bundle gerbe morphism, that is it commutes with the respective bundle gerbe products. So let ((an,aù, (a¡,az), (ur,gt)) be a point of. (tr;r Z x¡¡z1Y)[3] and let u¡nØuzs be an element of (zr¡1.Lø Q)(tu¡,vù,@u,yr)) and u¿jØunbe an element of (tr¡t LøQ)(@n,ar),(s¡,arÐ. Thus u23 Q Q@r,yr) and u¡x is an element of Q¡nþ¡t",y),u¡) and similarly for u12 and u¿¡. We show that q(u¡x Ø uzs)' rÌ(u¿¡ Ø un) : n((u¡*' u¿¡) Ø (uzsun))' We start with the left hand side and for convenience of notation, we denote by o both m and rî¿. The left hand side is equal to rl' o u p) ã(o g rl, f(u ¡ ¡" o u2s) ã,(o ¡ x, a ¡, a z)- l@n ¡ ¡¡, a ¿, t)- where the dot between the square brackets denotes multiplication in the bundle gerbe .L. Therefore the left hand side is equal to l(u¡¡oulæ)ã'(o¡t",a¡,az)-L)le(o¡t")o((u,¡oup)ã,(o¿¡,a¿,a))-r)Êu¡*(anoat)' Notice that since e(o¡n) is just the identity section, \¡/e can write e(o¡n) : e(o¡r)e(oin) and, since o is a bundie gerbe morphism, we get the above expression is equal to f(u¡¡ou2s)ô'(o¡t",U¡,az)-t)le(o¡x)o(u¿¡ourr)l[e(o¡r,)oã"(o¿¡,an,a)-Ll€¡¡r(ao"a). The right hand side is more complicated: RH s "l:,1'{,'r';l,t -, : î,y,::,:?:,' r; : T"'î:',:;','*'r,u, *, a ¿, s,) Next we use the fact that o commutes with bundle gerbe products to see that the expression in square brackets is equai to (0, (u¡¡ o u2) ((e(o ¡n) o u¿) r(a n)) o urr)' Now we re-express ((e(o¡n) ou¡)ê¿¡t(A)) oup by observing that it is equai to ((e(o¡n) ou¿)Ê¡¡¡(an)) o (upe(y)) and then using again the fact that o is a bundle gerbe morphism to get it equal to ((e(o¡t) o u¿¡) o uù(Ên¡t (an) o e(st)). Next we use the associator section ô to write (e(o¡x) o u¿¡) o u12 à,s L (e(o o (u¡¡ o ã'(o o o ã'(o ¡ r, a ¡, az)- ¡ *) up)) ¡ k, ii a ¡, U t)' L24 At this point, if we substitute all this into our original expression for the right hand side, then v/e see that if we can sho'w that ã,(o¡t , c¿j o U¿,U)(à¡x@¿) o e(g))ã(ou",A¿,At)-r (12.3) is equal to (e(o¡x) o ã(o¿¡,A¿,Aù-L)€r¡u(y¿ogr) then we are done. €u¡n(A) o e(Aù is equal to (ã'(o¡*, o¿¡, A)(o¿¡n o e(gr))) o e(A) : (ã(o¡n,o¿¡,U) o e(gù)((on¡* o e(Ai)) o e(AL)). Using the associator section â, we see that (o¿¡n o e(ù) o e(Aù is equal to ã,(o¡* o oij,Ui,aù-r(o¿¡t" " (e(an) o e(y:))a(o¿t ,g¡,Ut). Therefore €n¡r(an) o e(uL) : (â,(o¡n, o¿¡,a¿) o e(g1))(â( o¡k o c;¡ta¿,gt)-r (o¿¡t o ("(An) o e(y))ã(o¿n, A¿, At)). If we now substitute this into equation 12.3 then we get ã(o¡n,o¿j oA¿,gr)(ã,(o¡*,oij,Ai) o e(g))ã,(o¡ko o¿j,A¿,At)-L(o¿¡*o e(g¿"Aù), o get since e(3r¿) e(aL) equals e(a¿ " A). Using the coherency condition on ã,, we that this is equal to (e(o¡*) o â,(o¿¡, ai, aL)-r)ã(o¡x, or¡, u¿ o u)(o¿¡* o e(u¿ " aù), as required. Thus f is a bundle gerbe morphism. To complete the demonstration that .L trivialises the bundle 2-gerbe Q,we need to define the transformation section 0 and show that it is coherent with a. First we need to define the section á of the C" bundie (qr,rtr)-'L1 on Zs Xylty Yy x y¡tt Yrz. If (gr, Uzs, Ap) €. Zs x yys¡ Y4 X y¡s¡ Ytz, then i : (nr, nz)-L Lt has fibre equal to Q@noqs2soy72),o¿¿o((y;oszs)oyn)) at (g¿,Uzs,A1¡-). Let 1(3r¿) denote the identity section of .L evaluated at (An,An). Then we can define a section e of. i AV 0(au,arr,un):("(onn)oã(an,azs,an)-L)ã(ouu,u¿,azs"an)(L(y¿)-'o"(yrr"urù). (t2.4) Here 1(g¿)-1 means the inverse of the identity section 1(g¿) of L at g¿ caiculated in the bundle gerbe Q (recall Lfun,yn) : Q(o¿¿os¿,sn) so l(go)-t €. Q@n'n osn)). We need to show fhaf 0 descends to a section 0 of. J + Xl3), that is we need to show that ó("rtØ : n;'ê, where þ: nrri -, r21i is the descent isomorphism. Recall that / is constructed by choosing u¿¡ e Qçn¡oyn*r) : L(sn,y¡), 'uzs € Q@r"*Lr) arrd up € Q@,,r,ai") where (azs,aL) eY;|), (an,alz) eYISI and (e¿, a) e ZPI and setting Ô(o) : ûz(un¡ Ø uzt S urz)-1 ' u ' û{u¡¡ Ø uzs Ø u:r.) for z e J@¡,uLg,a'rz) and where the . denotes muitiplication in the bundle gerbe ,L. It is very tedious to prove that d commutes with the descent isomorphism / and so we will omit the caiculation. 725 The final step is to show that the descended section is coherent with a. Recall from Note 12.1 the defi.nitions of the bundie gerbe morphisms l¡: LaØQns¿+ Ltf.or i -- !,... ,5 and the sections in¡ of Un, f)-'L1 ovêr Zaxy¡+lYnsa,.Recall in particular that we had : Ø ã(Atn,Azs,Un)), f :,rz("n,au,Uzs,U:r-) ñ(L(rò Jzslz¿, Usa, Uzz, UD) : â(z¿,, As+, rn(Azz,, An)), Îrn(rn,as+, azs,ap) : 0(n?n,as¿).,azs,Un), ÎnuQn,usE, azz,an) : û(0(rn,,Us¿,Uzs) Ø e(an))-', jstla¿, as¿, Uzz, an) : 0(za,m(gsa,Uzt),up)-r, where (yo,,Atn,Uzz,An) e Za Xyl+1Ytzsa. We need to show that : r(un ((arn o o Îuriouitnirrir, " azs) E.'