nLab
transgression of bundle gerbes

Motivation

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

then transgression gives a map on cohomology

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

Example

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

$$\begin{array}{ccc}{H}^2(X,{\mathbb{Z}}_2)& \stackrel{\tau }{\to }& H^1(LX,{\mathbb{Z}}_2)\\ \xi & \mapsto & \tau (\xi )\end{array}$$\array{ H^2(X, \mathbb{Z}_2) &\stackrel{\tau}{\to}& H^1(L X, \mathbb{Z}_2) \\ \xi &\mapsto& \tau(\xi) }

where $\xi$ is the second Stiefel-Whitney class we have that $X$ has spin structure precisely if $\xi =0$ is the trivial class. This implies of course that also $\tau (\xi )$ vanishes. Atiyah showed that if the fundamental group ${\pi }_1(X)=1$ of $X$ vanishes, i.e. if $X$ is a simply connected space, that the also the converse holds: $X$ is spin if $\tau (\xi )$ vanishes in the cohomology of the loop space.

Questions

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

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

Transgression as a functor

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

we want to realize this as the connected components of a 2-groupoid ${\mathrm{Grb}}_A^{\nabla }(X)$ of bundle gerbes with connection on $X$.

Similarly we want to refine $H^1(LX,A)$ to a groupoid ${\mathrm{Bun}}_A^{\nabla }(LX)$ of connections on smooth $A$-principal bundles.

Jean-Luc Brylinski and MacLaughlin define a functor

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

by

$$𝒢\mapsto L𝒢{\mid }_{\beta }:={\mathrm{Hom}}_{{\mathrm{Grb}}_A^{\nabla }(S^1)}({\beta }^{*}𝒢,I_0)$$\mathcal{G} \mapsto L \mathcal{G}|_{\beta} := Hom_{Grb_A^\nabla(S^1)}(\beta^* \mathcal{G}, I_0)

for $\beta \in LX$ and where $I_0$ denotes the trivial gerbe on the circle.

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

Definition Let $P$ be an $A$-principal bundle over $LX$, then a fusion product on $P$ is a bundle isomorphism $\lambda$ that is fiberwise given for a triple of paths

$${\gamma }_i:x\to y,i\in \{1,2,3\}$$\gamma_i : x \to y \,,\;\;\;\;\; i \in \{1,2,3\}

$${\lambda }_{{\gamma }_1,{\gamma }_2,{\gamma }_3}:P_{{\overline{\gamma }}_2\star {\gamma }_1}\otimes P_{{\overline{\gamma }}_3\star {\gamma }_2}\to P_{{\overline{\gamma }}_3\star {\gamma }_1}$$\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 $(P,\lambda )$ is called

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

2. symmetrizing, if

   $$R_{\pi }(\lambda (q_1\otimes q_2))=\lambda (R_{\pi }(q_2)\otimes R_{\pi }(q_1)),$$R_\pi(\lambda(q_1 \otimes q_2)) = \lambda(R_\pi(q_2) \otimes R_\pi(q_1)) \,,

   where $R_{\pi }$ is a lift of the

   $$\begin{array}{ccc}P& \stackrel{}{\to }& P\\ \downarrow & & \downarrow \\ LX& \stackrel{r_{\pi }}{\to }& LX\end{array}$$\array{ P &\stackrel{}{\to}& P \\ \downarrow && \downarrow \\ L X &\stackrel{r_\pi}{\to}& L X }

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

   We can take $R$ 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 $\varphi \in LLX$, $\tilde{\varphi }:S^1\times S^1\to X$ has rank one, then ${\mathrm{Hol}}_P(\varphi )=1$;

   2. if ${\varphi }_1,{\varphi }_2\in LLX$ such that ${\tilde{\varphi }}_1,{\tilde{\varphi }}_2$ are rank-2-homotopic (i.e. think homotopic) then ${\mathrm{Hol}}_p({\varphi }_1)={\mathrm{Hol}}_p({\varphi }_2)$.

Definition An $A$-fusion bundle with connection over $LX$ is an $A$-principal bundle over $LX$ with fusion product and compatible, symmetrizing and superficial connection.

Lemma Transgression lifts

$$\begin{array}{ccccc}{\mathrm{Grb}}_A^{\nabla }(X)& & \stackrel{\tilde{K}}{\to }& & {\mathrm{FusBund}}_A^{\nabla }(LX)\\ & {}_L\searrow & & {\swarrow }_{\mathrm{forget}}\\ & & {\mathrm{Bun}}_A^{\nabla }(LX)\end{array}$$\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 $\tilde{L}$ is an equivalence of categories

Application: Spin structures and loop space orientation

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

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

the Steifel-Whitney class? of $X$. So spin structures on $X$ are in corresppndence with trivializations of $𝒢$.

On the other hand we have that orientations of $LX$ correspond to sections of $L𝒢$. Inside there are the fusion preserving sections, which by the above are equivalent to trivializations of $𝒢$.

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

References

• Konrad Waldorf, Transgression to Loop Spaces and its Inverse

  I: Diffeological Bundles and Fusion Maps (arXiv:0911.3212)

  II: Gerbes and Fusion Bundles with Connection (arXiv:1004.0031)