Massey products in differential cohomology via stacks
aa r X i v : . [ m a t h . A T ] S e p Massey products in differential cohomology via stacks
Daniel Grady and Hisham SatiSeptember 22, 2017
Abstract
We extend Massey products from cohomology to differential cohomology via stacks, organiz-ing and generalizing existing constructions in Deligne cohomology. We study the properties andshow how they are related to more classical Massey products in de Rham, singular, and Delignecohomology. The setting and the algebraic machinery via stacks allow for computations andmake the construction well-suited for applications. We illustrate with several examples fromdifferential geometry and mathematical physics.
Contents
Massey products were introduced in [Ma58] and further developed and generalized in [Kr66] [Ma69].The existence of (higher) Massey products indicates the complexity of the topology of a space. Theyalso determine whether and how various characterizing properties of a space might be related, in1articular how homotopy of a space might be related to its cohomology [GM13]. On the one hand,Massey products can be viewed as secondary cohomology operations associated with the primaryoperation given by the cup product. On the other hand, they can also be seen as higher orderproducts in homotopy ( A ∞ ) algebras (see [St63][BV73]).A differential graded algebra (DGA) is a (not a priori commutative) graded algebra A with amap d : A → A of degree +1 which satisfies the relations (up to sign conventions) dd = 0 and d ( ab ) = ( da ) b + ( − dim a a ( db ). Then the cohomology H ( A ) of A with respect to d is a gradedalgebra. It has further certain operations called (matrix) Massey products, the simplest of whichis a correspondence H ( A ) ⊗ H ( A ) ⊗ H ( A ) → H ( A ) (1.1)which is denoted by h a, b, c i , where a, b, c ∈ H ( A ). This has dimension dim( a )+dim( b )+dim( c ) −
1, isdefined only when ab = bc = 0 ∈ H ( A ), and is not well-defined but rather only defined modulo termsof the form ax + yb where x and y are some (auxiliary) elements of H ( A ). The indeterminacy may,however, sometimes be excluded, for example for dimension reasons, which occurs in applications.Generally, we have ab = dy and bc = dz for y, z ∈ A , so that h a, b, c i = yc + ( − dim a +1 az (1.2)is a cocycle, with the cohomology class defined modulo the indeterminacy given above.There are other notions of Massey products, but all are essentially variations on this principle.If the Massey product h a , ...., a n i exists, then all “lower” Massey products necessarily vanish,although the converse is not true in general. One may also apply a similar construction for matricesof elements, leading to matric Massey products [Ma69], where notions related to formal flatness ofthe connection become important.Differential cohomology has played an important role recently by combining geometric andtopological data, namely usual cohomology and differential forms, in a coherent way [CS85] [EV88][Br08] [DF99] [Ga97] [HS05] [SS08] [Bu12] [BS10] [Sc13] [BB14]. It is natural then to try to ex-tend Massey products, which exist in both of these ingredients, to differential cohomology. Masseyproducts have been considered in Deligne cohomology in [De95] [Sc02] [We00] [MT08]. We extendthe definitions and constructions to the level of stacks, which has the virtue of allowing for vastgeneralization to various settings and to a plethora of applications. We believe this formulation hasan advantage both for theory and for applications. In particular, we emphasize that desired proper-ties and behaviour of the Massey products are clearer and systematic in stacks, and computationsare generally doable and are more efficient there, making them quite suitable for applications. Theconstructions are based on the thesis [Gr15], but we have taken the opportunity here to sharpenthe results and add properties and applications. We emphasize that this paper is the first partof a bigger project, aimed at developing various concrete computational techniques for differentialcohomology theories.The paper is organized as follows. In Sec. 2, we provide the setting for the two main ingredientsthat we would like to combine, namely classical (generalized) Massey products in Sec. 2.1 anddifferential cohomology in Sec. 2.2. We set up the former in the general framework of [Ma69][Kr66] [BT00] and the latter in the language of stacks (see [FSS12] [SSS12] [Sc13]). Then we Throughout the paper, by stacks we mean simplicial sheaves as discussed, for instance, in [Lu09] [DI04] [Sc13]and recalled in section 2.2.
Massey product Chern-Simons theories . In Sec. 4.1, we illustrate how(stacky) Massey products arise in trivialization of higher structures, such as (differential) String,Fivebrane [SSS09], and Ninebrane structures [Sa15]. This gives natural trivializations of Chern-Simons theories at the level of (higher) bundles with connections. There are two expressionsthat involve three differential cohomology classes, namely the stacky Massey product and the tripleDeligne-Beilinson cup product. A natural question is whether these are related. Indeed, we proposesuch a relation via transfer in the context of cobordism.Then, in Sec. 4.2, we see how systems arising generally in anomaly cancellation lead naturallyto (stacky) Massey products. Finally, in settings inspired by type IIA and type IIB string theoryin Sec. 4.3 and Sec. 4.4, respectively, we illustrate how these lead to stacky Massey products.Interestingly, the latter gives rise to a quadruple Massey product. The reader need not be familiarwith these string theories in order to follow the discussion.
We recall some notions from [Ma69] [Kr66] [BT00]. This will be useful for the applications that wewill consider later as well as a starting point for comparison with our stacky constructions.Let ( A , d ) be a differential graded algebra over R endowed with augmentation. Let M ( A ) bethe set of all upper triangular half-infinite matrices with entries in A , zeroes on the diagonal andfinitely many nonzero entries, i.e. M ( A ) = { A = ( a ij ) , a ij ∈ A , a ij = 0 for j ≤ i and i, j ≥ n + 1 for some n with i, j ∈ N } . (2.1)The last condition distinguishes in M ( A ) a subset (which is in fact a subalgebra) M n ( A ) consistingof all ( n × n )-matrices with entries in A . The algebra M ( A ) is bigraded and endowed with abigraded Lie bracket. We introduce the differential d on M ( A ) as dA = ( da ij ) i,j ≥ . The algebra A admits an involution given by a a = ( − k a , which can be extended to an automorphism of M ( A ) as A = ( a ij ) i,j ≥ , with the differential d satisfying the generalized Leibnitz rule d ( AB ) =( dA ) B + A ( dB ). In [BT00], the Maurer-Cartan operator µ : M ( A ) −→ M ( A ) was defined as3 ( A ) = dA − A · A . Then a matrix A ∈ M ( A ) is said to be a matrix of formal connection if itsatisfies the Maurer-Cartan equation in A , dA − A · A ≡ A , (2.2)i.e. A is a formal connection if µ ( A ) ∈ ker A . Here ker A is a A -module generated by matrices 1 ij such that A · ij = 1 ij · A , where 1 ij denotes the matrix that has all zero entries except for 1 asthe ij -entry. Note that this implies that AB = BA for any matrix B ∈ ker A . The element µ ( A )is called the curvature of the formal connection A , and can be shown to be closed (see e.g. [BT00][Ch75]).Now comes the relation between Maurer-Cartan and Massey products. The generalized Masseyproducts are the cohomology classes of the curvature matrices of the formal connection A , i.e. if A is a solution to the Maurer-Cartan equation then the entries of the matrix [ µ ( A )] are the general-ized Massey products [BT00]. Geometrically, this means that the latter measure the deviation ofconnections from flat ones, so that the connection is flat if they vanish. Later we will make use ofthis approach in describing Massey products in stacks.Classical Massey products in integral cohomology H ∗ ( X ; Z ) arise by taking A to be an algebraover the commutative ring Z , with the multiplication being associative but not necessarily graded-commutative. Now let α, β, γ be the cohomology classes of closed elements a ∈ A p , b ∈ A q , and c ∈ A r . The triple Massey product h α, β, γ i is defined if one can solve the Maurer-Cartan equationwith the formal connection A = a ˜ f ∗ b ˜ g c . This is equivalent to the two separate equations d ˜ f = ( − p a ∧ b and d ˜ g = ( − q a ∧ c (2.3)and that implies that the Massey product is defined if and only if α ∪ β = β ∪ γ = 0 ∈ H ∗ ( A ) . (2.4)The matrix µ ( A ) has the form µ ( A ) = dA − A · A = τ and defines the Massey product [ µ ( A )] which is equal to the cohomology class h α, β, γ i = [ τ ] = h ( − p +1 a ∧ ˜ g + ( − p + q ˜ f ∧ c i . (2.5)Here [ a ] ∈ H ∗ ( A ) denotes the cohomology class of a closed element a ∈ A , and [ A ] = ([ a ij ]) i,j ≥ ∈ M ( H ∗ ( A ), for a closed matrix A ∈ M ( A ), denotes the corresponding matrix whose entries are thecohomology classes of the entries a ij of A . Since ˜ f and ˜ g are defined by expressions (2.3) up to closedelements from A , the triple Massey product h α, β, γ i is defined modulo α · H q + r ( A ) + γ · H p + q ( A ).4 .2 Differential cohomology There are several different approaches to differential cohomology. Initially, we will be concernedwith the construction as Deligne cohomology [Br08] [Ga97]. We will then move to the stacky setting,which illuminates the true nature of differential cohomology as a theory which counts isomorphismclasses of higher U (1)-gerbes with connection (generalizing the usual discussion for the gerbe casein [Br08]).The classical construction relies on hypercohomology of a complex of objects of an abeliancategory as an extension to complexes of the usual cohomology of an object. For n ∈ N , let Z ∞D [ n ]be the sheaf of chain complexes given by Z ∞D [ n ] := [ . . . → → Z ֒ → Ω → Ω → . . . → Ω n − ] , where Z is in degree n and Ω n − is the sheaf of real-valued ( n − X , the degree n sheaf hypercohomology with coefficients in Z ∞D [ n ] can be defined to bethe degree n differential cohomology of X : b H n ( X ; Z ) := H n ( X ; Z ∞D [ n ]) . (2.6)If X is paracompact, then these cohomology groups are given by the cohomology of the totalcomplex of the ˇCech-Deligne double complex corresponding to a good open cover of X . In whatfollows, we will always assume that X is paracompact, so that the hypercohomology groups can becomputed by either taking arbitrary injective resolutions, or via this more explicit ˇCech approach.In [SS08] (see also [Bu12]), it was observed that these cohomology groups fit nicely into an exacthexagon Ω n − ( X ) / im( d ) Ω n cl ( X ) H n − ( X ) b H n ( X ; Z ) H n dR ( X ) ,H n − ( X ; U (1)) H n ( X, Z ) da I R (2.7)where the bottom row is the Bockstein sequence and the diagonals are exact. The map R iscalled the curvature map and I is called the integration map. Notice that, by exactness, in thecase that the curvature of a differential cohomology class vanishes, the class lies in the image ofthe inclusion H n − ( X ; U (1)) ֒ → b H n ( X ; Z ). We call these classes flat , as they represent n -gerbeswith connections of vanishing curvature. Differential cohomology therefore detects the topologicalinformation – when the class is flat – and the differential geometric information encoded by the This is a descending grading, which is the opposite of the usual grading of the de Rham complex. That is, weare viewing this as a chain complex rather than a cochain complex. Furthermore, we take V [ n ] to denote the chaincomplex shifted by n , so that V is in degree n . n ∈ N , let C art S p be the category with objects convex open subsets of Cartesian space R n (hence diffeomorphic to R n ), and morphisms smooth functions. A smooth prestack is simply afunctor F : C art S p op → s S etwith target the category of simplicial sets. The passage from prestacks to stacks is achieved byimposing a sort of gluing condition on F . Roughly speaking, a stack F attaches an entire space(equivalently simplicial set) of data to each object in C art S p. This data should be viewed as being local data. The gluing condition then assembles this data into a geometric object, which is a stack.More precisely, we say that a prestack F satisfies descent if for each U ∈ C art S p and each opencover { U i } i ∈ I of U with contractible finite intersections U i i ...i k , we have a weak equivalence F ( U ) ≃ holim n . . . Q i,j,k F ( U ijk ) Q i,j F ( U ij ) Q i F ( U i ) o . (2.8)In particular, if F takes values in Kan complexes, this weak equivalence is part of an actual homotopy equivalence. The reader may notice the following. • If we change the target category to S et and impose the stronger condition that the strict limitover the diagram was isomorphic to F ( U ), we would recover the gluing condition for a sheaf. • If we change the target category to groupoids, then the above condition recovers the usualnotion of descent for classical stacks.In the latter, homotopy equivalence is simply categorical equivalence of groupoids. Hence the gluingcondition respects the correct notion of equivalence (which is weaker than isomorphism). We cantherefore view the equivalence (2.8) as the more general gluing condition for ∞ -groupoids (or Kancomplexes). We need the following (see [Lu09] [DI04] [Sc13]). Definition 1.
