Modularity, 4d mirror symmetry, and VOA modules of 4d $\mathcal{N} = 2$ SCFTs with $a = c$

TL;DR

This paper explores the modular properties of vertex operator algebras associated with a special class of 4d $ ext{N}=2$ superconformal field theories, revealing their structure, symmetries, and geometric interpretations through mirror symmetry.

Contribution

It derives the characters, modular matrices, and differential equations of VOAs linked to $a=c$ 4d $ ext{N}=2$ SCFTs with $SU(N)$ gauge groups, connecting them to mirror symmetry insights.

Findings

Derived the space of characters for the VOAs $V[T_{p,N}]$.

Computed the $S, T$-matrices and modular differential equations.

Established a map between module characters via flavored modular differential equations.

Abstract

The infinite series of 4d $\mathcal{N} = 2$ SCFTs with central charge relation $a_\text{4d} = c_\text{4d}$ are closely related to the $\mathcal{N}=4$ super Yang-Mills. In this paper we study the modular properties of their associated VOAs $\mathbb{V}[\mathcal{T}_{p,N}]$ where $\mathcal{T}_{p, N}$ are those $a = c$ theories with $SU(N)$ gauge group. We exploit the closed-form formula for the Schur index of the $\mathcal{N} = 4$ $SU(N)$ theories $\mathcal{T}_{SU(N)}$ to derive the space of characters of the VOA $\mathbb{V}[\mathcal{T}_{p,N}]$ and the $S, T$-matrices, and find the (non-monic) modular linear differential equations that constrain the module characters when possible. We investigate the geometric interpretation of some of these modular data through the view point of 4d mirror symmetry. Using insights from the flavored modular differential equation and defect index, we investigate a map between modules characters of $\mathbb{V}[\mathcal{T}_{SU(N)}]$ and those of $\mathbb{V}[\mathcal{T}_{p,N}]$.