  Given a two-dimensional quantum field theory with (0,2) supersymmetry, one
can construct a chiral (or vertex) algebra. The chiral algebra of a (0,2)
supersymmetric sigma model is, perturbatively, the cohomology of a sheaf of
chiral differential operators on a string K\"ahler manifold. However, it
vanishes in some cases when instantons are taken into account. I will discuss
the implication of this phenomenon for the H\"ohn-Stolz conjecture on the
Witten genus.
