Abelianisation and q-deformed Yang-Mills Theory

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2dimensional surfaces Σ and show that the method of Abelianisation, previously employed for trivial bundles Σ× S, can be adapted to this case. This reduces the non-Abelian theory onM 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. Dedicated to Bob Delbourgo on the occassion of his 65th Birthday e-mail: matthias.blau(at)unine.ch e-mail: thompson(at)ictp.it