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

TL;DR

This work extends Abelianisation of Chern-Simons theory to non-trivial circle bundles $M_{(g,p)}$, showing the 3D non-Abelian theory reduces to a 2D Abelian theory on the base $Σ_g$ that is equivalent to $q$-deformed Yang-Mills. It derives a finite-dimensional integral expression for the partition function, including Ray-Singer torsion factors and a sum over torsion line bundles, and demonstrates the equivalence between compact and non-compact formulations of the resulting YM-like theory. The results recover a surgery framing structure with $K^{(p)}=(TST)^p$ and connect to known S- and T-matrix data of the WZW model, providing a direct path-integral derivation of the Vafa–AOSV perspective on $q$-deformed YM. The paper also sketches generalisations to BF theory and other 3-manifolds, outlining potential Lagrangian realizations and future extensions.

Abstract

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $Σ$ and show that the method of Abelianisation, previously employed for trivial bundles $Σ\times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $Σ$ 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.