CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES. Notes by John Rognes. March 6th 2009

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1 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES Notes by John Rognes March 6th 2009 Gerbes and 2-vector bundles Let V be a bipermutative groupoid of finite-dimensional complex vector spaces, under direct sum and tensor product. For example, we may take V to have objects C n for n 0, so that C n has automorphism group GL n (C) Recall from Brylinski, Section 5.2: Definition. Let X be a smooth manifold, perhaps infinite-dimensional. A smooth C -gerbe over X is a sheaf of groupoids G that is locally equivalent to the sheaf of smooth functions to the groupoid GL 1 (V), with objects the 1-dimensional complex vector spaces and isomorphisms the complex linear isomorphisms between these. In more detail: (P1) for each local homeomorphism f: Y X there is a groupoid G(Y) = G(Y f X), (P2) for every further local homeomorphism g: Z Y there is a functor g 1 : G(Y) G(Z), (P3) and for every further local homeomorphism h: W Z there is a natural isomorphism θ g,h : h 1 g 1 = (gh) 1 of functors G(Y) G(W), such that (P4) for each further local homeomorphism k: T W the identity θ gh,k θ h,k = θ g,hk θ g,h holds. This makes G a presheaf of groupoids. For G to be a sheaf of groupoids, two descent conditions must hold: (D1) For each local homeomorphism f: Y X and each pair of objects P,Q G(Y), the assignment taking a further local homeomorphism g: Z Y to the morphism set G(Z)(g 1 P,g 1 Q) is a sheaf of sets on Y, denoted G(Y)(P,Q). (D2) For each open subset V X, each surjective local homeomorphism f: Y V, and each object P G(Y) equipped with a descent datum, that is, an isomorphism φ: p 1 2 P = p 1 1 P in G(Y X Y) such that p 1 12 φ p 1 23 φ φ (modulo θ s), there exists an object Q G(V) and an isomorphism p 1 13 ψ: f 1 (Q) = P in G(Y), such that φ (p 1 1 ψ)(p 1 2 ψ) 1 (modulo θ s). 1 Typeset by AMS-TEX

2 2 NOTES BY JOHN ROGNES For G to be a smooth C -gerbe, three gerbe conditions must hold: (G3) There is a surjective local homeomorphism f: Y X such that G(Y) is non-empty. (G2) For any local homeomorphism f: Y X and any two objects P and Q in G(Y), there exists a surjective local homeomorphism g: Z Y such that g 1 P and g 1 Q are isomorphic in G(Z). (G1) Givenanylocalhomeomorphismf: Y X andanyobjectp ofg(y), there is a preferred isomorphism from the sheaf G(Y)(P,P) of automorphisms of P to the sheaf C Y of smooth C -valued functions. ((Omitted: When are two gerbes equivalent?) Here is a definition from Baas Dundas Rognes. Definition. By an ordered open cover (U,I) of a topological space X, we mean a partially ordered indexing set I, and an open subset U α X for each α I, so that for each x X the partial ordering on I restricts to a total (= linear) ordering on the subset I x = {α I x U α }. It follows that for any subset J I of indices with α J U α, J is totally ordered. The following defines a charted 2-vector bundle of rank 1. We use the abbreviations U αβ = U α U β, U αβγ = U α U β U γ, etc. See also Murray s paper on bundle gerbes. Definition. Let X be a topological space, with an ordered open cover (U,I). A charted 2-line bundle L over X consists of (1) a complex line bundle L αβ over U αβ for each pair α < β in I, and (2) an isomorphism φ αβγ : L αβ L βγ = L αγ of complex line bundles over U αβγ for each triple α < β < γ in I, such that (3) the diagram L αβ (L βγ L γδ ) α (L αβ L βγ ) L γδ id φ βγδ φ αβγ id L αβ L βδ φ αβδ L αδ L αγ L γδ φ αγδ of line bundle isomorphisms over U αβγδ commutes for each chain α < β < γ < δ in I, where α is the coherent natural associativity isomorphism. The L αβ are the gluing line bundles and the φ αβγ are the coherence isomorphisms of the charted 2-line bundle. If X is a smooth manifold, each L αβ is smooth line bundle, and each φ αβγ is a smooth isomorphism, we get a smooth charted 2-line bundle. This notion is closely related to that of a bundle gerbe. ((Omitted: When are two 2-line bundles equivalent?) Here is the general definition, for n 1.

