Differential Geometry of Gerbes and Differential Forms

Differen tial Geometry of Gerb es and Differen tial F orms Lawrence Breen ∗ Institut Galil ´ ee Universit ´ e Paris 13 99, av enue J. - B. Cl ´ ement 93430 Villetaneuse, F rance breen@math .univ-paris1 3.fr T o Murr ay Gerstenhab er and J i m Stasheff Summary . W e discuss certa in aspects of the com binatorial approach to the d i ffer- entia l geometry of non-ab elian gerb es due to W. Messing and the author [5], and giv e a more direct deriv ation of th e associated co cycle equations. This leads u s to a more restrictive definition than in [5] of the corresp onding cob oundary relations. W e also show th at the d iagrammatic p roofs of certain local cu rving and cu rv ature equations may b e rep laced by computations with differential forms. 1 In tro duction It is a classical fact 1 that to a principa l G -bundle P on a scheme X , endow ed with a connection ǫ , is asso ciated a Lie ( G )-v alued 2-for m κ on P , the curv ature of the connectio n, satisfying a certain G -equiv ariance co ndition. While κ doe s not in general des c end to a 2 - form on X , the e quiv ariance co ndition may b e viewed as a des cent condition for κ fr o m a 2-form on P to a 2-form o n X , but now with v alues in the Lie algebra of the gauge group P ad of P . The connection on P also induces a connection µ on the group P ad , and the 2- form κ satisfies the Bianchi equatio n, an equation which ma y be expres sed in global ter ms as d κ + [ µ, κ ] = 0 (1.1) ([5] pro p o sition 1.7, [4] theorem 3.7 ). Cho os ing a lo ca l triv ia lization of the bundle P , on an op en cov er U := ` i ∈ I U i of X , the co nnec tion ǫ is describ ed ∗ Unit´ e Mixte de Recherc he CNRS 7539 1 at least in a d ifferenti al geometric setting, see [9 ], but the same construction can b e carried out within the context of algebraic geometry . 2 La wrence Breen by a family o f Lie ( G )-v a lued connection 1- forms ω i defined on the op en sets U i , and the asso cia ted curv ature κ corresp onds to a family of Lie ( G )-v alued 2- forms κ i defined, according to the so-calle d structural equation o f Elie Cartan, by the fo r mula 2 κ i = d ω i + 1 2 [ ω i , ω i ] (1.2) Equation (1.1) then r educes to the classica l Bia nchi iden tit y d κ i + [ ω i , κ i ] = 0 . (1.3) J. -L. Brylinski in tro duced in [7] the no tions of connection ǫ and curving K on an ab elia n G -gerb e P on a spac e X (where G was the multip licative group G m , or rather in his framework the gr oup U (1)), and show ed that to such co nnective data ( ǫ, K ) is asso ciated a c losed G m -v alued 3- form ω on X , the 3-c ur v ature. Mo re r ecently , W. Mess ing and the a uthor e x tended these concepts in [5] from ab elian to gene r al, not necessarily ab elian, gerb es P on a scheme X . The co efficients of such a ge rb e no longer co nstitute a sheaf of groups as in the principal bundle s ituation, but rather a mo noidal stack G on X , a s is to b e exp ected in that catego r ified s etting. In pa rticular, whe n the gerb e is asso ciated to a given non-ab elian gro up G (so that we refer to it as a G - gerb e), the co r resp onding co efficient stac k G is the monoidal stack asso ciated to the prestack deter mined by the cr ossed mo dule G − → Aut( G ), where Aut( G ) is the sheaf o f lo cal automorphisms of G . It may a lso b e describ ed more inv ariant ly as the mono idal s tack of G -bitor sors on X . Once more, to the gerb e P is asso ciated its g auge stack, a twisted form P ad := E q ( P , P ) of the given monoidal s tack G , and the connection o n P induces a connectio n µ on P ad . By analo gy with the principal bundle c a se, the corres po nding 3 - curv ature Ω , viewed as a glo ba l 3-form on X , now takes its v alues in the arrows of the stack P ad . There now a rises a new, and at first s ig ht so mewhat sur prising feature, but whic h is simply another facet o f the ca tegorifica tio n context in which we are op erating. The 3-for m Ω is acc o mpanied by an auxilia ry 2-form κ with v alues in the ob jects of the ga uge stack P ad , which we called in [5] the fake curvatur e of the given co nnective str ucture ( ǫ, K ). A first re la tion b etw een the forms Ω and κ comes fro m the very de finitio n [5] (4.1.2 0 ), (4.1.22) of Ω , and may b e stated as in [5](4.3.8) as the ca tegorica l equation tΩ + d κ + [ µ, κ ] = 0 (1.4) where t stands for “ ta rget” of a 1-a rrow with source the identit y ob ject I in the s tack of Lie( P ad )-v alued 3-forms on X . On the other hand, the 3-for m Ω is 2 The canonical divided pow er 1 / 2[ ω , ω ] of the 2-form [ ω , ω ] is also denoted ω ∧ ω or [ ω ] (2) . Differential Geometry of Gerbes and Differential F orms 3 no lo ng er closed, even in the µ -twisted sense described for principa l bundles by (1.1). It sa tisfie s instead the following more co mplica ted analog ue [5 ] (4.1.33 ) of the Bianchi iden tity (1.1): d Ω + [ µ, Ω ] + [ K , κ ] = 0 . (1.5) While the first tw o ter ms in this equation ar e s imila r to thos e of (1.1), the categorific a tion term K is an arrow in the sta ck o f 2- forms with v alues in the monoidal stac k E q ( P ad , P ad ) induced by the curving K . The pairing of K with κ is induced b y the ev alua tion o f the natural transformation K be t ween functors from P ad to itself on the ob ject κ of P ad . The price to b e paid for the co mpact form in which the g lobal curv a tur e equations (1.4) a nd (1.5) have b een stated is their rather a bstract nature, and it is of interest to descr ib e them in a more lo cal form in ter ms o f tra ditional group-v a lued differ e n tial forms, just a s was done in (1.3) for equation (1.1) . Such a lo cal descr iption was already obtained in [5 ], b oth for the co cycle conditions (1.4) and (1.5), and for the co rresp onding cob oundary equations which arise when alternate lo cal trivia lizations o f the gerb e hav e b een chosen. How ever, the determination of those lo ca l equa tions was ra ther indirect, as it req uir ed a third descr iption of a ger be, which w e hav e called the semi- lo cal description [6] § 4 , and which has a ls o app ear ed elsew he r e in a v arious situations [18], [14], [8 ]. The present text ma y b e viewed as a c ompanion piece to the author’s [6]. Its main purp ose is to provide a mo re transpa rent construc tio n than in [5] of the co c ycle conditions and related equations asso ciated to a gerb e with curving data summarized in [5] theorem 6.4. W e res tr ict our atten tio n, as in [6 ], to ger be s which are connected rather than lo cally co nnected, a s these determine ˇ Cech cohomology classes. A coc yclic descr iptio n in the general case requires hypercovers and could be dealt with along the lines dis c us sed in [3], but w ould no t shed any additiona l lig h t o n the phenomena b eing inv estiga ted here. Our main r esults are to b e found in sec tions 4 and 5, while section 3 reviews for the rea der’s co nv enience some asp ects of [5] and [6]. Sec tion 2 is a r eview of some o f the for mu las in the differen tial calculus of Lie ( G )-v alued forms, a few of which do not appe ar to b e well-known. Another aim of the prese n t work is to revisit the quite complicated cob oundary equa tio ns of [5] § 6.2. The co b o unda ry equations which a rise here are simpler, a nd mor e c o nsistent than those o f [5] with a non-ab elian ˇ Cech-de Rham interpretation. W e refer to remark 5.1 fo r a s p ecific comparis on b etw een the tw o notions. In order to make this compar ison easier, we hav e c hosen the orientations of o ur ar r ows co nsistenly with [5]. This a ccounts for example for the strange choice of orientation of the arrow B i in diagra m (4.13), or for the change of sign (4.2 8) for the arr ow γ ij . 