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

Dan Xie, Kazuya Yonekura

TL;DR

The paper develops a geometric framework to describe the full moduli space of vacua for 4d ${\cal N}=2$ class ${\cal S}$ theories, including Coulomb, Higgs, and mixed branches, via generalized Hitchin equations arising from twisted compactifications of 5d maximal SYM. A central role is played by holomorphic factorization of the Seiberg-Witten curve and by boundary data at punctures encoded as (nilpotent) orbits, with regular and irregular singularities treated distinctly. The authors provide a concrete algorithm to identify mixed-branch roots and compute the dimensions of Coulomb and Higgs factors, applied to a broad set of theories: ${\cal N}=2$ SQCD, the $T_N$ theory, Maldacena-Nunez theories, and various Argyres-Douglas theories. The results reproduce known field-theoretic conclusions (e.g., SQCD branch structures and AD points) and offer a unified framework for non-Lagrangian theories, enabling systematic exploration of vacua across regular and irregular puncture data and their coupled 5d SYM extensions.

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.

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

TL;DR

The paper develops a geometric framework to describe the full moduli space of vacua for 4d class theories, including Coulomb, Higgs, and mixed branches, via generalized Hitchin equations arising from twisted compactifications of 5d maximal SYM. A central role is played by holomorphic factorization of the Seiberg-Witten curve and by boundary data at punctures encoded as (nilpotent) orbits, with regular and irregular singularities treated distinctly. The authors provide a concrete algorithm to identify mixed-branch roots and compute the dimensions of Coulomb and Higgs factors, applied to a broad set of theories: SQCD, the theory, Maldacena-Nunez theories, and various Argyres-Douglas theories. The results reproduce known field-theoretic conclusions (e.g., SQCD branch structures and AD points) and offer a unified framework for non-Lagrangian theories, enabling systematic exploration of vacua across regular and irregular puncture data and their coupled 5d SYM extensions.

Abstract

We develop a systematic method to describe the moduli space of vacua of four dimensional class 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 SQCD, theory and Argyres-Douglas theories.

Paper Structure

This paper contains 55 sections, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General rules for finding mixed branches
  3. Pure Coulomb branch of class ${\cal S}$ theories
  4. Regular singularities.
  5. Irregular singularities.
  6. Mixed branches
  7. Step 1.
  8. Step 2.
  9. Step 3.
  10. Coulomb factor.
  11. Higgs factor.
  12. Vacuum structure of class ${\cal S}$ theories
  13. 5d maximal SYM and generalized Hitchin's equations
  14. Factorization of the Seiberg-Witten curve
  15. Constraint on factorization from Irregular singularity.
  16. ...and 40 more sections

Figures (4)

  • Figure 1: The vacuum structure of ${\cal N}=2$ theory: C: Coulomb branch; H: Higgs branch; M: Mixed branch.
  • Figure 2: The Higgs and mixed branch roots can be described as: the $SU(N)$ Hitchin system is decomposed into $SU(n_1)\times SU(n_2) \times\ldots\times SU(n_k) \times U(1)^{k-1}$ Hitchin systems, and the regular punctures of $SU(N)$ are decomposed into sums of regular punctures of lower rank Hitchin systems.
  • Figure 3: Upper: regular puncture are represented by Young diagrams. Bottom: Two types of irregular singularities are represented by Newton polygon, and a third type is represented by a collection of Young diagrams.
  • Figure 4: The factorization patter of $r$th mix branch of the $SU(N)$ theory with $2N$ flavors.