3 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 3 Definition. Let X be a topological space, with an ordered open cover (U,I). A charted 2-vector bundle E of rank n over X consists of (1) an n n matrix E αβ = (E αβ ij )n i,j=1 of complex vector bundles over U αβ, with dimension matrix dim(e αβ ) of determinant ±1 everywhere, for each pair α < β in I, and (2) an n n matrix φ αβγ = (φ αβγ ik )n i,k=1: E αβ E βγ = E αγ of vector bundle isomorphisms over U αβγ, where (E αβ E βγ ) ik = n j=1 E αβ ij E βγ jk for i,k = 1,...,n, for each triple α < β < γ in I, such that (3) the diagram E αβ (E βγ E γδ ) α (E αβ E βγ ) E γδ id φ βγδ φ αβγ id E αβ E βδ φ αβδ E αδ E αγ E γδ φ αγδ of n n matrices of vector bundle isomorphisms over U αβγδ commutes for each chain α < β < γ < δ in I, where α is the coherent natural associativity isomorphism. The E αβ are the gluing bundle matrices and the φ αβγ are the coherence isomorphism matrices of the charted 2-vector bundle. If X is a smooth manifold, each E αβ ij is smooth vector bundle, and each φ αβγ ik is a smooth isomorphism, we get a smooth charted 2-vector bundle. The following construction is adapted from Brylinski s Definition and Proposition 7.2.1, and the proof of surjectivity in his Theorem Proposition. To each smooth charted 2-line bundle L over X there is a naturally associated smooth C -gerbe G over X. Proof. We work in the smooth category. Let f: Y X be a local homeomorphism, and let Y α = f 1 (U α ), Y αβ = f 1 (U αβ ), etc. be open subsets of Y. Let G(Y) = G(Y f X) be the groupoid with objects consisting of (1) a complex line bundle P α over Y α for each α I, (2) an isomorphism ξ αβ : L αβ P β = P α of complex line bundles over Y αβ, for each pair α < β in I, such that

4 4 NOTES BY JOHN ROGNES (3) the diagram L αβ (L βγ P γ ) α (L αβ L βγ ) P γ id ξ βγ φ αβγ id L αβ P β ξ αβ P α L αγ P γ ξ αγ of isomorphisms over U αβγ commutes for each triple α < β < γ in I. The isomorphisms in G(Y) from (P α,ξ αβ ) to (Q α,η αβ ) consist of (1) isomorphisms ψ α : P α Q α over Y α for each α I, such that (2) the diagram L αβ P β ξ αβ id ψ β P α L αβ Q β η αβ Q α of isomorphisms over Y αβ commutes for each pair α < β in I. For each local homeomorphism g: Z Y the functor g 1 : G(Y) G(Z) is given on objects by pullback of the line bundles P α along Z α Y α and of the isomorphisms ξ αβ along Z αβ Y αβ, while on isomorphisms it is given by pullback of the isomorphisms ψ α along Z α Y α. This accounts for (P1) and (P2). We omit to discuss (P3), (P4), (D1), (D2), (G3) and (G2). The automorphisms of P = (P α,ξ αβ ) consist of isomorphisms ψ α : P α P α over Y α, for each α I, which amounts to multiplication by a (smooth) function a α : Y α C, and these satisfy a α = a β over Y αβ, hence these functions glue together to a (smooth) C -valued function a on Y. More generally this works for the pullback over any local homeomorphism g: Z Y, so we get the isomorphism of sheaves between G(Y)(P,P) and C Y, required in (G1). Definition. Let GL n (V) be the (smooth) groupoid with objects the n n matrices V = (V ij ) n i,j=1 of complex vector spaces with dimension matrix dim(v) = dim(v ij ) n i,j=1 of determinant ±1, and morphisms n n matrices φ: V W of complex linear isomorphisms between these. Matrix multiplication V W with (V W) ik = ψ α n V ij W jk j=1 defines a monoidal structure on GL n (V). There is one isomorphism class of objects in GL n (V), for each n n matrix D = (d ij ) n i,j=1 with non-negative integer entries and determinant ±1. These matrices form the monoid GL n (N 0 ) = Mat n (N 0 ) GL n (Z), whose invertible elements are exactly the permutation matrices Σ n GL n (N 0 ). The automorphisms in GL n (V) of the object C D = (C d ij ) n i,j=1 form the product group n GL D (C) = GL dij (C). i,j=1