4 La wrence Breen A final purp o s e of this text is to e xplain how the diagramma tic pro ofs of some of the lo ca l results of [5] ca n b e replaced by more classical computations inv olving Lie ( G )-v alued differential forms. F or this rea son, we have given t wo separate computations fo r certa in equa tions, one diagr ammatic and the other classical. W e do not assert that one o f the t wo metho ds of pro of is alwa ys preferable, though one might contend that diagrams pr ovide a b etter under- standing of the situation than the co rresp onding manipulation of differential forms. As the level of categor ification increase s , so will the dimension of the diagrams to b e considered, and it may not b e r ealistic to expect to tre ad along the diagr ammatic path muc h b eyond the hyp e rcub e pr o of [5] (4.1.33 ) of the hig he r Bianchi e q uation (1.5). The generality and algebra icit y of the formalism o f differen tial forms m ust then come into its own. In addition, it is our ho p e that the present appr oach, which extends to the gerb e context the traditional methods of differential geometry , will provide an acc e ssible p oint of entry into this topic. A num b er o f other author s have recently descr ib ed certain as pec ts of the differential ge o metry of gerb es in ter ms of differential forms, particularly [1], [12], and [16], [2]. I wis h to thank Bernard Julia a nd Ca mille Laur ent-Gengoux fo r enlight- ening discussions on re lated topics. The imp etus for the present work was provided by my co lla b oration with Wiliam Messing on our join t papers [4 ] and [5]. It is a pleasure to thank him here for our instructive a nd wide-rang ing discussions ov e r all these years. 2 Group-v alued differen tial forms 2.1 Let X be an S - scheme. W e assume fro m now on for simplicity that that the primes 2 and 3 are inv ertible in the r ing o f functions of S (for exa mple S = Sp ec( k ) where k is a field of c haracter istic 6 = 2 , 3). A relative differential n -form on an S -scheme X , with v a lues in a sheaf of O S -Lie alge br as g is defined as a glo bal section of the sheaf g ⊗ O S Ω n X/S on X . When X/S is smo oth, g ⊗ O S Ω n X/S ≃ Hom O X ( T n X/S , g X ) (2.1) where g X := g ⊗ O S O X and T n X/S is the n -th exterior p ower ∧ n T X/S of the relative tangen t sheaf T X/S , i.e the she a f of rela tive n - vector fields on X . Such an n - form is nothing else than an O X -linear ma p T n X/S − → g X . (2.2) In view of this definition, such a map is cla ssically called a g -v alued dif- ferential fo r m. A more geometric description of suc h forms is given in [4], Differential Geometry of Gerbes and Differential F orms 5 following the ideas of A. K o ck in the context of synthetic differ e n tial geom- etry [10], [11]. It is bas ed on the consideration, for any p os itive integer n , of the scheme ∆ n X/S of relative infinitesimal n -simplexes on X . F or any S - scheme T , a T -v alued p oint of ∆ n X/S consists of an ( n + 1)-tuple of T -v alued po int s ( x 0 , . . . , x n ) of X which ar e pairwise close to first o rder in an appr o- priate sense [4] (1.4.9 ). W e view ∆ n X as an X -scheme via the pr o jection p 0 of such p oints to x 0 . As n v aries, the schemes ∆ n X/S determine a simplicia l X -scheme ∆ ∗ X/S , whos e face and deg eneracy op era tions are induced by the usual pro jection and injection mor phisms X n − → X n ± 1 . Let G b e a flat S -gr o up sch eme, with O S -Lie a lgebra g . A relative g - v alued n -for m (2.2) on X / S may then b e iden tified by [4] prop ositio n 2.5 with a mo rphism o f S -schemes ∆ n X/S f − → G (2.3) whose restriction to the degenerate subsimplex s∆ n X/S of ∆ n X/S factors through the unit se c tion of G . When differen tial forms a re expre ssed in this combina- torial language, they deserve to be called G -v a lued differential forms, ev en though they a ctually coincide with the traditional g - v alued differen tial forms (2.1), (2.2). In the c ombinatorial cont ext, our notation will be multiplicativ e , and a dditiv e when we pass to the traditio na l la nguage o f differential forms. W e will now discuss some of the features o f these g -v a lued forms, and refer to [4] for further discussion. Fir st of a ll, let us recall tha t the action of the symmetric gr o up S n +1 on a com binatorial differential n - form ω ( x 0 , . . . , x n ) by per m utation of the v ariables is given b y ω ( x σ (0) , . . . , x σ ( n ) ) = ω ( x 0 , . . . , x n ) ǫ ( σ ) where ǫ ( σ ) is the s ignature o f σ . Also , the commutator pa iring [ g , h ] := g h g − 1 h − 1 on the gr oup G determines a bra ck et pairing o n g - v alued forms of degree ≥ 1, defined combinatorially by the rule ( g ⊗ O S Ω m X/S ) × ( g ⊗ O S Ω n X/S ) / / ( g ⊗ O S Ω m + n X/S ) (2.4) which sends ( ω , ω ′ ) to [ ω , ω ′ ], where [ ω , ω ′ ]( x 0 , . . . , x m + n ) := [ ω ( x 0 , . . . , x m ) , ω ′ ( x m , . . . , x m + n )] . This pairing is defined in classica l terms, by [ ω , ω ′ ] := [ Y , Y ′ ] ⊗ ( η ∧ η ′ ) 6 La wrence Breen for any pair of forms ω := Y ⊗ η and ω ′ := Y ′ ⊗ η ′ in g ⊗ O S Ω ∗ X/S . It endows g ⊗ O S Ω ∗ X/S with the str ucture of a gr aded O S -Lie algebra . In pa rticular, the brack et satisfies the graded c ommut ativity rule [ f , g ] = ( − 1 ) | f || g | +1 [ g , f ] , (2.5) where | f | is the degre e of the form f , so that [ f , f ] = 0 whenever | f | is even. The graded Jacobi iden tit y is expres sed (in additive notation) as: ( − 1) | f || h | [ f , [ g , h ]] + ( − 1) | f || g | [ g , [ h, f ]] + ( − 1) | g | | h | [ h, [ f , g ]] = 0 . In particular, [ f , [ f , f ]] = 0 (2.6) and, when | f | = | g | = 1, [ f , 1 2 [ g , g ]] = [[ f , g ] , g ] . Let Aut( G ) be the sheaf of lo c a l automorphisms o f G , whose gr oup of sections ab ove an S -scheme T is the group Aut T ( G T ) of a uto morphisms of the T -gr o up G T := G × S T . The definition (2.3) o f a co mbinatorial n -form still makes s ense when G is replace d by a shea f of gro ups F o n S , and the traditional descriptio n of such combinatorial n -for ms as n -for ms with v alues in the Lie algebr a of F remains v a lid by [4] propos ition 2.3 when F = Aut( G ). The ev aluation map Aut( G ) × G − → G ( u, g ) 7→ u ( g ) induces for all pair of p ositive integers a bilinear pairing (Lie (Aut( G )) ⊗ O S Ω m ) × ( g ⊗ O S Ω n X/S ) / / ( g ⊗ O S Ω m + n X/S ) (2.7) which sends ( u, g ) to [ u , g ], where [ u, g ]( x 0 , . . . , x m + n ) := u ( x 0 , . . . , x m )( g ( x m , . . . , x m + n )) g ( x m , . . . , x m + n ) − 1 . (2.8) This pairing is co mpatible with the pa irings (2.4) asso ciated to the S -gro ups G and Aut( G ) in the following sense. F or an y pair of g -v alued forms g , g ′ , and an Aut ( G )-v alued form u , [ i ( g ) , g ′ ] = [ g , g ′ ] and i ([ u, g ]) = [ u, i ( g )] (2.9) where i : G − → Aut( G ) is the inner conjugation map i ( γ )( g ) := γ g γ − 1 . More generally , an isomor phism r : G − → G ′ induces a morphism r from G -v alued Differential Geometry of Gerbes and Differential F orms 7 combinatorial n - fo rms to G ′ -v alued combinatorial n -for ms, compatible with the Lie bracket op eration (2 .4), and which cor resp onds in classical terms to the mor phism Lie( r ) ⊗ osc 1 : g ⊗ O S Ω n X/S − → g ′ ⊗ Ω n X/S . The functoriality of the brack e t (2.7) is expr essed by the for mu la r ([ u, g ] = [ r u, r ( g )] (2.10) where r u := r u r − 1 . When u is a n Aut( G )-v alued for m of deg ree m ≥ 1 a nd g is a G -v alued function, the definitio n o f a pair ing (Lie Aut( G ) ⊗ O S Ω m X/S ) × G − → g ⊗ O S Ω m X/S ( u, g ) 7→ [ u, g ] is still given by the formula (2 .8), but now with n = 0. This pairing are no longer line a r in g , but instead satisfies the eq ua tion [ u, g g ′ ] = [ u , g ] + g [ u, g ′ ] where for any G -v alued for m ω and any G -v a lued function g the adjoint left action g ω of a function g on a form ω is de fined co m binatoria lly b y ( g ω )( x 0 , . . . , x n ) := g ( x 0 ) ω ( x 0 , . . . , x n ) g ( x 0 ) − 1 , (and this expr ession is in fact e q ual to g ( x i ) ω ( x 0 , . . . , x n ) g ( x i ) − 1 for a ny 0 ≤ i ≤ n ). In classical nota tio n this cor resp onds, for ω = Y ⊗ η ∈ g ⊗ Ω n X/S , to the formula g ( Y ⊗ η ) = g Y ⊗ η for the a djo int left action of g on Y . The adjoint right action ω γ is defined by ω g := ( g − 1 ) ω so that ω g ( x 0 , . . . , x n ) = g ( x 0 ) − 1 ω ( x 0 , . . . , x n ) g ( x 0 ) . Similarly , when g is a G -v a lued a nd u an Aut( G )-v alued form, a pa iring [ g , u ] is defined by the combinatorial for m ula [ g , u ]( x 0 , . . . , x m + n ) := g ( x 0 , . . . , x m ) ( u ( x m , . . . , x