The chiral algebra of genus two class $$ \mathcal{S} $$ theory

Abstract

We construct the chiral algebra associated with the A1-type class $$ \mathcal{S} $$ S theory for the 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.

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

“…This Lagrangian perspective was touched upon briefly in [1] and has more recently been studied in detail in work of Kiyoshige and Nishinaka [10], which led to a proposal for the associated VOA presented in terms of the OPEs of a set of strong generators. For our purposes, the important features of this theory are not connected directly to its Lagrangian realisation, and instead pertain to the structure of the Higgs branch of moduli space, the corresponding low energy moduli space dynamics, and select data that can be extracted from anomalies and the superconformal index.…”

“…A list of strong generators of the genus two vertex algebra with two-dimensional conformal weight h 3 was determined in [1] on the basis of an analysis of the superconformal index. This list was subsequently shown to constitute a complete set of strong generators in the analysis of [10]. We reproduce this list in Table I along with our naming conventions for the generators added.…”

“…In this section we derive and present a free field realisation for the genus-two VOA. While an explicit form for the VOA in terms of strong generators has been given in [10], we will operate behind a veil of ignorance regarding results so as to help illustrate how the EFT-motivated free field Ansatz constrains the form of the free field realisation, ultimately leading to a quick and efficient (re-)derivation of the full OPEs. For reference, note that the complete set of OPEs (in our conventions) are given by canonical T and Z OPEs encoding dimensions and U(1) F charges, Equations ( 19), (20), and (25), and the Z 2 transformations of those with respect to the automorphism given in (24).…”

“…This is a Lagrangian gauge theory (with gauge group SU(2) × SU(2) × SU(2)), and the formulation of the VOA as a BRST quotient received some consideration in [1]. A presentation as a strongly finitely generated vertex algebra has recently been derived in [10] using a combination of BRST cohomology and VOA bootstrap methods. The result is fairly complicated, with seventeen strong generators organised into eleven irreducible representations of a novel SU(2) outer automorphism group.…”

“…Our realisation makes manifest a (surprisingly large) USp(4) outer automorphism symmetry that extends the previously identified SU(2) and is inherited directly from the symplectic fermions. We use this to give a more compact presentation of the VOA in terms of the OPEs of strong generators than that of [10]. In Section III we elaborate on a number of technical aspects of our construction.…”