  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.