m + n )( g ( x 0 , . . . , x m ) − 1 )) . (2.11) The pairing (2.11) satisfies the a nalogue [ g , u ] = ( − 1) | g | | u | +1 [ u, g ] of the gr a ded commut ativity rule (2.5), so that its prop erties may be deduced from tho se of the pair ing [ u , g ]. In pa rticular [ g − 1 , u ] = − [ u, g − 1 ] = [ u , g ] g . W e refer to app endix A of [5] for additional prop erties of these pair ings. 8 La wrence Breen 2.2 The de Rham differential map g ⊗ O S Ω n X/S d n X/S / / g ⊗ O S Ω n +1 X/S (2.12) is defined co mbinatorially for n ≥ 2, in Alex ander-Spanier fashion, b y d n X/S ω ( x 0 , . . . , x n +1 ) := n +1 Y i =0 ω ( x 0 , . . . , b x i , . . . ω n +1 ) ( − 1) i . (2.13) This definitio n agree s for n > 1 with the classica l definition of the G -v a lued de Rham differential: d n X/S ω := d X/S ω (2.14) where for ω = Y ⊗ η in g ⊗ Ω n X/S , d X/S ω := Y ⊗ d η . (2.15) In particular d n is an O S -linear map whenever n ≥ 2, and it follows from (2.15) that the comp osite d n +1 d n is trivia l . This also follows fro m the combinatorial definition o f d n , since for n ≥ 2 the facto rs in the express ion (2.13) for d n ω commute with each other. F or any section g o f G , we set d 0 X/S ( g ) := g ( x 0 ) − 1 g ( x 1 ) . (2.16) The map G X d 0 X/S − → g ⊗ O S Ω 1 X/S g 7→ g − 1 d g (2.17) is a crossed homomorphis m, for the adjoin t left action of G on g . Obser ve that the expression g − 1 d g is consistent with the combinatorial definition (2.16) of d 0 X/S ( g ). While this traditional expr ession of d 0 X/S ( g ) as a pro duct of the t wo terms g − 1 and d g doe s make sense whenev er G is a subgro up scheme of the linear group GL n,S , such a deco mpo sition is purely co nv entional for a general S -gr oup sc heme G . A co mpanion to d 0 X/S is the differential e d 0 : G − → g ⊗ O S Ω 1 X/S , defined b y e d 0 X/S ( g )( x 0 , x 1 ) := g ( x 1 ) g ( x 0 ) − 1 . The traditional notation for this expressio n is dg g − 1 . This notation is consis- ten t with such formu las (in additive notation) as Differential Geometry of Gerbes and Differential F orms 9 g ( g − 1 dg ) = d g g − 1 and − ( g − 1 d g ) = d g − 1 g . The different ial d 1 X/S is defined co mbinatorially b y (d 1 X/S ω )( x, y , z ) := ω ( x, y ) ω ( y , z ) ω ( z , x ) . (2.18) In classical ter ms, it follows (see [4] theore m 3.3) that d 1 X/S ω := d ω + 1 2 [ ω , ω ] . (2.19) W e will henceforth deno te d n X/S simply by d n for all n . The quadratic term 1 2 [ ω , ω ] implies that d 1 X/S is not a linear map, in fa ct it follows from (2.1 9), or the ele mentary combinatorial c a lculation of [4] lemma 3.2, that d 1 ( ω + ω ′ ) = d 1 ω + d 1 ω ′ + [ ω , ω ′ ] . In particular, d 1 ( − ω ) = − d 1 ( ω ) + [ ω , ω ] . It is immediate, from the combinatorial p oint of view, that d 1 d 0 ( g ) = d 1 ( g − 1 d g ) = 0 (2.20) for all g in G . The differential d 1 has a compa nion, which we will denote by e d 1 , defined b y e d 1 ( ω )( x, y , z ) := ω ( z , x ) ω ( y , z ) ω ( x, y ) . A comb inatoria l computation implies that e d 1 ω = d 1 ω − [ ω , ω ] = d ω − 1 2 [ ω , ω ] , and the ana logue e d 1 ( e d 0 ( g )) = e d 1 ( dg g − 1 ) = 0 of (2.2 0) is satisfied. Finally , it follows fro m (2.14) that the d n satisfy d i + j [ ω , ω ′ ] = [d i ω , ω ′ ] + ( − 1) i [ ω , d j ω ′ ] whenever i, j ≥ 2, and the corr esp onding for mula fo r the pairing [ u, g ] (2.8) is also v a lid. 10 La wrence Breen 2.3 W e now c ho ose, for any S -s cheme X and any S -gr oup s cheme G , a n Aut( G )- v alued 1-fo rm m on X . W e extend the definition o f the de Rham differ e ntials (2.17), (2.18) and (2.1 2) to the twisted differentials d n X/S, m : g ⊗ O S Ω n X/S − → g ⊗ O S Ω n +1 X/S (2.21) (or simply d n m ) defined c o mbinatorially b y the following form ulas: d 1 m ω ( x 0 , x 1 ) := ω ( x 0 , x 1 ) m ( x 0 , x 1 )( ω ( x 1 , x 2 )) m ( x 0 , x 1 ) m ( x 1 , x 2 )( ω ( x 2 , x 0 )) = ω ( x 0 , x 1 ) m ( x 0 , x 1 )( ω ( x 1 , x 2 )) ω ( x 0 , x 2 ) − 1 d n m ω ( x 0 , . . . , x n +1 ) := = m ( x 0 , x 1 )( ω ( x 1 , . . . x n +1 )) n +1 Y i =1 ω ( x 0 , . . . , b x i , . . . , x n +1 ) ( − 1) i when n > 1. When the Aut ( G )-v a lued fo r m m is the image i ( η ) under inner conjugation of a G -v alued form η , the expression d n i ( η ) ω will simply be denoted d n η ω . The corr e sp o nding deg ree zero ma p d 0 m : G − → g ⊗ O S Ω 1 X/S is defined by d 0 m ( g ) := g ( x 0 ) − 1 m ( x 0 , x 1 )( g ( x 1 )) , (and d 0 m ( g ) will also be denoted g − 1 d m ( g ), consistenly with (2.16)). It follows from elementary combinatorial computations that the differen- tials d n m can b e defined in class ic al terms by d n m ω = d n ω + [ m, ω ] (2.22) for all n , so that for any g -v alued 1-form η , d n m + i η ( ω ) = d n m ( ω ) + [ η , ω ] . (2.23) In particular, d 1 m ( ω ) = d 1 ω + [ m, ω ] = d ω + 1 2 [ ω , ω ] + [ m, ω ] . While the ma p d n m is linea r fo r n ≥ 2, d 1 m ( ω + ω ′ ) = d 1 m ω + d 1 m ω ′ + [ ω , ω ′ ] (2.24) so that d 1 m ( − ω ) = − d 1 m ( ω ) − [ ω , ω ] . (2.25) Differential Geometry of Gerbes and Differential F orms 11 Finally , for any section g o f Γ , g − 1 d m g = g − 1 d g + [ m, g ] . The compo site morphism d n +1 m d n m is in general non-trivial, and the previ- ous classical definitions o f d n m imply that d n +1 m d n m ω = [d 1 m, ω ] (2.26) whenever n ≥ 2. F or n = 0, the cor r esp onding for m ulas a re d 1 m d 0 m g = [ g − 1 , d 1 m ] and e d 1 m e d 0 m g = [d 1 m, g ] (2.27) so that, for n 6 = 1, we r ecov er the well-kno wn a ssertion that the v anishing of d 1 m = 0 implies that d n +1 d n = 0. One v erifies that for any 1- form ω d 2 m d 1 m ( ω ) = [d 1 m, ω ] + [d 1 m ω , ω ] (2.28 ) = [d 1 m, ω ] + [d 1 ω , ω ] + [[ m, ω ] , ω ] . (2.29) This reduces to the equatio n d 2 m d 1 m ( ω ) = [d 1 m, ω ] of t yp e (2.26) whenever d 1 m ω = 0. F or m = i ( ω ), equation (2.28) is equiv alent to the clas sical Bianchi iden tity [9] I I Theo rem 5.4 : d 2 ω d 1 ω = 0 . (2.30) W e now state the functoriality prop er ties of the differential (2.22) d n m for n ≥ 1 . W e define the twisted conjugate g ∗ ω of a G -v a lued 1 -form ω by g ∗ ω := ( p ∗ 0 g ) ω ( p ∗ 1 g ) − 1 = g ω + g d g − 1 (2.31) = ω + [ g , ω ] + g d g − 1 . It follows from the co m binatoria l definition (2.18) o f d 1 that g (d 1 ω ) = d 1 ( g ∗ ω ) . (2.32) More g enerally , for any G -v alued form ω o f degree n ≥ 1, and any s ection u of Aut( G ) on X , u (d n m ( ω )) = d n ( u ∗ m ) u ( ω ) (2.33) = d n ( u m ) u ( ω ) + [ u d u − 1 , u ( ω )] = d n m ( u ( ω )) + [[ u, m ] , u ( ω )] + [ u d u − 1 , u ( ω )] . (2.3 4) 12 La wrence Breen 3 Gerb es and their conn ective struct ures 3.1 Let P b e a gerb e 3 on a n S -scheme X . F o r simplicit y , in dis c ussing gerb es w e will make t wo additional assumptions: • P is a G -gerb e, for a given S - group s cheme G . • P is connected. The firs t ass umption gives us , for any o b ject x in the fibr e ca tegory P U ab ov e an op en set U ⊂ X , an isomor phism of sheav es on U G | U ∼ / / Aut P U ( x ) . (3.1) The second assumption a sserts that for any pair o f ob jects x, y ∈ ob( P U ) there exists an ar row x − → y in the categ ory P U . This ensures that the gerb e is describ ed b y a n element in the degree 2 ˇ Cech cohomolog y of X ra ther than by degree 2 co homology with resp ect to a h yp ercov er o f X . Let us c ho ose a family of lo c al ob jects x i ∈ P U i , for some op e n cov er U = ` i U i of X , and a family of arrows x j φ ij / / x i (3.2) in P U ij . Identifying elements of bo th Aut P ( x i ) and Aut P ( x j ) with the corre - sp onding sections o f G a bove U i and U j , these ar rows de ter mine a family of section λ ij ∈ Γ ( U ij , Aut( G )), defined by the commutativit y of the diagr ams x j γ / / φ ij   x j φ ij   x i λ ij ( γ ) / / x i (3.3) for ev ery γ ∈ G | U ij . In addition, the arrows φ ij determine a family of elements g ij k ∈ G | U ijk for all ( i, j, k ) by the commutativit y of the diagrams x k φ jk / / φ ik   x j φ ij   x i g ijk / / x i (3.4) 3 W e refer to [3] and [6] for the definition of a gerb e, and for additional details regarding the associated co cycle and cob oundary eq uations (3.7), (3.14). Differential Geometry of