  We explore two-dimensional sigma models with (0,2) supersymmetry through
their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have
a rich infinite-dimensional structure described by the cohomology of a sheaf of
chiral differential operators. Nonperturbatively, instantons can deform this
structure drastically. We show that under some conditions they even annihilate
the whole algebra, thereby triggering the spontaneous breaking of
supersymmetry. For a certain class of K\"ahler manifolds, this suggests that
there are no harmonic spinors on their loop spaces and gives a physical proof
of the H\"ohn-Stolz conjecture.
