Aspects of Massive Gauge Theories on Three Sphere in Infinite Mass Limit

TL;DR

This paper analyzes the behavior of the $S^3$ partition function of 3D $ ext{SU}(N)$ supersymmetric gauge theories with massive matter in the infinite mass limit, revealing the dominant Coulomb branch point and its relation to effective theories.

Contribution

It demonstrates how the infinite mass limit selects a specific Coulomb branch point, simplifying the partition function to that of an effective theory, and confirms this through large and small $N$ analyses.

Findings

The dominant Coulomb branch point is identified in the infinite mass limit.

Partition functions for small $N$ match large $N$ predictions in the infinite mass limit.

The infinite mass limit yields a partition function consistent with an interacting superconformal field theory.

Abstract

We study the $S^3$ partition function of three-dimensional supersymmetric $\mathcal{N}=4$ U($N$) SQCD with massive matter multiplets in the infinite mass limit with the so-called Coulomb branch localization. We show that in the infinite mass limit a specific point of the Coulomb branch is selected and contributes dominantly to the partition function. Therefore, we can argue whether each multiplet included in the theory is effectively massless in this limit, even on $S^3$, and conclude that the partition function becomes that of the effective theory on the specific point of the Coulomb branch in the infinite mass limit. In order to investigate which point of the Coulomb branch is dominant, we use the saddle point approximation in the large $N$ limit because the solution of the saddle point equation can be regarded as a specific point of the Coulomb branch. Then, we calculate the partition functions for small rank $N$ and confirm that their behaviors in the infinite mass limit are consistent with the conjecture from the results in the large $N$ limit. Our result suggests that the partition function in the infinite mass limit corresponds to that of an interacting superconformal field theory.