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

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

2006-01-11

arxiv.org/abs/hep-th/0601068v2

JHEP0605:003,2006

Physics

High Energy Physics

High Energy Physics - Theory

37 pages; v2: references added

Scientific paper

10.1088/1126-6708/2006/05/003

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.

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