Classification of vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one

TL;DR

This paper derives trace formulae for vertex operator algebras of class \(\mathcal{S}^4\), providing constraints on their structure and classifying those with minimal conformal weight one under certain conditions.

Contribution

It introduces trace formulae for compositions of adjoint actions and classifies \(\mathcal{S}^4\) VOAs with minimal conformal weight one.

Findings

Constraints on central charge and Lie algebra dimension for \(\mathcal{S}^4\) VOAs.

Classification of \(\mathcal{S}^4\) VOAs with minimal conformal weight one.

Derived trace formulae using Casimir elements.

Abstract

In this article, we describe the trace formulae of composition of several (up to four) adjoint actions of elements of the Lie algebra of a vertex operator algebra by using the Casimir elements. As an application, we give constraints on the central charge and the dimension of the Lie algebra for vertex operator algebras of class $\mathcal{S}^4$. In addition, we classify vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one under some assumptions.