We derive non-Abelian localization formulas for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold $M$, which is the three-dimensional space of a circle bundle over a two-dimensional Riemann surface $\Sigma$, by using the cohomological approach introduced by Källén. We find that the partition function and the vacuum expectation value of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find that the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on $S^1\times \Sigma$. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the Aharony-Bergman-Jafferis-Maldacena theory deduced from the localization.

• Received 16 July 2012

DOI:https://doi.org/10.1103/PhysRevD.86.105018