We call a smooth prestack F a smooth stack if it satisfies descent. We denote thefull subcategory of smooth stacks by S h ∞ ( C art S p) ֒ → [ C art S p op , s S et] , where the brackets denote the category of contravariant functors from C art S p to s S et , with mor-phisms that are natural transformations. Note that the above functor category is simplicially enriched in a natural way. Observe thatfor objects X and Y in any (locally small) category, hom( X, Y ) is always a set. This allows us toform the mapping space (i.e. simplicial set), which at level n is (cid:0) Map(
X, Y ) (cid:1) n := hom (cid:0) X × ∆[ n ] , Y (cid:1) , when X is fibrant and Y cofibrant (this requires a model structure). Here the operation × is theCartesian product in stacks, and the underline on ∆[ n ] denotes taking the locally constant stackassociated to ∆[ n ]. 6 emark 1. The inclusion functor admits a left adjoint L which preserves homotopy colimits (infact, a left Quillen adjoint [Sc13]). We call this functor L the stackification functor and call theimage of a prestack F under L the stackification of F . In [FSS12], the moduli stack of n -gerbes with connection, B n U (1) conn , was introduced. Thisstack was obtained as the stackification of the n -prestack obtained by applying the Dold-Kan map(see section 3.1) to the Deligne presheaf of chain complexes Z ∞D [ n + 1] := [ ... → → Z ֒ → Ω → Ω → ... → Ω n ] . These stacks are the differential analogues of Eilenberg-MacLane spaces and, for a fixed manifold X , there is a bijective correspondence (a “representation”) b H n +1 D ( X ; Z ) ≃ π Map( X, B n U (1) conn ) , (2.9)where the right hand side is the set of morphisms in the homotopy category of stacks. Remark 2.
In general the right hand side of the correspondence (2.9) may not be well-defined. Inorder to be able to take homotopy groups of the mapping space Y := Map( X, B n U (1) conn ) , Y hasto be a Kan complex, which is the case when X is confibrant and B n U (1) conn is fibrant. However,since B n U (1) conn satisfies descent, it is fibrant in a particular local model structure on presheaves(see [FSS12]). Even though X can be viewed as a stack, it is not cofibrant, and so we need tocofibrantly replace it. Indeed, if X is a (paracompact) manifold, thought of as a smooth stack, withgood open cover { U i } i ∈ I , then we can replace X by its ˇCech nerve C ( { U i } ) := hocolim n ` i U i ` i,j U ij ` i,j,k U ijk . . . o (2.10) which is both cofibrant and weak equivalent to X in the category of smooth stacks Sh ∞ ( C art S p) [DHI04]. For purely model category theoretic reasons it then follows that Map( C ( { U i } ) , B n U (1) conn ) is a Kan complex and we can take π , obtaining the set of morphisms in the homotopy category.This motivates the definition Map( X, B n U (1) conn ) := Map( C ( { U i } ) , B n U (1) conn ) . As explained in [FSS12], these stacks also have a nice geometric interpretation. The followingexample illustrates the point quite well.
Example 1.
Let X be a manifold. Let us calculate the set of vertices of the mapping space Map( X, B U (1) conn ) . Using the pointwise formula for the homotopy colimit [FSS12], we have hom( X, B U (1) conn ) = hom( C ( { U i } ) , B U (1) conn )= hom (cid:16) Z k ∈ ∆ ∆[ k ] × a α ,..,α k U α ,..,α k , B U (1) conn (cid:17) = Z k ∈ ∆ Y α ,..,α k hom(∆[ k ] × U α ,..,α k , B U (1) conn )= Y α ,..,α k Z k ∈ ∆ hom (cid:0) ∆[ k ] , B U (1) conn ( U α ,..,α k ) (cid:1) . (2.11)7 n element of the hom in the last line can be written out explicitly as a choice maps B α : ∆[0] → Y α B U (1)( U α ) A αβ : ∆[1] → Y αβ B U (1)( U αβ ) g αβγ : ∆[2] → Y αβγ B U (1)( U αβγ ) , (2.12) such that the face inclusions of each map are equal to their corresponding restrictions to higherintersections. Now since equivalent stacks will produce the same cohomology groups, we do notdistinguish between equivalent stacks. In particular, using the exponential quasi-isomorphism, wecould have equivalently defined B U (1) conn to be the stackification of the prestack given by applyingthe Dold-Kan functor to the presheaf of chain complexes [0 → . . . → C ∞ ( − , U (1)) d log −→ Ω → Ω ] . We can therefore describe the choices of B α , A αβ and g αβγ via the 2-simplex B α B γ B δ g αβγ A αβ A γδ A δα Here, g αβγ is a choice of smooth U (1) -valued function on triple intersections, A αβ is a choice of1-form on double intersections and B α is a choice of 2-form on open sets. Moreover, we have thatthese assignments must satisfy the conditions (i) g αβ g − γβ g γα = 1 ; (ii) g − αβγ dg αβγ = d log( g ) αβγ = A αβ − A γβ + A γα ; (iii) B β − B α = dA αβ .We identify this data as precisely giving a gerbe with connection [Br08]. Moreover, the fact that B n U (1) conn is a stack ensures that F α = dB α is a globally defined 3-form: the curvature of thegerbe. Notice that these are only the vertices in the mapping space. The entire mapping space keepstrack of more information, namely the homotopies and higher homotopies between gerbes. Theseencode automorphisms in the sense of gauge transformations (see [FSS13] [FSS15]). Example 2.
Let X be a paracompact manifold and C ( { U i } ) the ˇCech nerve of some good opencover. The maps L : C ( { U i } ) → B U (1) conn are in bijective correspondence with circle bundles on X equipped with a connection. In fact, us-ing the calculations in the above example shows that such a morphism gives the data U (1) -valued unctions g αβ on intersections satisfying g αβ g − βγ g γδ = 1 on triple intersections, along with 1-forms A α on open sets satisfying A α − A β = d log( g ) αβ on double intersections. If the homotopy classof L is trivial, then the circle bundle is trivializable . In fact, the trivializing map φ is nothingbut a homotopy φ : L → . To identify this homotopy, we use the Dold-Kan correspondence. Inparticular, an edge in Map( C ( { U i } ) , B U (1) conn ) is, by adjunction, an edge in the simplicial set Map( C ( { U i } ) , B n U (1) conn ) = DK(hom C h + ( N ( C ( { U i } )) , Z ∞D [2])) , (2.13) where N is the normalized Moore functor. Recall that this functor gives an equivalence of categories,from simplicial abelian groups s A b to chain complexes in non-negative degrees Ch + • (see [GJ09]).The hom in positively graded chain complexes is the truncated total complex of the ˇCech-Delignedouble complex [ . . . → tot C ( U , Z ∞D [2]) → Z (cid:0) tot C ( U , Z ∞D [2]) (cid:1) ] , where Z denotes the group of cocycles in that degree. Recalling that the differential is given by D := d + ( − k δ , where δ takes the alternating sum of restrictions, we identify an edge connecting L and as an assignment of ˇCech-Deligne cochain h of degree 1 such that ( d − δ ) h = L . Explicitly,this means a choice of U (1) -valued function h α on open sets such that (i) h α h − β = g αβ ; (ii) − ih − α dh α = d log( h α ) = A α .A straightforward calculation shows that the pattern continues and that null homotopies of n -gerbes(equivalently n -bundles, equivalently maps into B n U (1) conn ) can again be identified with trivializa-tions. Motivated by this last example, we will often refer to null homotopies as trivializations . Tosummarize, the mapping space Map( X, B n U (1) conn ) can be identified with the set of all n -gerbeswith connection, along with isomorphisms between these and higher homotopies between theseisomorphisms. Remark 3.
There are several other stacks related to B n U (1) conn which are useful for us and aredefined as follows (see [FSS12],[FSS13],[FSS15],[Sc13]): (i) If we forget about the connection on the these n -bundles, we obtain the bare moduli stack of n -gerbes B n U (1) . Explicitly, this stack is obtained by applying the Dold-Kan functor to the sheaf ofchain complexes C ∞ ( − , U (1))[ n ] : the sheaf of smooth U (1) -valued functions in degree n . (ii) We also define a stack which represents flat n -bundles with connection, ♭ B n U (1) . This stackis obtained by applying Dold-Kan to the sheaf of chain complexes disc U (1)[ n ] : the sheaf of locallyconstant U (1) valued functions in degree n . (iii) We have a stack representing the truncated de Rham complex ♭ dR B n U (1) obtained by applyingDold-Kan to the truncated de Rham sheaf of chain complexes Ω ≤ n cl := [ . . . → Ω → Ω → . . . → Ω n cl ] . Here “disc” refers to the underlying discrete topology. As an operation on stacks, disc is the composite functordisc : Sh ∞ ev ∗ / / s S et ( · ) / / Sh ∞ , where ev ∗ is the evaluation at a point and ( · ) takes the locally constant stackassociated to a simplicial set. For a smooth manifold X , the resulting stack disc( X ) is sometimes denoted instead by X δ . iv) Finally, we define the stack of closed n -forms Ω n cl to be the stack obtained by applying Dold-Kanto the sheaf of closed n -forms. One way to see that the second stack really does detect flat n -gerbes with connection is toobserve that, by Poincar´e lemma, one has a quasi-isomorphism of sheavesdisc( U (1))[ n ] ≃ [0 → . . . → C ∞ ( − , U (1)) d log −→ Ω → . . . → Ω n cl ] , where on the right we have closed n -forms in degree 0. These n -forms are to be interpreted asgiving the connection on the corresponding bundle. Hence, if the form is closed then the bundle isflat.The moduli stack B n U (1) conn is related to the stacks in Remark 3 in various ways. In [FSS12][Sc13],it was observed that B n U (1) conn is the homotopy pullback B n U (1) conn (cid:15) (cid:15) R / / Ω n +1cl ι (cid:15) (cid:15) B n U (1) θ / / ♭ dR B n +1 U (1) , (2.14)where the left composite B n U (1) conn → B n U (1) θ → ♭ dR B n +1 U (1) is homotopic to the mapcurv : B n U (1) conn → ♭ dR B n +1 U (1) (2.15)induced by the morphism of sheaves of chain complexes Z i / / i (cid:15) (cid:15) Ω / / d (cid:15) (cid:15) · · · d (cid:15) (cid:15) / / Ω nd (cid:15) (cid:15) Ω d / / Ω / / · · · / / Ω n +1cl . (2.16)This map gives the full de Rham data for the curvature of a bundle with connection. In fact, if onecalculates the sheaf hypercohomology in degree 0 of the bottom row, say via the ˇCech-de Rhamcomplex (as in [BT82]), one gets H n dR ( X ). Consequently, the map curv induces a mapcurv ∗ : π Map( X, B n U (1) conn ) −→ H n +1dR ( X ) , (2.17)which sends an ( n − Lemma 2.
The homotopy fiber of the map R : B n U (1) conn ։ Ω n +1cl can be identified with ♭ B n U (1) . roof. The map R is induced by the morphism of sheaves of chain complexes Z i / / (cid:15) (cid:15) Ω / / (cid:15) (cid:15) · · · (cid:15) (cid:15) / / Ω n − d (cid:15) (cid:15) d / / / / · · · / / Ω n cl . (2.18)Since this map is degree-wise surjective by Poincar´e lemma (traditionally in highest form-degree,and trivially in lower degrees), it is a fibration in the projective model structure on presheaves ofchain complexes. We can therefore calculate the homotopy fiber as the kernel of that map. Byinspection, the kernel is [ . . . → Z ֒ → Ω → Ω → . . . → Ω n cl ] , which, via the exponential map, is quasi-isomorphic to[ . . . C ∞ ( − , U (1)) d log −→ Ω → . . . → Ω n cl ] . Again, by Poincar´e lemma, this sheaf of chain complex is quasi-isomorphic to disc( U (1))[ n ] . Sincethe Dold-Kan functor is a right Quillen adjoint and preserves weak equivalences, it takes fibrationsequences to fibration sequences and we have the desired result. ✷ Using the above proposition along with diagram (2.14) and the pasting lemma for homotopypullbacks, we observe that we have the following iteration of homotopy pullbacks [Sc13] ♭ B n − U (1) / / (cid:15) (cid:15) B n U (1) / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∗ / / (cid:15) (cid:15) ♭ dR B n − U (1) / / (cid:15) (cid:15) ♭ B n U (1) / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∗ / / Ω ≤ n − / / (cid:15) (cid:15) B n U (1) conn / / / / (cid:15) (cid:15) Ω n +1cl (cid:15) (cid:15) ∗ / / B n U (1) / / ♭ dR B n +1 U (1) , (2.19)where 0 is the 0 map. From Lemma 2 along with this last diagram, we immediately get thefollowing: Proposition 3.
The based loop stack Ω B n U (1) conn can be identified with the stack ♭ B n − U (1) . Proof.
Consider the homotopy pullback square ♭ B n − U (1) / / (cid:15) (cid:15) B n U (1) / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∗ / / (cid:15) (cid:15) ♭ dR B n − U (1) / / (cid:15) (cid:15) ♭ B n U (1) (cid:15) (cid:15) ∗ / / Ω ≤ n − / / B n U (1) conn B n U (1) conn connecting the point inclusion ∗ → B n U (1) conn to itself: a loop. ✷ Note that Massey products in the homology of the based loop space is classically consideredin [St70] [Ch72]. The above discussions allows us to recast the “differential cohomology diamond”using our stacks.
Proposition 4.
The differential cohomology diagram (2.7) lifts to a diagram of stacks Ω ≤ n − Ω n cl ♭ dR B n − U (1) B n U (1) conn ♭ dR B n U (1) ♭ B n U (1) B n U (1) da Iβ j R (2.20) where the diagonals are fibration sequences. Proof.
