# Equivariant differential characters and Chern-Simons bundles

## TL;DR

This paper constructs Chern-Simons bundles as equivariant $U(1)$-bundles with connections over the space of connections, linking their holonomy to equivariant differential characters, and extends the theory to diffeomorphism actions and Teichmüller space.

## Contribution

It introduces a novel construction of Chern-Simons bundles as equivariant differential characters, generalizing to diffeomorphism actions and Teichmüller space.

## Key findings

- Chern-Simons bundles are classified by equivariant holonomies.
- The construction applies to compact even-dimensional manifolds.
- Extension to diffeomorphism actions and Teichmüller space is achieved.

## Abstract

We construct Chern-Simons bundles as $\mathrm{Aut}^{+}P$-equivariant $U(1)$ -bundles with connection over the space of connections $\mathcal{A}_{P}$ on a principal $G$-bundle $P\rightarrow M$. We show that the Chern-Simons bundles are determined up to an isomorphisms by means of its equivariant holonomy. The space of equivariant holonomies is shown to coincide with the space of equivariant differential characteres of second order. Furthermore, we prove that the Chern-Simons theory provides, in a natural way, an equivariant differential character that determines the Chern-Simons bundles. Our construction can be applied in the case in which $M$ is a compact manifold of even dimension and for arbitrary bundle $P$ and group $G$. The results are also generalized to the case of the action of diffeomorphisms on the space of Riemannian metrics. In particular, in dimension $2$ a Chern-Simons bundle over the Teichm\"{u}ller space is obtained.