# The hidden symmetry of the heterotic string

## TL;DR

This paper suggests that Borcherds' Fake Monster Lie algebra acts as a hidden symmetry in heterotic string theory compactified on specific tori, linking automorphic forms to BPS state counting and string symmetries.

## Contribution

It provides a concrete realization of the connection between generalized Kac-Moody algebras and supersymmetric string vacua through explicit BPS state counting.

## Key findings

- BPS instanton counting function controlled by automorphic form $\
- Degeneracies correspond to graded dimensions of a second-quantized Fock space
- Establishes a link between string symmetries and automorphic forms

## Abstract

We propose that Borcherds' Fake Monster Lie algebra is a broken symmetry of heterotic string theory compactified on $T^7 \times T^2$. As evidence, we study the fully flavored counting function for BPS instantons contributing to a certain loop amplitude. The result is controlled by $\Phi_{12}$, an automorphic form for $O(2, 26, \mathbb{Z})$. The degeneracies it encodes in its Fourier coefficients are graded dimensions of a second-quantized Fock space for this large symmetry algebra. This construction provides a concrete realization of Harvey and Moore's proposed relationship between Generalized Kac-Moody symmetries and supersymmetric string vacua.