Permutation orbifolds of Virasoro vertex algebras and $W$-algebras

TL;DR

This paper investigates permutation orbifolds of Virasoro vertex algebras, revealing their structure as specific W-algebras and providing new realizations of rational affine W-algebras, with detailed analysis of minimal models.

Contribution

It characterizes permutation orbifolds of Virasoro vertex algebras as W-algebras of specific types and constructs new realizations of rational affine W-algebras.

Findings

Permutation orbifolds of $ ext{Vir}_c^{ imes 3}$ are W-algebras of a specific type for all but finitely many c.

New realizations of rational affine W-algebras associated to principal nilpotent elements.

Analysis of permutation orbifolds of the $(2,5)$-minimal vertex algebra $ ext{L}_{-22/5}$.

Abstract

We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathcal{L}_{-\frac{22}{5}}$ is presented.