aa r X i v : . [ m a t h . C T ] F e b Differential Geometry of Gerbesand Differential Forms
Lawrence Breen ∗ Institut Galil´eeUniversit´e Paris 1399, avenue J. - B. Cl´ement93430 Villetaneuse, France [email protected]
To Murray Gerstenhaber and Jim Stasheff
Summary.
We discuss certain aspects of the combinatorial approach to the differ-ential geometry of non-abelian gerbes due to W. Messing and the author [5], andgive a more direct derivation of the associated cocycle equations. This leads us toa more restrictive definition than in [5] of the corresponding coboundary relations.We also show that the diagrammatic proofs of certain local curving and curvatureequations may be replaced by computations with differential forms.
It is a classical fact that to a principal G -bundle P on a scheme X , endowedwith a connection ǫ , is associated a Lie ( G )-valued 2-form κ on P , the curvatureof the connection, satisfying a certain G -equivariance condition. While κ doesnot in general descend to a 2-form on X , the equivariance condition may beviewed as a descent condition for κ from a 2-form on P to a 2-form on X ,but now with values in the Lie algebra of the gauge group P ad of P . Theconnection on P also induces a connection µ on the group P ad , and the 2-form κ satisfies the Bianchi equation, an equation which may be expressed inglobal terms as d κ + [ µ, κ ] = 0 (1.1)([5] proposition 1.7, [4] theorem 3.7). Choosing a local trivialization of thebundle P , on an open cover U := ` i ∈ I U i of X , the connection ǫ is described ∗ Unit´e Mixte de Recherche CNRS 7539 at least in a differential geometric setting, see [9], but the same construction canbe carried out within the context of algebraic geometry. Lawrence Breen by a family of Lie ( G )-valued connection 1-forms ω i defined on the open sets U i , and the associated curvature κ corresponds to a family of Lie ( G )-valued 2-forms κ i defined, according to the so-called structural equation of Elie Cartan,by the formula κ i = d ω i + 12 [ ω i , ω i ] (1.2)Equation (1.1) then reduces to the classical Bianchi identityd κ i + [ ω i , κ i ] = 0 . (1.3)J. -L. Brylinski introduced in [7] the notions of connection ǫ and curving K on an abelian G -gerbe P on a space X (where G was the multiplicativegroup G m , or rather in his framework the group U (1)), and showed that tosuch connective data ( ǫ, K ) is associated a closed G m -valued 3-form ω on X ,the 3-curvature. More recently, W. Messing and the author extended theseconcepts in [5] from abelian to general, not necessarily abelian, gerbes P ona scheme X . The coefficients of such a gerbe no longer constitute a sheaf ofgroups as in the principal bundle situation, but rather a monoidal stack G on X , as is to be expected in that categorified setting. In particular, when thegerbe is associated to a given non-abelian group G (so that we refer to it as a G -gerbe), the corresponding coefficient stack G is the monoidal stack associatedto the prestack determined by the crossed module G −→ Aut( G ), whereAut( G ) is the sheaf of local automorphisms of G . It may also be describedmore invariantly as the monoidal stack of G -bitorsors on X . Once more, tothe gerbe P is associated its gauge stack, a twisted form P ad := E q ( P , P ) ofthe given monoidal stack G , and the connection on P induces a connection µ on P ad . By analogy with the principal bundle case, the corresponding 3-curvature Ω , viewed as a global 3-form on X , now takes its values in thearrows of the stack P ad .There now arises a new, and at first sight somewhat surprising feature,but which is simply another facet of the categorification context in which weare operating. The 3-form Ω is accompanied by an auxiliary 2-form κ withvalues in the objects of the gauge stack P ad , which we called in [5] the fakecurvature of the given connective structure ( ǫ, K ). A first relation betweenthe forms Ω and κ comes from the very definition [5] (4.1.20), (4.1.22) of Ω ,and may be stated as in [5](4.3.8) as the categorical equation tΩ + d κ + [ µ, κ ] = 0 (1.4)where t stands for “target” of a 1-arrow with source the identity object I inthe stack of Lie( P ad )-valued 3-forms on X . On the other hand, the 3-form Ω is The canonical divided power 1 / ω, ω ] of the 2-form [ ω, ω ] is also denoted ω ∧ ω or [ ω ] (2) .ifferential Geometry of Gerbes and Differential Forms 3 no longer closed, even in the µ -twisted sense described for principal bundles by(1.1). It satisfies instead the following more complicated analogue [5] (4.1.33)of the Bianchi identity (1.1):d Ω + [ µ, Ω ] + [ K , κ ] = 0 . (1.5)While the first two terms in this equation are similar to those of (1.1), thecategorification term K is an arrow in the stack of 2-forms with values in themonoidal stack E q ( P ad , P ad ) induced by the curving K . The pairing of K with κ is induced by the evaluation of the natural transformation K betweenfunctors from P ad to itself on the object κ of P ad .The price to be paid for the compact form in which the global curvatureequations (1.4) and (1.5) have been stated is their rather abstract nature, andit is of interest to describe them in a more local form in terms of traditionalgroup-valued differential forms, just as was done in (1.3) for equation (1.1).Such a local description was already obtained in [5], both for the cocycleconditions (1.4) and (1.5), and for the corresponding coboundary equationswhich arise when alternate local trivializations of the gerbe have been chosen.However, the determination of those local equations was rather indirect, asit required a third description of a gerbe, which we have called the semi-local description [6] §
4, and which has also appeared elsewhere in a varioussituations [18], [14], [8].The present text may be viewed as a companion piece to the author’s [6].Its main purpose is to provide a more transparent construction than in [5]of the cocycle conditions and related equations associated to a gerbe withcurving data summarized in [5] theorem 6.4. We restrict our attention, asin [6], to gerbes which are connected rather than locally connected, as thesedetermine ˇCech cohomology classes. A cocyclic description in the general caserequires hypercovers and could be dealt with along the lines discussed in [3],but would not shed any additional light on the phenomena being investigatedhere. Our main results are to be found in sections 4 and 5, while section 3reviews for the reader’s convenience some aspects of [5] and [6]. Section 2 isa review of some of the formulas in the differential calculus of Lie ( G )-valuedforms, a few of which do not appear to be well-known.Another aim of the present work is to revisit the quite complicatedcoboundary equations of [5] § B i in diagram (4.13), or for thechange of sign (4.28) for the arrow γ ij . Lawrence Breen
A final purpose of this text is to explain how the diagrammatic proofs ofsome of the local results of [5] can be replaced by more classical computationsinvolving Lie ( G )-valued differential forms. For this reason, we have given twoseparate computations for certain equations, one diagrammatic and the otherclassical. We do not assert that one of the two methods of proof is alwayspreferable, though one might contend that diagrams provide a better under-standing of the situation than the corresponding manipulation of differentialforms. As the level of categorification increases, so will the dimension of thediagrams to be considered, and it may not be realistic to expect to treadalong the diagrammatic path much beyond the hypercube proof [5] (4.1.33)of the higher Bianchi equation (1.5). The generality and algebraicity of theformalism of differential forms must then come into its own. In addition, it isour hope that the present approach, which extends to the gerbe context thetraditional methods of differential geometry, will provide an accessible pointof entry into this topic. A number of other authors have recently describedcertain aspects of the differential geometry of gerbes in terms of differentialforms, particularly [1], [12], and [16], [2].I wish to thank Bernard Julia and Camille Laurent-Gengoux for enlight-ening discussions on related topics. The impetus for the present work wasprovided by my collaboration with Wiliam Messing on our joint papers [4]and [5]. It is a pleasure to thank him here for our instructive and wide-rangingdiscussions over all these years. Let X be an S -scheme. We assume from now on for simplicity that thatthe primes 2 and 3 are invertible in the ring of functions of S (for example S = Spec( k ) where k is a field of characteristic = 2 , n -form on an S -scheme X , with values in a sheaf of O S -Lie algebras g isdefined as a global section of the sheaf g ⊗ O S Ω nX/S on X . When X/S issmooth, g ⊗ O S Ω nX/S ≃ Hom O X ( T nX/S , g X ) (2.1)where