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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...