Gerbes and Differential F orms 13 ab ov e U ij k . By conjuga tion in the sense made clear by diag ram (3.3), it follo ws that the λ ij satisfy the co cycle condition λ ij λ j k = i ( g ij k ) λ ik . (3.5) By [6] lemma 5.1, the G -v alued co chains g ij k also satisfy the co cycle condition λ ij ( g j kl ) g ij l = g ij k g ikl . ( 3.6) These tw o co cycle equations may be wr itten mor e compactly as ( δ 1 λ ij = i ( g ij k ) δ 2 λ ij ( g ij k ) = 1 , (3.7) where δ 2 λ is the λ -twisted degree 2 ˇ Cech differential determined by eq ua tion (3.6). They may b e jointly view ed as the ( G − → Aut ( G ))-v alued ˇ Cech 1- co cycle 4 equations asso c iated to the gerb e P , the op en cover U of X , and the trivializing families of ob jects x i and a rrows φ ij in P . Let us choo se a second family of lo cal ob jects x ′ i in P U i , and of ar rows x ′ j φ ′ ij / / x ′ i (3.8) ab ov e U ij . T o these cor resp ond a new cocy c le pair ( λ ′ ij , g ′ ij k ). In order to compare this set of arrows with the previous one, we choo se (a fter a harmless refinement of the g iven op en cover U of X ) a family of arr ows x i χ i / / x ′ i (3.9) in P U i for all i . The arrow χ i induces by co njugation a sectio n r i in the g roup of sections Γ ( U i , Aut( G )), characteriz e d b y the co mm utativity of the sq ua re x i χ i   u / / x i χ i   x ′ i r i ( u ) / / x ′ i (3.10) for all u ∈ G . The lack of compa tibility be tw een these a r rows χ i and the arr ows φ ij , φ ′ ij (3.2), (3.8) is measured b y the family of sections ϑ ij ∈ Γ ( U ij , G ) determined by the c ommut ativity of the following diagram: 4 W e prefer to emphasize the fact that λ ij is a 1-co chain since this is more consistent with a simplicial defin ition of the associated cohomology , even though it is more customary to view the pair of eq uations (3.7) as a 2-cocycle equation, with (3.5) an auxiliary condition. 14 La wrence Breen x j φ ij / / χ j   x i χ i   x ′ i ϑ ij   x ′ j φ ′ ij / / x ′ i . (3.11) Under the identifications (3.1), diagr am (3 .1 1) induces by conjuga tion, in a sense made clea r by the definition (3.10) of the aur omorphism r i , a commu- tative diagram of g roup schemes a bove U ij G λ ij / / r j   G r i   G i ( ϑ ij )   G λ ′ ij / / G , whose commu tativity is express e d b y the eq ua tion λ ′ ij = i ( ϑ ij ) r i λ ij r − 1 j (3.12) in Aut( G ). Consider now the diag ram 5 x k φ jk { { v v v v v v v v v v v v v φ ik / / x i χ i   g ijk { { w w w w w w w w w w w w w x ′ i r i ( g ijk ) { { w w w w w w w w w w w w w ϑ ik   x j φ ij / / χ j   χ k x i χ i   x ′ i ϑ ij   x ′ j φ ′ ij / / ϑ jk     x ′ i λ ′ ij ( ϑ jk )   x ′ k φ ′ jk y y t t t t t t t t t t t t φ ′ ik / / x ′ i g ′ ijk z z t t t t t t t t t t t t t x ′ j φ ′ ij / / x ′ i . (3.13) 5 This diagram whose faces are five p entago ns and three squares (as well as t h ose in (4.9) and (4.25) b elo w) is the 1-skeleton of a Saneblidze-Umble cubical mod el [15], [13] for the Stasheff asso ciahedron K 5 [17]. Differential Geometry of Gerbes and Differential F orms 15 Both the top and the b ottom squares c o mm ute, s inc e these squares ar e o f t yp e (3.4). So do the back, the left and the top front vertical squares , since a ll three are o f type (3.11) . The same is true of the lower front s quare, a nd the upper right vertical squar e, since these tw o ar e resp ectively of the form (3.3) and (3.1 0). It follows that the rema ining low er right square in the diagram is also co mm utative, since all the a r rows in dia gram (3.13) are invertible. The commutativit y of this final s q uare is expr essed alg ebraically b y the eq uation g ′ ij k ϑ ik = λ ′ ij ( ϑ j k ) ϑ ij r i ( g ij k ) . W e say that tw o co cycle pairs ( λ ij , g ij k ) and ( λ ′ ij , g ′ ij k ) are cohomolog ous if w e are g iven a pair ( r i , ϑ ij ), with r i ∈ Γ ( U i , Aut( G )) a nd ϑ ij ∈ Γ ( U ij , G ), satisfying those tw o equations ( λ ′ ij = i ( ϑ ij ) r i λ ij r − 1 j g ′ ij k ϑ ik = λ ′ ij ( ϑ j k ) ϑ ij r i ( g ij k ) . (3.14) and display this a s ( λ ij , g ij k ) ( r i ,ϑ ij ) ∼ ( λ ′ ij , g ′ ij k ) . (3.15) The equiv alence clas s of the co cy cle pair ( λ ij , g ij k ) for this r elation is inde- pendent of the c hoices of ob jects x i and arrows φ ij by fro m which it was constructed. By definition, it determines an element in the first non-a be lian ˇ Cech co homology set ˇ H 1 ( U , G i − → Aut( G )) with co efficients in the cr ossed mo dule i : G − → Aut( G ). 3.2 In [5], the co mbinatorial des cription of differential forms is used in order to define the concepts of connections and curving s o n a gerb e. F o r any S -g roup scheme G , a (relativ e) connection o n a principal G -bundle P abov e the S - scheme X may b e defined as a morphism p ∗ 1 P ǫ / / p ∗ 0 P (3.16) betw een the tw o pullbacks of P to ∆ 1 X/S , whose r e striction to the diagonal subscheme ∆ : X ֒ → ∆ 1 X/S is the iden tity morphism 1 P . This type of definition o f a connection, as a vehicle for parallel tra nsp o rt, remains v alid for other structures than principal bundles. In par ticular, for any X -gro up scheme Γ , a connection on Γ is a mo rphism o f group schemes 16 La wrence Breen µ : p ∗ 1 Γ − → p ∗ 0 Γ (3.17) ab ov e ∆ 1 X/S whose res triction to the diagonal subscheme X ֒ → ∆ 1 X/S is the ident ity morphism 1 Γ . When Γ is the pullback to X o f an S -gr oup scheme G , the inv er se images p ∗ 1 G and p ∗ 0 G of G X ab ov e ∆ 1 X/S are ca nonically isomor- phic, so that the connection (3.17) is then describ ed by a Lie(Aut( G ))-v alue d 1-form m . A connection µ on a group Γ determines de Rham differ ent ials d n X/S, µ : Lie( Γ ) ⊗ O S Ω n X/S − → Lie( Γ ) ⊗ O S Ω n +1 X/S (or simply d n µ ) defined combinatorially by the for m ulas [5] (A.1.9)-(A.1.1 1) and their higher analog ue s . When Γ is the pullbac k of an S -gr oup scheme, d n µ is decrib ed in cla ssical ter ms as the deformation (2.22) d n µ := d n m of the de Rham differen tial d n determined by the asso ciated 1-form m . When the curv atur e d 1 m of the connection µ is trivial, the connectio n is said to b e int egra ble . In that case , it follows from (2.26) and (2.27) that the de Rham differentials satisfy the condition d n +1 m d n m = 0 for all n 6 = 1. The curv atur e o f a connection ǫ (3.16) on a principa l bundle P is the unique arrow κ ǫ : p ∗ 0 P − → p ∗ 0 P such that the following diagr am above ∆ 2 X/S commutes, with ǫ ij the pullbac ks of ǫ under the corr esp onding pro jections p ij : ∆ 2 X/S − → ∆ 1 X/S : p ∗ 2 P ǫ 12 / / ǫ 02   p ∗ 1 P ǫ 01   p ∗ 0 P κ ǫ / / p ∗ 0 P By co nstruction, κ ǫ is a r elative 2-form on X with v alues in the gauge group P ad := Isom G ( P, P ) of P . The connection ǫ on P induces a connection µ ǫ on the group P ad , deter- mined b y the co mm utativity of the squares p ∗ 1 P u / / ǫ   p ∗ 1 P ǫ   p ∗ 0 P µ ǫ ( u ) / / p ∗ 0 P (3.18) Differential Geometry of Gerbes and Differential F orms 17 for all sections u of p ∗ 1 ( P ad ). By [11], [5] prop osition 1.7, the cur v ature 2-form κ ǫ satisfies the Bianchi iden tity d 2 µ ǫ ( κ ǫ ) = 0 . (3.19) F or a given family o f lo ca l sections of P , with asso ciated G -v alue d 1-co cy cles g ij , the co nnection (3.16) is describ ed by a family of G -v alued 1-forms ω i ∈ g ⊗ Ω 1 U i /S , satisfying the g luing c ondition ω j = ω ∗ g ij i = ω g ij i + g − 1 ij d g ij (3.20) ab ov e U ij , for the a ction of G on g ⊗ O S Ω 1 U i /S induced by the adjoint right action of G on g . A 1 -form sa tisfying this equation is cla ssically known as a connection form. The induced curv ature κ is lo cally descr ib ed by the fa mily of 2-forms κ i := d 1 ω i = d ω i + 1 2 [ ω i , ω i ] , and these s a tisfy the simpler ˇ Cech (or g luing) co ndition κ j = κ g ij i . Equation (3.19) is r eflected a t the lo cal level in the equation d 2 ω i κ i = 0 , which is simply the cla ssical B ianchi iden tity (2 .30) for the 1-fo rm ω i . 