g X := g ⊗ O S O X and T nX/S is the n -th exterior power ∧ n T X/S of therelative tangent sheaf T X/S , i.e the sheaf of relative n -vector fields on X . Suchan n -form is nothing else than an O X -linear map T nX/S −→ g X . (2.2)In view of this definition, such a map is classically called a g -valued dif-ferential form. A more geometric description of such forms is given in [4], ifferential Geometry of Gerbes and Differential Forms 5 following the ideas of A. Kock in the context of synthetic differential geom-etry [10], [11]. It is based on the consideration, for any positive integer n ,of the scheme ∆ nX/S of relative infinitesimal n -simplexes on X . For any S -scheme T , a T -valued point of ∆ nX/S consists of an ( n + 1)-tuple of T -valuedpoints ( x , . . . , x n ) of X which are pairwise close to first order in an appro-priate sense [4] (1.4.9). We view ∆ nX as an X -scheme via the projection p of such points to x . As n varies, the schemes ∆ nX/S determine a simplicial X -scheme ∆ ∗ X/S , whose face and degeneracy operations are induced by theusual projection and injection morphisms X n −→ X n ± .Let G be a flat S -group scheme, with O S -Lie algebra g . A relative g -valued n -form (2.2) on X/S may then be identified by [4] proposition 2.5 witha morphism of S -schemes ∆ nX/S f −→ G (2.3)whose restriction to the degenerate subsimplex s∆ nX/S of ∆ nX/S factors throughthe unit section of G . When differential forms are expressed in this combina-torial language, they deserve to be called G -valued differential forms, eventhough they actually coincide with the traditional g -valued differential forms(2.1), (2.2). In the combinatorial context, our notation will be multiplicative,and additive when we pass to the traditional language of differential forms.We will now discuss some of the features of these g -valued forms, and referto [4] for further discussion. First of all, let us recall that the action of thesymmetric group S n +1 on a combinatorial differential n -form ω ( x , . . . , x n )by permutation of the variables is given by ω ( x σ (0) , . . . , x σ ( n ) ) = ω ( x , . . . , x n ) ǫ ( σ ) where ǫ ( σ ) is the signature of σ . Also, the commutator pairing[ g, h ] := g h g − h − on the group G determines a bracket pairing on g -valued forms of degree ≥ g ⊗ O S Ω mX/S ) × ( g ⊗ O S Ω nX/S ) / / ( g ⊗ O S Ω m + nX/S ) (2.4)which sends ( ω, ω ′ ) to [ ω, ω ′ ], where[ ω, ω ′ ]( x , . . . , x m + n ) := [ ω ( x , . . . , x m ) , ω ′ ( x m , . . . , x m + n )] . This pairing is defined in classical terms, by[ ω, ω ′ ] := [ Y, Y ′ ] ⊗ ( η ∧ η ′ ) Lawrence 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 structure of a graded O S -Lie algebra. In particular, thebracket satisfies the graded commutativity rule[ f, g ] = ( − | f || g | +1 [ g, f ] , (2.5)where | f | is the degree of the form f , so that[ f, f ] = 0whenever | f | is even. The graded Jacobi identity is expressed (in additivenotation) as:( − | f || h | [ f, [ g, h ]] + ( − | f || g | [ g, [ h, f ]] + ( − | g || h | [ h, [ f, g ]] = 0 . In particular, [ f, [ f, f ]] = 0 (2.6)and, when | f | = | g | = 1, [ f, [ g, g ]] = [[ f, g ] , g ] . Let Aut( G ) be the sheaf of local automorphisms of G , whose group ofsections above an S -scheme T is the group Aut T ( G T ) of automorphisms ofthe T -group G T := G × S T . The definition (2.3) of a combinatorial n -formstill makes sense when G is replaced by a sheaf of groups F on S , and thetraditional description of such combinatorial n -forms as n -forms with valuesin the Lie algebra of F remains valid by [4] proposition 2.3 when F = Aut( G ).The evaluation map Aut( G ) × G −→ G ( u, g ) u ( g )induces for all pair of positive integers a bilinear pairing(Lie (Aut( G )) ⊗ O S Ω m ) × ( g ⊗ O S Ω nX/S ) / / ( g ⊗ O S Ω m + nX/S ) (2.7)which sends ( u, g ) to [ u, g ], where[ u, g ]( x , . . . , x m + n ) := u ( x , . . . , x m )( g ( x m , . . . , x m + n )) g ( x m , . . . , x m + n ) − . (2.8)This pairing is compatible with the pairings (2.4) associated to the S -groups G and Aut( G ) in the following sense. For any pair of g -valued forms g, g ′ , andan Aut( G )-valued 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 − . Moregenerally, an isomorphism r : G −→ G ′ induces a morphism r from G -valued ifferential Geometry of Gerbes and Differential Forms 7 combinatorial n -forms to G ′ -valued combinatorial n -forms, compatible withthe Lie bracket operation (2.4), and which corresponds in classical terms tothe morphism Lie( r ) ⊗ osc g ⊗ O S Ω nX/S −→ g ′ ⊗ Ω nX/S . The functoriality ofthe bracket (2.7) is expressed by the formula r ([ u, g ] = [ r u, r ( g )] (2.10)where r u := r u r − .When u is an Aut( G )-valued form of degree m ≥ g is a G -valuedfunction, the definition of a pairing(Lie Aut( G ) ⊗ O S Ω mX/S ) × G −→ g ⊗ O S Ω mX/S ( u, g ) [ u, g ]is still given by the formula (2.8), but now with n = 0. This pairing are nolonger linear in g , but instead satisfies the equation[ u, g g ′ ] = [ u, g ] + g [ u, g ′ ]where for any G -valued form ω and any G -valued function g the adjoint leftaction g ω of a function g on a form ω is defined combinatorially by( g ω )( x , . . . , x n ) := g ( x ) ω ( x , . . . , x n ) g ( x ) − , (and this expression is in fact equal to g ( x i ) ω ( x , . . . , x n ) g ( x i ) − for any0 ≤ i ≤ n ). In classical notation this corresponds, for ω = Y ⊗ η ∈ g ⊗ Ω nX/S ,to the formula g ( Y ⊗ η ) = g Y ⊗ η for the adjoint left action of g on Y . The adjoint right action ω γ is defined by ω g := ( g − ) ω so that ω g ( x , . . . , x n ) = g ( x ) − ω ( x , . . . , x n ) g ( x ) . Similarly, when g is a G -valued and u an Aut( G )-valued form, a pairing[ g, u ] is defined by the combinatorial formula[ g, u ]( x , . . . , x m + n ) := g ( x , . . . , x m ) ( u ( x m , . . . , x m + n )( g ( x , . . . , x m ) − )) . (2.11)The pairing (2.11) satisfies the analogue[ g, u ] = ( − | g || u | +1 [ u, g ]of the graded commutativity rule (2.5), so that its properties may be deducedfrom those of the pairing [ u, g ]. In particular[ g − , u ] = − [ u, g − ] = [ u, g ] g . We refer to appendix A of [5] for additional properties of these pairings.
Lawrence Breen
The de Rham differential map g ⊗ O S Ω nX/S d nX/S / / g ⊗ O S Ω n +1 X/S (2.12)is defined combinatorially for n ≥
2, in Alexander-Spanier fashion, byd nX/S ω ( x , . . . , x n +1 ) := n +1 Y i =0 ω ( x , . . . , b x i , . . . ω n +1 ) ( − i . (2.13)This definition agrees for n > G -valuedde Rham differential: d nX/S ω := d X/S ω (2.14)where for ω = Y ⊗ η in g ⊗ Ω nX/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 followsfrom (2.15) that the composite d n +1 d n is trivial . This also follows from thecombinatorial definition of d n , since for n ≥ d n ω commute with each other.For any section g of G , we setd X/S ( g ) := g ( x ) − g ( x ) . (2.16)The map G X d X/S −→ g ⊗ O S Ω X/S g g − d g (2.17)is a crossed homomorphism, for the adjoint left action of G on g . Observe thatthe expression g − d g is consistent with the combinatorial definition (2.16) ofd X/S ( g ). While this traditional expression of d X/S ( g ) as a product of thetwo terms g − and d g does make sense whenever G is a subgroup schemeof the linear group GL n,S , such a decomposition is purely conventional for ageneral S -group scheme G . A companion to d X/S is the differential e d : G −→ g ⊗ O S Ω X/S , defined by e d X/S ( g )( x , x ) := g ( x ) g ( x ) − . The traditional notation for this expression is dg g − . This notation is consis-tent with such formulas (in additive notation) as ifferential Geometry of Gerbes and Differential Forms 9 g ( g − dg ) = d g g − and − ( g − d g ) = d g − g . The differential d X/S is defined combinatorially by(d X/S ω )( x, y, z ) := ω ( x, y ) ω ( y, z ) ω ( z, x ) . (2.18)In classical terms, it follows (see [4] theorem 3.3) thatd X/S ω := d ω + 12 [ ω, ω ] . (2.19)We will henceforth denote d nX/S simply by d n for all n .The quadratic term [ ω, ω ] implies that d X/S is not a linear map, in fact itfollows from (2.19), or the elementary combinatorial calculation of [4] lemma3.2, that d ( ω + ω ′ ) = d ω + d ω ′ + [ ω, ω ′ ] . In particular, d ( − ω ) = − d ( ω ) + [ ω, ω ] . It is immediate, from the combinatorial point of view, thatd d ( g ) = d ( g − d g ) = 0 (2.20)for all g in G . The differential d has a companion, which we will denote by e d , defined by e d ( ω )( x, y, z ) := ω ( z, x ) ω ( y, z ) ω ( x, y ) . A combinatorial computation implies that e d ω = d ω − [ ω, ω ]= d ω − [ ω, ω ] , and the analogue e d ( e d ( g )) = e d ( dg g − ) = 0of (2.20) is satisfied. Finally, it follows from (2.14) that the d n satisfyd i + j [ ω, ω ′ ] = [d i ω, ω ′ ] + ( − i [ ω, d j ω ′ ]whenever i, j ≥
2, and the corresponding formula for the pairing [ u, g ] (2.8)is also valid.