-)) in the bundle gerbe (fr,fù-tLt. This will imply the coherency condition ô(d) : ¿. Again I will omit the proof as it is long and cumbersome. Combining the above results we have the following proposition. PropositionL2.2. Let(Q,Y,X,M) be abundle 2-gerbe. ThenQ has zero four class in Ha(M;Z) if and onlg if Q i,s a tri.ui,al bundle 2-gerbe. Note 12.2. Throughout we have assumed that it is possible to find a section o¿¡ of. Y¡ : (s¿, s¡)-lY -- (Jü over (I¿¡. In general though 'fty : Y -- ylzl only admits locai sections and therefore it is only possible to find an open cover {Ufi}"ezri of. U¿¡ and local sections of;, : Ufi + Y¡ of Y¡ - U¿¡. It is possible to get around this restriction as we will now show. Assume lhat ry : Y ---+ X[2] only admits local sections. Then there is an open cover {Ufi}"e>ri of. U¿¡ such that there exist sections of,t : U$ --+ Y¿¡. Let Xfr : Xv¡. We can define maps ói¡ : Y,1*r, -- Y¡lxi, by sending y e Ylx\, to m(oit,g). As before the Si, extend to define bundle gerbe morphisms Óii t q¡*r, --+ Q¡l¡e with 6î, : (ôî,,óî,,ia¡ if we put óî,("n) : nþþi) 6 ur). Now we can use the þi, to define maps Xi¡: (ynxxY¡)1xi, -Vj'll*r,by xî¡(an,a¡): (óî¡fuu),y¡).Define line bundles Lfi on (Y xxY)lxA by Li¡ : (xi)-tQ¡. We now need to digress for a moment. Let Uüu : Uü I Ul Let pif be a section of the pullback bund.le (oî¡,of¡)-'Q4 over LIfiP. Observe that since(J¡¡ is contractible, the bundle gerbe (Qr¡,Y¡,u¿¡) is trivial a Ld so it follows that we may choosethe pif so that Ø pih: (12.5) ^eþfi pil onUfiP1 : Uü¡Uf¡ntlu1. This is simply the statement that the Dixmier-Douady class of Q;¡iszero. Nextchoosesections oifl otthepullbackbundle Aifl : @f¡"ofi,,o1)-rQ¡ or", tJ$Pn' . uere Ufif' : Uü nuft aUu'r. If. a' , B' and 7' are more indices of. E¿¡, E¡n U"'' 9't' and !¿¡ respectively, ,ü/€ can define a map hîf;"' : g?'flio' -t C' by . e''" ø : oro@íf''' ø (p,¡f o pîf h?P;o' , ^qþ71 "if,l) ) 126 where ¿¡?7rt"'ø''t' : uif"' nr4|f't' rr. a.", p" and.1" are more indices of E¿¡, E¡r, and E¿¡, then using Equation 12.5 for the various indices of I and the X, one can show that p''" p' " hfP;o' hij 'ya P"'r" P"'Y" k - 'oijt"^oþ'yo" Since (l¿¡¡, is contractible and {UiPr'}.rre¡,¡,€Ð¡rt€Ðon is a cover of. (J¿¡¡, it follows that there enst ci!{ t Uil'-+ C' such that h?F;o'p'''' : ¿/.f't'""Ê1-t. It follows that we can define the section, so that "ifl "t Aifl mq(p'll Ø : *o@î,f''' (pff pîf (12.6) "îf:l ø " )). Now we can define a section tif|" of the pullback bundle (oi,, @fu o o oi)-rQ¿ in the same fashion as in equation 11.1, nameiy we set "f) : qofil o 'if,lr' e(ofi))oiil'. Gz.T) Wealsodefineasection tif;* of thepullbackbundle (oit,o6nt"@fn"oi))-'Q¿¿inthe analogous way to equation 11.2, tfl,o' : ((e(o6r) . o7f^t¡oit'f (12.8) Finally, after equation 11.3, we define a map ¡îf;u" , U;l*|u" -' C' by tf;i* : o(of"¿,"fu,"Ðriff,r, . sif;ur,. (rz.s) Using equation 12.6, one can show that if e' , þ' ,'y' , õ' , r¡' and e' are some more indices of E, then sîf;6" - eif;,'''o'''; (12.10) Therefore the various maps gif;lo" are the restrictionsto UiPkTõq' of a map g;.¡r"t : (Iojm - C". It is not hard to see that g¿jta sa|isfres the cocycle condition : I j t"t^9 ¡¡rtrng t¡t .9 4Ir.9 tj t"t I and that g¿j*¿ is the Cech 3-cocycle constructed in Section 11.1. Returning to the problem at hand, we will use the Oif to define maps ,þîf , Li¡ - Lf, covering the identity on (X ,xY¡)lxy,nxf, as follows. Recall lhal Llj *îfl Ø : rne(me@fr Ø ñ,(e(ofi) Ø ø ( rz.r7) @Pjr "î,) "î¡)) eifo, @,)), L27 where eîfu" i, the section of the bundle (ö70,óP¡ro óî¡)-'Q* onYlxi1, given by €îfr'@,) : mq(ã@f ¡", o$,a) ø n@îf*' I e(sc))) where Ur eYlx;fl..We want to show that the various bundle gerbe products *ifl extend to define a bundle gerbe product m7 : L¡x * Lu -. Lu,. For this, we need to know that the foliowing diagram commutes: m.:"aß¡ tfrøLî¡3 L'l* tff ø+1' ,þll LlkØ^llt Lij --ì,Llk. 7lÆ The commutativity of this diagram foilows from equation 12.6. We also need to define the bundle gerbe morphism 4: rrrL@8 -* n;|L. Again we de4ne 4locally and then show that it is càmpatible with the glueinf isomorphisms $if . We define ffi : -- putting : (ui¡ u)ã.