VOAs and rank-two instanton SCFTs

TL;DR

This work shows that the VOAs associated with rank-two Deligne-Cvitanović instanton SCFTs arise from a uniform construction controlled solely by the dual Coxeter number $h^{\vee}$, and that diverse perspectives—from chiral algebra bootstrap to free-field realizations on Higgs-branch loci—converge to the same algebraic structure. The authors provide explicit VOAs for all $\mathfrak{g}$ in the DC series, including special handling of $H_0$ and $H_2$, and demonstrate that their vacuum characters obey a universal fourth-order modular differential equation, with the differential operator coefficients polynomials in $h^{\vee}$ and modular forms on $\Gamma^0(2)$. They further develop free-field realizations that illuminate the Higgs-branch relations and explore Higgs-branch physics via an affine uplift to $\mathcal{V}^{(2)}_{\mathfrak{g}}$, as well as Virasoro-building-block constructions for the $H_0$ case. Looking ahead, the results motivate a uniform, higher-rank program suggesting a complete set of strong generators for $\mathcal{V}^{(n)}_{\mathfrak{g}}$ and predicting higher-order modular differential equations, with concrete checks on rank-3 data and clear expectations for rank-$n$ generalizations.

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.