We now choose, for any S -scheme X and any S -group scheme G , an Aut( G )-valued 1-form m on X . We extend the definition of the de Rham differentials(2.17), (2.18) and (2.12) to the twisted differentialsd nX/S, m : g ⊗ O S Ω nX/S −→ g ⊗ O S Ω n +1 X/S (2.21)(or simply d nm ) defined combinatorially by the following formulas:d m ω ( x , x ) := ω ( x , x ) m ( x , x )( ω ( x , x )) m ( x , x ) m ( x , x )( ω ( x , x ))= ω ( x , x ) m ( x , x )( ω ( x , x )) ω ( x , x ) − d nm ω ( x , . . . , x n +1 ) :== m ( x , x )( ω ( x , . . . x n +1 )) n +1 Y i =1 ω ( x , . . . , b x i , . . . , x n +1 ) ( − i when n >
1. When the Aut( G )-valued form m is the image i ( η ) under innerconjugation of a G -valued form η , the expression d ni ( η ) ω will simply be denotedd nη ω . The corresponding degree zero map d m : G −→ g ⊗ O S Ω X/S is definedby d m ( g ) := g ( x ) − m ( x , x )( g ( x )) , (and d m ( g ) will also be denoted g − d m ( g ), consistenly with (2.16)).It follows from elementary combinatorial computations that the differen-tials d nm can be defined in classical terms byd nm ω = d n ω + [ m, ω ] (2.22)for all n , so that for any g -valued 1-form η ,d nm + i η ( ω ) = d nm ( ω ) + [ η, ω ] . (2.23)In particular, d m ( ω ) = d ω + [ m, ω ] = d ω + 12 [ ω, ω ] + [ m, ω ] . While the map d nm is linear for n ≥ m ( ω + ω ′ ) = d m ω + d m ω ′ + [ ω, ω ′ ] (2.24)so that d m ( − ω ) = − d m ( ω ) − [ ω, ω ] . (2.25) ifferential Geometry of Gerbes and Differential Forms 11 Finally, for any section g of Γ , g − d m g = g − d g + [ m, g ] . The composite morphism d n +1 m d nm is in general non-trivial, and the previ-ous classical definitions of d nm imply thatd n +1 m d nm ω = [d m, ω ] (2.26)whenever n ≥
2. For n = 0, the corresponding formulas ared m d m g = [ g − , d m ] and e d m e d m g = [d m, g ] (2.27)so that, for n = 1, we recover the well-known assertion that the vanishing ofd m = 0 implies that d n +1 d n = 0. One verifies that for any 1-form ω d m d m ( ω ) = [d m, ω ] + [d m ω, ω ] (2.28)= [d m, ω ] + [d ω, ω ] + [[ m, ω ] , ω ] . (2.29)This reduces to the equationd m d m ( ω ) = [d m, ω ]of type (2.26) whenever d m ω = 0. For m = i ( ω ), equation (2.28) is equivalentto the classical Bianchi identity [9] II Theorem 5.4:d ω d ω = 0 . (2.30)We now state the functoriality properties of the differential (2.22) d nm for n ≥
1. We define the twisted conjugate g ∗ ω of a G -valued 1-form ω by g ∗ ω := ( p ∗ g ) ω ( p ∗ g ) − = g ω + g d g − (2.31)= ω + [ g, ω ] + g d g − . It follows from the combinatorial definition (2.18) of d that g (d ω ) = d ( g ∗ ω ) . (2.32)More generally, for any G -valued form ω of degree n ≥
1, and any section u of Aut( G ) on X , u (d nm ( ω )) = d n ( u ∗ m ) u ( ω ) (2.33)= d n ( u m ) u ( ω ) + [ u d u − , u ( ω )]= d nm ( u ( ω )) + [[ u, m ] , u ( ω )] + [ u d u − , u ( ω )] . (2.34) Let P be a gerbe on an S -scheme X . For simplicity, in discussing gerbes wewill make two additional assumptions: • P is a G -gerbe, for a given S -group scheme G . • P is connected.The first assumption gives us, for any object x in the fibre category P U above an open set U ⊂ X , an isomorphism of sheaves on UG | U ∼ / / Aut P U ( x ) . (3.1)The second assumption asserts that for any pair of objects x, y ∈ ob( P U )there exists an arrow x −→ y in the category P U . This ensures that the gerbeis described by an element in the degree 2 ˇCech cohomology of X rather thanby degree 2 cohomology with respect to a hypercover of X .Let us choose a family of local objects x i ∈ P U i , for some open cover 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 both Aut P ( x i ) and Aut P ( x j ) with the corre-sponding sections of G above U i and U j , these arrows determine a family ofsection λ ij ∈ Γ ( U ij , Aut( G )), defined by the commutativity of the diagrams x j γ / / φ ij (cid:15) (cid:15) x jφ ij (cid:15) (cid:15) x i λ ij ( γ ) / / x i (3.3)for every γ ∈ G | U ij . In addition, the arrows φ ij determine a family of elements g ijk ∈ G | U ijk for all ( i, j, k ) by the commutativity of the diagrams x k φ jk / / φ ik (cid:15) (cid:15) x jφ ij (cid:15) (cid:15) x i g ijk / / x i (3.4) We refer to [3] and [6] for the definition of a gerbe, and for additional detailsregarding the associated cocycle and coboundary equations (3.7), (3.14).ifferential Geometry of Gerbes and Differential Forms 13 above U ijk . By conjugation in the sense made clear by diagram (3.3), it followsthat the λ ij satisfy the cocycle condition λ ij λ jk = i ( g ijk ) λ ik . (3.5)By [6] lemma 5.1, the G -valued cochains g ijk also satisfy the cocycle condition λ ij ( g jkl ) g ijl = g ijk g ikl . (3.6)These two cocycle equations may be written more compactly as ( δ λ ij = i ( g ijk ) δ λ ij ( g ijk ) = 1 , (3.7)where δ λ is the λ -twisted degree 2 ˇCech differential determined by equation(3.6). They may be jointly viewed as the ( G −→ Aut( G ))-valued ˇCech 1-cocycle equations associated to the gerbe P , the open cover U of X , and thetrivializing families of objects x i and arrows φ ij in P .Let us choose a second family of local objects x ′ i in P U i , and of arrows x ′ j φ ′ ij / / x ′ i (3.8)above U ij . To these correspond a new cocycle pair ( λ ′ ij , g ′ ijk ). In order tocompare this set of arrows with the previous one, we choose (after a harmlessrefinement of the given open cover U of X ) a family of arrows x i χ i / / x ′ i (3.9)in P U i for all i . The arrow χ i induces by conjugation a section r i in the groupof sections Γ ( U i , Aut( G )), characterized by the commutativity of the square x iχ i (cid:15) (cid:15) u / / x iχ i (cid:15) (cid:15) x ′ i r i ( u ) / / x ′ i (3.10)for all u ∈ G . The lack of compatibility between these arrows χ i and the arrows φ ij , φ ′ ij (3.2), (3.8) is measured by the family of sections ϑ ij ∈ Γ ( U ij , G )determined by the commutativity of the following diagram: We prefer to emphasize the fact that λ ij is a 1-cochain since this is more consistentwith a simplicial definition of the associated cohomology, even though it is morecustomary to view the pair of equations (3.7) as a 2-cocycle equation, with (3.5)an auxiliary condition.4 Lawrence Breen x j φ ij / / χ j (cid:15) (cid:15) x iχ i (cid:15) (cid:15) x ′ iϑ ij (cid:15) (cid:15) x ′ j φ ′ ij / / x ′ i . (3.11)Under the identifications (3.1), diagram (3.11) induces by conjugation, in asense made clear by the definition (3.10) of the auromorphism r i , a commu-tative diagram of group schemes above U ij G λ ij / / r j (cid:15) (cid:15) G r i (cid:15) (cid:15) G i ( ϑ ij ) (cid:15) (cid:15) G λ ′ ij / / G , whose commutativity is expressed by the equation λ ′ ij = i ( ϑ ij ) r i λ ij r − j (3.12)in Aut( G ).Consider now the diagram x kφ jk { { vvvvvvvvvvvvv φ ik / / x iχ i (cid:15) (cid:15) g ijk { { x ′ ir i ( g ijk ) { { ϑ ik (cid:15) (cid:15) x j φ ij / / χ j (cid:15) (cid:15) χ k x iχ i (cid:15) (cid:15) x ′ iϑ ij (cid:15) (cid:15) x ′ j φ ′ ij / / ϑ jk (cid:15) (cid:15) (cid:15) (cid:15) x ′ iλ ′ ij ( ϑ jk ) (cid:15) (cid:15) x ′ kφ ′ jk y y tttttttttttt φ ′ ik / / x ′ ig ′ ijk z z ttttttttttttt x ′ j φ ′ ij / / x ′ i . (3.13) This diagram whose faces are five pentagons and three squares (as well as thosein (4.9) and (4.25) below) is the 1-skeleton of a Saneblidze-Umble cubical model[15], [13] for the Stasheff associahedron K [17].ifferential Geometry of Gerbes and Differential Forms 15 Both the top and the bottom squares commute, since these squares are oftype (3.4). So do the back, the left and the top front vertical squares, since allthree are of type (3.11). The same is true of the lower front square, and theupper right vertical square, since these two are respectively of the form (3.3)and (3.10). It follows that