This is the same diagram as a portion of diagram (2.19) rotated. The top and bottomhorizontal maps in (2.20) are defined as the compositions d = Ra and β = jI . Fixing a manifold X , mapping into this diagram, and passing to connected components, i.e. taking π Map( X, − ), werecover the diamond diagram (2.7). Note that d in (2.20) recovers the usual exterior derivative,by the nature of R , and that β recovers the Beckstein by uniqueness of the latter as a cohomologyoperation. ✷ We now explain how to go the other direction, i.e. from stacks to Deligne cohomology. Wehave seen that for a manifold X , the mapping space Map( X, B n U (1) conn ) can be identified with thespace of n -gerbes equipped with connections (along with all isomorphisms and higher isomorphismsbetween them). It will be convenient to organize this mapping space itself into a stack. We definethe mapping stack to be the stackification of the prestack given by the assignment U Map( X × U, B n U (1) conn ) (2.21)for each U ∈ C art S p. We denote this stack by [ X, B n U (1) conn ]. Remark 4.
Notice the following: (i)
If we evaluate the mapping stack on the terminal object in C art S p (the point) and take π , werecover the usual differential cohomology groups from the correspondence (2.9) π [ X, B n U (1) conn ]( ∗ ) ≃ π Map( X × ∗ , B n U (1) conn ) ≃ b H n ( X, Z ) . Note that this is not to be confused with homotopy classes of maps as the notation might suggest. ii) Since the mapping stack is clearly functorial in both arguments and the stackification functorpreserves homotopy fibers (it is left exact), we can map into the diagram (2.19) to obtain the diagram (cid:2) X, Ω ≤ n − (cid:3) [ X, Ω ncl ] (cid:2) X, ♭ dR B n − U (1) (cid:3) [ X, B n U (1) conn ] [ X, ♭ dR B n U (1)][ X, ♭ B n U (1)] [ X, B n U (1)] da I R where the diagonals are again fibration sequences. If we evaluate this previous diagram at the pointand apply π , we indeed reproduce the usual differential cohomology diamond diagram (2.7) . Deligne [De71] and Beilinson [Be86] showed that differential cohomology admits a distinguishedcup product refining the usual cup product on singular cohomology. This product is defined onsections of Z ∞D [ n ] by the formula α ∪ DB β = αβ, deg( α ) = nα ∧ dβ, deg( α ) = 00 , otherwise . (2.22)Note that the grading here is such that the first case is simply multiplication by an integer. In fact,it is obvious from the definition that the Deligne-Beilinson (henceforth DB) cup product composedwith the natural inclusion Z [ n ] ֒ → Z ∞D [ n ]simply multiplies the two locally constant integer-valued functions. Since the sheaf cohomology ofthe locally constant sheaf Z , equipped with this product, is simply the ordinary cohomology ringwith integral coefficients, one immediately sees that this cup product does indeed refine the usualcup product.Equipped with this cup product, b H ∗ ( X ; Z ) becomes an associative and graded-commutativering [Br08]. This cup product structure also refines the wedge product of forms in the sense thatthe curvature map R : b H ∗ ( X ; Z ) → Ω ∗ cl defines a homomorphism of graded commutative rings[Bu12]. In particular this implies that the cup product of two classes of odd degree is flat. It canalso be shown [Bu12] that the cup product of a flat class with any other class is again flat and thatthe inclusion of H ∗ ( X, U (1)) into b H ∗ ( X ; Z ) is a two sided ideal.We now turn to the cup product, viewed as a morphism of stacks. In [FSS13] it was observedthat the lax monoidal structure of the Dold-Kan map gives rise to a cup product, exhibited as amorphism ∪ : B m U (1) conn × B n U (1) conn −→ B n + m +1 U (1) conn (2.23)13f stacks. This map is obtained by simply taking the DB cup product (2.22) ∪ DB : Z ∞D [ n + 1] ⊗ Z ∞D [ m + 1] −→ Z ∞D [ n + m + 2] , applying the Dold-Kan mapDK( ∪ DB ) : DK( Z ∞D [ n + 1] ⊗ Z ∞D [ m + 1]) −→ DK( Z ∞D [ n + m + 2]) , and using the lax monoidal structure ϕ of the map DK to get a map ∪ = DK( ∪ DB ) ◦ ϕ : DK( Z ∞D [ n +1]) × DK( Z ∞D [ n +1]) → DK( Z ∞D [ n +1] ⊗ Z ∞D [ m +1]) → DK( Z ∞D [ n + m +2]) . Applying the stackification functor then gives the desired map. This map then induces a map ofstacks (which we also denote as ∪ ) ∪ : [ X, B n U (1) conn ] × [ X, B m U (1) conn ] −→ [ X, B n + m +1 U (1) conn ] . (2.24)The following two propositions are implicit in [FSS13] [FSS15]. Proposition 5.
The DB cup product refines the singular cup product. That is, we have a commu-tative diagram B n U (1) conn × B m U (1) conn B n + m +1 U (1) conn B n +1 Z × B m +1 Z B n + m +2 Z . ∪ DB I × I ∪ I Proof.
Let p : Z ∞D [ n + 1] → Z [ n + 1] be the projection map Z id (cid:15) (cid:15) i / / Ω / / (cid:15) (cid:15) · · · / / (cid:15) (cid:15) Ω n (cid:15) (cid:15) Z d / / / / · · · / / . Then, by definition of the DB cup product, the diagram Z ∞D [ n + 1] ⊗ Z ∞D [ m + 1] Z ∞D [ n + m + 2] Z [ n + 1] ⊗ Z [ m + 1] Z [ n + m + 2] ∪ DB p ∪ p commutes in sheaves of chain complexes. Applying the Dold-Kan functor and using naturality ofthe lax monoidal structure map gives the result. ✷ roposition 6. The cup product refines the wedge product, and we have a commutative diagram B n U (1) conn × B m U (1) conn B n + m +1 U (1) conn Ω n +1cl × Ω m +1cl Ω n + m +2cl . ∪ DB R × R ∧ R Proof.
Let α and β be sections of Z ∞D [ n + 1] and Z ∞D [ m + 1], respectively. Applying the curvature R to the DB cup product (2.22) gives R ( α ∪ DB β ) = αd ( β ) if deg( α ) = nd ( α ) ∧ d ( β ) if deg( β ) = 00 otherwise , which is R ( α ) ∧ R ( β ). We therefore have a commuting diagram Z ∞D [ n + 1] ⊗ Z ∞D [ m + 1] Z ∞D [ n + m + 2]Ω n +1cl ⊗ Ω m +1cl Ω n + m +2cl . ∪ DB R ∧ R Applying the Dold-Kan map DK gives the result in stacks. ✷ The above results show that, in general, the Deligne-Beilinson cup product does not refine thede Rham wedge product for the whole de Rham complex, but does so only for the top and bottomdegrees. However, for the triple product the only cup products that arise are between degree zeroand degree one cocycles, so that nothing is missed in passing to ∪ DB . We will make this moreprecise in Prop. 17. Massey products in Deligne-Beilinson cohomology are described in [De95] [Sc02] [We00] [MT08]. Inthis section, we review the construction for hypercohomology found in [Sc02], with a slightly adaptedlanguage for later comparison and generalization. In section 3 we generalize this construction in twoways, which we describe. We use the Dold-Kan correspondence to establish these products in thestacky setting. We also use the machinery of May [Ma69] to exhibit these products as differentialmatric Massey products.Let R be a commutative ring and let C • ( n ), n ∈ N , be a sequence of positively graded chaincomplexes of R -modules. Moreover, let us assume that this sequence comes equipped with maps ∪ : C • ( n ) ⊗ C • ( m ) → C • ( n + m ) , which are associative in the sense that ∪ ◦ ( id ⊗ ∪ ) = ∪ ◦ ( ∪ ⊗ id ) . (2.25)15he maps ∪ induce an associative product on cohomology ∪ : H • ( n ) ⊗ H • ( m ) → H • ( n + m ) , called the cup product . Once a well-defined notion of a cup product is established, one can definethe Massey products via the following. Definition 7.
Let l ≥ and let n , · · · , n l and m , · · · , m l be integers. Define n s,t = t X i = s ( n i − and m s,t = t X i = s m i , for 1 ≤ s ≤ t ≤ l , and let ¯ a = ( − q +1 a denote the twist of a class a ∈ C q ( n ) . We define the l -fold Massey product asfollows: (i). Let a i ∈ H m i ( C • ( n i )) be cohomology classes. Suppose there exists cochains a s,t ∈ C m s,t +1 ( n s,t ) such that a i,i is a representative of a i and that da s,t = t − X i = s ¯ a s,i ∪ a i +1 ,t for 1 ≤ s ≤ t ≤ l, ( s, t ) = (1 , l ) . We call the collection M = { a s,t } a defining system for the l -fold Massey product. (ii). The cochain a ,l := l − X i =1 ¯ a ,i ∪ a i +1 ,l ∈ C m ,l +2 ( n ,l ) is a cocycle and represents a cohomology class m l . We call this class the l -fold Massey product ofthe elements a , .., a l with defining system M . In general, we would like to eliminate the dependance of the product on the defining system.The case of l = 3 will be the most important for us, and in this case we are indeed able to eliminatethis dependence. The following three examples are known, and we record them to highlight howMassey products arise in the different settings that we consider, and how stacks will provide, in asense, a unifying theme. Note that, while the above construction is fairly general, it is not obvioushow to generalize to other settings and how to do computations easily with it, and that is why welater use the stacky perspective. Example 3.
Let a , a and a be cohomology classes as above. Suppose we have a defining system M = { a s,t } . This means, by definition, that we have the relations da , = ¯ a , ∪ a , and da , = ¯ a , ∪ a , . Now a class m l representing the Massey product of this defining system has as a representing cocycle a ,l = ¯ a , ∪ a , + ¯ a , ∪ a , . Notice that, in this case, the class m l only depends on the defining system up to cocycles. That is,for another defining system N = { b s,t } , the classes a , − b , and a , − b , are cocycles. Moreover, f these cocycles are coboundaries, then the Massey products of both defining systems agree. We cantherefore define a Massey product, not depending on the defining system, as the quotient h a , a , a i ∈ H m + m + m − ( C • ( n + n + n )) H m + m − ( C • ( n + n )) ∪ a + a ∪ H m + m − ( C • ( n + n )) . Example 4.
Let X be a smooth manifold and let C ( n ) = Ω ∗ ( X ) for each n , where Ω ∗ ( X ) is thealgebra of differential forms on X . Let a, b and c be de Rham cohomology classes of degree p , q , r respectively, such that a ∧ b = 0 = b ∧ c . Choose representing closed forms α , β , γ for a , b , c respectively, and let η and ρ be cochains such that dη = α ∧ β and dρ = β ∧ γ . Then the combination η ∧ γ − ( − ) p α ∧ ρ is a closed form representing the triple Massey product of a , b and c corresponding to the definingsystem M = ( α, β, γ, ρ, η ) . Eliminating the dependence on M gives a well-defined class in thequotient group H p + q + r − ( X ) / (cid:16) a ∪ H q + r − ( X ) + c ∪ H p + q − ( X ) (cid:17) . The following constitutes our initial transition to differential cohomology, which we will developin stacks in the following section.
Example 5.
Consider the Deligne complex given by the sheaf of chain complexes Z ∞D [ n ] := [ Z ֒ → Ω → Ω → ... → Ω n − ] . Let X be a paracompact manifold with good open cover { U i } i ∈ I and let C ( n ) := tot C • ( { U i } , Z ∞D [ n ]) be the total complex of the ˇCech-Deligne double complex. The degree n cohomology of this totalcomplex calculates the differential cohomology of X : ˆ H n ( X ; Z ) = H n (cid:0) tot C • ( U , Z ∞D [ n ]) (cid:1) . The Deligne-Beilinson cup product is defined as a morphism ∪ DB : Z ∞D [ n ] ⊗ Z ∞D [ m ] −→ Z ∞D [ n + m + 1] , which on sections is given by the formula (2.22) . This map induces cup product morphisms on thetotal complexes C ( n ) which are associative in the sense of the identity in (2.25) . We can therefore use this cup product to define the Massey product in differential cohomology,viewed as the sheaf hypercohomology of the Deligne complex. Since our point of view will subsumethis construction, we will delay explicit examples until sections 3 and 4.
We provide our main construction of stacky Massey products in this section. We start with settingup the machinery needed. 17 .1 The Dold-Kan correspondence
The Dold-Kan correspondence will be an important component in defining the Massey product instacks. We will use the correspondence to organize the homotopies involved in certain homotopycommuting diagrams in an algebraic way.The classical Dold-Kan correspondence describes an equivalence of categories (see e.g. [GJ09])Γ : C h + s A b : N (3.1)between positively graded chain complexes and simplicial abelian groups. By post-composing withthe free-forgetful adjunction, one obtains an adjunctionDK := U Γ : C h + s S et : N F . (3.2)In fact, one can say more. This adjunction is a Quillen adjunction of model categories, with theprojective model structure on chain complexes and the Quillen model structure on simplicial sets.As such, it preserves the homotopy theories in both categories; it therefore comes as no surprisethat for a positively graded chain complex C • one has an isomorphism H n C • ≃ π n DK( C • ) . (3.3)For convenience, we remind the reader what the functor DK does to a chain complex, as this willbe a frequently used tool in producing abelian stacks.Let ∆ denote the category of linearly ordered sets of n elements with order preserving maps.Let C • be a positively graded chain complex. The degree n component of the simplicial abeliangroup DK( C • ) is given by DK( C • ) n = M [ n ] ։ [ k ] C k . Here the indexing set is taken to be all surjections [ n ] ։ [ k ]. It is a bit trickier to describe theface and degeneracy maps. Let d i : [ n − ֒ → [ n ] be a coface map in ∆. We want to define thecorresponding face map. To get a map out of the direct sum, it suffices to describe the map on eachfactor. Therefore, we need only define the face map on a term C k given by a surjection σ : [ n ] ։ [ k ].To see where to send this term, we form the composite σd i [ n − ֒ → [ n ] ։ [ k ]. Now this morphismneed not be surjective, so we factorize µσ ′ [ n − ։ [ m ] ֒ → [ k ] where the first map is a surjection andthe second map is an injection. Then σ ′ corresponds to a term C m ֒ → L [ n − → [ m ] A m = DK( C • ) n − .We send the factor C k to the factor C m by a map µ ′ : C k → C m . This map is given by µ ′ = id , µ = id , ( − k d, µ = d k , , otherwise. (3.4)A similar construction is used to define the degeneracy maps. The following example illustratesthe point quite well. Example 6.