(oi¡,A¿,at)-1 where uij e Lij rrt Lîj 8 I r;l Lij by nfi(uî¡ I z) " and u € Q. One can then show that the diagram below commutes: ñit rrr Lij Ø Q , r;r Lîj *;,l7f øq! [";,ørf 'Ît-''u,8Q---nltLlo. - L'¿: n¿¡ Hence the various 4fi glue together to form a bundle gerbe morphism r¡¡ : rrrLnr g Q + rrrL¿¡. Similarly, one can define the transformation d locally and show that it glues together to form a globally defined transformation which satisfies the required coherency condition. L2.2 Trivial stable bundle 2-gerbes Wefirstdefinethenotion of.atriuialstablebundie2-gerbe. Solet (T,Z,X) beabundle gerbe on X where r : X + M is a surjection admitting local sections. Form the bundle gerbe (ô(?), 6(Z),¡tzl) where ô(") : rlLTØlt;rT" and.where 6(2) : rlr Z x y¡,11r;'Z. We want to show that the quadruple (d(7), 6(2), X, M) defines a stable bundle 2-gerbe. We first need to define a stable morphism m: rrl (ô(r)) I ?rtl(ô(")) - n1'(ô(")) Such a stable morphism will correspond to a trivialisation (L,", ó^) of the bundie gerbe 1(ô(")) (?rf I ?rtl(ô("))). ø r2tQQ\. Let us use the notation of Chapter 7 to denote the bundle gerbe (n,,tT,TrrZ,Xl2l) as (T2,22,Xlzl¡ and the bundle gerbe (tr;rr;rT,n;rn;rZ,¡tsJ) as (Ts,Zz,¡lsl) and so on. Hence the bundle gerbe above can be rewritten in the new notation as (r, ø T; ØT2I ?i). s (73 s 7i) L28 'We seek a trivialisation (L^,ó^) of this bundle gerbe. So L,n will be a CX bundle on Zs Xn Zz Xn Zz Xn Zt xn Zs Xn 21 The obvious candidatef.or L* is 7s øTzØT{.It is easy to check that this trivialises the bundie gerbe. One can also check that there is a transformation a : n'L o (rn ø 1) + mo(78rn) of stable morphisms of bundle gerbes over X[a] which satisfies the coherency condition over X[5]. We now define what it means for a stable bundle 2-gerbe to be trivial. Definition L2.2. We say that a stable bundle 2-gerbe (Q,Y,X,M) is triui,al if there is a bundle gerbe (7, Z,X) on X and a stabie morphism f , Q -' ð(") over X[2]. We also require that there is a transformation of stable morphisms of bundle gerbe over X[3] as pictured in the following diagram. Tttre Ø nlte "ilJ8"'-t'f' 7tt1õ(T)g 7rt1ô(,-) In 0 7r;18 r;16(T). Finally we require that 0 satisfies the compatibility condition 6(0) : ø with a. Note 12.3. This defrnition suggests that we could define a stable bundle 2-gerbe mor- phism from a stable bundle 2-gerbe (P,Y, X, M) to a stable bundle 2-gerbe (Q, Z, X, M) to be a stable morphism f , P + Q from the bundle gerbe (P,Y,¡tzl) to the bun- dle gerbe (Q,2,¡tz1) together with a transformation 0¡ as pictured in the following diagram TlrP 8zr;1P ^ ' 7r;'P ";,rø,;,rJ à l_-,, rrrQØnl'QÈr;LQ. 9 is required to satisfy the compatibility condition õ(0) : aþ Ø aq. One couid then go further and show that any stable bundle 2-gerbe morphism was invertible up to a coherent'stable bundle 2-gerbe transformation'. The following Lemma is not difficult to prove. Lemma L2.3. The class ln H3(M;Cf;) represented bg the cocgcle g¿jg associated to a tri,uial stable bundle Z-gerbe (Q,,Y, X , M) ß zero. t29 130 Chapter 13 The relationship of bundle 2-gerbes with 2-gerbes 13.1 Simplicial gerbes and 2-gerbes We first recall the definition of a simplicial gerbe (see [8] and [9]). A simplicial gerbe is defined as follows. Definition 13.1 ([8], [9]). LetX: {Xo} beasimplicialmanifold. A si,mplicialgerbe on X consists of the following data. 1. A gerbe Q on Xy We will only be interested in the case where Q has band equal toCl_Al . 2. A morphism of gerbes m : d[Q Ø d;Q -- diQ over X2 where the d,¿ denote the face operators of. X.. 3. An invertible natural transformation d between morphisms of gerbes over X3 as pictured in the following diagram did,6Q Ø di@öa ø diØ 3 dö(döa ø a;q¡ Ø didiQ LØd;rnl I - ,1, ldö''81 did.öa Ø dädiQ dödiQ Ø drrdïQ _t : *e* di@,öa ø diØ di@.öa Ø dia) ¿;^lt* lor^ didia did,iQ. Here the equivalences of gerbes denoted by 3 are the equivalences of gerbes ô1,s of Lemma 6.1. 4. 0 is required to satisfy a coherency condition on Xa. This is 6(0) : 1, where we use the notation of Note 6.3. 