Equivariant prequantization bundles on the space of connections and characteristic classes

Abstract

We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group and the automorphisms group. The action of the elements in the connected component of the identity of the group generalizes known results in the literature. The action of the elements not connected with the identity is shown to be determined by a characteristic class by using differential characters and equivariant cohomology. We extend our results to the space of Riemannian metrics and the actions of diffeomorphisms. In dimension 2, a ${\Gamma}_{M}$-equivariant prequantization bundle of the Weil-Petersson symplectic form on the Teichm\"uller space is obtained, where ${\Gamma}_{M}$ is the mapping class group of the surface M.