Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold

TL;DR

This paper develops non-Abelian localization formulas for supersymmetric Yang-Mills-Chern-Simons theories on Seifert manifolds, simplifying the computation of partition functions and Wilson loop expectations to finite sums and integrals, and confirms results with previous predictions.

Contribution

It introduces a cohomological localization approach for these theories on Seifert manifolds, enabling exact calculations of partition functions and indices, and explores vacuum structures in ABJM theory.

Findings

Partition function reduces to finite-dimensional integrals and sums.

Exact evaluation of the supersymmetric index on S^1×Σ.

Agreement of the index with previous field theory and brane predictions.

Abstract

We derive non-Abelian localization formulae 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 Kallen. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find 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^1x\Sigma. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the localization.