131 Note that this differs from the definition of simplicial gerbe given in [S] and [9] in thatr m : d,[Q Ø d;Q - d,iQ is only a morphism of gerbes, rather than an equivalence of gerbes. We aiso recall the definition of a 2-gerbe (see [5], [S] and [9]). Definition L3.2 ([5],[8],[9]). A Z-gerbe on a manifold M consists of the following data: 1. For each open subset U C M there is an assignment of a bicategory ßu t'o U, U ¡- Bu. Given another open subset V C U c M there is a restriction bifunctor pu,v : ßu - 6v such that if we have a third open subset L4l with W c V C U C M then py,'¡r o Pu,v : Pu,w. Such an assignment is called a pre-sheaf of bicategories on M. 2. Various glueing laws for objects, 1-arrows and 2-arrows are required to hold. The glueing conditions for objects are as follows. Let U C M and let {U¡}¿çy be a covering of t/ by open subsets 14 of M. Suppose \¡/e are given objects A¿ in each ßu, together with l-arrows a¿j : pg¿,ry;,(An) - pu¡,y¿¡ between the restrictions of the objects A¿,A¡ to BUn,. So the a¿¡ belong lo ßUr,. Suppose also that we are given 2-arrows Ót¡ of.6u,jk such that : o Ôt¡ n Pu ¡ r,u;, r(o i r) Pur¡,(J¿¡ ¡"(on¡) + Pu ¿¡,tJ¿i ¡(on,r), where o denotes the application of the composition functor in Bu4,*. We try and indicate this in the following diagram which lives in Bun,o (we abuse notation and omit the restriction bifunctors p). We can now construct two 2-arrows a¡¿ o a¡n o a¿j è a¿¿ in ßUn¡*t ãS pictured in the foilowing diagram a¡¿o(o1noa¿j) oö¿jn (ooroa¡u)oati d.¡¿ o Q,¿¡ ó¡u Á A¡t o d;¡ Q¿l Finally we can state the glueing axiom for objects by saying that if these two 2-arrows coincide then there is an object A belongiug to Bu arLd a 1-arrow B in Hom(p4u,(A),A¿) which is compatible with the glueing l-arrows a¿¡ up to a coherent 2-arrow in the sense of Proposition 6.5. The glueing conditions for l-arrows and 2-arrows are as follows. For an open set U C M and objects A1 and Az of. Bu we require that Hom(At, Az) is a sheaf of categories. Thus, given an open cover {V}¿q of t/ by open sets V c tttt together with l-arrows o¿ , Atlu -- Azlu in Hom(,4r,Az)lyt and 2-arrows Ór¡, a¿è di r32 which satisfy the cocycle condition ó¡ró¿¡ : ó¿r, then there is a l-arrow o in Hom(.41, A2) which restricts to each a¿. The glueing conditions for 2-arrows are handled similarly. Such an assignment U * Bu of a pre-sheaf of bicategories to M which satisfies the effective descent or glueing conditions on objects, l-arrows and 2-arrows is called a 2-stack or a sheaf of bicategories. 3. For each point m €. M there is an open neighbourhood U of rn such that 6y is nonempty; that is, the set of objects of. Bu is nonempty. 4. Given an open subset U C M, any two objects A1 and A2 of. Bu can be locally joined by a 1-arrow. That is there is a coverin1 {U¿}¿q of t/ by open sets of M such that there exist 1-arrows o¿ : pv,r4n(At) t pu,ur(Az) in ßun. 5. We also require that 1-arrows are weakly invertible. This means that given an open set U C M and objects A1 and A2 of. Bu together with a l-arrow o : At - 42, there exists a l-arrow 0 , Az - At of. Bu together with 2-arrows ót þ"aè Id"arinB¡¡. 6. Finally we require that all two arrows are invertible. \Me will also need to have the notion of a 2-gerbe on M bound bV Çfu. This means that any two l-arrows of. Bu can be locally joined by a 2-arrow and for every l-arrow c there is a given isomorphism Cñ/ = Aut(a) which is required to be compatible with restrictions to smaller open sets and is compatible with 2-arrows. There is also a notion of equivalence of 2-gerbes will not spell this out but refer instead to [5]. One can associate to a 2-gerb e B on M bound bV Año a Õech three cocycle 7¿ju¿ ín n3g;A.ir¡. V/e will not do this here and refer to [5] and [S] for a discussion of this. There is an isomorphism between equivalence ciasses of 2-gerbes ana Hs(tW;Aiò via this cocycle. As is the case for gerbes, a 2-gerbe with zero class in Ut(U;Afo) is trivial precisely when it has a globai object. For mo¡e details we refer to [5]. Let U : {U¿}¿q be an open cover of. M such that each non-empty intersection Uioa. . .nU;,n is contractible. Let ß be a 2-gerbe on M bound bV Ai¿. Let B¿ denote the restriction of. B to U¿ for each z € -I. Since U¿ is assumed contractible, the 2-gerbe 6¿ has a global object A¿ sa!. Denote by A"lu, and A¡1u,, the restrictions of A¿ and A¡ to U¿¡ respectively. We have the presheaf of categories B¿¡ onU¿¡ where ß¿¡ has fibre at an open set 7 a Uu Bu¡(V) equal to Hom(Au¡r,A¡lv) (we ignore any complexities that may result from the objects A¿luu¡lv and A¿ly not being equal but isomorphic by a coherent l-arrow - otherwise we get that ß¡¡ is a fibred category ([5]) or a pre-sheaf of categories in the sense of [Z]). By definition of the glueing laws ín B, B¿¡ is a sheaf of categories. In fact, since 6 is bound by Ciø, B¿¡ is a gerbe on U¿¡ bound bV Ai¿. The composition functor m : Hom(Ajlur¡*, Axlunr) x Hom(A¿ lrr,r, Ajlunr.) * Hom(A¿la,n, Axlun,*) gives rise to a morphism B¡n, ßu -, ß* of gerbes 'bound' by the morphism of sheaves Cio * Aio + Air. One can