5 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 5 For example, if D is a permutation matrix, GL D (C) = GL 1 (C) n = (C ) n. ((Let D = n i,j=1 d ij. Then D n, with equality if and only if D is a permutation matrix. If A B = C then B C, with equality if and only if A is a permutation matrix. Hence a given matrix D can be factored in at most finitely many ways, up to permutation matrices (= units).)) Proposition. To each smooth charted 2-vector bundle E of rank n over X there is a naturally associated sheaf of groupoids P over X with a right action by the monoidal sheaf of smooth functions to GL n (V). ((Can say more about local structure, under further hypotheses on the ordered open cover.)) Proof. We work in the smooth category. Let f: Y X be a local homeomorphism, and let Y α = f 1 (U α ), Y αβ = f 1 (U αβ ), etc. be open subsets of Y. Let P(Y) = P(Y f X) be the groupoid with objects consisting of (1) an n n matrix P α = (P α ij )n i,j=1 of complex vector bundles over Y α, with dimension matrix dim(p α ) of determinant ±1 everywhere, for each α I, and (2) an n n matrix ξ αβ = (ξ αβ ik )n i,k=1: E αβ P β = P α of vector bundle isomorphisms over Y αβ, for each pair α < β in I, such that (3) the diagram E αβ (E βγ P γ ) α (E αβ E βγ ) P γ id ξ βγ φ αβγ id E αβ P β ξ αβ P α E αγ P γ ξ αγ of n n matrices of vector bundle isomorphisms over U αβγ commutes for each triple α < β < γ in I. The isomorphisms in P(Y) from (P α,ξ αβ ) to (Q α,η αβ ) consist of (1) an n n matrix ψ α = (ψ α ij) n i,j=1: P α Q α of vector bundle isomorphisms over Y α, for each α I, such that (2) the diagram E αβ P β ξ αβ id ψ β P α E αβ Q β η αβ Q α of n n matrices of vector bundle isomorphisms over Y αβ commutes, for each pair α < β in I. ψ α

6 6 NOTES BY JOHN ROGNES For each local homeomorphism g: Z Y the functor g 1 : P(Y) P(Z) is given on objects by pullback of the vector bundle matrix P α along Z α Y α and of the isomorphism matrix ξ αβ along Z αβ Y αβ, while on isomorphisms it is given by pullback of the isomorphism matrix ψ α along Z α Y α. This accounts for (P1) and (P2). The remaining presheaf conditions (P3) and (P4), and the sheaf conditions (D1) and (D2) should be clear. The value of the sheaf of smooth functions to GL n (V) at Y can be viewed as the groupoid of n n matrices R = (R ij ) n i,j=1 of complex vector bundles over Y, with dim(r) of determinant ±1, and n n matrices ζ: R S of vector bundle isomorphisms between these. The right action of R on an object (P α,ξ αβ ) gives the object consisting of (1) the n n matrix P α R of vector bundles over Y α, for each α I, and (2) the n n matrix of composite vector bundle isomorphisms E αβ (P β R) α (E αβ P β ) R ξαβ id P α R over Y αβ, for each pair α < β in I. The right action of ζ: R S on an isomorphism (ψ α ) from (P α,ξ αβ ) to (Q α,η αβ ) is the collection of isomorphisms for α I. When does P have objects locally? ψ α ζ: P α R = Q α S Proposition. Suppose that x X has an open neighborhood V such that the subset I V = {α I V U α } of I is finite and totally ordered. Suppose also that the V α = V U α for α I V, as well as all of their intersections, are contractible. ((Need a condition about extension to closures, to ensure that the union of two trivial bundles along a contractible open subspace is again trivial.)) Then P(V) is not the empty category, hence has objects. ((The missing