A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type $\mathfrak{a}_1$

TL;DR

This paper constructs a free field realization of the vertex operator algebra for the genus-two class  S theory of type , revealing an enhanced symmetry and simplifying its algebraic structure.

Contribution

It introduces a novel free field realization of the VOA for the genus-two  S theory, highlighting an  USp(4) automorphism and simplifying the algebraic framework.

Findings

Revealed an  USp(4) automorphism in the VOA

Provided a simplified algebraic structure for the genus-two  S theory

Connected the  subregular Drinfel'd-Sokolov    algebra to the principal  algebra

Abstract

We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent work of Kiyoshige and Nishinaka and significantly simplifies the structure of the algebra. Along the way, we also produce a realisation of the generic subregular Drinfel'd-Sokolov $\mathcal{W}$ algebra of type $\mathcal{c}_2$ in terms of the generic principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and a one-dimensional isotropic lattice vertex algebra.