show that this morphism in turn defines a morphism of gerbes B¡x Ø 6Ø - ß¿¡ç ovet U¿¡¡. The associator natural transformation in the bicategory Bryo,r, then provides a natural transformation / between morphisms of gerbes as pictured in the following diagram (we have omitted any restriction functors and 133 equivalences of gerbes that may result by passing between them for convenience): ßnt Ø B¡n Ø Uo, '*^, ß¡t Ø 86 _.,1 ã l"' ßr, å B¡n --- Bu. It follows that we have deflned a simplicial gerbe on the simplicial manifold X : {Xo} with Xo : U(J¿, Xt : Il,Uo¡ and so on. It is not hard to construct a Cech 3-cocycle g¿jr¿ representing a class in H3(M;çià : Hn(M;Z) associated to this simpliciai gerbe (compare with Theorem 5.7 of [8]). It is also easy to see that the class in H (M;Z) associated to this simplicial gerbe is equal to the class of the 2-gerbein H (M;Z). This simpliciai gerbe on X : {Xo} with xo :If.u¡o n...n u¿o is an example of a simplicial gerbe on the special class of simplicial manifolds formed by taking iterative fibre products of a surjectior: T : X - M. I had hoped to relate bundle 2-gerbes to 2-gerbes by frrst showing that a bundle 2-gerbe gave rise to a simplicial g.ib" on the simplicial manifold X : {Xr) with Xo : XIP+rl and then showing that any simplicial gerbe on X gave rise to a 2-gerbe on M. Unfortunately, as noted in Chapter 6 (Note 6.2), checking directly that the construction which associates a gerbe Ç(P) to a given bundle gerbe P commutes with equivalences of gerbes induced by successive tensor products and transitivity of pullbacks gets very messy and I was unable to produce a convincing proof that this was the case. Note that any simplicial gerbe O on the simplicial manifold X : {Xr} formed by taking iterated fibre products of a surjection z" : X --+ M adm\tting iocal sections gives rise to a 'local description' of a 2-gerbe as described above ie gerb ss Qni on U¿¡ and morphisms of gerbes rn : - gerbe on X to the simpliciai Q¡t ØQ¿¡ -+ Qu, and so on - by pulling back the simplicial gerbe on simpiicial manifold associated to the nerve of the open cover U : {U¿} using ylÀ, (so,(-),.--,sr,(m)). ihe simplicial map IJ.U¡, n...n ¡¡ù - (ia,...,io,m) - Hence every simplicial gerbe on X : {Xo} has a class in HL(U;V') associated to it. Lel n ; X + M be a surjection admitting local sections. Form the simplicial manifold X : {Xr| with Xo : yln+rl' Let (Q,m,g) be a simplicial gerbe on X' We define a sheaf of 2-categories 6 on M as follows. For an open set U c M, the objects of the 2-category ßu wíll be gerbes Tur on : on Xlll. Thus an object will consist of a gerbe Tu orL Xu "-r(i) which 'trivialise' Q Xy together with a morphism of gerbes r¡ : riTy Ø Q - riT plus an invertibie natural transformation X as pictured in the following diagram: r8ræ niriTu Ø riQ Ø r$Q > -înør I * 'ÍiT2 Ju Ø täQ _t x -+ T[triTu Ø r[Q îän r[r[qr. X is required to satisfy the compatibility condition ô(X) : I with the natural transfor- mation d - see Note 6.3 for the notation'ô'' L34 Given objects (Tt,qt,X1) and (Tz,nz,y2) al-arrorl, (d, ,o) t (Tt,rh,Xt) - (Tz,nz,Xz) consists of an equivalence of gerbes Ô : T -- Tz over Xy and an invertible natural transformation of gerbes r¿ as pictured in the following diagram: ¡riÁøQn' ,¡rTTt "îd@idl Z I"r, riTzØ QVr$72 We require that r4 satisfies õ(r): XzØyrt. A 2-arrow 0, (ór,ró) + (óz,ro): (Tr,rlt,Xr) * (Tz,qz,Xz) is an invertible natural transformation 0 as pictured in the following diagram: á is required to be compatible with 16, aîd16rin that ð(á) : rórrlr'. Given another open subset I/ with V cU C M, there is anatu¡alrestriction 2-functot pu,v:Bu --+ ßv. It follows that we have defined a presheaf of 2-categories. In actual fact the assignment U - Bu is a sheaf of 2-categories as we will now see. Firstly we check that the glueing a>rioms for objects are satisfied. Let U c M be an open set and let {Uu},rt be a covering of [/ by open subsets U¿ of. M. Let (T¿,\¿.,X) be objects of. ßu, for i, e I. Suppose we have l-arrows (ó;¡,rr ¡ : ([,n;,,X¿) - (T¡,rt¡,a¡), where for convenience of notation we have omitted the restriction 2-functors psn,so,. Suppose we also have 2-arrows 0¿¡¡ : (þ¡u o ó¿j,ró¡r o ror¡) è (ó¿*,rOn) which satisfy the glueing condition 0¿¡¿(0¡xL o id6o¡) : 1¿n¿(idør, o ?¿jr) of (2) in Definition 13.2 above and where id4n, is the identity 2-arrow at (þ¡¡,r6n,), that is id¿n, is the identity natural transformation from ó6 to itself. 