condition might be that X is a normal space (for the Tietze extension theorem), the gluing bundle matrices E αβ are defined over the closed intersections Ūα Ūβ, and that the coherence isomorphism matrices φ αβγ are defined over the closed intersections Ūα Ūβ Ūγ, with the usual compatibility. Furthermore, the V Ūα and all of their intersections, should be contractible.)) Proof. Write I V = {α < < ω}. For all β < γ in I V we (should) know that E βγ is trivial over the contractible open set V βγ. Choose an n n matrix of trivial vector bundles P ω over V ω. As the last step of a descending induction over I V, we shall assume that we have chosen (1) matrices of trivial vector bundles P β over V β for all β with α < β ω, and (2) matrices of isomorphisms ξ βγ : E βγ P γ P β over V βγ for all β and γ with α < β < γ ω, subject to the usual compatibility condition over V βγδ for all α < β < γ < δ ω. Then we can (1) form the matrix of trivial vector bundles E αβ P β over V αβ, for each β with α < β ω,

7 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 7 (2) these are related by a chain of matrices of isomorphisms E αβ (E βγ P γ ) id ξ βγ E αβ P β α (E αβ E βγ ) P γ φ αβγ id E αγ P γ over V αβγ, for all β and γ with α < β < γ ω, and (3) these chains of isomorphisms are strictly compatible over V αβγδ, for all α < β < γ < δ ω. Hence we can find a matrix of vector bundles P α over Ṽ α = V αβ V α α<β ω with isomorphisms ξ αβ : E αβ P β P α over V αβ for all α < β ω, such that the completed diagram E αβ (E βγ P γ ) α (E αβ E βγ ) P γ id ξ βγ φ αβγ id E αβ P β ξ αβ Pα E αγ P γ ξ αγ commutes over V αβγ. ((Now we argue that P α over Ṽα is a matrix of trivial bundles, hence can be extended to a matrix P α over V α of trivial bundles.)) When are objects of P locally isomorphic? ((Objects (P α,ξ αβ ) and (Q α,η αβ ) in P(V) can only be isomorphic if they have the same dimension matrices dim(p α ) = dim(q α ) over V α = V U α for all α. Assuming this, we can build a local isomorphism for small V, assuming that isomorphisms can be extended over inclusions Ṽα V α, as in the previous result.)) Connective structures on gerbes Let q 0 and let Y be a smooth manifold. The assignment taking a local homeomorphism g: Z Y to the complex vector space A q (Z;C) = Ω q (Z) C of complex alternating q-forms on Z, defines a complex vector space sheaf A q Y,C over Y. For each complex line bundle P over Y, the set of connections on P forms an affine space (a single free orbit) under A 1 (Y;C). For a more general complex vector bundle E, the connections form an affine space under the 1-forms in the endomorphism bundle of E, but when E = P is a line bundle, the endomorphism bundle is a trivial C 1 -bundle. Passing to restrictions along the local homeomorphisms g: Z Y, the collection of connections over the various Z, denoted Co(P), forms a torsor under A 1 Y,C. If ψ: P = Q is an isomorphism of complex line bundles over Y, there is an induced isomorphism ψ : Co(P) Co(Q) that takes a connection on P to the connection ψ ( ) on Q, determined on smooth sections s in Q Y by ψ ( ) w (s) = ψ( v (ψ 1 s))