Consider the chain complex A [1] , with the abelian group A in degree and ’s inall other degrees. We want to compute DK( A [1]) . Using the above formula, we see that the only onzero terms in degree n are given by the surjections [ n ] ։ [1] . Each surjection can be thought ofas being given by an element i ∈ [ n ] which divides the set into two subsets: those that go to andthose that go to . We therefore have n surjections and DK( A [1]) n = n M i =1 A .
For a coface map d j : [ n − → [ n ] , the corresponding face map d j is given as follows. Let A i denote the copy of A corresponding to the i th surjection. Then d j ( A i ) = (cid:26) A i − if i > j = 0 , nA i if i ≤ j = 0 , n , d ( A i ) = (cid:26) A i − if i = 00 if i = 0 , d n ( A i ) = (cid:26) A i if i = n if i = n. Notice that for j = 0 , n , the term corresponding to i = j and i = j + 1 both go to the same copy of A . We therefore have a map A × A → A extending the identity on each component. Hence, thismorphism is just group multiplication. From this, one can see that this simplicial abelian group isjust the delooping group BA . Another way to describe the simplicial set DK( C • ), which is perhaps more conceptual, is via alabeling of simplices with elements of the chain complex C • . A 2-simplex in DK( C • ), for example,is a simplex with face, edges and vertices labeled by elements of C • a a a c b b b such that dc = b + b − b and db ij = a j − a i . Here d is the chain complex differential. Notice that a 2-simplex in DK( C • ), defined as before, canbe identified as such a labeled simplex. To see this, let us calculate the data involved in specifying a2-simplex. First, observe that there is exactly one surjection 0 : [2] ։ [0], id : [2] ։ [2], and exactlytwo surjections σ i : [2] ։ [1]. Therefore, a 2-simplex is given by a quadruple ( a , b , b , c ),where a is in degree 1, while b , b are in degree 2 (corresponding to σ , σ , respectively), and c is in degree 3. To determine the edges, we evaluate d i on this quadruple. For i = 0, we havethe following epi-mono factorizationsid ◦ d = d ◦ id , σ ◦ d = d ◦ , σ ◦ d = id ◦ id . It follows from the formula, that the 0 face is b . For i = 1, we haveid ◦ d = d ◦ id , σ ◦ d = id ◦ id , σ ◦ d = id ◦ idand the 1 face is b + b . Finally, for i = 2, we haveid ◦ d = d ◦ id , σ ◦ d = id ◦ id , σ ◦ d = d ◦ dc + b . Forming the boundary of the simplex, we get ∂ ( a , b , b , c ) = b − ( b + b ) + ( dc + b ) = dc . That the edges of the simplex satisfy the second condition above is a straightforward calculationand will be omitted. In fact, it is a straightforward calculation to show that the boundary of ageneral n -simplex must be equal to d applied to the labeling on its n -face. Remark 5.
This second description provides a powerful conceptual advantage; namely, that thedifferential of the chain complex can be viewed as obstructing the chain from being a cycle. Forexample, if the resulting simplicial set were the nerve of a groupoid, then all simplices for n ≥ would be cycles. We are now ready to define Massey products in the category of stacks. We begin with a discussionon Massey triple products and then generalize to l -fold Massey products.The Massey triple product can be viewed as a homotopy built out of the associativity diagramof the cup product of three elements. In fact, suppose one is given a triple of higher gerbes withconnection on a manifold X . These gerbes are given by the data G i : X → B n i U (1), i = 1 , , G ∪ G and G ∪ G are homotopic to 0, with trivializing homotopies φ , and φ , . In this case, we can build a loop trivializing the triple product. To see this, consider the associativity diagram for the cup product B n U (1) conn × B n U (1) conn × B n U (1) conn B n + n +1 U (1) conn × B n U (1) conn B n U (1) conn × B n + n +1 U (1) conn B n + n + n +2 U (1) conn X ( ∪ × id )( id × ∪ ) ∪∪G × G × G Although the outer two maps agree, there is still nontrivial homotopy theoretic informationcontained in the diagram. To see this, suppose G ∪G and G ∪G are trivializable with trivializations20 , and φ , , respectively. Then we can add these homotopies to the diagram B n U (1) conn × B n U (1) conn × B n U (1) conn B n + n +1 U (1) conn × B n U (1) conn B n U (1) conn × B n + n +1 U (1) conn B n + n + n +2 U (1) conn X B n U (1) conn × B n U (1) conn B n U (1) conn × B n U (1) conn B n + n +1 U (1) conn B n + n +1 U (1) conn φ , φ , = ⇒ = ⇒ = ⇒ = ⇒ G ∪ φ , φ , ∪ G ( ∪ × id )( id × ∪ ) ∪∪G × G × G G × G G × G G ∪ G G ∪ G id × × id0000 0 These two choices of homotopies φ , and φ , make the entire diagram homotopy commutative,as the triple cup products (the two red arrows) are trivialized by the homotopies G ∪ φ , and φ , ∪ G . Since the cup product is strictly associative, the diagram in red commutes and we havethe homotopy commuting diagram B n + n + n +2 U (1) conn . X = ⇒ = ⇒ G ∪ φ , φ , ∪ G G ∪ G ∪ G (3.5)These two homotopies fit together to form a loop. Then, by Prop. 3 and the universal property ofthe homotopy pullback, we can equivalently describe this as a map X −→ Ω B n + n + n +2 U (1) conn ≃ ♭ B n + n + n +1 U (1) . (3.6) Lemma 8.
The homotopy class of the loop (3.6) is in the image of the inclusion of the group H n + n + n +1 ( X ; U (1)) into b H n + n + n +2 ( X ; Z ) . Proof.
Using the Dold-Kan adjunction along with a ˇCech resolution of X , we have the followingsequence of isomorphisms π Map(
X, ♭ B n + n + n +1 U (1)) ≃ H hom C h + ( N ( C ( { U i } ) , disc( U (1))[ n + n + n + 1]) ≃ H n + n + n +1 ( X ; U (1)) ֒ → ˆ H n + n + n +2 ( X ; Z ) . U (1) (i.e. this is the sheaf of locallyconstant U (1) valued functions). ✷ Remark 6. (i)
Notice that we could have equivalently taken the homotopy class of the loop directlyto get an element [ G ∪ φ , − φ , ∪ G ] in π Map( X, B n + n + n +2 U (1) conn ) ≃ H hom C h + (cid:0) C ( { U i } ) , Z ∞D [ n + n + n + 2] (cid:1) ≃ H n + n + n +1 ( X ; U (1)) ֒ → b H n + n + n +2 ( X ; Z ) . (ii) The above observations allow us to recover the usual definition of the Massey product as anelement in cohomology. In section 2.4, we observed that such a class is not completely well-definedpurely at the level of cohomology and there was some dependence on the chosen cochain representa-tives. Taking this point of view, one can see this dependence as a choice of trivializations φ , and φ , of the cup products. This definition works well for the triple product and gives a clear picture on how the tripleproduct is built out of the homotopies. However, to describe the higher triple products this waywould be cumbersome. Moreover, the algebraic nature of the products would not be transparent.For these reasons, we will use the language of simplicial homotopy theory to describe these homotopycommuting diagrams and the Dold-Kan correspondence to organize these homotopies in an algebraicway. To prepare the reader for this perspective, we first recast the triple product in this language.Notice that the triple product was described by two homotopies φ , and φ , connecting thebasepoint 0 to the double cup products. We can express this situation diagrammatically via thehorn-fillers ∂ ∆[1] (cid:2) X, B n + n +1 U (1) conn (cid:3) , ∆[1] (0 , G ∪ G ) φ , ∂ ∆[1] (cid:2) X, B n + n +1 U (1) conn (cid:3) . ∆[1] (0 , G ∪ G ) φ , Now we would like to use these homotopies to construct a loop. To do this, we need to manipulatealgebraically these homotopies. This motivates us to take the Moore complex of these diagrams inorder to translate the data into the language of sheaves of chain complexes. This gives the data Z ⊕ Z N (cid:0) [ X, B n + n +1 U (1) conn ] (cid:1) Z N (cid:0) [ X, B n + n +1 U (1) conn ] (cid:1) , (0 , G ∪ G ) (1 , − ∂φ , Z ⊕ Z N (cid:0) [ X, B n + n +1 U (1) conn ] (cid:1) Z N (cid:0) [ X, B n + n +1 U (1) conn ] (cid:1) , (0 , G ∪ G ) (1 , − ∂φ , where the subindices indicate the degree of the chain complex. Now we can represent these chain22omotopies succinctly in the upper triangular matrix A = G φ , ∗ G φ , G . By construction, this matrix satisfies the Maurer-Cartan equation dA − A · A = µ ( A ) ∈ Ker(A) . Moreover, µ ( A ) is of the form µ ( A ) = τ . Applying the differential d to τ and using the Leibniz rule, we get d ( τ ) = d ( G ∪ φ , − φ , ∪ G )= d ( G ) ∪ φ , + G ∪ d ( φ , ) − d ( φ , ) ∪ G + φ , ∪ d ( G )= G ∪ ( G ∪ G ) − ( G ∪ G ) ∪ G = 0 . At the level of sheaf hypercohomology, we have the following:
Proposition 9.
The cohomology class of the matrix cocycle µ ( A ) is the element [ µ ( A )] = [ G ∪ φ , − φ , ∪ G ] ∈ H n + n + n +1 ( X ; U (1)) . Proof.
We have the following sequence of isomorphisms[ µ ( A )] = [ G ∪ φ , − φ , ∪ G ] ∈ H hom C h + (cid:0) C ( U ) , B n + n + n +2 U (1) conn (cid:1) ≃ π Map (cid:0) C ( U ) , B n + n + n +2 U (1) conn (cid:1) ≃ π Map (cid:0) C ( U ) , ♭ B n + n + n +1 U (1) (cid:1) ≃ H n + n + n +1 ( X ; U (1)) . ✷ We would like to utilize the machinery of May [Ma69] which makes use of matrices. We willintroduce stacks labelled by two integers, which will be indexing the entries of the correspondingmatrices. To that end, let R ij , i, j ∈ N , be simplicial abelian stacks equipped with maps ∪ : R ij ⊗ R jk −→ R ik , which are associative in the sense that ∪ ◦ ( ∪ ⊗ id) = ∪ ◦ (id ⊗ ∪ ).23 emark 7. Let N denote the normalized Moore functor. It follows from the definition of thedifferential on the tensor product that the induced product ˜ ∪ : N ( R ij ) ⊗ N ( R jk ) ∼ −→ N ( R ij ⊗ R jk ) −→ N ( R ik ) must satisfy the Leibniz type rule d ( α ∪ β ) = d ( α ) ∪ β + ( − deg α ∪ d ( β ) on sections. We can now utilize an extension of the machinery of May [Ma69] locally to define the refinedmatric Massey products in our setting. To this end, we consider the set of all upper triangularhalf-infinite matrices M ( R ) = S n M ( R ) n , where (cf. (2.1)) M ( R ) n = { A = ( a ij ) | a ij ∈ N ( R ij ) , a ij = 0 for j ≤ i and i, j ≥ n + 1 for some n ∈ N } (3.7)is the subalgebra of n × n matrices. Notice that, with our definition, this set possesses morestructure. It becomes a sheaf of DGA’s with product given by matrix multiplication and differentialgiven by applying the differential on N ( R ij ) to each entry of the matrix. Just as in the case ofclassical Massey products, we have a filtration of presheaves of subalgebras M ( R ) ⊂ M ( R ) ⊂ . . . ⊂ M ( R ) n ⊂ . . . (3.8)and a bigrading M ( R ) = X p ≥ ,k ≥ M p,k , (3.9)where M p,k = span a i,i + p ; a i,i + p ∈ N ( R i,i + p ) . (3.10)We can define the following notions similarly to the classical case. Definition 10.
Let A be a matrix in M ( R ) . We define the (stacky version) of the Maurer-Cartanequation as dA − A · A ≡ A ) , and call a solution a formal connection with curvature µ ( A ) = dA − A · A .
We are now ready to define the stacky Massey product with a product on the bigraded sequenceof stacks.
Definition 11.