4, ó¿¡ and 0¿¡¡ provide 2-descent data for the 2-stack of gerbes on Xu - see [5][pages 32-33] and also Proposition 6.5. Consequently, there is a gerbe T on Xy which restricts to { on Xur. In other words there are equivalences of gerbes tþ¿ : Tlxn 3 4 on Xi which are compatible with the glueing l-arrows ón¡ ,tp to a coherent natural transformation {¿¡ (the coherency condition that the $¡ satisfy ts Q¿¡(0¿¡¡, o 7+) : €¡n(Ioin o {¿¡)). We have to show also that the gerbe morphisms 4¿ glue together to give a gerbe morphism r¡ : riT Ø Q - r[T and similarly that the X¿ glue together to give an invertible natural transformation X. The gerbe morphisms rìi : iri[ Ø Ql*p1 -- riT; give rise to morphisms of gerbes rt¡ : (lTiT Ø Q)lxf,t + (tr,T)lyet over X!21. f¿ is defined by the 135 following diagram (triT Ø Q)lxl,t \o - ("iT)lxpt niTlxø Ø Ql*,, ri(Tx,) Ø Qlr,o rg(Tlx¿) ni{;Øidç1 t xl2l x ";rlr'I riT Ø Qlx?t nn * rTT. We can define natural transformations i¿¡ : ñ¿ + ñ¡ by i¿¡ : Ti€øLrO,,ni$'. Because of thecoherencyconditionson röu¡,€;¡and0¿¡¡ onecanshowthat î¡xît¡: ã¿¡' Therefore gerbes q : riT over X[]l by Proposition 6.3 there is a morphism of riT Ø Q - "y! a natural transformation r¿ : rllxlrt + 4¿ between the morphisms of gerbes or", X!']. Similarly one can show that the natrual transformations X¿ giue together to give a natural transformation ¡ which is compatible with / in the required sense. We also have to check that the glueing conditions for 1-arrows and 2-arrows are satisfied. The glueing laws for 2-arrows are clearly satisfied, to show that the same is true for 1-arrows requires a little more work. So suppose that we have objects (Tr.,ry,X1) and (Tr,r'lr,Xz) of ßu and suppose that there are l-arrows (Ô¿,rO) from the restriction of. (T1,U,Xt) Lo XUo to the restriction of. (T2,Tz,Xz) to Xun for an open cover {U¡} of U and glueing 2-arrows 0¿¡: (ó¿,rö) + (Ó¡,ro).Thus we have equivalences of gerbes ó¿ : Ttlxn -- Tzlxo and natural transformations 0¿¡ , Ó¿lxn¡ + Ó¡lxu, which satisfy the glueing condition 0¡¡,04 - 0¿t". Therefore by Proposition 6.3 there is an equivalence of gerbes Ö , T ---+ T2 over Xy and natural transformations a¿ : þlxn + Ó¿ which are compatibie with the 0¿¡. We can use the natural transformations a¿ and r¿o to construct natural transformations fr between the morphisms of gerbes bounding the following diagram (trirt ø Q)l xl,t"t ":: þriT)l x:,t (,.ióøro)l*rzr I / l6ão)l*et ^¿+ 4¡n * " (niT, Ø Q)l*pt ,aS("irr)lyet. Because of the coherency conditions satisfied by ao and r4, with 0¿¡ one can show that lJxPt : i¡lxø and hence there is wîlh 16l*¡z) : i,¡. Again one can required coherency condition wit U -, ßu is a sheaf of 2-categories. We have the following conjecture. Conjecture 13.1. The sheaf of I-categories ß described aboue 'is actually a 2-gerbe. Furthermore the four class of the Ù-gerbe ß equals the four class of the s'impl'ic'ial gerbe a. Unfortunately I have been unable to give a concrete proof of this. However one can prove a version of this conjecture with gerbes replaced by bundle gerbes and gerbe 136 morphisms repiâced by stable morphisms (bundle gerbe morphisms will not do here as they do not glue together properly). Proposition 13.1. Let (Q,Y,X,M) be a sto,ble bundle ?-gerbe. Then there'is a 2- gerbe Q on M bound bA Ci, such that the class i,n Ha(M;Z) ol Q 'is equal to the class of Q i,n UL(tw;Z). Let U C M be an open subset of M. We define a bicategory Qu as follows. Qu has as objects the bundle gerbes (T,Z,Xu) on Xy such that there is a stable morphism f : r¡LT I I -, n;LT of bundle gerbes ouet Xl]l and a transformation ¡ between the stable morphisms of bundle gerbes orrer Xjjl as pictured in the following diagram. LØrn r;Lr;LTØnlt/Øn,'Q r;lrrrT Ø n;'Q I -;r.før I Y r;rr;rTØnr'Q+ r;Ln;LT Ø n;tQ I-,,, r;Ltr;LT Ø nltQ r;tn;LT "l'I r;Ln;Lr- Therefore x is a section of the C' bundle B on Xjfl such that Ln;'rono-tt;tyØm) : Ln;'lo1oi1.f8t"r-ro) Ø r-l B, where r : rlLrrr2, *li, nrtY x *at rsrY x *ut rsLn;LZ - Xffl is the projection. One can show, see Chapter 12, that there is an isomorphism 6(B) - A of C" bundles over X$1. We finally require that the induced section ô(X) of ô(B) maps to ¿ under this isomorphism. Given two such objects (7, Ír,¡7) and (S,.fs, Xs) u 1-arrow (7, fr,Xr) - (,9,,fs, Xs) consists of a stable morphism g : T --+ ,S and a transformation on between the stabie morphisms of bund,le gerbes oue, X[]l as pictured in the following diagram: nllrØQ r"n;'T *r'nø,l y l*r'n nrLSsQ=*r;'5. We also require that o, satisfies the