the remaining lower right square in the diagram isalso commutative, since all the arrows in diagram (3.13) are invertible. Thecommutativity of this final square is expressed algebraically by the equation g ′ ijk ϑ ik = λ ′ ij ( ϑ jk ) ϑ ij r i ( g ijk ) . We say that two cocycle pairs ( λ ij , g ijk ) and ( λ ′ ij , g ′ ijk ) are cohomologousif we are given a pair ( r i , ϑ ij ), with r i ∈ Γ ( U i , Aut( G )) and ϑ ij ∈ Γ ( U ij , G ),satisfying those two equations ( λ ′ ij = i ( ϑ ij ) r i λ ij r − j g ′ ijk ϑ ik = λ ′ ij ( ϑ jk ) ϑ ij r i ( g ijk ) . (3.14)and display this as ( λ ij , g ijk ) ( r i ,ϑ ij ) ∼ ( λ ′ ij , g ′ ijk ) . (3.15)The equivalence class of the cocycle pair ( λ ij , g ijk ) for this relation is inde-pendent of the choices of objects x i and arrows φ ij by from which it wasconstructed. By definition, it determines an element in the first non-abelianˇCech cohomology set ˇH ( U , G i −→ Aut( G )) with coefficients in the crossedmodule i : G −→ Aut( G ). In [5], the combinatorial description of differential forms is used in order todefine the concepts of connections and curvings on a gerbe. For any S -groupscheme G , a (relative) connection on a principal G -bundle P above the S -scheme X may be defined as a morphism p ∗ P ǫ / / p ∗ P (3.16)between the two pullbacks of P to ∆ X/S , whose restriction to the diagonalsubscheme ∆ : X ֒ → ∆ X/S is the identity morphism 1 P .This type of definition of a connection, as a vehicle for parallel transport,remains valid for other structures than principal bundles. In particular, forany X -group scheme Γ , a connection on Γ is a morphism of group schemes µ : p ∗ Γ −→ p ∗ Γ (3.17)above ∆ X/S whose restriction to the diagonal subscheme
X ֒ → ∆ X/S is theidentity morphism 1 Γ . When Γ is the pullback to X of an S -group scheme G ,the inverse images p ∗ G and p ∗ G of G X above ∆ X/S are canonically isomor-phic, so that the connection (3.17) is then described by a Lie(Aut( G ))-valued1-form m .A connection µ on a group Γ determines de Rham differentialsd nX/S, µ : Lie( Γ ) ⊗ O S Ω nX/S −→ Lie( Γ ) ⊗ O S Ω n +1 X/S (or simply d nµ ) defined combinatorially by the formulas [5] (A.1.9)-(A.1.11)and their higher analogues. When Γ is the pullback of an S -group scheme, d nµ is decribed in classical terms as the deformation (2.22)d nµ := d nm of the de Rham differential d n determined by the associated 1-form m . Whenthe curvature d m of the connection µ is trivial, the connection is said to beintegrable. In that case, it follows from (2.26) and (2.27) that the de Rhamdifferentials satisfy the condition d n +1 m d nm = 0 for all n = 1.The curvature of a connection ǫ (3.16) on a principal bundle P is theunique arrow κ ǫ : p ∗ P −→ p ∗ P such that the following diagram above ∆ X/S commutes, with ǫ ij the pullbacksof ǫ under the corresponding projections p ij : ∆ X/S −→ ∆ X/S : p ∗ P ǫ / / ǫ (cid:15) (cid:15) p ∗ P ǫ (cid:15) (cid:15) p ∗ P κ ǫ / / p ∗ P By construction, κ ǫ is a relative 2-form on X with values 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 by the commutativity of the squares p ∗ P u / / ǫ (cid:15) (cid:15) p ∗ P ǫ (cid:15) (cid:15) p ∗ P µ ǫ ( u ) / / p ∗ P (3.18) ifferential Geometry of Gerbes and Differential Forms 17 for all sections u of p ∗ ( P ad ). By [11], [5] proposition 1.7, the curvature 2-form κ ǫ satisfies the Bianchi identity d µ ǫ ( κ ǫ ) = 0 . (3.19)For a given family of local sections of P , with associated G -valued 1-cocycles g ij , the connection (3.16) is described by a family of G -valued 1-forms ω i ∈ g ⊗ Ω U i /S , satisfying the gluing condition ω j = ω ∗ g ij i = ω g ij i + g − ij d g ij (3.20)above U ij , for the action of G on g ⊗ O S Ω U i /S induced by the adjoint rightaction of G on g . A 1-form satisfying this equation is classically known as aconnection form. The induced curvature κ is locally described by the familyof 2-forms κ i := d ω i = d ω i + 12 [ ω i , ω i ] , and these satisfy the simpler ˇCech (or gluing) condition κ j = κ g ij i . Equation (3.19) is reflected at the local level in the equationd ω i κ i = 0 , which is simply the classical Bianchi identity (2.30) for the 1-form ω i . The notion of a connective structure on a G -gerbe P is a categorification of thenotion of a connection on a principal bundle, as we will now recall, following[5] §
4. To P is associated its gauge stack P ad . By definition this is the monoidalstack E q X ( P , P ) of self-equivalences of the stack P , the monoidal structurebeing defined by the composition of equivalences. A connection on a P is anequivalence between stacks p ∗ P ǫ / / p ∗ P (3.21)above ∆ X/S , together with a natural isomorphism between the restriction ∆ ∗ ǫ of ǫ to the diagonal subscheme X of ∆ X/S and the identity morphism1 P . Such a connection ǫ induces as in (3.18) a connection µ on the gauge stack P ad . A curving of ( P , ǫ ) is a natural isomorphism Kp ∗ P ǫ / / ǫ (cid:15) (cid:15) p ∗ P ǫ (cid:15) (cid:15) p ∗ P κ / / p ∗ P , K @ yyyy (3.22)for some morphism κ : p ∗ P −→ p ∗ P above ∆ X/S . It is determined by the choice of some explicit quasi-inverse ofthe connection ǫ . The arrow κ which arises as part of the definition of K iscalled the fake curvature associated to the connective structure ( ǫ, K ). It is aglobal object in the pullback to ∆ X/S of the gauge stack P ad .The connective structure ( ǫ, K ) determines a 2-arrow p ∗ P κ / / κ (cid:15) (cid:15) p ∗ P µ ( κ ) (cid:15) (cid:15) p ∗ P κ / / / / p ∗ P Ω v ~ tttttt This is the unique 2-arrow which may be inserted in diagram p ∗ P ǫ / / ǫ z z ttttttttt ǫ (cid:15) (cid:15) p ∗ P ǫ y y ssssssssss κ (cid:15) (cid:15) p ∗ P κ / / κ (cid:15) (cid:15) p ∗ P µ ( κ ) (cid:15) (cid:15) p ∗ P ǫ z z ttttttttt ǫ / / p ∗ P ǫ y y ssssssssss p ∗ P κ / / p ∗ P . K { (cid:3) (cid:127)(cid:127)(cid:127)(cid:127) K , bbb bbb K (cid:22) (cid:30) K / gggg Ω u } tttt M ( κ ) (cid:31) ' FFFFFF (3.23)so that the two composite 2-arrows p ∗ P µ ( κ ) κ ǫ * * * * ǫ ǫ ǫ p ∗ P (cid:11) (cid:19) which may be constructed by composition of 2-arrows in (3.23) coincide. ifferential Geometry of Gerbes and Differential Forms 19 This 2-arrow Ω may also be viewed as a 1-arrow above ∆ X/S in the gaugegroup P ad , or even as an arrow in the stack Lie( P ad ) ⊗ O S Ω X/S of relativeLie( P ad )-valued 3-forms on X . Returning to the combinatorial definition [5](A.1.10) of the de Rham differential, we may finally view Ω , by horizontalcomposition with appropriate 1-arrows, as a 1-arrow in P ad whose sourceobject is the identity arrow I P ad : I Ω / / d µ ( κ − ) . (3.24)Denoting the twisted differential d µ by the expression d + [ µ, ] to which itreduces when appropriate trivializations have been chosen, the 3-curvaturearrow Ω (3.24) is described by the equation (1.4). By [5] theorem 4.4 it sat-isfies another relation, described by the cubical pasting diagram [5] (4.1.24),and which may be expressed by the higher Bianchi identity (1.5). The pair ofequations (1.4) and (1.5) may now be thought of as a categorified version, sat-isfied by the pair of P ad -valued forms ( κ, Ω ), of the classical Bianchi identity(3.19), and can be written in symbolic form asd µ, K ( κ, Ω ) = 0 , where d nµ, K is the twisted de Rham differential on Lie( P ad )-valued n -forms de-termined by twisting data ( µ, K ) associated to the given connective structureon P . We observed in section 3.1 that a gerbe could be expressed in cocyclic terms,once local trivializations were chosen. We will now show that this is also thecase for the connection ǫ . We choose, for each i ∈ I , an arrow γ i : ǫp ∗ x i −→ p ∗ x i (4.1)in p ∗ P U i such that ∆ ∗ γ i = 1 x i . The arrow γ i determines by conjugation aconnection m i : p ∗ G | U i −→ p ∗ G | U i on the