Localization for Chern-Simons on circle bundles via loop groups

We consider Chern-Simons theory on a 3-manifold M that is the total space of a circle bundle over a 2d base Σ. We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level k central extension of the loop group LG. The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley-Witten about the equivariant localization of the Chern-Simons path integral. The main tool that we employ is the Caloron correspondence, originally due to Murray-Garland, that relates the space of gauge fields on M with a certain enlarged space of connections on an equivariant version of the loop space of the G-bundle. We show that the symplectic structure that Beasley-Witten found is related to a looped version of the Atiyah-Bott construction in 2-dimensional Yang-Mills theory. We also show that Wilson loops that wrap a single circle fiber are also described very naturally in this framework.