Lifting Problems and Transgression for Non-Abelian Gerbes

TL;DR

This paper develops a geometric framework for non-abelian gerbes and cohomology, introduces a transgression map linking manifold and loop space cohomology, and applies these to string structures and spin geometry.

Contribution

It provides a new geometric proof of Breen's exact sequence, defines a novel transgression map in non-abelian cohomology, and connects string structures with loop space spin structures.

Findings

Established a geometric formulation of Breen's long exact sequence.

Defined a transgression map relating non-abelian cohomology of manifolds and loop spaces.

Applied results to compare string structures and relate them to spin structures on loop spaces.

Abstract

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces - in case of a Lie 2-group with a single object - to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structure on the loop space.