Sheaves of N=2N=2 supersymmetric vertex algebras on Poisson manifolds

We construct a sheaf of N=2N=2 vertex algebras naturally associated to any Poisson manifold. The relation of this sheaf to the chiral de Rham complex is discussed. We reprove the result about the existence of two commuting N=2N=2 superconformal structures on the space of sections of the chiral de Rham complex of a Calabi–Yau manifold, but now calculated in a manifest N=2N=2 formalism. We discuss how the semi-classical limit of this sheaf of N=2N=2 vertex algebras is related to the classical supersymmetric non-linear sigma model.