Argyres-Douglas Theories, Modularity of Minimal Models and Refined Chern-Simons

TL;DR

This paper links Argyres-Douglas theories' Coulomb branch indices to modular transformations of minimal models and refined Chern-Simons theory, providing new computational tools and theoretical connections.

Contribution

It introduces a one-parameter generalization of minimal model modular matrices and connects Coulomb branch indices to refined Chern-Simons theory.

Findings

Proposed generalized modular transformation matrices for minimal models.

Established the relation between Coulomb branch indices and refined Chern-Simons partition functions.

Connected M-theory constructions to refined Chern-Simons theory.

Abstract

The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.