Two-dimensional twisted sigma models, the mirror chiral de Rham complex, and twisted generalised mirror symmetry

In this paper, we study the peturbative aspects of a "B-twisted" two-dimensional (0,2) heterotic sigma model on a holomorphic gauge bundle epsilon over a complex, hermitian manifold X. We show that the model can be naturally described in terms of the mathematical theory of "Chiral Differential Operator". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on X. In addition, one can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the (2, 2) locus, one can described the resulting half-twisted variant of the topological B-model in terms of a mirror "Chiral de Rham complex" (or CDR) defined by Malikov et al. in [1]. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-Kahler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR ( and its mirror) that are consistent with mathematically established results by Ben-Bassant in [2] on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.