Let R = {R ij } be a sequence of abelian stacks equipped with maps ∪ : R ij ⊗ R jk −→ R ik , which satisfy ∪ ◦ (id ⊗ ∪ ) = ∪ ◦ ( ∪ ⊗ id) . Let A be a formal connection with curvature µ ( A ) . Thenthe entries of the hypercohomology class [ µ ( A )] are called stacky Massey products . emark 8. The following examples of stacks satisfy the compatibility requirement of Def. 11and will be of particular interest to us. They are the mapping stacks corresponding to the stacksdescribed in Remark 3. Fix a manifold X and a sequence ( n i,j ) , i < j ≤ n , of integers satisfying n i,j + n j,k = n i,k ; (i) The stacks [ X, B n i,j − U (1) conn ] of higher bundles with connection, with the stacky cup productand ˇCech-Deligne differential. (ii) The stacks [ X, B n i,j Z ] of higher bundles, with the usual cup product and singular differential. (iii) The stacks [ X, ♭ dR B n i,j U (1) conn ] of differential forms of degrees ≤ n , with the wedge productand exterior derivative. We highlight the power of the above definitions in the following examples, where we are able todescribe all three of the differential, singular, and de Rham triple products.
Example 7. (Differential triple product)
Let G i , i = 1 , , , be bundles corresponding to mor-phisms ∆[0] → [ X, B n i,i +1 − U (1) conn ] . Suppose G ∪ G and G ∪ G represent trivial classes in π Map( X, B n i,j − U (1) conn ) . Choose a defining system A = G φ , ∗ G φ , G , where φ , and φ , are nondegenerate 1-simplices trivializing the cup products. Then the curvatureof A is µ ( A ) = τ , and the hypercohomology class [ τ ] is [ G ∪ φ , − φ , ∪ G ] . The latter is an element in π Map( X, B n , − U (1) conn ) ≃ π Map(
X, ♭ B n , − U (1)) ≃ H n , − ( X ; U (1)) , where we have n , = n , + n , = n , + n , + n , . Example 8. (Singular triple product)
Let X be a manifold, and let | X | be the topological spacedenoting its geometric realization. Let a i : | X | → K ( Z , n i,i +1 ) ≃ B n i,i +1 Z , i = 1 , , , be singularcochains with cup products vanishing in cohomology. Choose a defining system A = a f , ∗ a f , a . Since geometric realization is a left ∞ -adjoint the discrete stack functor disc [Sc13], these areequivalently given by maps of stacks ¯ a i : ∆[0] −→ [ X, B n i,i +1 Z ] , nd homotopies ¯ f i,i +1 : ∆[1] −→ [ X, B n i,i +2 Z ] trivializing the cup products, hence a defining system A = a ¯ f , ∗ a ¯ f , a . The hypercohomology class of the entry τ ∈ µ ( A ) is given by (cid:2) ¯ a ∪ ¯ f , − ¯ f , ∪ ¯ a (cid:3) , which is anelement in π Map( X, B n , Z ) ≃ π Map( | X | , K ( Z , n , )) ≃ π Map( | X | , K ( Z , n , − ≃ H n , − ( X, Z ) . Example 9. (de Rham triple product)
Let X be a manifold and let α i , i = 1 , , , be closedforms in different degrees. These forms are equivalently given by maps α i : ∆[0] −→ ♭ dR B n i,i +1 U (1) conn . Suppose that the wedge products α ∧ α and α ∧ α are trivial in cohomology. Then we can choosea defining system via A = α η , ∗ α η , α , where η , and η , are 1-simplices. The hypercohomology class of the entry τ ∈ µ ( A ) is given by [ α ∧ η , − η , ∪ α ] . The sheaf at each level in the complex Ω ≤ n , is acyclic (the sheaves are that of differential formsand so admit a partition of unity). Thus, we can calculate the hypercohomology as π Map( X, Ω ≤ n , ) ≃ H Ω ≤ n , ( X ) ≃ H n , − ( X ) . Our main result in this section relates Massey products for Deligne cocycles to correspondingones for higher bundles in the stacky sense.
Theorem 12.
Let ˆ a i , ≤ i ≤ l , be Deligne cocycles. Suppose the l -fold Massey product is defined.Let G i , ≤ i ≤ l , be n i,i +1 - bundles with connections G i : X −→ B n i,i +1 U (1) conn , representing the Deligne cocycles. Then there is a natural bijection between corresponding Masseyproducts hG , G , . . . , G l i ≃ h ˆ a , ˆ a , . . . , ˆ a l i . roof. Recall that B n U (1) conn := Γ( Z ∞D [ n + 1]) (see [FSS12]). Using the definition of the stackyhom, the fact that the counit ǫ : N Γ → id is a natural isomorphism and the lax monoidal structureon N , we have a homotopy equivalence for each test object U , N ([ X, B n U (1) conn ])( U ) = N ([ X, B n U (1) conn ]( U )) ≃ N (Map( C ( { U i } ) × U, B n U (1) conn )) ≃ hom C h + ( N ( C ( { U i } ) ⊗ N ( U ) , Z ∞D [ n + 1]) ≃ hom C h + ( N ( C ( { U i } ) , Z ∞D [ n + 1]( U )) ≃ C ( X, Z ∞D [ n + 1])( U ) , where the last line denotes the ˇCech resolution of the Deligne complex Z ∞D [ n + 1]. Hence, a definingsystem in the stacky sense is naturally equivalent to a defining system in the sense of [Sc02]. Sincethe set of Massey products is parametrized by the set of defining systems, it follows that indeed wehave a natural bijection hG , G , . . . , G l i ≃ h ˆ a , ˆ a , . . . , ˆ a l i . ✷ We will now consider properties of the stacky Massey products. Our setting allows for these to bequite attractive and natural. The most immediate of those are direct generalizations of classicalones. Later in this section we will see properties that are more peculiar to the differential setting.Among the properties that the classical Massey products satisfy are the following (see [Ma69][Kr66]): (i)
Dimension:
The dimension of h x , x , · · · , x n i is P deg( x i ) − n + 2. (ii) Naturality: If f : X → Y is a continuous map and y · · · , y k ∈ H ∗ ( Y ; R ) such that the k -foldMassey product h y , y , · · · , y k i is defined, then h x , · · · , x k i = h f ∗ ( y ) , · · · , f ∗ ( y k ) i is definedas a Massey product on the cohomology of X and f ∗ ( h y , · · · , y k i ) ⊂ h f ∗ ( y ) , · · · , f ∗ ( y k ) i . (iii) Definedness:
The vanishing of the the lower Massey products is only a necessary condition forthe k -fold Massey product to be defined for k >
3. For k = 3 the condition is both necessaryand sufficient. (iv) Slide relation:
If the Massey product h x , x , . . . , x n i is defined, then so is h x , x , . . . , rx i , . . . x n i for any r ∈ R . Moreover we have the relation r h x , x , . . . , x n i ⊂ h x , x , . . . , rx i , . . . x n i . These indeed extend to the stacky version.
Proposition 13.
The stacky Massey products satisfy the following properties: (i)
Dimension:
The dimension of hG , G , · · · , G l i is P deg( G i ) − l + 2 . ii) Naturality : If f : X → Y is a smooth map between manifolds and G · · · , G k ∈ ˆ H ∗D ( X ; Z ) suchthat the k -fold Massey product hG , G , · · · , G k i is defined, then hG , · · · , G k i = h f ∗ ( G ) , · · · , f ∗ ( G k ) i is defined as a Massey product on the differential cohomology of X and f ∗ ( hG , · · · , G k i ) ⊂ h f ∗ ( G ) , · · · , f ∗ ( G k ) i . (iii) Definedness : The vanishing of the the lower Massey products is only a necessary condition forthe k -fold Massey product to be defined for k > . For k = 3 the condition is both necessaryand sufficient. (iv) Slide relation:
If the Massey product hG , G , . . . , G n i is defined, then so is hG , G , . . . , m G i , . . . G n i for any m ∈ Z . Moreover we have the relation m hG , G , . . . , G n i ⊂ hG , G , . . . , m G i , . . . G n i . Proof.
Part 1 follows immediately from the definition. To prove part 2, note that the functor[ − , R ] is contravariant, sending a map f : X → Y to its pullback f ∗ : [ Y, R ij ] −→ [ X, R ij ] . Since the cup product is natural with respect to pullbacks, the induced morphism f ∗ : N ([ Y, R ij ]) → N ([ X, R ij ]) descends to a morphism of sheaves of DGA’s f ∗ : M ([ Y, R ij ]) −→ M ([ X, R ij ]) . It follows that if A is a formal connection in M ([ Y, R ij ]), then f ∗ ( A ) is a formal connection in M ([ X, R ij ]) satisfying the equation: df ∗ ( A ) − f ∗ ( A ) · f ∗ ( A ) = f ∗ ( µ ( A )) ∈ ker( f ∗ ( A )) . By definition of the k -fold Massey product, the claim follows. For part 3, we will show that for k = 3 the condition is both necessary and sufficient. From the proof, it will be clear that thiscannot be the case for higher products. Let G , G and G be bundles and suppose the tripleproduct hG , G , G i is defined. Then we have trivializations φ , and φ , such that dφ , = G ∪ G and dφ , = G ∪ G . Hence, both cup products are trivial. For the converse, it is clear that if both cup products aretrivial in cohomology, we can choose trivializing homotopies and form the Massey triple product.For higher products, the higher trivializations depend on the lower ones. In fact, for the fourfoldproduct, choose trivializations φ , , φ , and φ , of the cup products such that G ∪ φ , − φ , ∪ G ∈ hG , G , G i is trivializable. Then for the fourfold Massey product to be defined, the other triple product G ∪ φ , − φ , ∪ G ∈ hG , G , G i hG , G , G i contains 0. Finally, for part 4,let A be a formal connection of the form A = G φ , . . . ∗ G φ , . . . . . . . . . . . . G i − φ i − ,i G i φ i,i +1 G i +1 . . . φ n − ,n − G n − φ n − ,n G n . Then the matrix˜ A = G φ , . . . . . . mφ ,i . . . ∗ G φ , . . . . . . . . . . . . mφ i − ,i . . . G i − mφ i − ,i mφ i − ,i +1 . . .m G i mφ i,i +1 mφ i,i +2 . . . mφ i,n G i +1 . . .. . . φ n − ,n − G n − φ n − ,n G n is also a formal connection: that is, a defining system for the Massey product hG , . . . , m G i , . . . , G n i .Indeed, let us write the matrix A as a block matrix A = A A A . Then the second matrix can be written˜ A = A mA A . Now the Maurer-Cartan equation for ˜ A reads µ ( ˜ A ) = dA mdA dA − A mA A A mA A = dA mdA dA − A A m ( A A + A A )0 A A .
29e would like to show that µ ( ˜ A ) is in ker( ˜ A ). Since A satisfies the Maurer-Cartan equation up toan element in the kernel ker( A ) = . . . ∗ . . . (3.11)we must have dA = A · A and dA = A · A . Since A is a formal connection, we must also have µ ( A ) = dA − ( A A + A A )where µ ( A ) is the upper right block of µ ( A ) of dimension dim( A ). Since the only nonzero termof µ ( A ) is the cochain representative of the Massey product τ , located in the upper right corner of µ ( A ), we have that µ ( ˜ A ) = mdA − m ( A A + A A ) = mµ ( A ) has one nonzero element σ = mτ in the upper right corner. Therefore, ˜ A is indeed a formalconnection and, at the level of cohomology, the only nonzero term of the class [ µ ( A )] is [ σ ] = m [ τ ].Since [ τ ] was chosen to be an arbitrary element of the Massey product hG , . . . , G n i , we have m hG , . . . , G n i ⊂ hG , . . . , m G i , . . . G n i . ✷ We now discuss the relationship between the stacky Massey product and the singular Masseyproduct. The following parametrizes how forgetting the differential data on the Massey productis not quite the same as taking the Massey product of cohomology classes after forgetting thedifferential data on these.
Proposition 14.
Let G i : ∆[0] → [ X, B n i,i +1 U (1) conn ] , ≤ i ≤ l , be higher bundles on X withdefined Massey product. Then precomposition with the forgetful morphism I : B n U (1) conn −→ B n +1 Z , induced by the map Z (cid:15) (cid:15) i / / Ω (cid:15) (cid:15) / / · · · (cid:15) (cid:15) / / Ω n − d (cid:15) (cid:15) Z d / / / / · · · / / , yields singular cocycles with defined Massey product. Furthermore, we have I hG , G , G i ⊂ h I ( G ) , I ( G ) , I ( G ) i . Proof.
For simplicitiy, we denote the sheaf of matrix algebras M diff := M ([ X, B ∗ U (1) conn ]) M sing := M ([ X, B ∗ Z ])30ccording to the corresponding cohomology theories for these matrices. It is clear by definitionthat I respects the cup product structure, hence I induces a morphism of sheaves of DGA’s I ∗ : M diff → M sing . It follows immediately from the definition of the Maurer-Cartan equationDef. 10, that formal connections are sent to formal connections. Then passing to hypercohomologygives the result. ✷ Remark 9. (i)
It follows from the proposition that if the classical Massey product h I ( G ) , I ( G ) , I ( G ) i is zero then certainly the left hand side is zero, i.e. hG , G , G i is in the kernel of the forgetfulmorphism I . From the sequence Ω n − / Im( d ) → ˆ H n I −→ H n we have that hG , G , G i will be an ( n − -form. However, it is important to note that this is not quite the ( n − -form given by theclassical Massey product. (ii) A related question is to ask whether the differential Massey product completely refines thesingular Massey product. That is: do we have a bijection , I hG , G , G i ≃ h I ( G ) , I ( G ) , I ( G ) i ? Unfortunately, this cannot be possible. Essentially, this is because the map I ∗ : M diff → M sing hasa nontrivial kernel. Hence we cannot expect the Maurer-Cartan equation to hold after refining. (iii) However, this does help explain the nature of differential Massey products. In fact, since theseproducts are always flat, it follows from diagram (2.7) that if the refinement of a singular formalconnection is again a formal connection, then the singular Massey product must have been torsion. We will show that the failure of the refinement to satisfy the Maurer-Cartan equation can bemeasured by the de Rham Massey product.
