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Mirror symmetry for circle compactified 4d $A_1$ class-$S$ theories

Yiwen Pan, Wenbin Yan

Abstract

In this letter, we propose a 4d mirror symmetry for the class-$\mathcal{S}$ theories which relates the representation theory of the chiral quantization of the Higgs branch and the geometry of the Coulomb branch. We study the representation theory by using the 4d/VOA correspondence, (defect) Schur indices and (flavor) modular differential equations, and match the data with the fixed manifolds of the Hitchin moduli spaces. This correspondence extends the connection between Higgs and Coulomb branch of Argyres-Douglas theories, and can provide systematic guidance for the study of the representation theory of vertex operator algebras by exploiting results from Hitchin systems.

Mirror symmetry for circle compactified 4d $A_1$ class-$S$ theories

Abstract

In this letter, we propose a 4d mirror symmetry for the class- theories which relates the representation theory of the chiral quantization of the Higgs branch and the geometry of the Coulomb branch. We study the representation theory by using the 4d/VOA correspondence, (defect) Schur indices and (flavor) modular differential equations, and match the data with the fixed manifolds of the Hitchin moduli spaces. This correspondence extends the connection between Higgs and Coulomb branch of Argyres-Douglas theories, and can provide systematic guidance for the study of the representation theory of vertex operator algebras by exploiting results from Hitchin systems.

Paper Structure

This paper contains 6 sections, 1 theorem, 13 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Ordinary modules of Vgn
  5. Generalizations to higher rank
  6. Outlook

Key Result

Corollary 1

All OMs of $V_{g,n}$ correspond to FMs in the subset ${\cal M}^\text{ord}_{g,n}\equiv\{M_{d,(1,1,\cdots,1)}\}_{-n+\sum_i\alpha_i<d\leq g-1-n/2}$ (resp. ${\cal M}^\text{ord}_{g,n}\equiv\{M_{d,(0,1,\cdots,1)}\}_{-n+1-\alpha_1+\sum_{i>1}\alpha_i<d\leq g-1-(n-1)/2}$) of ${\cal M}^T_{g,n}$ when $n$ is ev The Jordan type of $T^\text{ord}$ (resp. $(STS)^\text{ord}$) is $[\dim M+g+2]_{M\in {\cal M}^\text{

Theorems & Definitions (4)

  • Conjecture 1
  • Conjecture 2
  • Corollary 1
  • Conjecture 3