Equivariant holonomy for bundles and abelian gerbes

TL;DR

This paper extends the equivariant Chern character to abelian gerbes, introducing a local higher Hochschild complex to explicitly compute 2-holonomy and its derivatives, with implications for topological field theories.

Contribution

It introduces a new local higher Hochschild complex framework to explicitly compute equivariant holonomy for abelian gerbes, linking to topological field theories.

Findings

Explicit construction of equivariantly closed forms for abelian gerbes.

Calculation of 2-holonomy along closed surfaces.

Derivation of the differential of 2-holonomy.

Abstract

This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe along any closed oriented surface, as well as the derivative of 2-holonomy, which in the case of a torus fits into a sequence of higher holonomies and their differentials.