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transgression of bundle gerbes

Contents

Motivation

Let XX be a smooth manifold, Write LX=C (S 1,X)L X = C^\infty(S^1,X) for the free loop space.

then transgression gives a map on cohomology

τ:H k(X)H k1(LX) \tau : H^k(X) \to H^{k-1}(L X)

Example

2\mathbb{Z}_2-coefficients, k=2k = 2

H 2(X, 2) τ H 1(LX, 2) ξ τ(ξ) \array{ H^2(X, \mathbb{Z}_2) &\stackrel{\tau}{\to}& H^1(L X, \mathbb{Z}_2) \\ \xi &\mapsto& \tau(\xi) }

where ξ\xi is the second Stiefel-Whitney class we have that XX has spin structure precisely if ξ=0\xi = 0 is the trivial class. This implies of course that also τ(ξ)\tau(\xi) vanishes. Atiyah showed that if the fundamental group π 1(X)=1\pi_1(X) = 1 of XX vanishes, i.e. if XX is a simply connected space, that the also the converse holds: XX is spin if τ(ξ)\tau(\xi) vanishes in the cohomology of the loop space.

Questions

  1. What is the relation between ξ\xi and τ(ξ)\tau(\xi) in general, that would make τ\tau a bijection.

  2. What are relation between trivializations of ξ\xi and those of τ(ξ)\tau(\xi) that would make τ\tau a functor – such that this makes transgression an equivalence of categories.

Transgression as a functor

Let AA be an abelian Lie group. Write H 2(X,A)H^2(X,A) for the abelian sheaf cohomology.

we want to realize this as the connected components of a 2-groupoid Grb A (X)Grb_A^\nabla(X) of bundle gerbes with connection on XX.

Similarly we want to refine H 1(LX,A)H^1(L X, A) to a groupoid Bun A (LX)Bun_A^\nabla(L X) of connections on smooth AA-principal bundles.

Jean-Luc Brylinski and MacLaughlin define a functor

L:Grb A (X)Bun A (LX). L : Grb_{A}^\nabla(X) \to Bun_A^\nabla(L X) \,.

by

𝒢L𝒢 β:=Hom Grb A (S 1)(β *𝒢,I 0) \mathcal{G} \mapsto L \mathcal{G}|_{\beta} := Hom_{Grb_A^\nabla(S^1)}(\beta^* \mathcal{G}, I_0)

for βLX\beta \in L X and where I 0I_0 denotes the trivial gerbe on the circle.

We want to understand the image of this transgression map, i.e. to characterize those bundles over LXL X that can be obtained by transgression of a gerbe on XX.

Definition Let PP be an AA-principal bundle over LXL X, then a fusion product on PP is a bundle isomorphism λ\lambda that is fiberwise given for a triple of paths

γ i:xy,i{1,2,3} \gamma_i : x \to y \,,\;\;\;\;\; i \in \{1,2,3\}
λ γ 1,γ 2,γ 3:P γ¯ 2γ 1P γ¯ 3γ 2P γ¯ 3γ 1 \lambda_{\gamma_1, \gamma_2,\gamma_3} : P_{\bar\gamma_2 \star \gamma_1} \otimes P_{\bar \gamma_3 \star \gamma_2} \to P_{\bar \gamma_3 \star \gamma_1}

Brylinske-MacLaughlin have a similar fusion product but over figur-8s of paths. This however gives associativity only up to homotopy. Here we are aiming for a product that is strictly associative.

Definition A connection on the fusion bundle (P,λ)(P,\lambda) is called

  1. compactible if λ\lambda is connection-preserving;

  2. symmetrizing, if

    R π(λ(q 1q 2))=λ(R π(q 2)R π(q 1)), R_\pi(\lambda(q_1 \otimes q_2)) = \lambda(R_\pi(q_2) \otimes R_\pi(q_1)) \,,

    where R πR_\pi is a lift of the

    P P LX r π LX \array{ P &\stackrel{}{\to}& P \\ \downarrow && \downarrow \\ L X &\stackrel{r_\pi}{\to}& L X }

    lifts the loop rotation operation by an angle π\pi from loop space to the bundle over loop space.

    We can take RR to be the parallel transport of the connection on loop space along the canonical path in loop space that connects a loop to its rotated loop.

  3. superficial (German: oberflächlich – this is a joke with translations) if it behaves like a surface holonomy in that

    1. if ϕLLX\phi \in L L X, ϕ˜:S 1×S 1X\tilde \phi : S^1 \times S^1 \to X has rank one, then Hol P(ϕ)=1Hol_P(\phi) = 1;

    2. if ϕ 1,ϕ 2LLX\phi_1, \phi_2 \in L L X such that ϕ˜ 1,ϕ˜ 2\tilde \phi_1, \tilde \phi_2 are rank-2-homotopic (i.e. think homotopic) then Hol p(ϕ 1)=Hol p(ϕ 2)Hol_p(\phi_1) = Hol_p(\phi_2).

Definition An AA-fusion bundle with connection over LXL X is an AA-principal bundle over LXL X with fusion product and compatible, symmetrizing and superficial connection.

Lemma Transgression lifts

Grb A (X) K˜ FusBund A (LX) L forget Bun A (LX) \array{ Grb_A^\nabla(X) &&\stackrel{\tilde K}{\to}&& FusBund_A^\nabla(L X) \\ & {}_{\mathllap{L}}\searrow && \swarrow_{\mathrlap{forget}} \\ && Bun_A^\nabla(L X) }

Theorem Lifted transgression L˜\tilde L is an equivalence of categories

Application: Spin structures and loop space orientation

Assume 𝒢\mathcal{G} is the 2\mathbb{Z}_2-lifting gerbe for spin structure on XX whose characteristic class is

[𝒢]=ξH 3(X, 2) [\mathcal{G}] = \xi \in H^3(X, \mathbb{Z}_2)

the Steifel-Whitney class? of XX. So spin structures on XX are in corresppndence with trivializations of 𝒢\mathcal{G}.

On the other hand we have that orientations of LXL X correspond to sections of L𝒢L \mathcal{G}. Inside there are the fusion preserving sections, which by the above are equivalent to trivializations of 𝒢\mathcal{G}.

So we find that in general spin structures on XX are not in bijection to just all orientations of LXL X, but precisely ot the fusion-compatible ones.

References

Revised on June 4, 2010 13:42:18 by David Corfield (86.139.50.131)