Vertex operator expressions for Lie algebras of physical states

TL;DR

This paper explores the Lie algebra of physical states linked to specific vertex operator algebras, using string theory techniques to express Lie brackets, and addresses a question related to the Moonshine module and Monstrous moonshine.

Contribution

It provides a new vertex operator algebra-based expression for Lie brackets of physical states, resolving a question posed by Borcherds for the Moonshine module.

Findings

Expressed Lie brackets in terms of vertex algebra operations

Connected the structure to the Monstrous moonshine conjecture

Provided a new perspective on the Lie algebra of physical states

Abstract

We study the Lie algebra of physical states associated with certain vertex operator algebras of central charge 24. By applying the no-ghost theorem from string theory we express the corresponding Lie brackets in terms of vertex algebra operations. In the special case of the Moonshine module this result answers a question of Borcherds, posed in his paper on the Monstrous moonshine conjecture.