3.3 The notion of a connectiv e s tructure o n a G -gerb e P is a categorificatio n of the notion of a connection on a principal bundle, as w e will now reca ll, following [5] § 4. T o P is as s o ciated its ga uge stack P ad . By definitio n this is the mo no idal stack E q X ( P , P ) of self-equiv a lences of the stack P , the monoidal s tructure being defined by the comp ositio n of equiv alence s . A connection o n a P is an equiv alence b e tw een stacks p ∗ 1 P ǫ / / p ∗ 0 P (3.21) ab ov e ∆ 1 X/S , together with a natural isomor phism betw een the r estriction ∆ ∗ ǫ o f ǫ to the diag onal subscheme X of ∆ 1 X/S and the identit y mo rphism 1 P . Suc h a connection ǫ induces as in (3.18) a connection µ on the ga uge stack P ad . 18 La wrence Breen A curving of ( P , ǫ ) is a natural isomor phism K p ∗ 2 P ǫ 12 / / ǫ 02   p ∗ 1 P ǫ 01   p ∗ 0 P κ / / p ∗ 0 P , K 8 @ y y y y (3.22) for some mor phism κ : p ∗ 0 P − → p ∗ 0 P ab ov e ∆ 2 X/S . It is determined by the choice of some explicit qua si-inv e r se of the connectio n ǫ . The arr ow κ which arises a s part o f the definition of K is called the fake curvatur e asso ciated to the connective structure ( ǫ, K ). It is a global o b ject in the pullbac k to ∆ 2 X/S of the ga uge s tack P ad . The co nnective structure ( ǫ, K ) determines a 2-arr ow p ∗ 0 P κ 013 / / κ 023   p ∗ 0 P µ 01 ( κ 123 )   p ∗ 0 P κ 012 / / / / p ∗ 0 P Ω v ~ t t t t t t This is the unique 2- a rrow which may b e inser ted in diag r am p ∗ 3 P ǫ 13 / / ǫ 03 z z t t t t t t t t t ǫ 23   p ∗ 1 P ǫ 01 y y s s s s s s s s s s κ 123   p ∗ 0 P κ 013 / / κ 023   p ∗ 0 P µ 01 ( κ 123 )   p ∗ 2 P ǫ 02 z z t t t t t t t t t ǫ 12 / / p ∗ 1 P ǫ 01 y y s s s s s s s s s s p ∗ 0 P κ 012 / / p ∗ 0 P . K 123 {      K 013 , 4 b b b b b b K 023   4 4 4 4 K 012 / 7 g g g g Ω u } t t t t M 01 ( κ 123 )  ' F F F F F F (3.23) so that the tw o compo site 2 -arrows p ∗ 3 P µ 01 ( κ 123 ) κ 013 ǫ 03 * * * * ǫ 01 ǫ 12 ǫ 23 4 4 4 4 p ∗ 0 P   which may be cons tructed b y compo sition of 2-a rrows in (3 .23) coincide. Differential Geometry of Gerbes and Differential F orms 19 This 2-ar row Ω may als o b e viewed a s a 1 - arrow a b ove ∆ 3 X/S in the ga uge group P ad , or even a s a n arrow in the stack Lie( P ad ) ⊗ O S Ω 3 X/S of relative Lie( P ad )-v alued 3-forms on X . Returning to the combinatorial definition [5] (A.1.10) of the de Rham differential, we may finally vie w Ω , by horiz o ntal comp osition with a ppr opriate 1-a rrows, as a 1-arr ow in P ad whose so ur ce ob ject is the identit y arrow I P ad : I Ω / / d 2 µ ( κ − 1 ) . (3.24) Denoting the twisted differential d 2 µ by the expre s sion d + [ µ, ] to which it reduces when appro priate trivia liz ations hav e b een c hosen, the 3-curv ature arrow Ω (3.24) is describ ed by the equatio n (1.4). By [5 ] theor em 4.4 it s a t- isfies ano ther rela tion, describ ed by the cubica l pa sting diag r am [5] (4.1.24 ), and which ma y b e expres sed b y the higher Bianchi identit y 6 (1.5). The pa ir of equations (1.4) and (1.5) ma y now b e thought of as a categor ified v er sion, sat- isfied by the pair of P ad -v alued forms ( κ, Ω ), of the classical Bianchi iden tit y (3.19), and ca n be written in symbolic form as d 2 µ, K ( κ, Ω ) = 0 , where d n µ, K is the twisted de Rham differen tia l on Lie( P ad )-v alued n -forms de- termined b y t wis ting da ta ( µ, K ) asso ciated to the given connective structur e on P . 4 ˇ Cec h-de Rham co c ycles 4.1 W e o bs erved in s ection 3.1 that a ger be could b e e x pressed in coc yclic terms, once lo cal trivia liz a tions were chosen. W e will now show that this is also the case for the co nnection ǫ . W e choose , for each i ∈ I , an arrow γ i : ǫp ∗ 1 x i − → p ∗ 0 x i (4.1) in p ∗ 0 P U i such that ∆ ∗ γ i = 1 x i . The arr ow γ i determines by co njuga tion a connection m i : p ∗ 1 G | U i − → p ∗ 0 G | U i on the pullback G | U i of the gr oup G ab ov e the op en set U i ⊂ X . The arrow m i is describ ed, fo r a ny section g ∈ Γ ( ∆ 1 X/S U i , p ∗ 1 G ), by the commutativit y of the diagram 6 See [5] (4.1.28) for a pro of of this identity . 20 La wrence Breen ǫp ∗ 1 x i ǫ ( g ) / / γ i   ǫp ∗ 1 x i γ i   p ∗ 0 x i m i ( g ) / / p ∗ 0 x i . (4.2) The pair ( φ ij , γ i ) determines a family of arr ows γ ij in the pullback G ∆ 1 U ij of G , defined by the commutativit y o f the diagr am ǫp ∗ 1 x j γ j / / ǫp ∗ 1 φ ij   p ∗ 0 x j p ∗ 0 φ ij   p ∗ 0 x i γ ij   ǫp ∗ 1 x i γ i / / p ∗ 0 x i (4.3) By co njugation, this determines a co mmu tative diagram p ∗ 1 G m j / / p ∗ 1 λ ij   p ∗ 0 G p ∗ 0 λ ij   p ∗ 0 G i ( γ ij )   p ∗ 1 G m i / / p ∗ 0 G (4.4) so that the equa tion i ( γ ij ) ( p ∗ 0 λ ij ) m j ( p ∗ 1 λ ij ) − 1 = m i . (4.5) of [5] (6 .1 .2) is satisfied. W e may restate (4.5 ) as i ( γ ij ) [( p ∗ 0 λ ij ) m j ( p ∗ 0 λ ij ) − 1 ] = m i [ p ∗ 1 λ ij ( p ∗ 0 λ − 1 ij )] , (4.6) an equation all of whose facto rs are Aut( G )-v alued 1-forms on U ij and there- fore commute with each other. In the notation intro duced in (2.31), equation (4.6) c a n be r ewritten a s λ ij ∗ m j = m i − i ( γ ij ) , ( 4.7) or more classically a s λ ij m j = m i − λ ij d λ − 1 ij − i ( γ ij ) . (4.8) This is is the a nalogue for the Aut ( G )-v alued forms m i and λ ij of the clas s ical expression (3.20) for a connectio n for m, but now catego rified b y the ins ertion of an a dditional s ummand − i ( γ ij ). Differential Geometry of Gerbes and Differential F orms 21 Consider now the following dia gr in P ∆ 1 U ijk : ǫp ∗ 1 x k ǫp ∗ 1 φ jk y y s s s s s s s s s s s s s s ǫp ∗ 1 φ ik / / ǫp ∗ 1 x i γ i   ǫp ∗ 1 g ijk y y t t t t t t t t t t t t t t p ∗ 0 x i m i ( p ∗ 1 g ijk ) y y t t t t t t t t t t t t t t O O γ ik ǫp ∗ 1 x j ǫp ∗ 1 φ ij / / γ j   ǫp ∗ 1 x i γ i   p ∗ 0 x i O O γ ij p ∗ 0 x j p ∗ 0 φ ij / / O O γ jk γ k   p ∗ 0 x i O O λ ij ( γ jk ) p ∗ 0 x k p ∗ 0 φ jk w w p p p p p p p p p p p p p p ∗ 0 φ ik / / p ∗ 0 x i p ∗ 0 g ijk w w p p p p p p p p p p p p p p ∗ 0 x j p ∗ 0 φ ij / / p ∗ 0 x i (4.9) Of the eight faces of this cub e, seven ar e known to b e commutativ e. I t follows that the remaining low er square on the right v er tical side is also commutativ e. This is the squa re p ∗ 0 x i p ∗ 0 g ijk / / γ ik   p ∗ 0 x i λ ij ( γ jk )   p ∗ 0 x i γ ij   p ∗ 0 x i m i ( p ∗ 1 g ijk ) / / p ∗ 0 x i , (4.10) whose commu tativity corr esp onds to the equation γ ij ( p ∗ 0 λ ij ( γ j k )) = m i ( p ∗ 1 g ij k ) γ ik ( p ∗ 0 g ij k ) − 1 in other words to the equation [5] (6.1.7), all of whose factors are G -v alued 1-forms on U ij k . W e may rewrite this as γ ij p ∗ 0 λ ij ( γ j k ) = ( m i ( p ∗ 1 g ij k ) p ∗ 0 g − 1 ij k ) ( p ∗ 0 g ij k γ ik p ∗ 0 g − 1 ij k ) so that, ta king in to a ccount the equa tion (3 .5 ), we finally obta in (in additive notation) γ ij + λ ij ( γ j k ) − λ ij λ j k ( λ − 1 ik ( γ ik )) = dg ij k g − 1 ij k + [ m i , g ij k ] , with brack et defined b y (2.7) an equation whic h can be written in abbreviated form a s δ 1 λ ij ( γ ij ) = d m i g ij k g − 1 ij k . (4.11) 22 La wrence Breen 4.2 W e now des crib e in similar ter ms the curv ing K and the fake cur v ature κ o f diagram (3.22). Just as we asso cia ted to the connec tion ǫ (3.21) a family of arrows γ i (4.1), we now choose, for each i ∈ I , an a rrow κp ∗ 0 x i δ i / / p ∗ 0 x i (4.12) in the categ ory P ∆ 2 U i , whos e restrictio n to the degener ate subsimplex s∆ 2 U i of ∆ 2 U i is the identit y . T o the cur ving K is as so ciated a family o f “ B -field” g -v alued 2- forms B i ∈ g ⊗ Ω 2 U i , characterized by the comm uta tivit y of the following diagram 7 in whic h a n expression such as γ 12 i is the pullback of γ i by the corresp onding pro jection p 12 : ∆ 2 X/S − → ∆ 1 X/S : ǫ 01 ǫ 12 ( p ∗ 2 x i ) ǫ 01 γ 12 i   K ( p ∗ 2 x i ) / / κǫ 02 ( p ∗ 2 x i ) κγ 02 i   ǫ 01 ( p ∗ 1 x i ) γ 01 i   κp ∗ 0 x i δ i   p ∗ 0 x i o o B i p ∗ 0 x i (4.13) Let us now define a family of G -v alued 2-forms ν i on U i by the e q uations ν i := d 1 m i − i ( B i ) (4.14) in Lie Aut( G ) ⊗ Ω 2 U i , in other words by the commutativit y o f the diagr am p ∗ 2 G m 12 i   p ∗ 2 G m 02 i   p ∗ 1 G m 01 i   p ∗ 0 G ν i   p ∗ 0 G i o o i ( B i ) p ∗ 0 G . (4.15) By co mparing dia gram (4.15) with the conjugate of dia gram (4 .1 3), we see that ν i is simply the co njugate of the arr ow δ i . It can there fo re describ ed by the comm utativity of the diagram κp ∗ 0 x i κ ( g ) / / δ i   κp ∗ 0 x i δ i   p ∗ 0 x i ν i ( g ) / / p ∗ 0 x i (4.16) 7 The chose n orientation of the arro w B i is consisten t with that in [5 ]. Differential Geometry of Gerbes and Differential F orms 23 for all g ∈ Γ ( ∆ 2 U i /S , p ∗ 0 G ), just as the connection m i was describe d b y diagram (4.2). W e also define a family of 2-forms δ ij by the commutativit y of the diagram p ∗ 0 x i λ ij ( B j ) / / δ ij   p ∗ 0 x i γ 01 ij   p ∗ 0 x i γ 02 ij   p ∗ 0 x i m 01 i ( γ 12 ij )   p ∗ 0 x i B i / / p ∗ 0 x i , (4.17) i.e., since a ll terms co mmu te, by the equation δ ij := λ ij ( B j ) − B i − d 1 m i ( − γ ij ) in Lie( G ) ⊗ Ω 2 U i /S . In ˇ Cech-de Rham notation, this is δ ij := δ 0 λ ij ( B i ) − d 1 m i ( − γ ij ) , (4.18) and in cla s sical notation δ ij := λ ij ( B j ) − B i + d γ ij − 1 2 [ γ ij , γ ij ] + [ m i , γ ij ] . Here is a nother characteriza tion of δ ij : Lemma 4 .1. F or every p air ( i, j ) ∈ I , the analo gu e κp ∗ 0 x j δ j / / κp ∗ 0 φ ij   p ∗ 0 x j p ∗ 0 φ ij   p ∗ 0 x i δ ij   κp ∗ 0 x i δ i / / p ∗ 0 x i . (4.19) of diagr am (4.3) is c ommu tative. 