BPS Algebras in 2D String Theory

TL;DR

This paper explores 1+1 dimensional string compactifications with factorized internal CFTs, revealing that BPS states form modules of Borcherds-Kac-Moody algebras and establishing their role as symmetries of genus zero BPS amplitudes.

Contribution

It demonstrates that BPS states in these models form modules for BKM (super)algebras and proves these algebras are symmetries of genus zero BPS string amplitudes.

Findings

BPS states form modules for Borcherds-Kac-Moody algebras.

BKM (super)algebras are symmetries of genus zero BPS amplitudes.

Supersymmetric indices relate to automorphic forms and Borcherds-Weyl-Kac denominators.

Abstract

We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs of the form $V_1\otimes \bar V_2$, where $V_1, V_2$ are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds-Kac-Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds-Weyl-Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications.