The Chiral Algebra of Genus Two Class $\mathcal{S}$ Theory

TL;DR

This paper constructs and analyzes the chiral algebra for genus two class S theory of type A1, revealing a novel SU(2) symmetry that constrains operator interactions and confirming the algebra's completeness through index comparisons.

Contribution

It provides the explicit construction of the chiral algebra for genus two class S theory and uncovers a new SU(2) symmetry influencing its structure.

Findings

Constructed the chiral algebra for genus two class S theory.

Identified a novel SU(2) symmetry acting on the algebra.

Confirmed the algebra's completeness via S-duality index comparison.

Abstract

We construct the chiral algebra associated with the $A_{1}$-type class $\mathcal{S}$ theory for genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of $SU(2)$ which is not associated with any conserved one-form current in four dimensions. We find that this novel $SU(2)$ strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.