The Z_2 Orbifold of the Universal Affine Vertex Algebra

TL;DR

This paper explicitly describes the minimal generating set for the Z_2 orbifold of the universal affine vertex algebra associated with a simple Lie algebra, and identifies special levels where this description does not hold.

Contribution

It provides the first explicit minimal strong generating set for the orbifold of universal affine vertex algebras under the Cartan involution for generic levels.

Findings

Explicit minimal strong generating set for generic levels

Identification of nongeneric levels for  case

Enhanced understanding of orbifold vertex algebras

Abstract

Let $\gg$ be a simple, finite-dimensional complex Lie algebra, and let $V^k(\gg)$ denote the universal affine vertex algebra associated to $\gg$ at level $k$. The Cartan involution on $\gg$ lifts to an involution on $V^k(\gg)$, and we denote by $V^k(\gg)^{\mathbb{Z}_2}$ the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for $V^k(\gg)^{\mathbb{Z}_2}$ for generic values of $k$. In the case $\gg = \gs\gl_2$, we also determine the set of nongeneric values of $k$, where this set does not work.