  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.
