# VOAs and rank-two instanton SCFTs

## TL;DR

This paper characterizes the vertex operator algebras associated with certain N=2 superconformal field theories from D3-branes probing F-theory singularities, revealing a uniform structure linked to the dual Coxeter number.

## Contribution

It provides a complete identification and free field realizations of these VOAs, and derives modular differential equations for their vacuum characters, unifying the description across all cases.

## Key findings

- VOAs are uniformly described by the dual Coxeter number.
- Explicit free field realizations of the VOAs are constructed.
- Fourth-order modular differential equations for vacuum characters are derived.

## Abstract

We analyze the N=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.