Let be a smooth manifold, Write for the free loop space.
then transgression gives a map on cohomology
where is the second Stiefel-Whitney class we have that has spin structure precisely if is the trivial class. This implies of course that also vanishes. Atiyah showed that if the fundamental group of vanishes, i.e. if is a simply connected space, that the also the converse holds: is spin if vanishes in the cohomology of the loop space.
What is the relation between and in general, that would make a bijection.
What are relation between trivializations of and those of that would make a functor – such that this makes transgression an equivalence of categories.
Transgression as a functor
Let be an abelian Lie group. Write for the abelian sheaf cohomology.
we want to realize this as the connected components of a 2-groupoid of bundle gerbes with connection on .
Similarly we want to refine to a groupoid of connections on smooth -principal bundles.
Jean-Luc Brylinski and MacLaughlin define a functor
for and where denotes the trivial gerbe on the circle.
We want to understand the image of this transgression map, i.e. to characterize those bundles over that can be obtained by transgression of a gerbe on .
Definition Let be an -principal bundle over , then a fusion product on is a bundle isomorphism 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 is called
compactible if is connection-preserving;
where is a lift of the
lifts the loop rotation operation by an angle from loop space to the bundle over loop space.
We can take 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 , has rank one, then ;
if such that are rank-2-homotopic (i.e. think homotopic) then .
Definition An -fusion bundle with connection over is an -principal bundle over with fusion product and compatible, symmetrizing and superficial connection.
Lemma Transgression lifts
Theorem Lifted transgression is an equivalence of categories
Application: Spin structures and loop space orientation
Assume is the -lifting gerbe for spin structure on whose characteristic class is
the Stiefel-Whitney class of . So spin structures on are in corresppndence with trivializations of .
On the other hand we have that orientations of correspond to sections of . Inside there are the fusion preserving sections, which by the above are equivalent to trivializations of .
So we find that in general spin structures on are not in bijection to just all orientations of , but precisely ot the fusion-compatible ones.