Partition function of $\mathcal{N}=2$ supersymmetric gauge theory and two-dimensional Yang-Mills theory

TL;DR

This paper explores the connection between four-dimensional $ =2$ supersymmetric gauge theories and two-dimensional Yang-Mills theory, revealing simplified partition functions and operator correspondences in special parameter regimes.

Contribution

It establishes a precise relation between the partition function of 4D $ =2$ gauge theory and 2D Yang-Mills theory, including Wilson loop operator correspondence.

Findings

Partition function simplifies at special parameter points.

Partition function of 4D theory relates to 2D Yang-Mills on $S^2$.

Wilson loop operators correspond between theories.

Abstract

We study four-dimensional $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory with $2N$ fundamental hypermultiplets in the self-dual $\Omega$-background. The partition function simplifies at special points of the parameter space and is related to the partition function of two-dimensional Yang-Mills theory on $S^{2}$. We also consider the insertion of a Wilson loop operator in two-dimensional Yang-Mills theory and find the corresponding operator in the four-dimensional $\mathcal{N}=2$ gauge theory.