24 La wrence Breen Pro of : Consider the dia gram κǫ 02 ( κp ∗ 2 x j ) κγ 02 j / / κp ∗ 0 x j κp ∗ 0 φ ij   δ j / / p ∗ 0 x j p ∗ 0 φ ij   ǫ 01 ǫ 12 ( p ∗ 2 x j ) ǫ 01 ǫ 12 ( p ∗ 2 φ ij )   γ 12 j / / K ( p ∗ 2 x j ) 8 8 p p p p p p p p p p p κǫ 02 ( p ∗ 2 φ ij )   ǫ 01 ( p ∗ 1 x j ) γ 01 j / / ǫ 01 ( p ∗ 1 φ ij )   p ∗ 0 x j v v B j m m m m m m m m m m m m m m m p ∗ 0 φ ij   p ∗ 0 x i δ ij   p ∗ 0 x i v v λ ij ( B j ) m m m m m m m m m m m m m m m κp ∗ 0 x i δ i / / p ∗ 0 x i ν i ( γ 02 ij )   ǫ 01 ( p ∗ 1 x i ) γ 01 i / / p ∗ 0 x i   γ 01 ij κǫ 02 ( p ∗ 2 x i ) κγ 02 i / / κp ∗ 0 x i δ i / /   κγ 02 ij p ∗ 0 x i ǫ 01 ǫ 12 ( p ∗ 2 x i ) K ( p ∗ 2 x i ) 7 7 p p p p p p p p p p p ǫ 01 ( γ 12 i ) / / ǫ 01 ( p ∗ 1 x i )   ǫ 01 ( γ 12 ij ) γ 01 i / / p ∗ 0 x i   m 01 i ( γ 12 ij ) v v B i m m m m m m m m m m m m m m m . (4.20) Diagrams (4.13), (4.17) and (4.16) imply that all sq uares in (4.20) a re co mmu - tative 8 , except possibly the rear right upp er one. This remaining square (4.19) is therefor e also co mm utative. ✷ Conjugating diagram (4.19), we o btain as in (4.4) a square p ∗ 0 G ν j / / p ∗ 0 λ ij   p ∗ 0 G p ∗ 0 λ ij   p ∗ 0 G i δ ij   κp ∗ 0 G ν i / / p ∗ 0 G , whose commu tativity is express e d algebraica lly as i ( δ ij ) ( p ∗ 0 λ ij ) ν j = ν i ( p ∗ 0 λ ij ) . (4.21) 8 This is true for diagram (4.17) since ν i ( γ 02 ij ) = γ 02 ij . Differential Geometry of Gerbes and Differential F orms 25 In additive notation, this is eq uation λ ij ν j = ν i − i ( δ ij ) , (4.22 ) in other words δ 0 λ ij ν i = − i ( δ ij ) . It is instructive to note that this equation can be derived directly from equa- tion (4.8) and the definitions (4.14) and (4.18) o f ν i and δ ij . Firs t of a ll, observe that b y (2.32) d 1 ( λ ij ∗ m i ) = λ ij (d 1 m i ) . (4.23) One then co mputes λ ij ν j = λ ij (d 1 ( m j ) − i B j ) = d 1 ( λ ij ∗ m j ) − i ( λ ij ( B j )) = d 1 ( m i − i ( γ ij )) − i ( B i + d 1 m i ( − γ ij ) + δ ij ) = d 1 m i − d 1 ( i ( γ ij )) − [ m i , γ ij ] − i ( B i ) − i (d 1 m i ( − γ ij )) − i ( δ ij ) . Since the homomor phism i comm utes with d 1 m and [ m i , i ( γ ij )] = i ([ m i , γ ij ]), the summands i (d 1 m ( − γ ij )) a nd d 1 ( i ( γ ij )) + [ m i , γ ij ] cancel out. The first t wo remaining summands describ e ν i , so tha t equatio n (4.2 2) is satisfied. In the sa me vein, the analogue for the fake curv atur e κ of (4.1 0) is the following asser tion. Lemma 4 .2. The diagr am p ∗ 0 x i p ∗ 0 g ijk / / δ ik   p ∗ 0 x i λ ij ( δ jk )   p ∗ 0 x i δ ij   p ∗ 0 x i ν i ( p ∗ 0 g ijk ) / / p ∗ 0 x i (4.24) is c ommutative. 26 La wrence Breen Pro of: By (4.19), (3.4) and (4 .16), all squa res in the diagr am κp ∗ 0 x k κp ∗ 0 φ jk { { w w w w w w w w w w w w w w w w κp ∗ 0 φ ik / / κp ∗ 0 x i δ i   κp ∗ 0 g ijk { { w w w w w w w w w w w w w w w w p ∗ 0 x i ν i ( p ∗ 0 g ijk ) { { w w w w w w w w w w w w w w w w O O δ ik κp ∗ 0 x j κp ∗ 0 φ ij / / δ j   δ k κp ∗ 0 x i δ i   p ∗ 0 x i O O δ ij p ∗ 0 x j p ∗ 0 φ ij / / O O δ jk   p ∗ 0 x i O O λ ij ( δ jk ) p ∗ 0 x k p ∗ 0 φ jk y y s s s s s s s s s s s s s s p ∗ 0 φ ik / / p ∗ 0 x i p ∗ 0 g ijk y y t t t t t t t t t t t t t t p ∗ 0 x j p ∗ 0 φ ij / / p ∗ 0 x i (4.25) are commutativ e, except p os sibly the lower right-hand one. It follows that the latter one, whic h is simply (4.24), a lso commut es. ⊓ ⊔ The co mm utativity of (4.2 4) co rresp onds to equa tion δ ij ( p ∗ 0 λ ij )( δ j k ) = ν i ( p ∗ 0 g ij k ) δ ik ( p ∗ 0 g ij k ) − 1 , an equation whose ter ms are G - v alued 2-forms on U ij k . By the sa me reasoning as for (4.1 1), this ca n b e written additively as δ ij + λ ij ( δ j k ) − λ ij λ j k ( λ − 1 ik ( δ ik )) = [ ν i , g ij k ] , or, in the co mpact for m of [5] (6.1.15 ), as δ 1 λ ij ( δ ij ) = [ ν i , g ij k ] . (4.26) Just w e w er e able to derive (4.22) directly from (4.8 ) a nd the definitions (4.14) and (4.18), w e now sho w that it is p ossible to deduce (4 .2 6) from (4.18),(4.14) and (4.1 1). Fir st of a ll, δ 1 λ ij ( δ ij ) = δ 1 λ ij ( δ 0 λ ij ( B i ) − d 1 m i ( − γ ij )) = δ 1 λ ij δ 0 λ ij ( B i ) − δ 1 λ ij d 1 m i ( − γ ij ) . (4.27) W e now wish to a s sert that the ˇ Cech differential δ 1 λ ij and de Rham dif- ferential d 1 m i in (4.27 ) comm ute with eac h other, despite the fact that the Differential Geometry of Gerbes and Differential F orms 27 1-form γ ij takes its v alues in a non-commutativ e g roup G , and that d 1 m i is not a ho momorphism. F o r this we simplify our notatio n, b y setting e γ ij := − γ ij ∈ g ⊗ Ω 1 U ij (4.28) and λ ij k := λ ij λ j k λ − 1 ik ∈ Γ ( U ij k , Aut( G i )) . Equation (4.11) ca n be re stated a s δ 1 λ ij e γ := e γ ij + λ ij ( e γ j k ) − λ ij k ( e γ ik ) = − d g ij k g − 1 ij k − [ m i , g ij k ] . (4.2 9) Lemma 4 .3. The fol lowing e qu ality b etwe en G -value d 2-forms ab ove U ij k is satisfie d: d 1 m i δ 1 λ ij ( e γ ij ) = δ 1 λ ij d 1 m i ( e γ ij ) . (4 .30) Pro of: W e co mpute the left-hand s ide of the equa tio n (4.30), taking into account the quadraticity equation (2.24) d 1 m i δ 1 λ ij ( e γ ij ) = d m i ( e γ ij ) + d 1 m i ( λ ij ( e γ j k )) + d 1 m i ( − λ ij k ( e γ ik )) + + [ e γ ij , λ ij ( e γ j k )] − [ e γ ij , λ ij k ( e γ ik )] − − [ λ ij ( e γ j k ) , λ ij k ( e γ ik )] = d m i ( e γ ij ) + d 1 m i ( λ ij ( e γ j k )) − d 1 m i ( λ ij k ( e γ ik )) + + [ λ ij k ( e γ ik ) , λ ij k ( e γ ik )] + [ e γ ij , λ ij ( e γ j k )] − − [ e γ ij + λ ij ( e γ j k ) , λ ij k ( e γ ik )] . W e now compute the right-hand side of (4.30): δ 1 λ ij d 1 m i ( e γ ij ) = d 1 m i ( e γ ij ) + λ ij (d 1 m j ( e γ j k )) − λ ij k (d 1 m i ( e γ ik )) . (4.31) By (4.7) and b y the functoriality prop erty (2.32), we find that λ ij (d 1 m j ( e γ j k )) = d 1 λ ij ∗ m j ( λ ij ( e γ j k )) = d 1 m i ( λ ij ( e γ j k )) + [ e γ ij , λ ij ( e γ j k )] and by (2.34) λ ij k (d 1 m i ( e γ ik )) = d 1 λ ijk ∗ m i ( λ ij k ( e γ ik )) = d 1 m i ( λ ij k ( e γ ik )) + [[ λ ij k , m i ] , λ ij k ( e γ ik )]+ + [ λ ij k d λ − 1 ij k , λ ij k ( e γ ik )] . Inserting these expressio ns for λ ij (d 1 m j ( e γ j k )) a nd λ ij k (d 1 m i ( e γ ik )) into the righ t- hand side o f (4.31) we find the following expressio n for δ 1 λ ij d 1 m i ( e γ ij ): 28 La wrence Breen δ 1 λ ij d 1 m i ( e γ ij ) = d 1 m i ( e γ ij ) + d 1 m i ( λ ij e γ j k ) + [ e γ ij , λ ij ( e γ j k )] − − d 1 m i ( λ ij k ( e γ ik ) − [[ λ ij k , m i ] , λ ij k ( e γ ik )] − [ λ ij k d λ − 1 ij k , λ ij k ( e γ ik )] − − d 1 m i ( λ ij k )( e γ ik ) − [ λ ij k d λ − 1 ij k , λ ij k ( e γ ik )] . Comparing this with the expres sion (4.31) for d 1 m i δ 1 λ ij ( e γ ij ), we see that the equation (4.30) is satis fie d if and only if [ e γ ij + λ ij ( e γ j k ) − λ ij k ( e γ ik ) , λ ij k ( e γ ik )] = [[ λ ij k , m i ] , λ ij k ( e γ ik )]+ + [ λ ij k d λ − 1 ij k , λ ij k ( e γ ik )] . By (2.9), this is simply a consequence of (4.29), since λ ij k = i ( g ij k ) . ✷ W e now return to our computation (4.2 7): δ 1 λ ij ( δ ij ) = δ 1 λ ij δ 0 λ ij ( B i ) − δ 1 λ ij d 1 m i ( − γ ij ) = δ 1 λ ij δ 0 λ ij ( B i ) − d 1 m i δ 1 λ ij ( − γ ij ) = [ g ij k , B i ] − d 1 m i ( g ij k d m i ( g − 1 ij k )) = [ g ij k , i B i − dm i ] by (2.27) = [ ν i , g ij k ] . This finishes the sec o nd pro of o f equation (4 .26) . ✷ W e now