S-duality for the large $N=4$ superconformal algebra

TL;DR

This paper proves conjectures about vertex algebras related to gauge theory and the geometric Langlands program, revealing new dualities and algebraic structures at specific central charges.

Contribution

It introduces the conjectural kernel vertex algebra for a specific duality in $SU(2)$ gauge theory and determines branching rules for large and small $N=4$ superconformal algebras.

Findings

Determined branching rules for superconformal algebras at specific central charges.

Identified affine vertex superalgebra of $rak{osp}(1|2)$ as a quantum Hamiltonian reduction.

Found a coincidence hinting at the nature of a duality wall.

Abstract

We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the $S T^2 S$ duality transformation in $SU(2)$ gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small $N=4$ superconformal algebra at central charge $-9$ as well as for the generic large $N=4$ superconformal algebra at central charge $-6$. Moreover we obtain the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ superconformal algebra times a free fermion as Quantum Hamiltonian reductions of the large $N=4$ superconformal algebras at $c=-6$.