The chiral algebras of two-dimensional sigma models with (0,2)
supersymmetry are infinite-dimensional analogs of the chiral rings of
(2,2) models. Perturbatively, they enjoy rich mathematical structures
described by sheaves of chiral differential operators.
Nonperturbatively, however, they vanish completely for certain (0,2)
models with no left-moving fermions. In this talk, I will explain how
this vanishing phenomenon takes places. The vanishing of the chiral
algebra of a (0, 2) model implies that supersymmetry is spontaneously
broken, which in turn suggests that no harmonic spinors exist on the
loop space of the target space. This provides a physics proof of a
special case of the Hohn-Stolz conjecture.