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

Matthias Blau·George Thompson
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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.

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Matthias Blau, George Thompson. 2006-02-05. Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory. https://doi.org/10.1088/1126-6708%2F2006%2F05%2F003

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