Chern-Simons theory onS1-bundles: abelianisation and q-deformed Yang-Mills theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation,previously employed for trivial bundles $\Sigma \times S^1$, can be adapted tothis case. This reduces the non-Abelian theory on $M$ to a 2-dimensionalAbelian 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 thoseobtained by Beasley and Witten using the method of non-Abelian localisation,and determine the surgery and framing presecription implicit in this pathintegral evaluation. We also comment on the extension of these methods to BFtheory and other generalisations.Comment: 37 pages; v2: references adde