8 8 NOTES BY JOHN ROGNES whentψ: TP TQtakesthetangentvectorv tow. Whenψ istheautomorphism given by multiplication by a smooth function a: Y C, we can compute that ψ ( ) w (s) = a v (a 1 s) = a a 1 w (s)+a d(a 1 )(w) s = w (s) a 1 da(w) s so that ψ ( ) = a 1 da = dloga. In a C -gerbe, the local groupoids are non-canonically equivalent to the groupoid of complex line bundles, and a connective structure specifies corresponding torsors under the complex vector space sheaf A 1 Y,C. Definition. Let G be a smooth C -gerbe over X. A connective structure Co on G consists of (1) a rule that for each local homeomorphism f: Y X and each object P G(Y) associates an A 1 Y,C-torsor Co(P), (2) an isomorphism co g : g 1 Co(P) = Co(g 1 P) of A 1 Z,C-torsors for each further local homeomorphism g: Z Y, and (3) an isomorphism ψ : Co(P) Co(Q) ofa 1 Y,C-torsorsforeachisomorphismψ: P QinG(Y). Thesearerequired to be functorial and compatible, in the sense that (4) co gh co h co g (modulo canonical isomorphisms and θ s) for further local homeomorphisms h: W Z, (5) (ψ ψ) = ψ ψ for further isomorphisms ψ : Q R in G(Y), and (6) the diagram g 1 Co(P) g 1 (ψ ) g 1 Co(Q) co g Co(g 1 P) co g (g 1 ψ) Co(g 1 Q) commutes. Finally, the isomorphisms ψ behave as expected for gauge automorphisms, meaning that (7) whenψ: P P istheautomorphisminducedbymultiplicationbyasmooth function a: Y C, we have ψ ( ) = a 1 da for all in the A 1 Y,C-torsor Co(P). See Brylinski s Definition He writes α g for our co g, and labels (6) as (R2) and (7) as (R1). The following combines Brylinski s Proposition and

9 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 9 Proposition. Let G be a smooth C -gerbe over a smooth manifold X. Then G admits connective structures, and the isomorphism classes of torsors under A 1 X,C act freely and transitively on the isomorphism classes of connective structures on G. Definition. Let X be a smooth manifold, with an ordered open cover (U,I). Let L X be a smooth charted 2-line bundle, with gluing line bundles L αβ and coherence isomorphisms φ αβγ. A connective structure on L is (1) a connection αβ on L αβ over U αβ, for each α < β in I, such that (2) the isomorphism φ αβγ : L αβ L βγ = L αγ over U αβγ takes the tensor product connection αβ + βγ on L αβ L βγ to the connection αγ on L αγ, so that (φ αβγ ) ( αβ + βγ ) = αγ over U αβγ. For sections s in L αβ and t in L βγ, we have ( αβ + βγ )(s t) = αβ (s) t+s βγ (t) as 1-forms of sections in L αβ L βγ over U αβγ. The following is close to the first part of the proof of Brylinski s Proposition For an ordered open cover (U,I) of X and an abelian sheaf A over X, the ordered Čech complex consists of the abelian groups Č p (U;A) = α 0 < <α p A(U α0...α p ). An ordered Čech p-cochain η associates to each sequence α 0 < < α p an element η(α 0,...,α p ) A(U α0...α p ). Its coboundary dη is the ordered Čech (p+1)- cochain that associates to each sequence α 0 < < α p+1 the alternating sum p+1 i=0 ( 1)i η(α 0,..., α i,...,α p+1 ), interpreted via various restriction maps to lie in A(U α0...α p+1 ). Its cohomology is denoted Ȟ (U;A). The colimit over all ordered open covers (U,I) defines the Čech cohomology Ȟ (X;A). Proposition. Let L be a smooth charted 2-line bundle over a smooth manifold X. There exists a connective structure on L. Proof I. First choose arbitrary connections αβ on L αβ for each α < β in I. Then (φ αβγ ) ( αβ + βγ ) = αγ +ω αβγ for a unique complex 1-form ω αβγ on U αβγ. Letting α < β < γ vary, we get a Čech 2-cochain ω in Č2 (U;A 1 X,C). By condition (3) in the definition of a charted 2-line bundle, this Čech 2-cochain is a Čech 2-cocycle. As discussed in Brylinski s Section 1.4, the sheaves A q X,C are soft, so that the restriction maps A q (X;C) A q (Z;C) are surjective for all closed Z X, hence acyclic. So for reasonable X, Ȟ p (U;A q X,C ) = H p (U;A q X,C ) = 0 for p > 0. Murray gives a more direct proof if this fact. In particular, Ȟ 2 (U;A 1 X,C) = 0, so the Čech 2-cocycle ω is a Čech 2-coboundary. ((Does it matter that we consider ordered Čech cochains?))

