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

Abstract

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.

37 pages; v2: references added

References in corpus (2)

• Two Dimensional Yang-Mills, Black Holes and Topological Strings
• A DK Phase Transition in q-Deformed Yang-Mills on S^2 and Topological Strings

Cited by in corpus (18)

• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory
• Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions
• Chern-Simons Invariants from Ensemble Averages
• Keeping matter in the loop in dS${}_3$ quantum gravity
• G/G gauged WZW model and Bethe Ansatz for the phase model
• Self-dual Strings and 2D SYM
• Large N reduction for Chern-Simons theory on S^3
• A Novel Large-N Reduction on S^3: Demonstration in Chern-Simons Theory
• Large N Expansion of q-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras
• Topological Strings, Two-Dimensional Yang-Mills Theory and Chern-Simons Theory on Torus Bundles
• Remarks on Wilson Loops and Seifert Loops in Chern-Simons Theory
• New phase in Chern-Simons theory on lens space
• Topological phase, spin Chern-Simons theory and level rank duality on lens space
• Contact 4d Chern-Simons theory: Generalities
• 3 Definitions of BF Theory on Homology 3-Spheres
• Some Lower Dimensional Quantum Field Theories Reduced from Chern-Simons Gauge Theories
• Massive Ray-Singer Torsion and Path Integrals