Vanishing Chiral Algebras and H？hn-Stolz Conjecture

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.