pullback G | U i of the group G above the open set U i ⊂ X . The arrow m i is described, for any section g ∈ Γ ( ∆ X/SU i , p ∗ G ), by the commutativityof the diagram See [5] (4.1.28) for a proof of this identity.0 Lawrence Breen ǫp ∗ x i ǫ ( g ) / / γ i (cid:15) (cid:15) ǫp ∗ x iγ i (cid:15) (cid:15) p ∗ x i m i ( g ) / / p ∗ x i . (4.2)The pair ( φ ij , γ i ) determines a family of arrows γ ij in the pullback G ∆ Uij of G , defined by the commutativity of the diagram ǫp ∗ x j γ j / / ǫp ∗ φ ij (cid:15) (cid:15) p ∗ x jp ∗ φ ij (cid:15) (cid:15) p ∗ x iγ ij (cid:15) (cid:15) ǫp ∗ x i γ i / / p ∗ x i (4.3)By conjugation, this determines a commutative diagram p ∗ G m j / / p ∗ λ ij (cid:15) (cid:15) p ∗ G p ∗ λ ij (cid:15) (cid:15) p ∗ G i ( γ ij ) (cid:15) (cid:15) p ∗ G m i / / p ∗ G (4.4)so that the equation i ( γ ij ) ( p ∗ λ ij ) m j ( p ∗ λ ij ) − = m i . (4.5)of [5] (6.1.2) is satisfied.We may restate (4.5) as i ( γ ij ) [( p ∗ λ ij ) m j ( p ∗ λ ij ) − ] = m i [ p ∗ λ ij ( p ∗ λ − ij )] , (4.6)an equation all of whose factors are Aut( G )-valued 1-forms on U ij and there-fore commute with each other. In the notation introduced in (2.31), equation(4.6) can be rewritten as λ ij ∗ m j = m i − i ( γ ij ) , (4.7)or more classically as λ ij m j = m i − λ ij d λ − ij − i ( γ ij ) . (4.8)This is is the analogue for the Aut( G )-valued forms m i and λ ij of the classicalexpression (3.20) for a connection form, but now categorified by the insertionof an additional summand − i ( γ ij ). ifferential Geometry of Gerbes and Differential Forms 21 Consider now the following diagr in P ∆ Uijk : ǫp ∗ x kǫp ∗ φ jk y y ssssssssssssss ǫp ∗ φ ik / / ǫp ∗ x iγ i (cid:15) (cid:15) ǫp ∗ g ijk y y tttttttttttttt p ∗ x im i ( p ∗ g ijk ) y y tttttttttttttt O O γ ik ǫp ∗ x j ǫp ∗ φ ij / / γ j (cid:15) (cid:15) ǫp ∗ x iγ i (cid:15) (cid:15) p ∗ x i O O γ ij p ∗ x j p ∗ φ ij / / O O γ jk γ k (cid:15) (cid:15) p ∗ x i O O λ ij ( γ jk ) p ∗ x kp ∗ φ jk w w ppppppppppppp p ∗ φ ik / / p ∗ x ip ∗ g ijk w w ppppppppppppp p ∗ x j p ∗ φ ij / / p ∗ x i (4.9)Of the eight faces of this cube, seven are known to be commutative. It followsthat the remaining lower square on the right vertical side is also commutative.This is the square p ∗ x i p ∗ g ijk / / γ ik (cid:15) (cid:15) p ∗ x iλ ij ( γ jk ) (cid:15) (cid:15) p ∗ x iγ ij (cid:15) (cid:15) p ∗ x i m i ( p ∗ g ijk ) / / p ∗ x i , (4.10)whose commutativity corresponds to the equation γ ij ( p ∗ λ ij ( γ jk )) = m i ( p ∗ g ijk ) γ ik ( p ∗ g ijk ) − in other words to the equation [5] (6.1.7), all of whose factors are G -valued1-forms on U ijk . We may rewrite this as γ ij p ∗ λ ij ( γ jk ) = ( m i ( p ∗ g ijk ) p ∗ g − ijk ) ( p ∗ g ijk γ ik p ∗ g − ijk )so that, taking into account the equation (3.5), we finally obtain (in additivenotation) γ ij + λ ij ( γ jk ) − λ ij λ jk ( λ − ik ( γ ik )) = dg ijk g − ijk + [ m i , g ijk ] , with bracket defined by (2.7) an equation which can be written in abbreviatedform as δ λ ij ( γ ij ) = d m i g ijk g − ijk . (4.11) We now describe in similar terms the curving K and the fake curvature κ ofdiagram (3.22). Just as we associated to the connection ǫ (3.21) a family ofarrows γ i (4.1), we now choose, for each i ∈ I , an arrow κp ∗ x i δ i / / p ∗ x i (4.12)in the category P ∆ Ui , whose restriction to the degenerate subsimplex s∆ U i of ∆ U i is the identity. To the curving K is associated a family of “ B -field” g -valued 2-forms B i ∈ g ⊗ Ω U i , characterized by the commutativity of thefollowing diagram in which an expression such as γ i is the pullback of γ i bythe corresponding projection p : ∆ X/S −→ ∆ X/S : ǫ ǫ ( p ∗ x i ) ǫ γ i (cid:15) (cid:15) K ( p ∗ x i ) / / κǫ ( p ∗ x i ) κγ i (cid:15) (cid:15) ǫ ( p ∗ x i ) γ i (cid:15) (cid:15) κp ∗ x iδ i (cid:15) (cid:15) p ∗ x i o o B i p ∗ x i (4.13)Let us now define a family of G -valued 2-forms ν i on U i by the equations ν i := d m i − i ( B i ) (4.14)in Lie Aut( G ) ⊗ Ω U i , in other words by the commutativity of the diagram p ∗ G m i (cid:15) (cid:15) p ∗ G m i (cid:15) (cid:15) p ∗ G m i (cid:15) (cid:15) p ∗ G ν i (cid:15) (cid:15) p ∗ G i o o i ( B i ) p ∗ G . (4.15)By comparing diagram (4.15) with the conjugate of diagram (4.13), we seethat ν i is simply the conjugate of the arrow δ i . It can therefore described bythe commutativity of the diagram κp ∗ x i κ ( g ) / / δ i (cid:15) (cid:15) κp ∗ x iδ i (cid:15) (cid:15) p ∗ x i ν i ( g ) / / p ∗ x i (4.16) The chosen orientation of the arrow B i is consistent with that in [5].ifferential Geometry of Gerbes and Differential Forms 23 for all g ∈ Γ ( ∆ U i /S , p ∗ G ), just as the connection m i was described by diagram(4.2).We also define a family of 2-forms δ ij by the commutativity of the diagram p ∗ x i λ ij ( B j ) / / δ ij (cid:15) (cid:15) p ∗ x iγ ij (cid:15) (cid:15) p ∗ x iγ ij (cid:15) (cid:15) p ∗ x im i ( γ ij ) (cid:15) (cid:15) p ∗ x i B i / / p ∗ x i , (4.17)i.e., since all terms commute, by the equation δ ij := λ ij ( B j ) − B i − d m i ( − γ ij )in Lie( G ) ⊗ Ω U i /S . In ˇCech-de Rham notation, this is δ ij := δ λ ij ( B i ) − d m i ( − γ ij ) , (4.18)and in classical notation δ ij := λ ij ( B j ) − B i + d γ ij −
12 [ γ ij , γ ij ] + [ m i , γ ij ] . Here is another characterization of δ ij : Lemma 4.1.
For every pair ( i, j ) ∈ I , the analogue κp ∗ x j δ j / / κp ∗ φ ij (cid:15) (cid:15) p ∗ x jp ∗ φ ij (cid:15) (cid:15) p ∗ x iδ ij (cid:15) (cid:15) κp ∗ x i δ i / / p ∗ x i . (4.19) of diagram (4.3) is commutative. Proof : Consider the diagram κǫ ( κp ∗ x j ) κγ j / / κp ∗ x jκp ∗ φ ij (cid:15) (cid:15) δ j / / p ∗ x jp ∗ φ ij (cid:15) (cid:15) ǫ ǫ ( p ∗ x j ) ǫ ǫ ( p ∗ φ ij ) (cid:15) (cid:15) γ j / / K ( p ∗ x j ) ppppppppppp κǫ ( p ∗ φ ij ) (cid:15) (cid:15) ǫ ( p ∗ x j ) γ j / / ǫ ( p ∗ φ ij ) (cid:15) (cid:15) p ∗ x j v v B j mmmmmmmmmmmmmmm p ∗ φ ij (cid:15) (cid:15) p ∗ x iδ ij (cid:15) (cid:15) p ∗ x i v v λ ij ( B j ) mmmmmmmmmmmmmmm κp ∗ x i δ i / / p ∗ x iν i ( γ ij ) (cid:15) (cid:15) ǫ ( p ∗ x i ) γ i / / p ∗ x i (cid:15) (cid:15) γ ij κǫ ( p ∗ x i ) κγ i / / κp ∗ x i δ i / / (cid:15) (cid:15) κγ ij p ∗ x i ǫ ǫ ( p ∗ x i ) K ( p ∗ x i ) ppppppppppp ǫ ( γ i ) / / ǫ ( p ∗ x i ) (cid:15) (cid:15) ǫ ( γ ij ) γ i / / p ∗ x i (cid:15) (cid:15) m i ( γ ij ) v v B i mmmmmmmmmmmmmmm . (4.20)Diagrams (4.13), (4.17) and (4.16) imply that all squares in (4.20) are commu-tative , except possibly the rear right upper one. This remaining square (4.19)is therefore also commutative. ✷ Conjugating diagram (4.19), we obtain as in (4.4) a square p ∗ G ν j / / p ∗ λ ij (cid:15) (cid:15) p ∗ G p ∗ λ ij (cid:15) (cid:15) p ∗ G i δij (cid:15) (cid:15) κp ∗ G ν i / / p ∗ G , whose commutativity is expressed algebraically as i ( δ ij ) ( p ∗ λ ij ) ν j = ν i ( p ∗ λ ij ) . (4.21) This is true for diagram (4.17) since ν i ( γ ij ) = γ ij .ifferential Geometry of Gerbes and Differential Forms 25 In additive notation, this is equation λ ij ν j = ν i − i ( δ ij ) , (4.22)in other words δ 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) of ν i and δ ij . First of all,observe that by (2.32) d ( λ ij ∗ m i ) = λ ij (d m i ) . (4.23)One then computes λ ij ν j = λ ij (d ( m j ) − i B j )= d ( λ ij ∗ m j ) − i ( λ ij ( B j ))= d ( m i − i ( γ ij )) − i ( B i + d m i ( − γ ij ) + δ ij )= d m i − d ( i ( γ ij )) − [ m i , γ ij ] − i ( B i ) − i (d m i ( − γ ij )) − i ( δ ij ) . Since the homomorphism i commutes with d m and [ m i , i ( γ ij )] = i ([ m i , γ ij ]),the summands i (d m ( − γ ij )) and d ( i ( γ ij )) + [ m i , γ ij ] cancel out. The firsttwo remaining summands describe ν i , so that equation (4.22) is satisfied.In the same vein, the analogue for the fake curvature κ of (4.10) is thefollowing assertion. Lemma 4.2.