set ω i := d 2 m i ( B i ) . (4.32) Since the combinatorial de finitio n o f the twisted de Rham different ial d 2 ([4] (3.3.1)) matches the globa l geometr ic definition (3.2 3) o f the 3 -curv atur e Ω , this 3-curv atur e Ω is lo cally describ ed by the G -v alued 3-forms ω i . It follows fro m the definitions (4 .14) and (4.32) of the fo rms ν i and ω i , and from (2 .26), that d 3 m i ( ω i ) = d 3 m i d 2 m i ( B i ) = [d 1 m i , B i ] = [ ν i , B i ] + [ B i , B i ] so that the lo ca l 3 - curv ature form ω i satisfies the higher B ianchi iden tity d 3 m i ( ω i ) = [ ν i , B i ] . (4.33) Differential Geometry of Gerbes and Differential F orms 29 A seco nd r elation b etw een the for ms ν i and ω i follows from their definitions and the Bia nchi identit y fo r the Aut( G )-v a lued 1-form m i : i ( ω i ) = d 2 m i i ( B i ) = d 2 m i (d 1 m i − ν i ) = d 2 m i ( − ν i ) , in other words d 2 m i ν i + i ( ω i ) = 0 . (4.3 4) This equation is the lo cal for m of equatio n(1.4), just as (4.33) was the lo cal form o f (1.5). W e will now show that the equa tion (4 .18) for the 2 - forms B i , which we write her e as δ 0 λ ij ( B i ) = d 1 m i ( − γ ij ) + δ ij , induces the corresp onding gluing equation fo r the lo ca l 3- forms ω i . F rom the definition of λ ij ( ω j ) and (2.33), it follows that λ ij ( ω j ) = λ ij (d 2 m j ( B j )) = d 2 λ ij ∗ m j λ ij ( B j ) and by the g luing laws (4.8 ) and (4.18) for m i and B i , this can b e stated as λ ij ( ω j ) = d 2 m i − i ( γ ij ) ( B i + δ ij + d 1 m i ( − γ ij )) = d 2 m i ( B i ) + d 2 m i ( δ ij ) + d 2 m i d 1 m i ( − γ ij ) − [ γ ij , B i + δ ij + d 1 m i ( − γ ij )] . By (2.28), this last equality can be re w r itten as λ ij ( ω j ) = ω i + d 2 m i ( δ ij ) + [d 1 m i , − γ ij ] − [ γ ij , B i ] − [ γ ij , δ ij ] = ω i + d 2 m i ( δ ij ) + [ γ ij , d 1 m i − B i ] − [ γ ij , δ ij ] and by (4.21) this prov es the gluing law fo r the 3- forms ω i [5] (6 .1.23): λ ij ( ω j ) = ω i + d 2 m i ( δ ij ) + [ γ ij , ν i ] − [ γ ij , δ ij ] . By combining this with the gluing law (4.22) for ν i , we see tha t (4.35) ca n finally b e r e written in the more compact form λ ij ( ω j ) + [ λ ij ν j , γ ij ] = ω i + d 2 m i ( δ ij ) (4.35) 30 La wrence Breen 5 ˇ Cec h-de Rham cob oundaries 5.1 W e saw in sectio n 2 how a change in the c hoice trivializing data ( x i , φ ij ) in a gerb e P could b e measured by a pair ( r i , θ ij ) (3.10),(3.11) inducing a cob ound- ary relation (3.1 5) b etw een the corr esp onding co cyc le pair s ( λ ij , g ij k ). W e will now examine how the terms ( m i , γ ij ), ( ν i , δ ij ) and B i int ro duced in sectio n 4 v ary w hen the arrows γ i (4.1) and δ i (4.12) which determine them have b een mo dified. The difference betw een the arrow γ i and an analogous ar row γ ′ i is meas ured by a 1-for m e i ∈ Lie ( G ) ⊗ Ω 1 U i , defined by the commutativit y of the follo wing diagram: ǫp ∗ 1 x i ǫp ∗ 1 χ i / / γ i   ǫp ∗ 1 x ′ i γ ′ i   p ∗ 0 x i p ∗ 0 χ i / / p ∗ 0 x ′ i e i / / p ∗ 0 x ′ i (5.1) This conjugates to a commutativ e dia gram p ∗ 1 G p ∗ 1 r i / / m i   p ∗ 1 G m ′ i   p ∗ 0 G p ∗ 0 r i / / p ∗ 0 G i ( e i ) / / p ∗ 0 G so that m ′ i = i ( e i ) ( p ∗ 0 r i ) m i ( p ∗ 1 r i ) − 1 = i ( e i ) [ p ∗ 0 r i m i p ∗ 0 r i − 1 ] [ p ∗ 0 r i p ∗ 1 r i − 1 ] In classical ter ms, this is expr essed as an equation m ′ i = r i m i + r i d r − 1 i + i ( e i ) (5.2) = r i ∗ m i + i ( e i ) . (5.3) which co mpares the connections m i and m ′ i induced on the group G by the arrows γ i and γ ′ i . Differential Geometry of Gerbes and Differential F orms 31 W e now consider the following dia gram in P U ij : p ∗ 0 x ′ i e i   p ∗ 0 x ′ i r i ( γ ij ) o o p ∗ 0 θ ij   p ∗ 0 x ′ i m ′ i ( p ∗ 1 θ ij )   p ∗ 0 x ′ i λ ′ ij ( e j )   p ∗ 0 x ′ i p ∗ 0 x ′ i . γ ′ ij o o (5.4) Prop ositio n 5 .1. The diagr am (5.4) is c ommut ative. Pro of: Consider the diagram ǫp ∗ 1 x j ǫp ∗ 1 φ ij z z t t t t t t t t t t t t t t t t t γ j / / p ∗ 0 x j p ∗ 0 χ j   p ∗ 0 φ ij x x q q q q q q q p ∗ 0 x i γ ij x x q q q q q q q p ∗ 0 χ i   ǫp ∗ 1 x i ǫp ∗ 1 χ i   γ i / / ǫp ∗ 1 χ j   p ∗ 0 x i p ∗ 0 χ i   p ∗ 0 x ′ i p ∗ 0 θ ij   p ∗ 0 r i ( γ ij ) y y r r r r r r r p ∗ 0 x ′ j p ∗ 0 φ ′ ij y y r r r r r r r e j   p ∗ 0 x ′ i e i   p ∗ 0 x ′ i p ∗ 0 λ ′ ij ( e j )   ǫp ∗ 0 x ′ j ǫp ∗ 1 φ ′ ij u u u u u u u u z z u u u u u u u u γ ′ j / / p ∗ 0 x ′ j p ∗ 0 φ ′ ij y y r r r r r r r ǫp ∗ 1 x ′ i γ ′ i / / ǫp ∗ 1 θ ij   p ∗ 0 x ′ i m ′ i ( p ∗ 1 θ ij )   p ∗ 0 x ′ i γ ′ ij x x r r r r r r r ǫp ∗ 1 x ′ i γ ′ i / / p ∗ 0 x ′ i (5.5) The low er front square of the right-hand face of this cube is just the squar e (5.4). Since we know tha t all the other squares in this diagra m commut e, so do es the square (5.4). ⊓ ⊔ The commut ativity of (5.4) is equiv alent to the equatio n m ′ i ( p ∗ 1 θ ij ) e i r i ( γ ij ) = γ ′ ij λ ′ ij ( e j ) p ∗ 0 θ ij . (5.6) 32 La wrence Breen This may be rewr itten in class ical notatio n as: ( γ ′ ij − θ ij r i ( γ ij )) + ( λ ′ ij ( e j ) − θ ij e i ) = d m ′ i θ ij θ − 1 ij . (5.7) W e now choo se a family of arrows δ ′ i : κp ∗ 0 x ′ i − → p ∗ 0 x ′ i . The families δ ′ i and γ ′ i determine a s in (4.13) a family of g -v alued 2-form B ′ i ab ov e U i . The latter in turn determines, tog ether with the pa ir of form ( m ′ i , γ ′ ij ) (5.2), (5.7), a new pair o f 2-fo r ms ( ν ′ i , δ ′ ij ) and a 3 -form ω ′ i satisfying the corr esp onding equations (4.22), (4.34), (4.26), (4.33) a nd (4.3 5). T he families δ i and δ ′ i are compared by the following analogue of diag ram (5.1): κp ∗ 0 x i κp ∗ 0 χ i / / δ i   κp ∗ 0 x ′ i δ ′ i   p ∗ 0 x i p ∗ 0 χ i / / p ∗ 0 x ′ i n i / / p ∗ 0 x ′ i . (5.8) W e will now compare the 2- fo rms B i and B ′ i . W e consider the diagram ǫ 01 ǫ 12 ( p ∗ 2 x i ) ǫ 01 ǫ 12 ( p ∗ 2 χ i ) t t j j j j j j j j j j j j j j j j K ( p ∗ 2 x i ) / / ǫ 01 ( γ 12 i ) κǫ 02 ( p ∗ 2 x i ) κ ǫ 02 ( p ∗ 2 χ i ) u u k k k k k k k k k k k k k k κ ( γ 02 i )   ǫ 01 ǫ 12 ( p ∗ 2 x ′ i ) ǫ 01 γ ′ 12 i   K ( p ∗ 2 x ′ i ) / /   κ ǫ 02 ( p ∗ 2 x ′ i ) κ ( γ ′ 02 i )   ǫ 01 p ∗ 1 x i ǫ 01 p ∗ 1 χ i v v l l l l l l l l γ 01 i   κ p ∗ 0 x i κp ∗ 0 χ i v v n n n n n n n n δ i   ǫ 01 p ∗ 1 x ′ i v v m m m γ ′ 01 i   κp ∗ 0 x ′ i δ ′ i   κ ( e 02 i ) w w o o o ǫ 01 p ∗ 0 x ′ i γ ′ 01 i   κp ∗ 0 x ′ i δ ′ i   p ∗ 0 x i o o B i p ∗ 0 χ i w w n n n n p ∗ 0 x i p ∗ 0 χ i w w p p p p ∗ 0 x ′ i o o r i ( B i ) e i 01 y y t t p ∗ 0 x ′ i n i { { v v p ∗ 0 x ′ i v v m m m m m p ∗ 0 x ′ i ν ′ i ( e 02 i ) w w o o o o p ∗ 0 x ′ i o o B ′ i p ∗ 0 x ′ i (5.9) in which the upp er and low er unlab elled ar rows are re sp e ctively ǫ 01 ( p ∗ 1 e 12 i ) and m ′ i 01 ( e 12 i ). Differential Geometry of Gerbes and Differential F orms 33 The front square (o r ra ther hex agon) o f the b ottom face p ∗ 0 x ′ i o o r i ( B i ) e i 01   p ∗ 0 x ′ i p ∗ 0 n i   p ∗ 0 x ′ i m ′ i 01 ( e 12 i )   p ∗ 0 x ′ i ν ′ i ( e 02 i )   p ∗ 0 x ′ i o o B ′ i p ∗ 0 x ′ i is commut ative, since all other squares in diagram (5.9) are. Equiv alently , since the action of the Aut( G )-v a lued 2-form ν ′ i on e 02 i is trivial, this pr oves that the e quation B ′ i = r i ( B i ) − d 1 m ′ i ( − e i ) − n i . (5.10) is satisfied. In particula r for g iven B i and e i , the 2-forms B ′ i and n i actually determine each other. By co njugation, diagram (5 .8) induces a commutativ e dia gram p ∗ 0 G p ∗ 0 r i / / ν i   p ∗ 0 G ν ′ i   p ∗ 0 G p ∗ 0 r i / / p ∗ 0 G i n i / / p ∗ 0 G equiv alent to the equation i ( n i ) p ∗ 0 r i ν i = ν ′ i p ∗ 0 r i . In classical ter ms, this is the simpler analog ue ν ′ i = r i ν i + i ( n i ) (5.11) for ν i of the equation (5 .2) for m i . W e will now show that this co bo undary equa tion for ν i can b e derived from the definition (4.14) of ν i , and the cob oundary equations (5.2 ) and (5.10) for m i and B i : ν ′ i = d 1 m ′ i − i ( B ′ i ) = d 1 ( r i ∗ m i + i ( e i )) − i ( r i ( B i ) + n i + d 1 m ′ i ( − e i )) = r i d 1 m i + i (d 1 e i ) + [ r i ∗ m i , i ( e i )] − i ( r i ( B i )) + i (d 1 m ′ i ( − e i )) + i ( n i ) = r i (d 1 m i − i ( B i )) + i ( n i ) + i (d 1 m ′ i ( − e i ) + d 1 e i + [ r i ∗ m i , e i ]) 34 La wrence Breen In order to prov e (5.11), it now suffices to verify that