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

Abstract

We develop a systematic method to describe the moduli space of vacua of four dimensional $\mathcal{N}=2$ class $\mathcal{S}$ 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.

33 pages, 4 figures

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• A slow review of the AGT correspondence
• Magnetic quivers for rank 2 theories
• ADE Little String Theory on a Riemann Surface (and Triality)
• Four dimensional superconformal theories from M5 branes
• Compact Conformal Manifolds
• Hilbert series and mixed branches of $T[SU(N)]$ theory
• 4d $\mathcal{N}=2$ SCFTs and lisse W-algebras
• Cluster Algebras and Supersymmetric Gauge Theories