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We consider Chern-Simons theory on 3-manifold $M$ that is the total space ofa circle bundle over a 2d base $\Sigma$. We show that this theory is equivalentto a new 2d TQFT on the base, which we call Caloron BF theory, that can beobtained by an appropriate type of push-forward. This is a gauge theory on abundle with structure group given by the full affine level $k$ centralextension of the loop group $LG$. The space of fields of this 2d theory isnaturally symplectic, and this provides a new formulation of a result ofBeasley-Witten about the equivariant localization of the Chern-Simons pathintegral. 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 theloop space of the $G$-bundle. We show that the symplectic structure thatBeasley-Witten found is related to a looped version of the Atiyah-Bottconstruction in 2-dimensional Yang-Mills theory. We also show that Wilson loopsthat wrap a single circle fiber are also described very naturally in thisframework.

Localization for Chern-Simons on Circle Bundles via Loop Groups