Schreiber
principal 2-bundle with connection

This entry discusses in the general context of differential nonabelian cohomology the special case of GG-principal 2-bundles with connection: differential cocycles in H¯(X,BG)\bar \mathbf{H}(X, \mathbf{B}G) for GG a 2-group.

entry under construction. See blog discussion here

Contents

Idea

For GG a 2-group, a GG-principal 2-bundle is the first step in the generalization of principal bundles to principal ∞-bundles.

For instance

  • for G=BAG = \mathbf{B}A the delooping of an abelian group AA, GG-principal 2-bundles are equivalent to AA-bundle gerbes;

  • for G=AUT(H)G = AUT(H) the automorphism 2-group of an ordinary group HH, GG-principal bundles are equivalent to HH-gerbes.

But 2-bundles are a bit more general than these examples. For instance for GG the string Lie 2-group, GG-principal 2-bundles are smooth realizations of String structures on a space, in analogy to how ordinary Spin(n)Spin(n)-principal bundles are realizations of Spin structures. Part of the interest in principal 2-bundles derives from this example, which is believed to be an ingredient in geometric models for elliptic cohomology.

Principal 2-bundles had also been the testing ground for definitions of higher generalizations of the notion of connection on a bundle in the nonabelian situation. Proposals for definitions of the notion of connections on general 2-bundles or at least gerbes were put forward in BrMe and AsJu . But it remained unclear how to define parallel transport for these. In BaSc it was suggested that parallel transport in a 2-bundle should be a refinement of the cocycle XBGX \to \mathbf{B}G of the bundle to a cocylce P 2(X)BG\P_2(X) \to \mathbf{B}G on the path 2-groupoid of XX. It was shown in BaSc, ScWaiII that this is equivalent to the data proposed by BrMe and AsJuexcept that the 2-form curvature? appearing was constrained to vanish.

This led to some puzzlement, which in turn eventually led to the idea put forward here – at theory of differential nonabelian cohomology – , that a fully general differential cocycle is not an ordinary cocycle on a path n-groupoid, but instead a cocycle in twisted cohomology measuring the obstruction to having a cocycle on the full path ∞-groupoid Π(X)\Pi(X).

Below we will first recall the ordinary cocycles on the path 2-groupoid from BaSc and ScWaIII . Then we unwrap the full machinery of differential nonabelian cohomology and show how this induces a notion of connectoin data on principal 2-bundles which has no constraints on the curvature? forms and is essentially the data proposed by BrMe and AsJu .

Context and notation

Throughout this page we fix the following choices and notation

For most of the development we can assume, for simplicity and definiteness, that H\mathbf{H} is presented by the model structure on simplicial presheaves on Diff – and even simpler and even more definite, all of the objects that we will encounter are strict 2-groupoids or at worst Gray-groupoids (semistrict 3-groupoids) internal to diffeological spaces.

Details

Recall from differential cohomology - nonabelian case? that a differential refinement in H¯(X,BG)\bar \mathbf{H}(X,\mathbf{B}G) of a cocycle g:YBGg : Y \to \mathbf{B}G of a principal 2-bundle for some cover YXY \stackrel{\simeq}{\to} X is a diagram

Y g BG Π(Y) tra ϵBG Π(X) ΣBG \array{ Y &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow && \downarrow \\ \Pi(Y) &\stackrel{tra}{\to}& \epsilon \mathbf{B}G \\ \downarrow && \downarrow \\ \Pi(X) &\to& \Sigma \mathbf{B}G }

where ϵBG\epsilon \mathbf{B}G is the cone on BG\mathbf{B}G and ΣBG\Sigma \mathbf{B}G its suspension.

Expressing this in terms of ∞-Lie algebroid valued differential forms by extracting the corresponding Cartan-Ehresmann ∞-connection data yields a diagram of dg-algebras

Ω (Y) A vert CE(𝔤) Ω (Y) (A,F A) W(𝔤) Ω (X) inv(𝔤). \array{ \Omega^\bullet(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\leftarrow& inv(\mathfrak{g}) } \,.

To express the explicit data encoded by a connection on a principal 2-bundle, we will in the following describe the data encoded by the horizontal morphism Π(Y)ϵ𝔤\Pi(Y) \to \epsilon \mathfrak{g} and then characterize the constraints imposed on it by the requirement that it sits in a diagram as shown.

strict structure 2-group

We concentrate first on the special case that the structure group GG is a strict 2-group.

group- and Lie-theoretic data

Let GG is a strict structure 2-group, i.e. BG\mathbf{B}G is a strict ∞-Lie groupoid. This is equivalently encoded in a crossed module

(G 2δG 1,G 1αAut(G 2)) (G_2 \stackrel{\delta}{\to} G_1, G_1 \stackrel{\alpha}{\to} Aut(G_2))

in diffeological spaces.

