Bundle gerbes and moduli spaces

TL;DR

This paper constructs and compares geometric and analytic bundle gerbes related to Dirac operators and applies these to moduli spaces of Riemann surfaces, revealing their topological invariants.

Contribution

It introduces the caloron bundle gerbe and demonstrates its isomorphism with the index bundle gerbe, extending the geometric understanding of moduli space invariants.

Findings

Constructed the index bundle gerbe refining Segal's work.

Defined the caloron bundle gerbe with natural connection and curving.

Identified the 3-curvature as generators of third de Rham cohomology groups.

Abstract

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.