A Loop Space Formulation for Geometric Lifting Problems

TL;DR

This paper introduces a novel geometric formulation of lifting problems using loop space theory, linking bundle gerbes, their transgression, and spin structures on manifolds.

Contribution

It combines two aspects of bundle gerbe theory to provide a new loop space-based approach to lifting problems and spin structures.

Findings

New loop space formulation of lifting problems

Clarifies relation between spin structures and loop space orientations

Integrates transgression of gerbes with lifting bundle gerbes

Abstract

We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.