Lemma 15.
Let F ij → R ij ։ S ij be a fibration sequence of abelian prestacks for each i and j .Suppose, moreover, that we have commuting diagrams F ij ⊗ F jk F ik R ij ⊗ R jk R ik S ij ⊗ S jk S ik . ∪∪∪ i ⊗ ip ⊗ p ip Then the induced sequence → M ( F ) → M ( R ) ։ M ( S ) → is a short exact sequence of DGA’s. Proof.
Since the normalized Moore functor is right Quillen and preserves equivalences, it follows31hat it sends fiber sequences to fiber sequences. Hence, we have a diagram N ( F ij ) ⊗ N ( F jk ) N ( F ik ) N ( R ij ) ⊗ N ( R jk ) N ( R ik ) N ( S ij ) ⊗ N ( S jk ) N ( S ik ) , ∪∪∪ i ⊗ ip ⊗ p ip where the right hand side is a short exact sequence of presheaves of chain complexes. By definition,it follows that we have a short exact sequence0 → M ( F ) → M ( R ) ։ M ( S ) → ✷ It follows from the lemma along with diagram (2.19), that there is a short exact sequence ofpresheaves of bigraded rings0 −→ M ([ X, Ω ≤∗ ]) −→ M diff −→ M sing −→ . (3.12)Hence, M form := M ([ X, Ω ≤∗ ]) is a two-sided ideal in M diff .Now, by definition of ker( A ) along with the above observation, we haveker( ˆ A ) ⊂ \ ker( A ) , where b denotes a choice of differential refinement. In fact, for a matrix C ∈ ker( ˆ A ) and C ′ ∈ \ ker( A ′ ), we have that the difference C − C ′ = B ∈ M form . It is this lack of commutativity betweentaking kernels and taking differential refinements that leads to a nontrivial structure than mightotherwise be anticipated.Summarizing the previous observations gives the following theorem. Theorem 16.
Let A be a formal connection for M sing , and let ˆ A be a differential refinement of A with µ ( A ) a solution to the Maurer-Cartan equation. Then any differential refinement [ µ ( A ) satisfies the twisted Maurer-Cartan equation [ µ ( A ) = d ˆ A − ˆ A · ˆ A ≡ B mod ker( ˆ A ) , (3.13) where B is some matrix in the ideal M form . Proof.
Since A is a formal connection, µ ( A ) satisfies µ ( A ) = dA − A · A ≡ A ) . [ µ ( A ) = D ˆ A − ˆ A · ˆ A ≡ \ ker( A ) , where D = d + ( − ∗ δ is the ˇCech-Deligne differential on M diff . Now by sequence (3.12), we seethat this is equivalent to existence of a matrix of forms B satisfying (3.13). ✷ In general, the Deligne-Beilinson cup product does not refine the de Rham wedge product forthe whole de Rham complex, but does so only for the top and bottom degrees, as we have seen inProp. 5 and Prop. 6. However, for the triple product the only cup products that arise are betweendegree zero and degree one cocycles, so that nothing is missed in passing to ∪ DB . Consequently, forthe case of the triple product, the matrix B in the above example encodes the information neededto define the de Rham Massey product. More precisely, we have the following. Proposition 17.
Let a i ∈ H ∗ ( X, Z ) , i = 1 , , , and let ι ( a ) i ∈ H ∗ dR ( X ) denote the inclusions intode Rham cohomology. Let A = a φ , ∗ a φ , a . in M sing be a matrix of singular cochains defining a formal connection and let µ ( A ) be the corre-sponding solution to the corresponding Maurer-Cartan equation. Then for any differential refine-ment [ µ ( A ) of µ ( A ) , the curvature R ( [ µ ( A )) is a de Rham Massey product in h ι ( a ) , ι ( a ) , ι ( a ) i . If,in addition, [ µ ( A ) is a solution to the differential Maurer-Cartan equation, then R ( [ µ ( A )) = 0 and µ ( A ) represents a torsion class. Proof.
Let a i , i = 1 , ,
3, be singular cochains of degree n i,i +1 . Suppose that the triple product isdefined, and choose a defining system A = a φ , ∗ a φ , a . Let ˆ A = a ˆ φ , ∗ a ˆ φ , a be a refinement. Then we know that the refinement [ µ ( A ) satisfies the equation D ˆ A = ˆ A · ˆ A + B up to some element in ker( ˆ A ). Explicitly, letting B = ( η ij ), we haveˆ A = D ˆ φ , ∗ D ˆ φ , = η ˆ a ∪ ˆ a + ˆ η ˆ a ∪ ˆ φ , − ˆ φ , ∪ ˆ a η ˆ a ∪ ˆ a + ˆ η η . A ) forces the equations η = 0 , D ˆ φ , = ˆ a ∪ ˆ a + η ,η = 0 , D ˆ φ , = ˆ a ∪ ˆ a + η . At the level of connections, the data provided by the right two equations reduces to dφ , = b ∧ a + η (3.14) dφ , = b ∧ a + η , (3.15)where b and b are forms representing the connections with curvatures a and a .Now forming [ µ ( A ) gives the matrix [ µ ( A ) = a ∪ ˆ φ , − ˆ φ , ∪ ˆ a . Finally, applying the curvature map R to the only nonzero term gives R (cid:16) ˆ a ∪ ˆ φ , − ˆ φ , ∪ ˆ a (cid:17) = R (ˆ a ) ∧ R ( ˆ φ , ) − R ( ˆ φ , ) ∧ R (ˆ a )= R (ˆ a ) ∧ R (ˆ a ∪ ˆ a ) + R (ˆ a ) ∧ η − ( η ∧ R (ˆ a ) + R (ˆ a ∪ ˆ a ) ∧ R (ˆ a ))= R (ˆ a ∪ (ˆ a ∪ ˆ a )) + R (ˆ a ) ∧ η − η ∧ R (ˆ a ) − R ((ˆ a ∪ ˆ a ) ∪ ˆ a )= R (ˆ a ) ∧ η − η ∧ R (ˆ a )= a ∧ η − η ∧ a . Notice that it follows from equations (3.14) and (3.15) that the last line represents a de RhamMassey product (simply apply d to both sides of those equations). This proves the first claim.For the second, observe that if ˆ µ solves the Maurer-Cartan equation, then we can choose B = ( η ij ) = 0, and the curvature calculated above must vanish. ✷ We will discuss our applications in this section, both from geometry and mathematical physics.We will show how Massey products arise in various settings, both classically and then in the newlyconstructed stacky form.
In this section we will consider Massey products arising from characteristic classes, hence associatedwith bundles or (higher) abelian gerbes. The refined Massey products will be associated withbundles or (higher) abelian gerbes together with connections on them. We consider examplesinvolving the Deligne derivative D , which in the setting of the ˇCech-Deligne double complex, isgiven by D = d + ( − k δ . 34 xample 10. Let π : E → M be a vector bundle equipped with connection ∇ . Let ˆ c ( E, ∇ ) be theˇCech-Deligne cochain representing the differential refinement of the charateristic form correspondingto the connection (see [Bu12]). Suppose that ˆ c ( E, ∇ ) is trivializable as a ˇCech-Deligne cochain andthat moreover that there are cochains ˆ a and ˆ b such that ˆ c ( E, ∇ ) = ˆ a ∪ ˆ b . Since the class of ˆ c ( E ) vanishes in differential cohomology, there is a ˇCech-Deligne cocycle ˆ A , with curvature A , such that D ˆ A = ˆ c ( E, ∇ ) = ˆ a ∪ ˆ b. (4.1) It was shown by Gomi [Go08] that for a differential cohomology classes ˆ a of odd degree n , wehave the formula [ˆ a ∪ ˆ a ] = ji ( Sq n − (¯ a )) . (4.2) Here, i is the map on cohomology induced via the representation as the square roots of unity i : Z / → U (1) (see [Go08][Bu12]), and j denotes the inclusion into differential cohomology via themap in diagram (2.7) which raises the degree by 1. Let us assume that a is divisible by 2 so thatthe mod 2-reduction is trivial and choose a trivializing ˇCech cochain φ . Write ϕ = ji ( φ ) . In thiscase, (4.2) implies the equation D ( ϕ ) = jiδ ( φ ) = ˆ a ∪ ˆ a . (4.3) Now the following matrix organizes the defining system given by equations (4.1) and (4.3) : a ϕ a ˆ A b . Then an element of the Massey product h ˆ a, ˆ a, ˆ b i is given by the class of the ˇCech-Deligne cochain ˆ a ∪ ˆ A − ϕ ∪ ˆ b , which is an element in b H ( E ; Z ) . The previous example can be generalized to higher Chern classes.
Example 11.
Let E → M be a vector bundle with connection ∇ . Suppose that at the level ofˇCech-Deligne cochains, we have ˆ c n − ( E, ∇ ) = ˆ a n − ∪ ˆ b n − and D ˆ A (4 n − = ˆ c n − ( E, ∇ ) = ˆ a n − ∪ ˆ b n − , (4.4) so that ˆ c n − ( E, ∇ ) is trivializable as a bundle equipped with connection. We also assume that Sq n − (¯ a n − ) = 0 , where ¯ a is the mod 2 reduction of a . Then, as in example 10 we have ˆ a n − ∪ ˆ a n − = Dϕ , for some cochain ϕ [Go08]. We have D ( ϕ ) = ji ( Sq n − (¯ a n − )) = ˆ a n − ∪ ˆ a n − . (4.5) Now the following matrix organizes the defining system given by equations (4.4) and (4.5) : a n − ϕ a n − ˆ A (2 n − b n − , and an element of the Massey product h ˆ a n − , ˆ a n − , ˆ b n − i is given by the class ˆ a n − ∪ ˆ A (2 n − − ϕ ∪ ˆ b n − .
35e now consider the more interesting trivializations of String, Fivebrane [SSS09] and Ninebranestructures [Sa15]. In fact, what we will consider are slightly weaker versions, i.e. the vanishing ofthe p i , i = 1 , ,
3, where p i is the i th Pontrjagin class rather than the vanishing of the precise frac-tional classes. These differ from p i -structures by the fact that we still require the lower Pontrjaginclasses to vanish (see [Sa15] for more discussion). We will then in turn consider differential re-finements of these structures, leading to Massey products representing geometric String, Fivebraneand Ninebrane structures, respectively.
Example 12 (Differential String structures and Chern-Simons theory) . On a smooth manifold M ,viewed as a stack, consider a Spin bundle E with connection ∇ characterized by a morphism ofstacks ∇ : M → B Spin( n ) conn , to the moduli stack of bundles of rank n Spin bundles with Spinconnections. At the level of classifying spaces, the fractional Pontrjagin class appears as a map p : B Spin( n ) −→ B U (1) ≃ K ( Z , which obstructs String orientability. There is a unique differential refinement of the first Spincharacteristic class p denoted b p which gives a map at the level of moduli stacks b p : B Spin( n ) conn −→ B U (1) conn . and captures the data of Chern-Simons theory (see [FSS12] [SSS12] [FSS13] [FSS15] [Bu11] [Wa13][Re11] [CJMSW]). Composing this map with with a map ∇ : M → B Spin conn giving a Spin bundle,equipped with connection and resolving M by its ˇCech nerve gives a ˇCech-Deligne cochain b p ( ∇ ) on M . Suppose that the Spin bundle trivializes as a bundle with connection, i.e. that we have b p ( ∇ ) = 0 as a differential cohomology class. There are two interesting cases that can arise inpractice and we will treat these cases separately. Suppose that b p ( ∇ ) decomposes as a square of aˇCech-Deligne cochain. That is, we have b p ( ∇ ) = ˆ a ∪ DB ˆ a . (4.6) Diagrammatically, we have & & M (ˆ a, ˆ a ) * * / / ∇ / / B Spin( n ) conn c p / / B U (1) conn , B U (1) conn × B U (1) conn ∪ DB q y ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (4.7) where, by the trivialization condition (4.6) , the lower diagram commutes strictly, and when we passto connected components π Map( M, B U (1) conn ) the map b p is trivial, so that the upper part of thediagram commutes up to homotopy. A choice of homotopy is precisely a trivializing ˇCech-Deligne3-cochain ˆ B . Given two such cochains ˆ B and ˆ C , the difference is necessarily a cocycle since D ( ˆ B − ˆ C ) = b p − b p = 0 . onsider the defining system a ˆ B a ˆ C a . The corresponding Massey product then takes the form h ˆ a, ˆ a, ˆ a i = ˆ B ∪ DB ˆ a − ˆ a ∪ DB ˆ C = ˆ a ∪ DB ( ˆ B − ˆ C ) . (4.8) Thus we can identify the Massey product as a flat bundle which is built entirely out of the trivializa-tions of the
Spin bundle with connection ∇ . Another interesting case happens when b p decomposesas ˆ a ∪ DB ˆ b . In this case, if the class of both ˆ a ∪ DB ˆ a and b p vanish in differential cohomology,choosing local trivialization ˆ B and ˆ C of ˆ a ∪ DB ˆ a and b p (respectively) lead to the defining system a ˆ B a ˆ C b , and we get the Massey product h ˆ a, ˆ a, ˆ b i = ˆ B ∪ DB ˆ b − ˆ a ∪ DB ˆ C .
