# Contents

## Motivation

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

then transgression gives a map on cohomology

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

Example

${ℤ}_{2}$-coefficients, $k=2$

$\begin{array}{ccc}{H}^{2}\left(X,{ℤ}_{2}\right)& \stackrel{\tau }{\to }& {H}^{1}\left(LX,{ℤ}_{2}\right)\\ \xi & ↦& \tau \left(\xi \right)\end{array}$\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 $\tau \left(\xi \right)$ vanishes. Atiyah showed that if the fundamental group ${\pi }_{1}\left(X\right)=1$ of $X$ vanishes, i.e. if $X$ is a simply connected space, that the also the converse holds: $X$ is spin if $\tau \left(\xi \right)$ vanishes in the cohomology of the loop space.

Questions

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

2. What are relation between trivializations of $\xi$ and those of $\tau \left(\xi \right)$ 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 ${H}^{2}\left(X,A\right)$ for the abelian sheaf cohomology.

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

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

Jean-Luc Brylinski and MacLaughlin define a functor

$L:{\mathrm{Grb}}_{A}^{\nabla }\left(X\right)\to {\mathrm{Bun}}_{A}^{\nabla }\left(LX\right)\phantom{\rule{thinmathspace}{0ex}}.$L : Grb_{A}^\nabla(X) \to Bun_A^\nabla(L X) \,.

by

$𝒢↦L𝒢{\mid }_{\beta }:={\mathrm{Hom}}_{{\mathrm{Grb}}_{A}^{\nabla }\left({S}^{1}\right)}\left({\beta }^{*}𝒢,{I}_{0}\right)$\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

${\gamma }_{i}:x\to y\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}i\in \left\{1,2,3\right\}$\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 $\left(P,\lambda \right)$ is called

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

2. symmetrizing, if

${R}_{\pi }\left(\lambda \left({q}_{1}\otimes {q}_{2}\right)\right)=\lambda \left({R}_{\pi }\left({q}_{2}\right)\otimes {R}_{\pi }\left({q}_{1}\right)\right)\phantom{\rule{thinmathspace}{0ex}},$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

$\begin{array}{ccc}P& \stackrel{}{\to }& P\\ ↓& & ↓\\ LX& \stackrel{{r}_{\pi }}{\to }& LX\end{array}$\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 $\varphi \in LLX$, $\stackrel{˜}{\varphi }:{S}^{1}×{S}^{1}\to X$ has rank one, then ${\mathrm{Hol}}_{P}\left(\varphi \right)=1$;

2. if ${\varphi }_{1},{\varphi }_{2}\in LLX$ such that ${\stackrel{˜}{\varphi }}_{1},{\stackrel{˜}{\varphi }}_{2}$ are rank-2-homotopic (i.e. think homotopic) then ${\mathrm{Hol}}_{p}\left({\varphi }_{1}\right)={\mathrm{Hol}}_{p}\left({\varphi }_{2}\right)$.

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

$\begin{array}{ccccc}{\mathrm{Grb}}_{A}^{\nabla }\left(X\right)& & \stackrel{\stackrel{˜}{K}}{\to }& & {\mathrm{FusBund}}_{A}^{\nabla }\left(LX\right)\\ & {}_{L}↘& & {↙}_{\mathrm{forget}}\\ & & {\mathrm{Bun}}_{A}^{\nabla }\left(LX\right)\end{array}$\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 $\stackrel{˜}{L}$ is an equivalence of categories

## Application: Spin structures and loop space orientation

Assume $𝒢$ is the ${ℤ}_{2}$-lifting gerbe for spin structure on $X$ whose characteristic class is

$\left[𝒢\right]=\xi \in {H}^{3}\left(X,{ℤ}_{2}\right)$[\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

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