coherency condition ô(ar) : XrAX! over X$1. Finally a2-arrow 0 , (g,d) + (h,on), (7,Ír,Xr) - (S,,fs,Xs) ir a transformation 0 : g + å, between stable morphisms of bundle gerbes over Xu which satisfies the coherency condition õ(0) : an Ø ai ouer Xljl. By Proposition 3.9, Qu is a bicategory. The proof that [/ ¡+ Bu is a sheaf of 2-categories given above carries through to the case of bundle gerbes and stable morphisms (although now Q is a sheaf of bicategories). Therefore we have only have to show that the sheaf of bicategories I is (i) Iocally non- empty, (ii) any two objects of. Qu may locally be joined by a l-arrow, (iii) 1-arrows are weakly invertible and (iv) 2-arrows are invertible. r37 Proof. We first prove that the sheaf of bicategories O is locally non-empty. Choose an open cover tl : {(l¿l¿et of M such that there exist local sections s¿ : U¿ -+ X of. n : X -+ M. Form the maps Ê¿: X; -- ylzl sending r € X¿ to (r,s¿(zr(ø))). Let Q¿: 3¿1Q. So 8u : (Q¿,Y,X¡) is a bundle gerbe on X¿, Now pullback the stable morphism m : r¡tQ Ø nltQ '-- tr;rq using the map X:') - ¡[sl, (zr, ,r) - (rr,rz,s¿(rn)) where rn:7T(rr) : T(r2). Let f¡ denote this pullback stable morphism f¿ : rtrQ¿ø Q - rlLQ¿. To construct a transformation X¿ : ¡rlLÍ¿ o (l,o¡r",¡tqn I rn) + nltfo o (nr'Ío 611,-1a) we pullback the transformation o-1 using the map ylal, : : X:t) - (rr,rz,rs) - (ør, 12,r,z,s¿(nz)) where again m: r(rt) T(r2) T(rs). The coherency condition ô(ø) : 1 satisfied by ø ensures that ô(¡¿) : ¿. Therefore for each i e I we have constructed objects (Qn, Íu,X) of Quo. Therefore I is locally non-empty. We now show that g is locally connected. Let (7, f ,y) and (S,g,p) be objects of Qu. So ? and,S are bundle gerbes (T,Z,X) and (S,W,X). We need to show that at least locally on [/ these two objects are connected by a l-arrow. We have stable morphisms f : r¡LT 88 -' n;rT and g : zr'¡1^9 ØQ -. n;rs corresponding to trivialisåtiott. (L¡,ôà and (tre, ô) ofthe bundle gerbes (n1,'T 88)- ØrrrT and' (rritS g 8)- g nr-1,S respectively. ró.* the bundle gerbe (ni1(^9. I ?))- Ø nlt(.g..^P ii* (0. S 8) This is trivialised by the C' bundle Lr Ø Li which lives over V xnYl2), where v : ¡rrr(z x,w) x* n;LØ x^w). Consider the C' bundle L : L¡ S ¿; S Q* which lives over V xn Y[2]. One can show that there is a section $ of 6y¡r1(i) satistying the coherency condition 6ypt(Ó) : L. Ó corresponds to the isomorphism given fibrewise by composition L¡(r,at, z') ø L[(w,gz,w') Ø Q.(u,uz) æ L¡(r,yt,z') ø L|(w,az,w') ØQ"(h,gg) a Q*(ar,aa) ØQ*(an,ar) N L¡(r,as, z') Ø L[(w,a¿,w') ø Q.(ys,uò. One can show this isomorphism satisfres a coherency condition given a third point (r,r',u,u',gs,Aa) of.V xnYt2l lying in the same fibre as the points (r,"',u,u',Ut,Uz) Ànd (2, ,' ,õ,w',as,u¿). Therefore by Lemma 3.2 L descends to a CX bundle L on and, is not difficult to show that.L trivialises the bundle gerbe 8?). A V it ",ft(^9- 7r;'(5.8?) over y[2]. Therefore the trivialisation .L corresponds to a stable morphism h : n;1(5. A 7) ---+ n;L(S" El 7). One can also show that there is a transformation 0 : r¡th o r;rh + r;rh satisfying the coherency condition ô(d) : t. We will omit the proof of this as the calculation gets rather messy. The calculation exploits the fact that the transformations X and p satisfy ô(X) : a and õ(p) :ø and so ô(x8P*) : 1 in a sense one has to make precise. Therefore r'¡¡e can apply Proposition 6.7 to conclude there is a bundle gerbe (P,Xp,U) on U and a stable morphism q: t-rP -+ ,S* 8? over Xy corresponding to a trivialisation (L,1,ó;. There is also a transformation T/. U if necessary, rffe may assume € t ó " ¡rlL\ + rrtr¡ compatible with By shrinking gerbe P is trivial say P 6(J). Therefore I7 is trivial as well. that the bundle - - ^9. In fact, if one considers the C' bundie r-LJ* Ø L, on n-rXp XnW xn Z lhen one can show that there is a coherent section $ of. õ*-ryr(n-'J* Ø Lr) and hence r-LJ* Ø L, 138 descends to a C* bundle Ln on W xn Z . One can also show that there is a trivialisation ,S* So we have a stable : þ¡: 6(L¡) -t I7. morphism k ^S -+ 7. As well, one can show that the transformations { give rise to transformations a.k: T;Llçog + f o(zr¡1kø1) and which satisfy the compatibility condition with X and p. Therefore (k,o,r): (S,g,p)--- (7,Í,¡) is a l-arrow. So O is locally connected. Since the bicategory of bundle gerbes and stable morphisms is a bigroupoid (Propo- sition 3.9), all l-arrows of. Q are coherently invertible and all 2-arrows of. Q are