Two-Dimensional Twisted Sigma Models, the Mirror Chiral de Rham Complex, and Twisted Generalised Mirror Symmetry

In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". 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 describe the resulting half-twisted variant of the topological B-model in terms of a $\it{mirror}$ "Chiral de Rham complex" (or CDR) defined by Malikov et al. in \cite{GMS1}. 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-K\"ahler 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-Bassat in \cite{ben} 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.