# BPS Algebras, Genus Zero, and the Heterotic Monster

## TL;DR

This paper explores the connection between heterotic string compactifications, Monstrous moonshine, and BPS states, revealing new algebraic structures and interpretations related to the genus zero property and Monstrous Lie algebras.

## Contribution

It constructs heterotic string models linked to Monstrous moonshine and clarifies the module structure of BPS states over Monstrous Lie algebras, connecting physics and algebraic structures.

## Key findings

- Heterotic string compactifications relate to Monstrous moonshine.
- BPS state space forms a module over Monstrous Lie algebras.
- Provides a vertex operator algebra perspective on the results.

## Abstract

In this note, we expand on some technical issues raised in \cite{PPV} by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0+1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras $\mathfrak{m}_g$---some of the first and most prominent examples of Generalized Kac-Moody algebras---constructed by Borcherds and Carnahan. In particular, we clarify the structure of the module present in the second-quantized string theory. We also sketch a proof of our methods in the language of vertex operator algebras, for the interested mathematician.