  We consider Chern-Simons theory on 3-manifold $M$ that is the total space of
a circle bundle over a 2d base $\Sigma$. 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.
