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
Example
$\mathbb{Z}_2$-coefficients, $k = 2$
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
What is the relation between $\xi$ and $\tau(\xi)$ in general, that would make $\tau$ a bijection.
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.
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
by
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
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
compactible if $\lambda$ is connection-preserving;
symmetrizing, if
where $R_\pi$ is a lift of the
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.
superficial (German: oberflächlich – this is a joke with translations) if it behaves like a surface holonomy in that
if $\phi \in L L X$, $\tilde \phi : S^1 \times S^1 \to X$ has rank one, then $Hol_P(\phi) = 1$;
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
Theorem Lifted transgression $\tilde L$ is an equivalence of categories
Assume $\mathcal{G}$ is the $\mathbb{Z}_2$-lifting gerbe for spin structure on $X$ whose characteristic class is
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.
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)