Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory

TL;DR

This paper extends Abelianisation techniques for Chern-Simons theory on circle-bundled 3-manifolds, connecting it to q-deformed Yang-Mills theory and comparing with non-Abelian localisation results.

Contribution

It adapts Abelianisation to non-trivial circle bundles, linking 3D Chern-Simons to 2D q-deformed Yang-Mills, and clarifies the surgery and framing prescriptions involved.

Findings

Reduction of non-Abelian Chern-Simons to Abelian theory on the base surface

Identification of the reduced theory with q-deformed Yang-Mills

Comparison with non-Abelian localisation methods and framing prescriptions

Abstract

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.