The corresponding Lie 2-algebra 𝔤\mathfrak{g} is equivalent to a differential crossed module

(:𝔤 2𝔤 1,ρ:𝔤 1Der(𝔤 2) (\partial : \mathfrak{g}_2 \to \mathfrak{g}_1, \rho : \mathfrak{g}_1 \to Der(\mathfrak{g}_2)

and is characterized by its sheaf of ∞-Lie algebroid valued differential forms as

Ω flat (,𝔤):U((A,B)Ω 1(U,𝔤 1)×Ω 2(U,𝔤 2)F A:=dA+[AA]=δ *B;H:=d AB=0). \Omega_{flat}^\bullet(-,\mathfrak{g}) : U \mapsto \left( (A,B) \in \Omega^1(U, \mathfrak{g}_1) \times \Omega^2(U, \mathfrak{g}_2) | F_A := d A + [A \wedge A] = - \delta_* B; H := d_A B = 0 \right) \,.

the cone structure group

A useful realization of the cone object ϵBG\epsilon \mathbf{B}G as a Gray-groupoid BINN(G)\mathbf{B}INN(G) was given in [RoSc].

Its Lie 3-algebra is inn(𝔤)inn(\mathfrak{g}) with CE(inn(𝔤))=W(𝔤)CE(inn(\mathfrak{g})) = W(\mathfrak{g}).

This Lie 3-algebra is such that

Hom(CE(inn(𝔤)),Ω (X)) ={(A,B,β,H)Ω 1(X,𝔤 1)×Ω 2(X,𝔤 2)×Ω 2(X,𝔤 1)×Ω 3(X,𝔤 2)β=dA+[AA]+B;H=dB+ρ(AB)} Ω 1(X,𝔤 1)×Ω 2(X,𝔤 2). \begin{aligned} Hom(CE(inn(\mathfrak{g})), \Omega^\bullet(X)) &= \left\{ (A,B,\beta,H) \in \Omega^1(X,\mathfrak{g}_1) \times \Omega^2(X,\mathfrak{g}_2) \times \Omega^2(X,\mathfrak{g}_1) \times \Omega^3(X,\mathfrak{g}_2) | \beta = d A + [A \wedge A] + \partial B; H = d B + \rho(A \wedge B) \right\} \\ &\simeq \Omega^1(X,\mathfrak{g}_1) \times \Omega^2(X,\mathfrak{g}_2) \end{aligned} \,.

a truncation of the path \infty-groupoid

Using the Gray-groupoid model BINN(G)\mathbf{B}INN(G) for cone(BG)cone(\mathbf{B}G) we may without loss generality replace morphisms Π(Y)BINN(G)\Pi(Y) \to \mathbf{B}INN(G) out of the path ∞-groupoid by morphisms out of a Gray-groupoid truncation Π 3(Y)\Pi_3(Y). An explicit such path 3-groupoid P 3(Y)P_3(Y) has been described in [MaPi], we obtain Π 3(Y)\Pi_3(Y) from this by identifying 3-morphisms that are given by homotopic maps.

Also described explicitly there is how Lie 3-algebra valued form data such as our Ω (Y)CE(inn(𝔤))\Omega^\bullet(Y) \leftarrow CE(inn(\mathfrak{g})) integrates, using the above discussion, to morphisms Π 3(Y)BINN(G)\Pi_3(Y) \to \mathbf{B}INN(G).

the cover of the path \infty-groupoid

Half of the work of extracting the explicit data of a differential cocycle is done by exhibiting a generators-and-relations presentation of the cover Π(Y)Π(X)\Pi(Y) \stackrel{\simeq}{\to} \Pi(X) of the path ∞-groupoid Π(X)\Pi(X) that is induced from the given cover YXY \stackrel{\simeq}{\to} X.

For P 2(Y)P 2(X)P_2(Y) \to P_2(X) this is done in great detail in [BaSc, ScWaIII] and the description of Π 3(Y)\Pi_3(Y) is entirely analogous. In fact, since by the very nature of the diagrams that define our cocycles all degree 3-generators of Π 3(Y)\Pi_3(Y) that are not given by 3-dimensional paths in UU will be mapped to the 2-groupoid BG\mathbf{B}G and hence to identies, we can ignore all of them. The remaining generators and relations are precisely those of P 2(Y)P_2(Y) together with the 3-dimensional paths in UU.

Here is a list of the generators

  • for each point (x,i,j,k)U iU jU k(x,i,j,k) \in U_{i} \cap U_j \cap U_k in a triple overlap a 2-morphism

    (x,j) (x,i) (x,k) \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) }
  • for each path (γ,i,j)P(U iU j)(\gamma,i,j) \in P (U_i \cap U_j) in a double overlap a 2-morphism

    (x,i) (x,j) (γ i) γ j (y,i) (y,j) \array{ (x,i) &\to& (x,j) \\ \downarrow^{(\gamma_i)} &\Downarrow& \downarrow^{\gamma_{j}} \\ (y,i) &\to& (y,j) }
  • for each 2-path (Σ,i)P 2U i(\Sigma, i) \in P^2 U_i in a single patch a 2-morphism