In this case the trivialization of the
Spin bundle and the trivialization of the square ˆ a ∪ DB ˆ a combineto give a flat bundle representing the Massey product. Remark 10. (i)
Note that the above example can be extended to the case when the Spin bundle hasa different rank than the dimension of the manifold. In particular, this holds for the stable case. (ii)
Note that (4.6) implies, in particular, that at the level of de Rham cohomology we locallyhave dB = CS ( ∇ ) , where B is the connection on the bundle ˆ B . This then can be viewed asa generalization of local trivialization of Chern-Simons theory. Hence the Massey product is sbundle on E that is built out of the trivializations, including those of Chern-Simons. Furthermore,the structure of the Massey product (12) indicates that, even though we have a trivialization ofChern-Simon theory, we still have some secondary structure. (iii) Note that Example 12 generalizes in a similar fashion to the cases of differential Fivebrane[SSS12] and differential Ninebrane structures [Sa15] with trivializing conditions on the charac-teristic classes given by ˆ p ( ∇ ) = D ˆ B = D ˆ C and ˆ p ( ∇ ) = D ˆ B = D ˆ C , respectively, withtrivializing bundles ˆ B i = ˆ C i of degree i . If ˆ p ( ∇ ) decomposes as the square ˆ a ∪ DB ˆ a , the diagram (4.7) will have the obvious modifications in degrees with the middle entry being replaced by theappropriate structure, e.g. B String conn for the case of a Fivebrane structure. The trivialization ofthese structures a priori give rise to Chern-Simons theories in dimension 7 and 11, respectively,as highlighted in [SSS09] [Sa15]. In the current setting, we will have trivializations of the Chern-Simons theories themselves at the level of complete data of bundles with connections, and governedby the corresponding Massey products, which would read the same as (12) but with obvious changesin degrees. Note that Chern-Simons theory by itself can be viewed in a sense as a secondary structure, so the above is asecondary structure (in one sense) on some other secondary structure. We plan to make this precise elsewhere. emark 11 (Transfer of Massey products) . (i) A natural question is whether one can relate thestacky Massey triple product to the triple Deligne-Beilinson cup product. To that end, we recall thefollowing from [KS05] (the argument there was for specific dimensions but it extends evidently toany dimension). Consider Z n +1 as obtained from gluing two cobordisms together, i.e. Z n +1 is anorientable compact manifold and Y n is a submanifold of codimension 1 such that Z n +1 − Y n hastwo connected components, each of which is a cobordism. Then from the Mayer-Vietoris sequence,there is a connecting (or transfer) map T : H k ( Y n ) −→ H k +1 ( Z n +1 ) . (4.9) Now let a, b, c ∈ H ∗ ( Z n +1 ) with restrictions a ′ , b ′ , c ′ ∈ H ∗ ( Y n ) , and suppose further that the cupproducts vanish a ′ ∪ b ′ = b ′ ∪ c ′ = 0 ∈ H ∗ ( Y n ) so that the Massey product is defined. Then, byconsidering the Poincar´e dual chains, one has that the transfer of the Massey product gives thetriple product [KS05] T h a ′ , b ′ , c ′ i = a ∪ b ∪ c mod indeterminacy (4.10) where the Massey product is taken in H ∗ ( Y n ) , the product in H ∗ ( Z n +1 ) . The indeterminacy canbe taken as a ∪ z + x ∪ c where z, x are cocycles in the opposite connected components of Z n +1 − Y n .We propose generalizing this to our stacky setting of differential cohomology. We expect that theconnecting homomorphism for differential cohomology takes the form T : H k − ( Y n , U (1)) −→ ˆ H k +1 ( Z n +1 ) , and sends the differential Massey product to the triple DB cup product (modulo indeterminacy). (ii) The Deligne-Beilinson triple cup product arises in the description of certain Chern-Simons typefield theories in [FSS13] [FSS15]. The above then would be applied to these theories, giving thatthe Massey triple product of three differential cohomology elements on Z n +1 transfers to a triplecup product Chern-Simons theory (in the sense of [FSS13] [FSS15]) on Y n . We leave the detailsof checking this for the future. Presence of anomalies in a physical theory parametrizes to which extent certain entities are not(well) defined. Cancellation of these anomalies amounts to defining physical entities in the rightmathematical setting. The process often requires an extension of a topological or geometric settingto a more refined one. For example, to be able to talk about spinors, one has to set up the problemin the Spin bundle as opposed to the tangent bundle. This requirement is obstructed by thesecond Stiefel-Whitney class, and the structure itself leads to interesting geometry and topology.One important instance of this is the Green-Schwarz anomaly cancellation condition required forconsistency of string theory, which from the mathematical point of view essentially requires workingon manifolds with a (twisted) String structure. See [Fr00] [SSS09] [SW] for readable accounts aimedat mathematicians.A generic situation is as follows. Consider a bundle P with curvature F on a manifold M . Let c i ( P ) be a characteristic class of degree i and let c i ( F ) be the corresponding characteristic form.Consider the conditions in cohomology c i ( P ) ∪ c j ( P ) = 0 and c j ( P ) ∪ c k ( P ) = 0. Then at the levelof characteristic forms we have the trivializations via differential forms α and β of the indicateddegrees c i ( F ) ∧ c j ( F ) = dα ( i + j − , c j ( F ) ∧ c k ( F ) = dβ ( j + k − . (4.11)38e build the composite differential form µ = c i ( F ) ∧ β ( j + k − + ( − i − α ( i + j − ∧ c k ( F ) ∈ Ω i + j + k − ( M ) , which is directly verified to be closed. This then allows us to form the Massey triple product of thecorresponding cohomology classes h c i ( P ) , c j ( P ) , c k ( P ) i ∈ H i + j + k − ( M ; Z ) . Notice that we can consider conditions analogous to (4.11) in differential cohomologyˆ c i ( F ) ∪ ˆ c j ( F ) = D ˆ α ( i + j − , ˆ c j ( F ) ∪ ˆ c k ( F ) = D ˆ β ( j + k − , (4.12)requiring not only that the characteristic forms vanish, but that the corresponding bundles trivializeas bundles with connection. In this case, we can form the bundle (differential cochain)ˆ µ = ˆ c i ( F ) ∪ ˆ β ( j + k − + ( − i − ˆ α ( i + j − ∪ ˆ c k ( F ) ∈ Map( M, B i + j + k − U (1) conn ) , (4.13)which is an element in h ˆ c i ( P ) , ˆ c j ( P ) , ˆ c k ( P ) i ∈ ˆ H i + j + k − ( M ; Z ) . (4.14)We summarize the above. Proposition 18.
Given a system (4.12) of trivializations of products of differential characteristicclasses, we can build the stacky Massey product given by (4.13) . We now provide an application of this direct but fairly general observation. Consider a 10-dimensional manifold X with metric g on which there is a vector bundle with connection A . Onecan consider the setting in families, i.e. take a bundle E with fiber X and base a parameter spaceand then integrate over the fiber to get a class on the parameter space (see [Fr00] for beautifulconstructions). We will not do all this but simply just set up integral expressions which will sufficefor our purposes. The Green-Schwarz anomaly polynomials are given as I = p ( g ) − ch ( A ) ,I = − ch ( A ) + p ( g )ch ( A ) − p ( g ) + p ( g ) . In [SSS12] the first polynomial I is interpreted as giving rise to a twisted String structure, andthe indecomposable terms p ( g ) and ch ( A ) in I are interpreted as giving rise (essentially) to aFivebrane structure and its twist, respectively. Their trivializations H and H provide trivializa-tions of String and Fivebrane structures, respectively. A question remained on how to interpret thedecomposable terms in I , namely p ( g )ch ( A ) and − p ( g ) . We provide one interpretation ofthe corresponding trivializations, which fits well within our context. Consider the situation when[ p ∪ ch ] = 0 = [ p ∪ p ], i.e.ch ( A ) ∧ p ( g ) = dα ( A, g ) , p ( g ) ∧ p ( g ) = dβ ( g ) , (4.15)and build the differential form µ = ch ( A ) ∧ β ( g ) − α ( A, g ) ∧ p ( g ) . (4.16)39his form is closed by virtue of (4.15). Therefore, we can form the Massey triple product h ch , p , p i ∈ H ( E ; Z ) . (4.17)As expected, the previous discussion refines to differential cohomology. Let X be as before. Sincewe are fixing a Riemannian metric on X and equipping the vector bundle with a connection A ,it follows by uniqueness of characteristic forms (see [Bu12] [SS08]) that we have unique differentialrefinements ˆ I = ˆ p ( g ) − ˆch ( A ) , (4.18)ˆ I = − ˆch ( A ) + ˆ p ( g ) ˆch ( A ) − ˆ p ( g ) + ˆ p ( g ) . (4.19)We now consider the situation when these bundles trivialize as bundles with connections: [ˆ p ∪ ˆch ] =0 = [ˆ p ∪ ˆ p ], so that expressions (4.15) get replaced byˆch ( A ) ∧ ˆ p ( g ) = D ˆ α ( A, g ) , ˆ p ( g ) ∧ ˆ p ( g ) = D ˆ β ( g ) . We then build the bundleˆ µ = ˆch ( A ) ∪ ˆ β ( g ) − ˆ α ( A, g ) ∪ ˆ p ( g ) ∈ Map( X , B U (1) conn ) , (4.20)which is a representative of the Massey triple product h ˆch , ˆ p , ˆ p i ∈ ˆ H ( E ; Z ) . (4.21)Therefore, we have the following Proposition 19.
The mixed terms in the Green-Schwarz anomaly polynomials (4.18) (4.19) giverise to a stacky Massey product given by the top class (4.21) . It is interesting to note the form of the connection on the bundle ˆ µ . Using the formula for theDB cup product, we see that the connection is CS ( A ) ∧ CS ( g ) ∧ p ( g ) − α ( A, g ) ∪ p ( g ) , (4.22)which we will make use of below (see Prop. 21). Fiber integration of Massey products and anomaly line bundles
In [FSS13] [FSS15], afiber integration map was defined by taking the usual fiber integration in cohomology, lifting todifferential cohomology and then lifting to the internal hom in sheaves of positively graded chaincomplexes to produce a map Z Σ k − : [ N ( C ( { U i } ) , Z ∞D [ n ]] −→ Z ∞D [ n − k ] . Here Σ k is a paracompact manifold of dimension k and C ( { U i } ) is the ˇCech nerve correspondingto a good open cover of Σ k . The lifts are provided by the construction of Gomi and Terashimain [GT00]. Post-composing with the quasi-isomorphism provided by the exponential and applyingthe Dold-Kan functor gives a morphism of stacks in the form of holonomyhol Σ k := exp (cid:18) πi Z Σ k − (cid:19) : [Σ k , B n U (1) conn ] −→ B n − k U (1) conn . (4.23)40gain in [FSS13] [FSS15], it was observed that the abelian Chern-Simons action functional can bedescribed by post-composing the cup product morphism with this holonomy map. In particular,for a manifold Σ k +3 , this composite induces an intersection pairing on differential cohomology(ˆ x, ˆ y ) −→ exp (cid:18) πi Z Σ k +3 ˆ x ∪ ˆ y (cid:19) . (4.24)For k = 0 and ˆ y = ˆ x , this pairing gives the usual Chern-Simons action. We now would liketo describe how to lift this morphism to the Massey product (when defined). In fact, when thedifferential Massey product is defined, we have a map h ˆ x, ˆ y, ˆ z i U : Σ k × U −→ B n + n + n +2 U (1) conn , (4.25)which is natural in any test space U . Hence, we can apply the fiber integration map. Since Masseyproducts necessarily define flat bundles, we see immediately that we have the following. Proposition 20.
