The moduli space of vacua of N=2 class S theories

TL;DR

This paper introduces a systematic approach to describe the moduli space of vacua in 4D N=2 class S theories, including Coulomb, Higgs, and mixed branches, using generalized Hitchin equations and Seiberg-Witten curve factorization.

Contribution

It provides a new method to determine the structure and dimensions of various branches of the moduli space in class S theories, incorporating puncture data and singularity reduction.

Findings

Derived Higgs and mixed branch roots for class S theories.

Calculated dimensions of Coulomb and Higgs components.

Applied method to examples like SQCD, T_N, and Argyres-Douglas theories.

Abstract

We develop a systematic method to describe the moduli space of vacua of four dimensional $\mathcal{N}=2$ class ${\cal S}$ theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin's equations obtained from twisted compactification of 5d maximal Super-Yang-Mills, with local degrees of freedom at punctures given by (nilpotent) orbits. The crucial thing is the holomorphic factorization of the Seiberg-Witten curve and reduction of singularity at punctures. We illustrate our method by many examples including ${\mathcal N}=2$ SQCD, $T_N$ theory and Argyres-Douglas theories.