    (γ,i,j) (x,i,j) (Σ,i,j) (y,i,j) (γ,i,j) \array{ & \nearrow \searrow^{(\gamma,i,j)} \\ (x,i,j) &\Downarrow^{(\Sigma,i,j)}& (y,i,j) \\ & \searrow \nearrow_{(\gamma,i,j)} }
  • and some others

cocycle data

We shall assume now for definiteness that the object YY covering our based space XX by a weak equivalence YXY \stackrel{\simeq}{\to} X is the Cech nerve

Y:=( ijkU ijk ijU ij iU i) Y := \left( \cdots \coprod_{i j k} U_{i j k} \stackrel{\stackrel{\to}{\to}}{\to} \coprod_{i j} U_{i j} \stackrel{\to}{\to} \coprod_i U_i \right)

of a Cech cover U= iU iXU = \coprod_i U_i \to X. We could also consider more general covers and in particular could consider hypercovers, but this only makes the notation less transparent without yielding new insights for our purpose of extracting the nature of differential cocycle data on a principal 2-bundle.

So then let

g:YBG g : Y \to \mathbf{B}G

be cocycle on YY for some GG-principal 2-bundle.

vanishing 2-form curvature

Before describing the general case, it is useful to recall the cocycle data of connections on principal 2-bundles for which the 2-form curvature? vanishes. Writing P 2(Y)P_2(Y) for the path 2-groupoid, these are given by extensions of the cocycle gg of the form

Y g BG = P 2(Y) tra (A,B),g BG. \array{ Y &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow && \downarrow^= \\ P_2(Y) &\stackrel{tra_{(A,B), g}}{\to}& \mathbf{B}G } \,.

Detailed description of this is in [BaSc, ScWaIII].

general case

We shall now demonstrate that the cocycle data for a general connection on a principal 2-bundle, in the sense of a general element in H¯(X,BG)\bar \mathbf{H}(X,\mathbf{B}G) is given by the same data as above, the only difference being that the 2-form curvature? is not required to vanish.

Proposition

The cocycle data of an element in H¯(X,BG)\bar \mathbf{H}(X,\mathbf{B}G) is like that above, except that the 2-form β\beta is not required to vanish.

Proof

On the generators of Π 3(Y)\Pi_3(Y) cominmg from paths in UU the morphism tra (A,B),g:Π 3(Y)BINN(G)tra_{(A,B),g} : \Pi_3(Y) \to \mathbf{B}INN(G) comes from differential form data

(A,B), (A,B),

with unconstrained curvature? form

(β=F A+B,H=d AB) (\beta = F_A + \partial B, H = d_A B)

The first \infty-Ehresmann condition says that there is a commuting square

Y g BG Π(Y) (A,B) g BINN(G) \array{ Y &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow && \downarrow \\ \Pi(Y) &\stackrel{\nabla_{(A,B)}^g}{\to}& \mathbf{B} INN(G) }

let

(x,j) (x,i) (x,k) \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) }

be one of the generating 2-cells in P 3(Y)P_3(Y). Then this condition says that evaluated on this 2-cell tra (A,B),gtra_{(A,B),g} has to reproduce the cocycle gg, with values in BG\mathbf{B}G.

The second Ehresmann condition says that all curvature characteristic forms? obtained from invariant polynomials Pinv()P \in inv(\mathcal{g}) by P(β,H)Ω (U)P(\beta,H) \in \Omega^\bullet(U) descent to forms on XX.

Ω (Y) W(𝔤) Ω (X) inv(𝔤) \array{ \Omega^\bullet(Y) &\leftarrow& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\leftarrow & inv(\mathfrak{g}) }

This implies that for any 2-morphism generator coming from a surface in a double overlap with 2-morphism generators coming from the corresponding paths in double overlaps

(γ,i,j) (x,i,j) (Σ,i,j) (y,i,j) (γ,i,j) \array{ & \nearrow \searrow^{(\gamma,i,j)} \\ (x,i,j) &\Downarrow^{(\Sigma,i,j)}& (y,i,j) \\ & \searrow \nearrow_{(\gamma,i,j)} }

we have that P(β i,H i)=P(β j,H j)P(\beta_i,H_i) = P(\beta_j, H_j). By the very definintion of ing()W()ing(\mathcal{g}) \hookrightarrow W(\mathcal{g}) this is the case if the transformation relating (A i,B i)(A_i, B_i) with (A j,B j)(A_j, B_j) is given by transition functions in the image of the inclusion GINN(G)G \hookrightarrow INN(G).

This transformation is given by conjugating with the value of (A,B) g\nabla_{(A,B)}^{g} on 2-cells of the form

(x,i) (x,j) (γ i) γ j (y,i) (y,j). \array{ (x,i) &\to& (x,j) \\ \downarrow^{(\gamma_i)} &\Downarrow& \downarrow^{\gamma_{j}} \\ (y,i) &\to& (y,j) } \,.

So the second Ehresmann condition says that that (A,B) g\nabla_{(A,B)}^g also colors these 2-cells with elements in the image of the inclusion BGBINN(G)\mathbf{B}G \hookrightarrow \mathbf{B} INN(G)

weak structure 2-group

Revised on September 29, 2009 17:30:08 by Urs Schreiber (195.37.209.182)