# Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in $M_{23}$

## TL;DR

This paper constructs ten specific Borcherds-Kac-Moody algebras associated with elements in M_{23} by realizing them as BRST cohomology of vertex algebras, providing a natural construction for these algebras.

## Contribution

It provides a uniform construction of ten Borcherds-Kac-Moody algebras via BRST cohomology, answering a question posed by Borcherds.

## Key findings

- Ten Borcherds-Kac-Moody algebras can be realized as BRST cohomology of vertex algebras.
- The construction confirms the naturality of these algebras.
- The approach links automorphic products with vertex algebra theory.

## Abstract

Borcherds-Kac-Moody algebras generalise finite-dimensional, simple Lie algebras. Scheithauer showed that there are exactly ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of square-free level. These belong to a larger class of Borcherds-Kac-Moody (super)algebras Borcherds obtained by twisting the denominator identity of the Fake Monster Lie algebra. Borcherds asked whether these Lie (super)algebras admit natural constructions. For the ten Lie algebras from the classification we give a positive answer to this question, i.e. we prove that they can be realised uniformly as the BRST cohomology of suitable vertex algebras.