10 10 NOTES BY JOHN ROGNES Hence there exists a Čech 1-cochain η with dη = ω. This η consists of complex 1-forms η αβ over U αβ for each α < β, such that η βγ η αγ +η αβ = ω αβγ over U αβγ. Let αβ = αβ η αβ be a new connection on L αβ, for all α < β in I. Then so is a connective structure on L. (φ αβγ ) ( αβ + βγ ) = αγ, Proof II. First choose 1-forms η αβ on U α β for all α < β with α+1 = β, so that β is the successor of α. For α+1 = β and β +1 = γ the 1-form η αγ is determined on U αβγ from the relation η βγ η αγ +η αβ = ω αβγ. We (should then be able to) choose an extension of η αγ over U αβγ U αγ. For α+1 = β, β +1 = γ and γ +1 = δ, the 1-form η αδ is determined on U αβδ by and on U αγδ by η βδ η αδ +η αβ = ω αβδ η γδ η αδ +η αγ = ω αγδ. These 1-forms agree on U αβγδ, as a consequence of the relation ω βγδ ω αγδ +ω αβδ ω αβγ = 0m so they define a 1-form on Ũαδ = U αβδ U αγδ. We (should then be able to) choose an extension of η αδ over Ũαδ U αδ. Continuing like this, we determine η αβ for all α < β where α+k = β for finite k. (Classify the choices of connective structure.) Proposition. Let and be two connective structures on L over X. Then ((ETC)) Proof. There are unique complex 1-forms η αβ on U αβ so that αβ = αβ +η αβ. This defines a Čech 1-cochain η. Since both and are connective structures, η αβ + η βγ = η αγ, so η is a Čech 1-cocycle. Since Ȟ1 (U;A 1 X,C) = 0, this Čech 1-cocycle is a Čech 1-coboundary. ((ETC)) Proposition. To each connective structure on a smooth charted 2-line bundle L over X there is a naturally associated connective structure Co on the associated smooth C -gerbe G over X. Proof. Consider a local homeomorphism f: Y X and an object P G(Y), consisting of complex line bundles P α over Y α = f 1 (U α ) for each α I, and isomorphisms ξ αβ : L αβ P β = P α over Y αβ = f 1 (U αβ ) for each α < β. We need to define the A 1 Y,C-torsor Co(P) of connections on P, and its restrictions over further local homeomorphisms g: Z Y. Over Y the elements D = (D α ) α I of Co(P) consist of (1) connections D α on P α over Y α for all α I, such that (2) the isomorphism ξ αβ : L αβ P β = P α takes the tensor product connection αβ +D β to D α, for all α < β, so that over Y αβ. (ξ αβ ) ( αβ +D β ) = D α

11 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 11 Given another such element, D = ( Dα ) α I, we can write D α = D α + ǫ α for a unique ǫ α A 1 (Y α ;C), for all α I. By condition (2) for D and D, it follows that ǫ α = ǫ β over Y αβ, so that the ǫ α are all restrictions of a global complex 1-form ǫ A 1 (Y;C). This makes it clear that A 1 (Y;C) acts freely and transitively on the elements of Co(P) over Y. The extension to a sheaf is clear, as is the conclusion that Co(P) is a torsor under A 1 Y,C. ((Explain rest of requirements?)) Proposition. To each smooth charted 2-line bundle L over X, there is an associated Čech cohomology class [L] in Ȟ2 (X;C X) = H 2 (X;C X) = H 3 (X;Z). Proof. After possibly refining the ordered cover (U, I), we may assume that the gluinglinebundlesl αβ admitsmoothsectionss αβ,givingtrivializationsu αβ C = L αβ. Comparing these trivializations via φ αβγ, there is a unique smooth function h αβγ : U αβγ C such that φ αβγ (s αβ s βγ ) = h αβγ s αγ. Letting α < β < γ vary, the h αβγ combine to a Čech 2-cochain h Č2 (U;C X). By property (3) in the definition of a charted 2-line bundle, h is a Čech 2-cocycle, and [L] is its cohomology class in Ȟ2 (X;C X). ((Should check independence of choice of sections s αβ.)) (For a suitable notion of equivalence of smooth charted 2-line bundles, this rule induces an isomorphism between the group of equivalence classes of smooth charted 2-line bundles over X and the cohomology group H 3 (X;Z).) Associated to the complex 0 C X dlog A 1 X,C 0 of abelian sheaves over X, and an ordered open cover (U,I) of X, we can associate the Čech bicomplex 0 Č (U;C X) dlog Č (U;A 1 X,C) 0 with Čp (U;C X) in bidegree (p,0) and Čp (U;A 1 X,C) in bidegree (p,1). The cohomology of the corresponding total complex is the hypercohomology Ȟ (U;C X dlog A 1 X,C). Its colimit over all ordered open covers is the Čech hypercohomology Ȟ (X;C X dlog A 1 X,C), which for reasonable X agrees with sheaf hypercohomology H. There is a long exact sequence H p (X;C X dlog A 1 X,C ) Hp (X;C X ) dlog H p (X;A 1 X,C ).... Here H p (X;A 1 X,C) = 0 for p 0, so there are isomorphisms for all p 2. H p (X;C X dlog A 1 X,C) = H p (X;C X) = H p+1 (X,Z)

12 12 NOTES BY JOHN ROGNES Proposition. To each smooth charted 2-line bundle L over X with connective structure, there is an associated Čech cohomology class [L, ] in Ȟ 2 (X;C X dlog A 1 X,C). (In view of the isomorphisms above, this says that the choice of connective structure is unique up to a suitable notion of equivalence.) Proof. After possibly refining the ordered cover (U,I), we may choose sections s αβ in the complex line bundles L αβ over U αβ. The connections αβ from the connective structure on L act on these sections, so that αβ (s αβ ) = η αβ s αβ for unique complex 1-forms η αβ on U αβ. These combine to a Čech 1-cochain η Č 1 (U;A 1 X,C). It is not a Čech 1-cocycle, but its coboundary has a well-understood form, as we shall now see. By assumption, the connections αβ + βγ on L αβ L βγ and αγ on L αγ are compatible under φ αβγ over U αβγ. Since ( αβ + βγ )(s αβ s βγ ) = (η αβ +η βγ )(s αβ s βγ ) we get αγ (h αβγ s αγ ) = (η αβ +η βγ )(h αβγ s αγ ). At the same time αγ (h αβγ s αγ ) = dh αβγ s αγ +h αβγ αγ (s αγ ) = dh αβγ s αγ +h αβγ η αγ s αγ so that (η αβ +η βγ )h αβγ = dh αβγ +h αβγ η αγ which we can rewrite as η βγ η αγ +η αβ = (h αβγ ) 1 dh αβγ = dlogh αβγ. Hence the pair (h, η) forms a Čech 2-cocycle in the total complex Č (U;C X dlog A 1 X,C). We let [L, ] be its associated Čech cohomology class. ((Should check independence of choice of sections s αβ.))