The diagram p ∗ x i p ∗ g ijk / / δ ik (cid:15) (cid:15) p ∗ x iλ ij ( δ jk ) (cid:15) (cid:15) p ∗ x iδ ij (cid:15) (cid:15) p ∗ x i ν i ( p ∗ g ijk ) / / p ∗ x i (4.24) is commutative. Proof:
By (4.19), (3.4) and (4.16), all squares in the diagram κp ∗ x kκp ∗ φ jk { { κp ∗ φ ik / / κp ∗ x iδ i (cid:15) (cid:15) κp ∗ g ijk { { p ∗ x iν i ( p ∗ g ijk ) { { O O δ ik κp ∗ x j κp ∗ φ ij / / δ j (cid:15) (cid:15) δ k κp ∗ x iδ i (cid:15) (cid:15) p ∗ x i O O δ ij p ∗ x j p ∗ φ ij / / O O δ jk (cid:15) (cid:15) p ∗ x i O O λ ij ( δ jk ) p ∗ x kp ∗ φ jk y y ssssssssssssss p ∗ φ ik / / p ∗ x ip ∗ g ijk y y tttttttttttttt p ∗ x j p ∗ φ ij / / p ∗ x i (4.25)are commutative, except possibly the lower right-hand one. It follows that thelatter one, which is simply (4.24), also commutes. ⊓⊔ The commutativity of (4.24) corresponds to equation δ ij ( p ∗ λ ij )( δ jk ) = ν i ( p ∗ g ijk ) δ ik ( p ∗ g ijk ) − , an equation whose terms are G -valued 2-forms on U ijk . By the same reasoningas for (4.11), this can be written additively as δ ij + λ ij ( δ jk ) − λ ij λ jk ( λ − ik ( δ ik )) = [ ν i , g ijk ] , or, in the compact form of [5] (6.1.15), as δ λ ij ( δ ij ) = [ ν i , g ijk ] . (4.26)Just we were able to derive (4.22) directly from (4.8) and the definitions (4.14)and (4.18), we now show that it is possible to deduce (4.26) from (4.18),(4.14)and (4.11). First of all, δ λ ij ( δ ij ) = δ λ ij ( δ λ ij ( B i ) − d m i ( − γ ij ))= δ λ ij δ λ ij ( B i ) − δ λ ij d m i ( − γ ij ) . (4.27)We now wish to assert that the ˇCech differential δ λ ij and de Rham dif-ferential d m i in (4.27) commute with each other, despite the fact that the ifferential Geometry of Gerbes and Differential Forms 27 γ ij takes its values in a non-commutative group G , and that d m i isnot a homomorphism. For this we simplify our notation, by setting e γ ij := − γ ij ∈ g ⊗ Ω U ij (4.28)and λ ijk := λ ij λ jk λ − ik ∈ Γ ( U ijk , Aut( G i )) . Equation (4.11) can be restated as δ λ ij e γ := e γ ij + λ ij ( e γ jk ) − λ ijk ( e γ ik ) = − d g ijk g − ijk − [ m i , g ijk ] . (4.29) Lemma 4.3.
The following equality between G -valued 2-forms above U ijk issatisfied: d m i δ λ ij ( e γ ij ) = δ λ ij d m i ( e γ ij ) . (4.30) Proof:
We compute the left-hand side of the equation (4.30), taking intoaccount the quadraticity equation (2.24)d m i δ λ ij ( e γ ij ) = d m i ( e γ ij ) + d m i ( λ ij ( e γ jk )) + d m i ( − λ ijk ( e γ ik )) ++ [ e γ ij , λ ij ( e γ jk )] − [ e γ ij , λ ijk ( e γ ik )] −− [ λ ij ( e γ jk ) , λ ijk ( e γ ik )]= d m i ( e γ ij ) + d m i ( λ ij ( e γ jk )) − d m i ( λ ijk ( e γ ik )) ++ [ λ ijk ( e γ ik ) , λ ijk ( e γ ik )] + [ e γ ij , λ ij ( e γ jk )] −− [ e γ ij + λ ij ( e γ jk ) , λ ijk ( e γ ik )] . We now compute the right-hand side of (4.30): δ λ ij d m i ( e γ ij ) = d m i ( e γ ij ) + λ ij (d m j ( e γ jk )) − λ ijk (d m i ( e γ ik )) . (4.31)By (4.7) and by the functoriality property (2.32), we find that λ ij (d m j ( e γ jk )) = d λij ∗ m j ( λ ij ( e γ jk ))= d m i ( λ ij ( e γ jk )) + [ e γ ij , λ ij ( e γ jk )]and by (2.34) λ ijk (d m i ( e γ ik )) = d λijk ∗ m i ( λ ijk ( e γ ik ))= d m i ( λ ijk ( e γ ik )) + [[ λ ijk , m i ] , λ ijk ( e γ ik )]++ [ λ ijk d λ − ijk , λ ijk ( e γ ik )] . Inserting these expressions for λ ij (d m j ( e γ jk )) and λ ijk (d m i ( e γ ik )) into the right-hand side of (4.31) we find the following expression for δ λ ij d m i ( e γ ij ): δ λ ij d m i ( e γ ij ) = d m i ( e γ ij ) + d m i ( λ ij e γ jk ) + [ e γ ij , λ ij ( e γ jk )] −− d m i ( λ ijk ( e γ ik ) − [[ λ ijk , m i ] , λ ijk ( e γ ik )] − [ λ ijk d λ − ijk , λ ijk ( e γ ik )] −− d m i ( λ ijk )( e γ ik ) − [ λ ijk d λ − ijk , λ ijk ( e γ ik )] . Comparing this with the expression (4.31) for d m i δ λ ij ( e γ ij ), we see that theequation (4.30) is satisfied if and only if[ e γ ij + λ ij ( e γ jk ) − λ ijk ( e γ ik ) , λ ijk ( e γ ik )] = [[ λ ijk , m i ] , λ ijk ( e γ ik )]++ [ λ ijk d λ − ijk , λ ijk ( e γ ik )] . By (2.9), this is simply a consequence of (4.29), since λ ijk = i ( g ijk ) . ✷ We now return to our computation (4.27): δ λ ij ( δ ij ) = δ λ ij δ λ ij ( B i ) − δ λ ij d m i ( − γ ij )= δ λ ij δ λ ij ( B i ) − d m i δ λ ij ( − γ ij )= [ g ijk , B i ] − d m i ( g ijk d m i ( g − ijk ))= [ g ijk , i B i − dm i ] by (2.27)= [ ν i , g ijk ] . This finishes the second proof of equation (4.26) . ✷ We now set ω i := d m i ( B i ) . (4.32)Since the combinatorial definition of the twisted de Rham differential d ([4](3.3.1)) matches the global geometric definition (3.23) of the 3-curvature Ω ,this 3-curvature Ω is locally described by the G -valued 3-forms ω i .It follows from the definitions (4.14) and (4.32) of the forms ν i and ω i , andfrom (2.26), that d m i ( ω i ) = d m i d m i ( B i )= [d m i , B i ]= [ ν i , B i ] + [ B i , B i ]so that the local 3-curvature form ω i satisfies the higher Bianchi identityd m i ( ω i ) = [ ν i , B i ] . (4.33) ifferential Geometry of Gerbes and Differential Forms 29 A second relation between the forms ν i and ω i follows from their definitionsand the Bianchi identity for the Aut( G )-valued 1-form m i : i ( ω i ) = d m i i ( B i )= d m i (d m i − ν i )= d m i ( − ν i ) , in other words d m i ν i + i ( ω i ) = 0 . (4.34)This equation is the local form of equation(1.4), just as (4.33) was the localform of (1.5).We will now show that the equation (4.18) for the 2-forms B i , which wewrite here as δ λ ij ( B i ) = d m i ( − γ ij ) + δ ij , induces the corresponding gluing equation for the local 3-forms ω i . From thedefinition of λ ij ( ω j ) and (2.33), it follows that λ ij ( ω j ) = λ ij (d m j ( B j ))= d λij ∗ m j λ ij ( B j )and by the gluing laws (4.8) and (4.18) for m i and B i , this can be stated as λ ij ( ω j ) = d m i − i ( γ ij ) ( B i + δ ij + d m i ( − γ ij ))= d m i ( B i ) + d m i ( δ ij ) + d m i d m i ( − γ ij ) − [ γ ij , B i + δ ij + d m i ( − γ ij )] . By (2.28), this last equality can be rewritten as λ ij ( ω j ) = ω i + d m i ( δ ij ) + [d