The integration over the fiber of the differential Massey product (4.25) can beidentifies with a map e ( πi R Σ k h ˆ x, ˆ y, ˆ z i ) U : U −→ B n + n + n +2 − k U (1) conn , which is natural in U . Moreover, this map defines a flat bundle on U , and the map factors throughthe inclusion j : ♭ B n + n + n +2 − k U (1) ֒ → B n + n + n +2 − k U (1) conn . Remark 12. (i)
The above construction can be generalized to higher Massey products, as we canfiber integrate any differential cohomology class of any degree, including those that are Masseyproducts. (ii)
The notation e ( πi R Σ k h ˆ x, ˆ y, ˆ z i ) is slightly abusive, since this map may not be well-defined on theentire Massey product (due to indeterminacy). What we really mean here is an element of theMassey product. In particular, when ˆ x , ˆ y and ˆ z come as characteristic forms, they are given by morphisms ofstacks; e.g. ˆ x : [Σ k , B G conn ] −→ [Σ k , B n U (1) conn ] , which gives a natural assignment of differential cohomology classes as we vary the G -principalbundle with connection on Σ k [FSS13]. In this case, after choosing trivialization of ˆ x ∪ DB ˆ y andˆ y ∪ DB ˆ z , fiber integration gives the morphism of stacks e ( πi R Σ k : h ˆ x, ˆ y, ˆ z i ) : [Σ k , B G conn ] −→ ♭ B n + n + n +2 − k U (1) . (4.26)One interesting instance of this morphism comes from the previous example of Green-Schwarzanomaly polynomials. That is, we are interested in the triple product h ˆch , ˆ p , ˆ p i . In this case, weget a morphism e ( πi R X h ˆch , ˆ p , ˆ p i ) : [ X , B G conn ] −→ ♭ B − U (1) = U (1) δ , (4.27)from the moduli stack of bundles on X equipped with connection to smooth U (1)-valued functions.It is useful to unwind this map at the level of connections. Indeed, noting (4.22), we have at thatlevel: 41 roposition 21. The connection on the bundle prescribed by (4.26) is given by the form Z X CS ( A ) ∧ CS ( g ) ∧ p ( g ) − α ( A, g ) ∪ p ( g ) . Remark 13. (i)
The exponential of the functional on the right, being built out of Chern-Simonsforms, is indeed in U (1) . (ii) As the structure of the functional in the proposition involves a product of two Chern-Simonsforms, this suggests a formulation where X is viewed as a manifold of corners of codimensiontwo, in the sense of the setting in [Sa11] [Sa14]. We hope to take up this point of view elsewhere. We consider the Ramond-Ramond (RR) fields in type IIA string theory on a ten-dimensionalmanifold X with a B-field, whose curvature is a closed three-form H . The RR fields of variousdegrees can be combined into the expression F = P i =0 F i , and satisfy the twisted Bianchi d F n + H ∧ F n − = 0. In components, H ∧ F = − dF , H ∧ F = − dF , H ∧ F = − dF ,H ∧ F = − dF , H ∧ F = − dF , dF = 0 = dF . (4.28) Remark 14.
From these we will build expressions of degree ten. (i)
Considering the first and fifth expressions in (4.28) , we can set up the top differential form µ = F ∧ F + F ∧ F . This is closed by dimension reasons, so that we can form the triple Massey product h F , H , F i ∈ H ( X ; Z ) . (ii) Considering the second and fourth expressions in (4.28) , we build the top form µ ′ = F ∧ F + F ∧ F . This is closed again by dimension reasons, and we can build the triple Massey product h F , H , F i ∈ H ( X ; Z ) . We now would like to refine the previous discussion to differential cohomology. Notice thatsince dF i = 0, we cannot simply put hats everywhere and expect the equations to hold at the levelof ordinary differential cohomology. Consequently, there are two directions we can go. First, wecould try to form Massey products in twisted differential cohomology, which is outside the scopeof the present paper. Second, we can view the F i ’s as improved gauge invariant field strengthscorresponding to potentials C i − with curvatures G i , which are not gauge invariant. We willexpand on this latter point of view. To this end, we require that the potentials C i − satisfy dC n + H ∧ C n − = 0 (4.29)42otice that this equation implies that the improved field strengths F i vanish, by definition. Com-bining the potentials into the single potential C = P i =0 C i − we have, by assumption, the equations H ∧ C = − dC , H ∧ C = − dC , H ∧ C = − dC . (4.30)These equations can be viewed as conditions on the connections for differential refinements of thefield strengths G i . Indeed, the full differentially refined equations readˆ H ∪ ˆ G = − D ˆ G , ˆ H ∪ ˆ G = − D ˆ G , ˆ H ∪ ˆ G = − D ˆ G . (4.31) (i) Considering the first and third expressions in (4.31), we can form the bundleˆ µ = ˆ G ∪ ˆ G + ˆ G ∪ ˆ G , with higher connection C ∧ G + C ∧ G = C ∧ H ∧ C + C ∧ H ∧ C . (4.32)This bundle is an element in the stacky Massey triple product h ˆ G , ˆ H , ˆ G i ∈ ˆ H ( X ; Z ) . (4.33) (ii) Considering instead the first and second expressions in (4.31), we form the higher bundleˆ µ ′ = ˆ G ∪ ˆ G + ˆ G ∪ ˆ G . with higher connection C ∧ G + C ∧ G = C ∧ H ∧ C + C ∧ H ∧ C . (4.34)This is an element in the stacky triple Massey product h ˆ G , ˆ H , ˆ G i ∈ ˆ H ( X ; Z ) . (4.35) Proposition 22.
The system of twisted Bianchi identities for the differential RR fields leads totwo higher bundles with connections (4.32) and (4.34) which are elements in the stacky Masseyproducts in top degree (4.33) and (4.35) , respectively.
It would be interesting to investigate the implications of these expressions to string theory. Fornow we just observe that, essentially and up to signs, µ and µ are part of the couplings thatarise in calculating the topological partition function of the RR fields (in the case when H = 0)[DMW00] [BM06]. While we do not pursue this here, we expect ˆ µ and ˆ µ to be relevant for thecalculation of the partition function in the twisted differential case, ˆ H = 0, as well, extending thetwisted topological case in [MS03] [MS04]. 43 .4 Quadruple Massey products We now consider a setting inspired by type IIB string theory. The main feature of this theory thatconcerns us here is that it has fields of odd degree, where the degree three play a somewhat specialrole. Consider four fields as cohomology classes h ( i )3 ∈ H ( X ; Z ), i = 1 , · · · ,
4, on a ten-dimensionalmanifold X , and consider analogues of three composite (Ramond-Ramond) fields F ( j )5 , j = 1 , , h (1)3 ∧ h (2)3 = − dF (3)5 , h (2)3 ∧ h (3)3 = − dF (1)5 , h (3)3 ∧ h (4)3 = − dF (2)5 . Then there are further composite (again analogues of Ramond-Ramond) fields F ( i )7 , i = 1 , · · · , F (3)5 ∧ h (3)3 = − dF (3)7 , h (1)3 ∧ F (1)5 = − dF (1)7 ,F (1)5 ∧ h (4)3 = − dF (4)7 , h (2)3 ∧ F (2)5 = − dF (2)7 . Then we will end up (see below) having the Massey quadruple product as the integer h h (1)3 , h (2)3 , h (3)3 , h (4)3 i := − F (3)7 ∧ h (4)3 − F (1)7 ∧ h (4)3 − F (3)5 ∧ F (2)5 + h (1)3 ∧ F (4)7 + h (1)3 ∧ F (2)7 ∈ H ( X ; Z ) ∼ = Z . We now elaborate on the above. We first start with the triple Massey product in the current IIBstring theory inspired context. Let [ h ( i )3 ] ∈ H ( X ) ( i = 1 , ,
3) be non-zero cohomology classessuch that [ h (1)3 ] ∪ [ h (2)3 ] = 0 and [ h (2)3 ] ∪ [ h (3)3 ] = 0. For the cocycle representatives h ( i )3 , write h (1)3 ∪ h (2)3 = dF (1)5 and h (2)3 ∪ h (3)3 = dF (2)5 . (4.36)Notice that from these two equations one gets immediately that d ( F (1)5 ∪ h (3)3 + h (1)3 ∪ F (2)5 ) = 0 by astraightforward application of the Leibnitz rule. This can then be used to define the triple Masseyproduct as the subset of H ( X ) given by (cid:10) [ h (1)3 ] , [ h (2)3 ] , [ h (3)3 ] (cid:11) = nh F (1)5 ∪ h (3)3 + h (1)3 ∪ F (2)5 io , (4.37)where h and F run over all possible choices above. The indeterminacy in the choice of therepresentative w = h (1)3 ∪ F (2)5 + F (1)5 ∪ h (3)3 for the triple product lies in the ideal (cid:16) [ h (1)3 ] , [ h (2)3 ] (cid:17) .In order to connect with the Massey 4-fold product, it is good to rewrite the triple product inmatrix form. The classes [ h ( i )3 ] and the elements F ( i )5 can be encoded in a matrix form a a a a a = h (1)3 F (1)5 h (2)3 F (2)5 h (3)3 . (4.38)44he defining properties of the Massey triple product can be expressed in a matrix multiplicationas d h (1)3 F (1)5 h (2)3 F (2)5 h (3)3 = h (1)3 h (2)3 h (2)3 h (3)3 , h (1)3 F (1)5 h (2)3 F (2)5 h (3)3 = h (1)3 h (2)3 h (1)3 F (2)5 + F (1)5 h (3)3 h (2)3 h (3)3 . (4.39)Now in order to go one step further to the quadruple (or 4-fold) product, we need to satisfycertain conditions on the triple product, in analogy to saying that higher obstructions arise onlyonce the lower ones vanish. So in our case, we first need to assume that we can complete our setby adding two more elements, a fourth h (4)3 and a third F (3)5 , such that dF (3)5 = h (3)3 ∪ h (4)3 . (4.40)Besides the above representative w , we then have a second representative for the triple product andis given by z = h (2)3 ∪ F (3)5 + F (2)5 ∪ h (4)3 , namely representing (cid:10) [ h (2)3 ] , [ h (3)3 ] , [ h (4)3 ] (cid:11) . The conditionto be able to define the quadruple product is that both triple products vanish simultaneously , i.e.that both cohomology representatives w and z can be chosen as coboundaries, which we write as w = dF (1)7 and z = dF (2)7 .We are now ready to define the 4-fold or quadruple Massey product. In analogy to the tripleproduct, we start with the equations (4.36) and (4.40), and then write the two cocycles of degreeeight dF (1)7 = h (1)3 ∪ F (2)5 + F (1)5 ∪ h (3)3 and dF (1)7 = h (2)3 ∪ F (3)5 + F (2)5 ∪ h (4)3 , from which we get a cocycle x = h (1)3 ∪ F (2)7 + F (1)5 ∪ F (2)5 + F (1)7 ∪ h (4)3 (4.41)of degree ten. Remark 15. (i)
Again, we define the quadruple Massey product (cid:10) [ h (1)3 ] , [ h (2)3 ] , [ h (3)3 ] , [ h (4)3 ] (cid:11) as acollection of all cohomology classes [ x ] ∈ H ( X ) that we can obtain by the above procedure. (ii) The indeterminacy is best presented as the matrix triple product of certain elements, namelyof ( h (1)3 , H ( X )) , h (2)3 H ( X )0 h (3)3 ! , and H ( X ) h (4)3 ! . We now generalize this construction to differential cohomology. To produce the desired prod-ucts, we again view the F ( i )5 and F ( i )7 as improved, gauge invariant field strengths and denote the45orresponding potentials as C ( i )4 and C ( i )6 , with curvatures G ( i )5 and G ( i )7 . We now lift everything tothe level of differential cohomology, which yields the equationsˆ h (1)3 ∪ ˆ h (2)3 = − D ˆ G (3)5 , ˆ h (2)3 ∪ ˆ h (3)3 = − D ˆ G (1)5 , ˆ h (3)3 ∪ ˆ h (4)3 = − D ˆ G (2)5 , (4.42)and ˆ G (3)5 ∪ ˆ h (3)3 = − D ˆ G (3)7 , ˆ h (1)3 ∪ ˆ G (1)5 = − D ˆ G (1)7 , ˆ G (1)5 ∪ ˆ h (4)3 = − D ˆ G (4)7 , ˆ h (2)3 ∪ ˆ G (2)5 = − D ˆ G (2)7 . (4.43)The connection on the higher bundle is calculated as follows. Set A := − C (3)6 ∧ h (4)3 − C (1)6 ∧ h (4)3 − C (3)4 ∧ G (2)5 + b (1)2 ∪ G (4)7 + b (1)2 ∪ G (2)7 = − C (3)6 ∧ h (4)3 − C (1)6 ∧ h (4)3 . Then, by writing the higher components in the last three terms via lower components, we get A = − C (3)6 ∧ h (4)3 − C (1)6 ∧ h (4)3 − C (3)4 ∧ b (3)2 ∧ h (4)3 + b (1)2 ∧ C (1)4 ∧ h (4)3 + b (1)2 ∧ b (2)2 ∧ b (3)2 ∧ h (4)3 . (4.44)Here b ( i )2 denotes a local potentials for the forms h ( i )3 . Therefore, we have the following descriptionas phase or holonomy. Proposition 23.
The system (4.42) (4.43) leads to the stacky Massey quadruple product h ˆ h (1)3 , ˆ h (2)3 , ˆ h (3)3 , ˆ h (4)3 i := − ˆ G (3)7 ∪ ˆ h (4)3 − ˆ G (1)7 ∪ ˆ h (4)3 − ˆ G (3)5 ∪ ˆ G (2)5 +ˆ h (1)3 ∪ ˆ G (4)7 +ˆ h (1)3 ∪ ˆ G (2)7 ∈ ˆ H ( X ; Z ) , viewed as a higher bundle whose connection A is given by (4.44) . The discussion using matric Massey products carries over to differential cohomology in a similarfashion. We also leave the discussion on the physical impact of the above constructions to a separatetreatment.
Acknowledgement
The authors would like to thank Domenico Fiorenza and Urs Schreiber for very useful discussionsand comments, Chris Kapulkin for a useful comment on the first version of the manuscript, andthe referee for a careful reading of the manuscript and for many useful suggestions.
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