Permutation Orbifolds of the Heisenberg Vertex Algebra $\mathcal{H}(3)$

TL;DR

This paper analyzes the structure and generators of the $S_3$-orbifold of a rank three Heisenberg vertex algebra, revealing its minimal generating set and exploring modular properties of its modules.

Contribution

It provides a detailed description of the generators and relations of the orbifold algebra, including the minimal generating set and conformal weights, using invariant theory.

Findings

The orbifold algebra has a minimal strongly generating set with weights 1 to 6.

The structure of the cyclic $ ext{Z}_3$-orbifold is explicitly determined.

Modular properties of module characters are studied.

Abstract

We study the $S_3$-orbifold of a rank three Heisenberg vertex algebras in terms of generators and relations. By using invariant theory we prove that the orbifold algebra has a minimal strongly generating set of vectors whose conformal weights are $1,2,3,4,5,6^2$ (two generators of degree $6$). The structure of the cyclic $\mathbb{Z}_3$-oribifold is determined by similar methods. We also study modular properties of characters of modules for these vertex algebras.