m i , − γ ij ] − [ γ ij , B i ] − [ γ ij , δ ij ]= ω i + d m i ( δ ij ) + [ γ ij , d m i − B i ] − [ γ ij , δ ij ]and by (4.21) this proves the gluing law for the 3-forms ω i [5] (6.1.23): λ ij ( ω j ) = ω i + d 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 that (4.35) canfinally be rewritten in the more compact form λ ij ( ω j ) + [ λ ij ν j , γ ij ] = ω i + d m i ( δ ij ) (4.35) We saw in section 2 how a change in the choice trivializing data ( x i , φ ij ) in agerbe P could be measured by a pair ( r i , θ ij ) (3.10),(3.11) inducing a cobound-ary relation (3.15) between the corresponding cocycle pairs ( λ ij , g ijk ). We willnow examine how the terms ( m i , γ ij ), ( ν i , δ ij ) and B i introduced in section 4vary when the arrows γ i (4.1) and δ i (4.12) which determine them have beenmodified.The difference between the arrow γ i and an analogous arrow γ ′ i is measuredby a 1-form e i ∈ Lie ( G ) ⊗ Ω U i , defined by the commutativity of the followingdiagram: ǫp ∗ x i ǫp ∗ χ i / / γ i (cid:15) (cid:15) ǫp ∗ x ′ iγ ′ i (cid:15) (cid:15) p ∗ x i p ∗ χ i / / p ∗ x ′ i e i / / p ∗ x ′ i (5.1)This conjugates to a commutative diagram p ∗ G p ∗ r i / / m i (cid:15) (cid:15) p ∗ G m ′ i (cid:15) (cid:15) p ∗ G p ∗ r i / / p ∗ G i ( e i ) / / p ∗ G so that m ′ i = i ( e i ) ( p ∗ r i ) m i ( p ∗ r i ) − = i ( e i ) [ p ∗ r i m i p ∗ r i − ] [ p ∗ r i p ∗ r i − ]In classical terms, this is expressed as an equation m ′ i = r i m i + r i d r − i + i ( e i ) (5.2)= r i ∗ m i + i ( e i ) . (5.3)which compares the connections m i and m ′ i induced on the group G by thearrows γ i and γ ′ i . ifferential Geometry of Gerbes and Differential Forms 31 We now consider the following diagram in P U ij : p ∗ x ′ ie i (cid:15) (cid:15) p ∗ x ′ ir i ( γ ij ) o o p ∗ θ ij (cid:15) (cid:15) p ∗ x ′ im ′ i ( p ∗ θ ij ) (cid:15) (cid:15) p ∗ x ′ iλ ′ ij ( e j ) (cid:15) (cid:15) p ∗ x ′ i p ∗ x ′ i . γ ′ ij o o (5.4) Proposition 5.1.
The diagram (5.4) is commutative.
Proof:
Consider the diagram ǫp ∗ x jǫp ∗ φ ij z z ttttttttttttttttt γ j / / p ∗ x jp ∗ χ j (cid:15) (cid:15) p ∗ φ ij x x qqqqqqq p ∗ x iγ ij x x qqqqqqq p ∗ χ i (cid:15) (cid:15) ǫp ∗ x iǫp ∗ χ i (cid:15) (cid:15) γ i / / ǫp ∗ χ j (cid:15) (cid:15) p ∗ x ip ∗ χ i (cid:15) (cid:15) p ∗ x ′ ip ∗ θ ij (cid:15) (cid:15) p ∗ r i ( γ ij ) y y rrrrrrr p ∗ x ′ jp ∗ φ ′ ij y y rrrrrrr e j (cid:15) (cid:15) p ∗ x ′ ie i (cid:15) (cid:15) p ∗ x ′ ip ∗ λ ′ ij ( e j ) (cid:15) (cid:15) ǫp ∗ x ′ jǫp ∗ φ ′ ij uuuuuuuu z z uuuuuuuu γ ′ j / / p ∗ x ′ jp ∗ φ ′ ij y y rrrrrrr ǫp ∗ x ′ i γ ′ i / / ǫp ∗ θ ij (cid:15) (cid:15) p ∗ x ′ im ′ i ( p ∗ θ ij ) (cid:15) (cid:15) p ∗ x ′ iγ ′ ij x x rrrrrrr ǫp ∗ x ′ i γ ′ i / / p ∗ x ′ i (5.5)The lower front square of the right-hand face of this cube is just the square(5.4). Since we know that all the other squares in this diagram commute, sodoes the square (5.4). ⊓⊔ The commutativity of (5.4) is equivalent to the equation m ′ i ( p ∗ θ ij ) e i r i ( γ ij ) = γ ′ ij λ ′ ij ( e j ) p ∗ θ ij . (5.6) This may be rewritten in classical notation as:( γ ′ ij − θ ij r i ( γ ij )) + ( λ ′ ij ( e j ) − θ ij e i ) = d m ′ i θ ij θ − ij . (5.7)We now choose a family of arrows δ ′ i : κp ∗ x ′ i −→ p ∗ x ′ i . The families δ ′ i and γ ′ i determine as in (4.13) a family of g -valued 2-form B ′ i above U i . Thelatter in turn determines, together with the pair of form ( m ′ i , γ ′ ij ) (5.2), (5.7),a new pair of 2-forms ( ν ′ i , δ ′ ij ) and a 3-form ω ′ i satisfying the correspondingequations (4.22), (4.34), (4.26), (4.33) and (4.35). The families δ i and δ ′ i arecompared by the following analogue of diagram (5.1): κp ∗ x i κp ∗ χ i / / δ i (cid:15) (cid:15) κp ∗ x ′ iδ ′ i (cid:15) (cid:15) p ∗ x i p ∗ χ i / / p ∗ x ′ i n i / / p ∗ x ′ i . (5.8)We will now compare the 2-forms B i and B ′ i . We consider the diagram ǫ ǫ ( p ∗ x i ) ǫ ǫ ( p ∗ χ i ) t t jjjjjjjjjjjjjjjj K ( p ∗ x i ) / / ǫ ( γ i ) κǫ ( p ∗ x i ) κ ǫ ( p ∗ χ i ) u u kkkkkkkkkkkkkk κ ( γ i ) (cid:15) (cid:15) ǫ ǫ ( p ∗ x ′ i ) ǫ γ ′ i (cid:15) (cid:15) K ( p ∗ x ′ i ) / / (cid:15) (cid:15) κ ǫ ( p ∗ x ′ i ) κ ( γ ′ i ) (cid:15) (cid:15) ǫ p ∗ x iǫ p ∗ χ i v v llllllll γ i (cid:15) (cid:15) κ p ∗ x iκp ∗ χ i v v nnnnnnnn δ i (cid:15) (cid:15) ǫ p ∗ x ′ i v v mmm γ ′ i (cid:15) (cid:15) κp ∗ x ′ iδ ′ i (cid:15) (cid:15) κ ( e i ) w w ooo ǫ p ∗ x ′ iγ ′ i (cid:15) (cid:15) κp ∗ x ′ iδ ′ i (cid:15) (cid:15) p ∗ x i o o B i p ∗ χ i w w nnnn p ∗ x ip ∗ χ i w w ppp p ∗ x ′ i o o r i ( B i ) e i y y tt p ∗ x ′ in i { { vv p ∗ x ′ i v v mmmmm p ∗ x ′ iν ′ i ( e i ) w w oooo p ∗ x ′ i o o B ′ i p ∗ x ′ i (5.9)in which the upper and lower unlabelled arrows are respectively ǫ ( p ∗ e i )and m ′ i ( e i ). ifferential Geometry of Gerbes and Differential Forms 33 The front square (or rather hexagon) of the bottom face p ∗ x ′ i o o r i ( B i ) e i (cid:15) (cid:15) p ∗ x ′ ip ∗ n i (cid:15) (cid:15) p ∗ x ′ im ′ i ( e i ) (cid:15) (cid:15) p ∗ x ′ iν ′ i ( e i ) (cid:15) (cid:15) p ∗ x ′ i o o B ′ i p ∗ x ′ i is commutative, since all other squares in diagram (5.9) are. Equivalently,since the action of the Aut( G )-valued 2-form ν ′ i on e i is trivial, this provesthat the equation B ′ i = r i ( B i ) − d m ′ i ( − e i ) − n i . (5.10)is satisfied. In particular for given B i and e i , the 2-forms B ′ i and n i actuallydetermine each other.By conjugation, diagram (5.8) induces a commutative diagram p ∗ G p ∗ r i / / ν i (cid:15) (cid:15) p ∗ G ν ′ i (cid:15) (cid:15) p ∗ G p ∗ r i / / p ∗ G i ni / / p ∗ G equivalent to the equation i ( n i ) p ∗ r i ν i = ν ′ i p ∗ r i . In classical terms, this is the simpler analogue ν ′ i = r i ν i + i ( n i ) (5.11)for ν i of the equation (5.2) for m i .We will now show that this coboundary equation for ν i can be derived fromthe definition (4.14) of ν i , and the coboundary equations (5.2) and (5.10) for m i and B i : ν ′ i = d m ′ i − i ( B ′ i )= d ( r i ∗ m i + i ( e i )) − i ( r i ( B i ) + n i + d m ′ i ( − e i ))= r i d m i + i (d e