An Atiyah-Singer theorem for gerbes

TL;DR

This paper aims to establish an Atiyah-Singer index theorem for gerbes, extending classical topological results to the setting where gerbes classify obstructions like the second Stiefel-Whitney class.

Contribution

It introduces a generalized Atiyah-Singer theorem applicable to gerbes, providing a new framework for topological and geometric analysis involving gerbes.

Findings

Formulation of an Atiyah-Singer theorem for gerbes

Connection between gerbes and topological invariants

Potential applications to spinor and index theory

Abstract

Let M be a riemannian manifold. The existence of a spin structure on M, enables to study the topology of M. The obstruction to the existence of the spin structure is given by the second Stiefel-Whitney class. This class is the classifying cocycle of a gerbe. One may expect that the study of this gerbe may have topological applications, for example, one may try to generalize the spinors Lichnerowicz theorem in this setting. On this purpose, we must first prove an Atiyah-Singer theorem for gerbes which is the main goal of this paper.