Differential characters and cohomology of the moduli of flat Connections

TL;DR

This paper develops a homology map using Chern-Simons differential characters to study the cohomology of the moduli space of flat connections, linking it to topological quantum field theories and geometric holonomy.

Contribution

It introduces a new homology map based on differential characters for the moduli space of flat connections, connecting it to Chern-Simons theory and geometric holonomy interpretations.

Findings

Defined a homology map using differential characters for flat connection moduli

Related first order characters to Dijkgraaf-Witten action

Interpreted second order characters as holonomy of line bundle connections

Abstract

Let $\pi\colon P\to M$ be a principal bundle and $p$ an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map $\chi^{k} : H_{2r-k-1}(M)\times H_{k}(\mathcal{F}/\mathcal{G})\to \mathbb{R}/\mathbb{Z}$, for $k<r-1$, where $\mathcal{F} /\mathcal{G}$ is the moduli space of flat connections of $\pi$ under the action of a subgroup $\mathcal{G}$ of the gauge group. The differential characters of first order are related to the Dijkgraaf-Witten action for Chern-Simons Theory. The second order characters are interpreted geometrically as the holonomy of a connection in a line bundle over $\mathcal{F}/\mathcal{G})$. The relationship with other constructions in the literature is also analyzed.