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.
Highlights
The structure of moduli space of vacua plays a crucial role in studying the dynamics of supersymmetric field theory
We develop a systematic method to describe the moduli space of vacua of four dimensional N = 2 class S theories including Coulomb branch, Higgs branch and mixed branches
There is usually a pure Coulomb branch, which we denote as C; sometimes, there is a pure Higgs branch H which touches with the Coulomb branch at a single point
Summary
The structure of moduli space of vacua plays a crucial role in studying the dynamics of supersymmetric field theory. The full structure of moduli spaces including mixed branches was studied only in very few examples such as N = 2 SQCD [4, 10] using the non-renormalization theorem and exact solution on the Coulomb branch. Four dimensional theory is derived by twisted compactification of the 6d theory on a Riemann surface C, and one get a Higgs field Φ in the canonical (or cotangent) bundle of C which describes the Coulomb branch deformations. The pure Higgs branch is described by turning on only φ deformations (we can get another scalar from Bμν field on M5 brane [16].).
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