Localization for Chern-Simons on Circle Bundles via Loop Groups

TL;DR

This paper establishes a new 2D topological quantum field theory called Caloron BF theory, derived from Chern-Simons theory on circle bundle 3-manifolds, using loop group techniques and the Caloron correspondence.

Contribution

It introduces Caloron BF theory as a novel 2D TQFT equivalent to Chern-Simons on circle bundles, utilizing the Caloron correspondence and loop group structures.

Findings

Caloron BF theory provides a new formulation of Chern-Simons on circle bundles.

The symplectic structure relates to the Atiyah-Bott construction in 2D Yang-Mills.

Wilson loops wrapping fibers are naturally described in this framework.

Abstract

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.