i ) + [ r i ∗ m i , i ( e i )] − i ( r i ( B i )) + i (d m ′ i ( − e i )) + i ( n i )= r i (d m i − i ( B i )) + i ( n i ) + i (d m ′ i ( − e i ) + d e i + [ r i ∗ m i , e i ]) In order to prove (5.11), it now suffices to verify that the 3 terms in the lastsummand of the final equation cancel each other out:d m ′ i ( − e i ) + d ( e i ) + [ r i ∗ m i , e i ] = d ( − e i ) − [ m ′ i , e i ] + d e i + [ r i ∗ m i , e i ]= d ( − e i ) + d e i − [ e i , e i ]= 0 . ✷ The other equation satisfied by the forms n i is the counterpart of equation(5.6). It is obtained by considering the following diagram, analogous to (5.5): κp ∗ x jκp ∗ φ ij z z uuuuuuuuuuuuuuuuu δ j / / p ∗ x jp ∗ χ j (cid:15) (cid:15) p ∗ φ ij y y rrrrrrr p ∗ x iδ ij y y rrrrrrr p ∗ χ i (cid:15) (cid:15) κp ∗ x iκp ∗ χ i (cid:15) (cid:15) δ i / / κp ∗ χ j (cid:15) (cid:15) p ∗ x ip ∗ χ i (cid:15) (cid:15) p ∗ x ′ ip ∗ θ ij (cid:15) (cid:15) r i ( δij ) y y sssssss p ∗ x ′ jp ∗ φ ′ ij y y sssssss n j (cid:15) (cid:15) p ∗ x ′ in i (cid:15) (cid:15) p ∗ x ′ ip ∗ λ ′ ij ( n j ) (cid:15) (cid:15) κp ∗ x ′ jκp ∗ φ ′ ij vvvvvvvv z z vvvvvvvv δ ′ j / / p ∗ x ′ jp ∗ φ ′ ij y y sssssss κp ∗ x ′ i δ ′ i / / κp ∗ θ ij (cid:15) (cid:15) p ∗ x ′ iν ′ i ( p ∗ θ ij ) (cid:15) (cid:15) p ∗ x ′ iδ ′ ij y y sssssss κp ∗ x ′ i δ ′ i / / p ∗ x ′ i . (5.12)The lower front square on the right-hand face p ∗ x ′ i p ∗ θ ij / / r i ( δ ij ) (cid:15) (cid:15) p ∗ x ′ i p ∗ λ ′ ij ( n j ) / / p ∗ x ′ iδ ′ ij (cid:15) (cid:15) p ∗ x ′ i n i / / p ∗ x ′ i ν ′ i ( p ∗ θ ij ) / / p ∗ x ′ i of diagram (5.12) is commutative, since all other squares in this diagram are. ifferential Geometry of Gerbes and Differential Forms 35 This proves that equation ν ′ i ( p ∗ θ ij ) n i r i ( δ ij ) = δ ′ ij p ∗ λ ′ ij ( n j ) p ∗ θ ij in Lie ( G ) ⊗ Ω U i /S is satisfied. Regrouping the various terms in this equationas we did above for equation (5.6), we find that it is equivalent, in additivenotation, to ( δ ′ ij − r i ( δ ij )) + ( λ ′ ij ( n j ) − θ ij n i ) = [ ν ′ i , θ ij ] , an equation for 2-forms very similar to equation (5.7) for 1-forms.We will now examine the effect of the chosen transfomations( λ ij , g ijk , m i , γ ij ) −→ ( λ ′ ij , g ′ ijk , m ′ i , γ ′ ij ) (5.13)and B i −→ B ′ i (5.10) on the 3-curvature 3-forms ω i (4.32). For this, it willbe convenient to set¯ e i := r − i ( e i ) and ¯ n i := r − i ( n i ) . It follows from (2.23), (2.10), and the transformation formula (5.3) thatd nm ′ i ( r i ( η )) = r i (d nm i ( η ) + [¯ e i , η ]) (5.14)for any G -valued n -form η with n >
1. In particulard m ′ i ( − e i ) = d ri ∗ m i ( − e i ) − [ e i , e i ]= r i (d m i ( − ¯ e i ) − [¯ e i , ¯ e i ])so that (5.10) may be expressed as B ′ i = r i ( B i − d m i ( − ¯ e i ) + [¯ e i , ¯ e i ] − ¯ n i ) . Applying once more the formula (5.14), we find that ω ′ i = d m ′ i ( B ′ i )= d m ′ i ( r i ( B i − d m i ( − ¯ e i ) + [¯ e i , ¯ e i ] − ¯ n i ))= r i (d m i ( B i − d m i ( − ¯ e i ) + [¯ e i , ¯ e i ] − ¯ n i )) ++ [¯ e i , B i − d m i ( − ¯ e i ) + [¯ e i , ¯ e i ] − ¯ n i ] . (5.15)We now make use of (2.28) in order to compute the value of the expressiond m i d m i ( − ¯ e i ) which arises when we expand the first summand of the lastequation (5.15):d m i d m i ( − ¯ e i ) = [d m i , − ¯ e i ] + [d ( − ¯ e i ) , − ¯ e i ] + [[ m i , − ¯ e i ] , − ¯ e i ]= − [d m i , ¯ e i ] + [d ¯ e i , ¯ e i ] + [[ m i , ¯ e i ] , ¯ e i ] . Inserting this expression into (5.15), we find that ω ′ i = r i ( ω i + [d m i , ¯ e i ] − [d ¯ e i , ¯ e i ] − [[ m i , ¯ e i ] , ¯ e i ] − d m i (¯ n i ) ++ d m i [¯ e i , ¯ e i ] + [¯ e i , B i ] − [¯ e i , d m i ( − ¯ e i )] − [¯ e i , ¯ n i ]) . (5.16)The four terms − [d ¯ e i , ¯ e i ] − [[ m i , ¯ e i ] , ¯ e i ] + d m i [¯ e i , ¯ e i ] − [¯ e i , d m i ( − ¯ e i )]cancel each other out, so that we are left in (5.16) with ω ′ i = r i ( ω i + [d m i , ¯ e i ] − d m i (¯ n i ) + [¯ e i , B i ] − [¯ e i , ¯ n i ])= r i ( ω i + [d m i − i ( B i ) , ¯ e i ] + [ ¯ n i , ¯ e i ] − d m i (¯ n i ))= r i ( ω i ) + r i ([ ν i , ¯ e i ]) + r i ([ ¯ n i , ¯ e i ]) − r i (d m i (¯ n i ))= r i ( ω i ) + [ r i ν i , e i ] + [ n i , e i ] − d ri ∗ m i ( n i ) (5.17)where in the last line we made use of the functoriality property (2.10) of thebracket operation. Amalgamating the last two summands, we may finally writethe coboundary transformation for the 3-curvature form ω i in the compactform ω ′ i = r i ( ω i ) + [ r i ν i , e i ] − d m ′ i ( n i ) . If instead we amalgamate the second and third term in (5.17), we find theequivalent formulation ω ′ i = r i ( ω i ) + [ ν ′ i , e i ] − d ri ∗ m i ( n i ) . (5.18) Remark 5.1 (Comparison with [5]):
The coboundary equation (5.18) is compatible with equation (6.2.19) of[5], but neither is a special case of the other. Here we allowed both the trivial-izing data ( x i , φ ij ) for the gerbe and the expressions ( γ i , δ i , B i ) for the curv-ing data to vary, whereas in the coboundary equations of [5] the gerbe data( x i , φ ij ) was fixed and only the ( γ i , δ i , B i ) varied. This restriction amountedto setting ( r i , θ ij ) = (1 ,
1) in our equation (5.7). On the other hand, a notionof equivalence between cocycles was introduced in [5] which was more exten-sive than the one considered here. In order for these to be comparable, onemust suppose that the arrow h in diagram (4.2.1) of [5] is the identity map,i.e. that the pair of differential forms ( π i , η ij ) associated to h in loc. cit § . h could re-ally be termed a gauge transformation, rather than a coboundary term. With ifferential Geometry of Gerbes and Differential Forms 37 this additional condition, the last two summands in equation (6.2.19) of [5]vanish, so that this equation reduces to ω ′ i = ω i + δ m i ( α i ) − [ ν ′ i , E i ] . (5.19)This simplified equation is compatible with our equation (5.18) with r i = 1,under the correspondence e i := − E i and n i := − α i . References
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