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) sigma model is, perturbatively, the cohomology of a sheaf of chiral differential operators on a string Kähler manifold. However, it vanishes in some cases when instantons are taken into account. I will discuss the implications of this phenomenon for the geometry of loop spaces and the Höhn-Stolz conjecture on the Witten genus.