  We explore the nonperturbative aspects of the chiral algebras of N = (0,2)
sigma models, which perturbatively are intimately related to the theory of
chiral differential operators (CDOs). The grading by charge and scaling
dimension is anomalous if the first Chern class of the target space is nonzero.
This has some nontrivial consequences for the chiral algebra. As an example, we
study the case where the target space is CP^1, and show that worldsheet
instantons trivialize the chiral algebra entirely. Consequently, supersymmetry
is spontaneously broken in this model. We then turn to a closer look at the
supersymmetry breaking from the viewpoint of Morse theory on loop space. We
find that instantons interpolate between pairs of perturbative supersymmetric
states with different fermionic numbers, hence lifting them out of the
supersymmetric spectrum. Our results reveal that a "quantum" deformation of the
geometry of the target space leads to a trivialization of the kernels of
certain twisted Dirac operators on CP^1.