13 CONNECTIVE STRUCTURES ON GERBES AND TWO-VECTOR BUNDLES 13 Connective structures on 2-vector bundles Definition. Let X be a smooth manifold, with an ordered open cover (U,I). Let E X be a smooth charted 2-vector bundle, with gluing bundle matrices E αβ and coherence isomorphism matrices φ αβγ. A connective structure on E is (1) a matrix of connections αβ = ( αβ ij )n i,j=1 on Eαβ over U αβ, for each α < β in I, such that (2) the matrix of isomorphisms φ αβγ : E αβ E βγ = E αγ over U αβγ takes the matrix product of connections αβ id+id βγ on E αβ Eβγ to the matrix of connections αγ on E αγ, so that over U αβγ. (φ αβγ ) ( αβ id+id βγ ) = αγ For matrices of sections s = (s ij ) n i,j=1 in Eαβ and t = (t jk ) n j,k=1 in Eβγ, with (s t) ik = n j=1 s ij t jk in n j=1 Eαβ ij E βγ, we have ( αβ id+id βγ )(s t) = αβ (s) t+s βγ (t) as 1-forms of matrices of sections in E αβ E βγ over U αβγ. In the (i,k)-th entry, this asserts that jk ( αβ id+id βγ ) ik (s t) ik = n ( αβ ij (s ij) t jk +s ij βγ jk (t jk)). j=1 Do 2-vector bundles admit connective structures? Proposition. Assume ((ETC)). Then E admits a connective structure. Proof. Give each E αβ a matrix αβ of connections. For each α < β < γ we get a unique End(E αγ )-valued 1-form ω αβγ on U αβγ, such that (φ αβγ ) ( αβ id+id βγ ) = αγ +ω αβγ. Here ω αβγ ik is a 1-form with values in the endomorphism bundle of E αγ ik, for each i,k = 1,...,n. We can change the connections on E αβ by End(E αβ )-valued 1-forms η αβ on U αβ, to αβ = αβ η αβ. Here η αβ ij is a 1-form with values in the endomorphism bundle of E αβ ij, for each i,j = 1,...,n. Then ( αβ id+id βγ ) ik = ( αβ id+id βγ ) ik n j=1 (η αβ ij id E βγ jk +id E αβ ij η βγ jk ) and αγ ik = αγ ik ηαγ ik.

14 14 NOTES BY JOHN ROGNES For the αβ to specify a connective structure on E, we need to have η αγ ik +(φαβγ ik ) n j=1 (η αβ ij id E βγ jk +id E αβ ij η βγ jk ) = ωαβγ ik, which we can rewrite as (φ αβγ ) (id E αβ η βγ ) η αγ +(φ αβγ ) (η αβ id E βγ) = ω αβγ. The left hand side gives a generalized Čech-style coboundary operator Ω 1 (U αβ ;End(E αβ )) d α<β α<β<γ Ω 1 (U αβγ ;End(E αγ )) taking the family of 1-forms η = (η αβ ) to dη. There is a similar operator α<β<γ Ω 1 (U αβγ ;End(E αγ )) d α<β<γ<δ Ω 1 (U αβγδ ;End(E αδ )) taking a general family of 1-forms ω = (ω αβγ ) to the family with value (φ αβδ ) (id E αβ ω βγδ ) ω αγδ +ω αβδ (φ αγδ ) (ω αβγ id E γδ) at α < β < γ < δ. Then dd = 0 and dω = 0, when ω is defined as at the beginning of the proof. ((Spell out the proofs.)) We would like to have ω = dη. ((Is the complex exact?)) ((ETC)) ((Easier to give a Proof II like in the 2-line bundle case?)) ((Can we specify a connective structure Co on the stack P with right GL n (V)- action, associated to a 2-vector bundle E?))