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

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.

23 pages. v2: minor improvements and references added