$S_3$-Permutation Orbifolds of Virasoro Vertex Algebras

TL;DR

This paper constructs and analyzes the $S_3$-orbifold subalgebra of the tensor product of three Virasoro vertex algebras, identifying generators, structure, and special cases related to minimal models and affine $W$-algebras.

Contribution

It provides a minimal generating set for the $S_3$-orbifold of three Virasoro VOAs and explores specific cases at central charges $c=1/2$ and $c=-22/5$, revealing new algebraic structures.

Findings

Generated the $S_3$-orbifold algebra with explicit types of generators.

Identified the $c=1/2$ case as a new unitary $W$-algebra.

Proved the $c=-22/5$ case is isomorphic to an affine $W$-algebra of type $rak{g}_2$.

Abstract

In this paper, a continuation of \cite{MPS}, we investigate the $S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we consider the $S_3$-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of $c$. More precisely, we show that this vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$. We also investigate two prominent examples of simple $S_3$-orbifold algebras corresponding to central charges $c=\frac12$ (Ising model) and $c=-\frac{22}{5}$ (i.e. $(2,5)$-minimal model). We prove that the former is a new unitary $W$-algebra of type $(2,4,6,8)$ and the latter is isomorphic to the affine simple $W$-algebra of type $\frak{g}_2$ at non-admissible level $-\frac{19}{6}$. We also provide another version of this isomorphism using the affine $W$-algebra of type $\frak{g}_2$ coming from a subregular nilpotent element.