Fun with $F_{24}$

TL;DR

This paper explores the structure of the $F_{24}$ superconformal field theory, constructing new Borcherds-Kac-Moody superalgebras from it and relating it to orbifold models based on the $E_8$ lattice.

Contribution

It introduces new Borcherds-Kac-Moody superalgebras derived from $F_{24}$ and classifies models based on supercurrents linked to semisimple Lie algebras of dimension 24.

Findings

Eight Lie superalgebras of physical states are constructed.

All these superalgebras have Borcherds-Kac-Moody structure.

$F_{24}$ can be obtained via orbifolding from an $E_8$ lattice-based SCFT.

Abstract

We study some special features of $F_{24}$, the holomorphic $c=12$ superconformal field theory (SCFT) given by 24 chiral free fermions. We construct eight different Lie superalgebras of "physical" states of a chiral superstring compactified on $F_{24}$, and we prove that they all have the structure of Borcherds-Kac-Moody superalgebras. This produces a family of new examples of such superalgebras. The models depend on the choice of an $\mathcal{N}=1$ supercurrent on $F_{24}$, with the admissible choices labeled by the semisimple Lie algebras of dimension 24. We also discuss how $F_{24}$, with any such choice of supercurrent, can be obtained via orbifolding from another distinguished $c=12$ holomorphic SCFT, the $\mathcal{N}=1$ supersymmetric version of the chiral CFT based on the $E_8$ lattice.