invert- ible. We now have to show that the 2-gerbe I is bound by the sheaf of abelian groups Cfr.. For this, we fi.rst need to show that any two l-arrows can be locally connected by a 2- arrow. Solet (S,g,p) and (7, Í,X)be objects of Qujoinedby 1-arrows (ä,a¿), (k,a¡): (S,g,p) -- (7,Í,X). So /¿ and k correspond to trivialisations (.L¿,da) and (Lr,ón) respectively of .9. I ?. Therefore there is a C" bundle D on Xy such that Ln - Ln Ø r-rb. It is easy to check lhat a¡ and a¿ provide a section at" Ø ai of the bundle 6(D) over Xljl . The coherency conditions satisfied by an and o¡ ensure that ô(o* ø ai) :1 and hence by Lemma 3.1 the bundle D descends to a C' bundle D on U. By shrinking U if necessary we may assume Lhat D is trivial. Hence there is a section of D which lifts to a section 0 of D which is compatible with at Ø ai. In other words, d is a 2-arrow (h,on) + (k,a¿). In a similar rü/ay one can show that for a given 1-arrow (h,on) the sheaf of 2-arrows (h,a¡) + (h,a¿) is isomorphic ro C.i¡ and that this isomorphism is natural with respect to composition of 2-arrows. Therefore Q is a 2-gerbe on M bound AV gio. If one calculates the Õech cocycle gijlc¿ representing the class of Q in n31U;C,¡¡ then it is not hard to show that 9¿j¡¿ represents the same class as the four class of the stable bundle 2-gerbe Q. ¡ As a consequence of this Proposition we have the following corollary: Corollary 13.1. A stable bundle 2-gerbe (Q,Y, X, M) has zero class in Hn(M;Z) pre- ciselg when i,t is triai,al. Proof. We have already seen (Lemma 12.3) that a trivial stable bundle 2-gerbe has zero class in HA(M;Z). Conversely, given a stabie bundle 2-gerbe with zero class in Hn(M : Z), the associated 2-gerbe also has zero class in Ha(M;Z) and hence has a global object. This global object trivialises the stable bundle 2-gerbe Q. ¡ Let (Q,Y,X,M) be a stable bundle 2-gerbe. We can form a new stable bundle 2-gerbe (Q*,Y, X, M) from Q which we could think of as the 'dual.' of Q by letting the bundle gerbe (Q*,Y,¡tzl) be the dual of the bundle gerbe (Q,Y, Xl2)). So the C' bundle Q* - ylzl is Q with the action of C* changed to its inverse. It is clear that the stable morphism m : r¡rQØnlrQ -- T;LQ induces a stable morphism m* : r¡rQ*8zr¡lQ* -- T;tQ* and it is not difficult to show rn* satisfies the requirements of Definition 7.4. It is also not difficult to show that the four class of the stable bundle 2-gerbe 8* is the negative of the four class of the stable bundle 2-gerbe Q. Similarly, given stable bundle 2-gerbes (Qt,Yr, Xu M) and (Q2,Yz, Xz, M) there is a notion of their product. This is the stable bundle 2-gerbe (8r I Qz,Ytz,Xn,M) where Xt2 : X1 x¡4 X2, Yn: @?)-'n **o, @?\-tY, (hu." pt: Xt xu Xz -. Xr and, p2: X1 x¡4 X2 -+ X2 denote the projections on the first and second factors respectively). 8r A 8z is the C* bundle oY]ll given uv (ql'l)-tQr I Øfl)-'Q, where noïu Ç1 ,Yß)-* rf21 and 139 qz , Yßl - Vj2) are the projections on the first and second factors respectively. Again it is not difficult to show that 818 8z is a stable bundle 2-gerbe and that the four class of QtØ Qz is the sum of the fou¡ classes of the stabie bundle 2-gerbes Qt and Qz- Let us now apply the preceding discussion to the following situation. Suppose we are given stable bundle 2-gerbes (Qr,Y'., Xu M) and (Q2, Yz, Xz, M) wifh four classes Da(Qù and, D¿(Q2) respectively. Suppose moreover thar Da(Qt) : Dn(Qz)' Form the stable bundle 2-gerbe Qt Ø Qi. Then by the above discussion Dn(Qt Ø QÐ : 0. Therefore by the corollary above 8r A Qi i" trivial. Hence there is a bundle gerbe (7,2,X12) on Xp aîd a stable morphism Í , Qt ØQi -- ô(?) satisfying the requirements of Definilíon 12.2. Therefore / induces a stable bundle 2-gerbe morphism (see Note I2.3) QLØQiØQz -+ ô(?) ØQyThe stable bundle 2-gerbe QiØQz can then ùe written ur'411'¡ utrã .o we have a stable bundle 2-gerbe morphism 8r A õ(T') '-> Qz Ø 6Q). It should then be the case that this notion of 'stable 2-isomorphism' for stable bundle 2-gerbes introduces an equivalence relation on the set of all stable bundle 2-gerbes onM. Sinceeveryclassin HL(M;Z)appearsasthefourclassof somestable bundle 2-gerbe (for example the stable bundle 2-gerbe arising from the strict bundle 2- gerbe associated to the BBC" bundle with the four class in question - see Section 9.1) this leads us to the following result which we will state as a conjecture since we have not given a definite proof. Conjecture 13.2. 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