N=1 Chern-Simons-matter theory and localization

TL;DR

This paper studies a specific three-dimensional N=1 Chern-Simons-matter theory with global symmetry, showing it admits localization on T^3 and analyzing its partition function contributions.

Contribution

It introduces a more general N=1 Chern-Simons-matter theory with an extra free parameter and performs partial localization analysis on T^3.

Findings

The theory admits localization on T^3.

Saddle points with zero gauge connection give trivial partition function contribution.

Bosonic and fermionic contributions cancel exactly.

Abstract

We consider the most general, classically-conformal, three-dimensional $\mathcal{N}=1$ Chern-Simons-matter theory with global symmetry $Sp(2)$ and gauge group $U(N)\times U(N)$. We show that the Lagrangian in the on-shell formulation of the theory admits one more free parameter as compared to the theory formulated in off-shell $\mathcal{N}=1$ superspace. The theory on $T^3$ can be formally localized. We partially carry out the localization procedure for the theory on $T^3$ with periodic boundary conditions. In particular we show that restricting to the saddle points with vanishing gauge connection gives a trivial contribution to the partition function, i.e. the bosonic and fermionic contributions exactly cancel each other.