Equivariant differential characters and Chern-Simons bundles

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.