nLab transgression of bundle gerbes

Motivation

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

then transgression gives a map on cohomology

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

Example

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

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

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 $Grb_A^\nabla(X)$ of bundle gerbes with connection on $X$.

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

Jean-Luc Brylinski and MacLaughlin define a functor

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

by

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

for $\beta \in L X$ 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 $L X$ that can be obtained by transgression of a gerbe on $X$.

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

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

$$\lambda_{\gamma_1, \gamma_2,\gamma_3} : P_{\bar\gamma_2 \star \gamma_1} \otimes P_{\bar \gamma_3 \star \gamma_2} \to P_{\bar \gamma_3 \star \gamma_1}$$

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

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

   where $R_\pi$ is a lift of the

   $$\array{ P &\stackrel{}{\to}& P \\ \downarrow && \downarrow \\ L X &\stackrel{r_\pi}{\to}& L X }$$

   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 $\phi \in L L X$, $\tilde \phi : S^1 \times S^1 \to X$ has rank one, then $Hol_P(\phi) = 1$;

   2. if $\phi_1, \phi_2 \in L L X$ such that $\tilde \phi_1, \tilde \phi_2$ are rank-2-homotopic (i.e. think homotopic) then $Hol_p(\phi_1) = Hol_p(\phi_2)$.

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

Lemma Transgression lifts

$$\array{ Grb_A^\nabla(X) &&\stackrel{\tilde K}{\to}&& FusBund_A^\nabla(L X) \\ & {}_{\mathllap{L}}\searrow && \swarrow_{\mathrlap{forget}} \\ && Bun_A^\nabla(L X) }$$

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

Application: Spin structures and loop space orientation

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

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

the Stiefel-Whitney class of $X$. So spin structures on $X$ are in corresppndence with trivializations of $\mathcal{G}$.

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

So we find that in general spin structures on $X$ are not in bijection to just all orientations of $L X$, 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)

For an extension of this formalism to include D-branes, see

• Severin Bunk, Konrad Waldorf, Transgression of D-branes, Adv. Theor. Math. Phys. 25(5):1095-1198 (2021), (arXiv:1808.04894)