the 3 terms in the last summand of the final equa tio n canc e l each other out: d 1 m ′ i ( − e i ) + d 1 ( e i ) + [ r i ∗ m i , e i ] = d 1 ( − e i ) − [ m ′ i , e i ] + d 1 e i + [ r i ∗ m i , e i ] = d 1 ( − e i ) + d 1 e i − [ e i , e i ] = 0 . ✷ The other equatio n satisfied by the forms n i is the counterpart of equation (5.6). It is obtained by considering the following dia gram, analo g ous to (5.5): κp ∗ 0 x j κp ∗ 0 φ ij z z u u u u u u u u u u u u u u u u u δ j / / p ∗ 0 x j p ∗ 0 χ j   p ∗ 0 φ ij y y r r r r r r r p ∗ 0 x i δ ij y y r r r r r r r p ∗ 0 χ i   κp ∗ 0 x i κp ∗ 0 χ i   δ i / / κp ∗ 0 χ j   p ∗ 0 x i p ∗ 0 χ i   p ∗ 0 x ′ i p ∗ 0 θ ij   r i ( δij ) y y s s s s s s s p ∗ 0 x ′ j p ∗ 0 φ ′ ij y y s s s s s s s n j   p ∗ 0 x ′ i n i   p ∗ 0 x ′ i p ∗ 0 λ ′ ij ( n j )   κp ∗ 0 x ′ j κp ∗ 0 φ ′ ij v v v v v v v v z z v v v v v v v v δ ′ j / / p ∗ 0 x ′ j p ∗ 0 φ ′ ij y y s s s s s s s κp ∗ 0 x ′ i δ ′ i / / κp ∗ 0 θ ij   p ∗ 0 x ′ i ν ′ i ( p ∗ 0 θ ij )   p ∗ 0 x ′ i δ ′ ij y y s s s s s s s κp ∗ 0 x ′ i δ ′ i / / p ∗ 0 x ′ i . (5.12) The lower fron t s quare on the r ig ht-hand face p ∗ 0 x ′ i p ∗ 0 θ ij / / r i ( δ ij )   p ∗ 0 x ′ i p ∗ 0 λ ′ ij ( n j ) / / p ∗ 0 x ′ i δ ′ ij   p ∗ 0 x ′ i n i / / p ∗ 0 x ′ i ν ′ i ( p ∗ 0 θ ij ) / / p ∗ 0 x ′ i of diagra m (5.12) is co mmu tative, since all other squares in this diagram ar e. Differential Geometry of Gerbes and Differential F orms 35 This prov es that equation ν ′ i ( p ∗ 0 θ ij ) n i r i ( δ ij ) = δ ′ ij p ∗ 0 λ ′ ij ( n j ) p ∗ 0 θ ij in Lie ( G ) ⊗ Ω 2 U i /S is sa tisfied. Regro uping the v a rious ter ms in this eq uation as we did ab ov e for equa tion (5.6), we find that it is e quiv alent, in additive notation, to ( δ ′ ij − r i ( δ ij )) + ( λ ′ ij ( n j ) − θ ij n i ) = [ ν ′ i , θ ij ] , an eq uation fo r 2 -forms very similar to equa tion (5.7 ) for 1 -forms. W e will now exa mine the effect of the chosen transfomatio ns ( λ ij , g ij k , m i , γ ij ) − → ( λ ′ ij , g ′ ij k , m ′ i , γ ′ ij ) (5.13) and B i − → B ′ i (5.10) on the 3 - curv ature 3- forms ω i (4.32). F o r this, it will be co nv enient to set ¯ e i := r − 1 i ( e i ) and ¯ n i := r − 1 i ( n i ) . It follows from (2 .23), (2.10), and the transfor mation formula (5.3) that d n m ′ i ( r i ( η )) = r i (d n m i ( η ) + [ ¯ e i , η ]) (5.14) for an y G -v alued n -form η with n > 1. In particular d 1 m ′ i ( − e i ) = d 1 r i ∗ m i ( − e i ) − [ e i , e i ] = r i (d 1 m i ( − ¯ e i ) − [ ¯ e i , ¯ e i ]) so that (5 .10) ma y be expr essed as B ′ i = r i ( B i − d 1 m i ( − ¯ e i ) + [ ¯ e i , ¯ e i ] − ¯ n i ) . Applying once more the for m ula (5.1 4), we find that ω ′ i = d 2 m ′ i ( B ′ i ) = d 2 m ′ i ( r i ( B i − d 1 m i ( − ¯ e i ) + [ ¯ e i , ¯ e i ] − ¯ n i )) = r i (d 2 m i ( B i − d 1 m i ( − ¯ e i ) + [ ¯ e i , ¯ e i ] − ¯ n i )) + + [ ¯ e i , B i − d 1 m i ( − ¯ e i ) + [ ¯ e i , ¯ e i ] − ¯ n i ] . (5 .15) W e now make use o f (2.28) in orde r to co mpute the v a lue of the expressio n d 2 m i d 1 m i ( − ¯ e i ) which arises when we expand the first summand of the la st equation (5.15): d 2 m i d 1 m i ( − ¯ e i ) = [d 1 m i , − ¯ e i ] + [d 1 ( − ¯ e i ) , − ¯ e i ] + [[ m i , − ¯ e i ] , − ¯ e i ] = − [d 1 m i , ¯ e i ] + [d 1 ¯ e i , ¯ e i ] + [[ m i , ¯ e i ] , ¯ e i ] . 36 La wrence Breen Inserting this expression into (5.1 5), we find that ω ′ i = r i ( ω i + [d 1 m i , ¯ e i ] − [d 1 ¯ e i , ¯ e i ] − [[ m i , ¯ e i ] , ¯ e i ] − d 2 m i ( ¯ n i ) + + d 2 m i [ ¯ e i , ¯ e i ] + [ ¯ e i , B i ] − [ ¯ e i , d 1 m i ( − ¯ e i )] − [ ¯ e i , ¯ n i ]) . (5.16) The four ter ms − [d 1 ¯ e i , ¯ e i ] − [[ m i , ¯ e i ] , ¯ e i ] + d 2 m i [ ¯ e i , ¯ e i ] − [ ¯ e i , d 1 m i ( − ¯ e i )] cancel each other o ut, so tha t we a re left in (5.16) with ω ′ i = r i ( ω i + [d 1 m i , ¯ e i ] − d 2 m i ( ¯ n i ) + [ ¯ e i , B i ] − [ ¯ e i , ¯ n i ]) = r i ( ω i + [d 1 m i − i ( B i ) , ¯ e i ] + [ ¯ n i , ¯ e i ] − d 2 m i ( ¯ n i )) = r i ( ω i ) + r i ([ ν i , ¯ e i ]) + r i ([ ¯ n i , ¯ e i ]) − r i (d 2 m i ( ¯ n i )) = r i ( ω i ) + [ r i ν i , e i ] + [ n i , e i ] − d 2 r i ∗ m i ( n i ) (5.17) where in the last line we made use of the functoria lit y prop erty (2.10) of the brack et op er a tion. Amalgamating the last tw o summands, we ma y finally write the cob oundary transforma tion for the 3-cur v ature form ω i in the compact form ω ′ i = r i ( ω i ) + [ r i ν i , e i ] − d 2 m ′ i ( n i ) . If ins tead we ama lg amate the seco nd and third term in (5.1 7), we find the equiv alent formulation ω ′ i = r i ( ω i ) + [ ν ′ i , e i ] − d 2 r i ∗ m i ( n i ) . (5.18) Remark 5. 1 (Comparison with [5]): The cob oundary equation (5 .18) is compatible with eq ua tion (6.2.19 ) of [5], but neither is a sp ecial case o f the o ther. Here we allo wed b o th the trivial- izing data ( x i , φ ij ) for the g erb e and the expressions ( γ i , δ i , B i ) for the curv- ing data to v a ry , whereas in the cob oundary equa tions of [5] the g erb e data ( x i , φ ij ) was fixed and only the ( γ i , δ i , B i ) v aried. This r estriction amounted to setting ( r i , θ ij ) = (1 , 1) in our equation (5.7). On the other hand, a notion of equiv alence be tw een co cycles was int ro duced in [5] which was more exten- sive than the o ne co nsidered here. In o rder for these to b e c omparable, one m ust supp o se that the arrow h in dia g ram (4.2.1 ) of [5] is the ident ity map, i.e. that the pair of differ ent ial forms ( π i , η ij ) asso cia ted to h in lo c. cit § 6 . 2 is trivial. This is a reas onable as sumption, since a non-tr iv ial arr ow h could re- ally be ter med a ga uge trans formation, r ather than a cob ounda r y term. With Differential Geometry of Gerbes and Differential F orms 37 this additional condition, the la st tw o summands in equa tion (6.2.19 ) of [5] v anish, so tha t this equa tio n re duce s to ω ′ i = ω i + δ 2 m i ( α i ) − [ ν ′ i , E i ] . (5.19) This s implified equation is co mpatible with our eq ua tion (5.18) with r i = 1 , under the co r resp ondence e i := − E i and n i := − α i . References 1. Aschieri, P ., Can tini, L., Jurˇ co B.: N onab elian Bun dle Gerb es, their Differ- entia l Geometry and Gauge Theory , Comm. in Math. Phys. 254 , 367–400 (2005). 2. Baez J. C. and Sc h reiber U.: Higher Gauge Theory . Preprin t, arXiv:math/0511710. 3. Breen, L.: Classification of 2-gerb es and 2-stacks Ast´ erisque 225 , So ci ´ et´ e Math´ e matique de F rance (1994) 4. Breen, L., Messing, W.: Com binatorial Differential F orms. Adv ances in Math. 164 , 203–282 (2001) 5. Breen, L., Messing, W.: Differentia l Geometry of Gerb es. Adv ances in Math. 198 , 732–846 (2005) 6. Breen, L.: Notes on 1- and 2-gerb es: arXiv :math/0611317 , to app ear in the Proceedings of the IMA Meeting “ n -Categories: F oundations and Applica- tions” (Minneap olis, 2004) 7. Brylinski, J.-L.: Lo op spaces, chara cteristic classe s and geometric quantiza- tion. Progress in Math. 107 , Birkh¨ auser, Boston, Basel, Berlin (1993) 8. Hitchin, N.: Lectures on sp ecial lagrangian submanifolds, preprint arXiv:math.DG/990703 4 (1999). 9. Kobay ashi S., Nomizu K.: F ound ations of differential geometry , Inters cience T racts in Pure and A pplied Mathematics 15 (1969). 10. Ko ck, A.: Differen tial forms with v alues in groups. Bull. Austral. Math. Soc 25 , 357–386 (1982) 11. Ko ck, A.: Com binatorics of curv ature, and the Bianchi iden tity . Theory and Applications of Categories 2 , 69–89 (1996). 12. Laurent-Gengoux C., Stenion M., Xu P .: Non A b elian D ifferentia l Gerb es. Preprint, arXiv:math.DG/0511696 . 13. Lod a y J.-L.: The d iagonal of the Stasheff p olytop e. This volume, preprint arXiv:0710.05 72 . 14. Murray , M. K .: Bundle Gerbes. J. of the London Math. S o ciety 54 , 403– 416 (1996). 15. Saneblidze S ., Um ble, R.: A diagonal on the A sso ciahedra. Preprin t , arXiv:math.A T/0011065 16. Schreiber U.: F rom Loop Space Mechanics to Nonab elian Strings. Preprint, hep-th/0509163. 17. Stasheff J.: Homotop y associativity of H - sp aces I. T rans. A mer. Math. Soc. 108 , 275–29 2 (1963). 18. Ulbrich, K. - H.: On cocycle bitorsors and gerbes o ver a Grothendiec k top os. Math. Proc. Cam bridge Phil. So c. 110 , 49-55 (1991)