  In this paper, we study the perturbative aspects of a twisted version of the
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 in terms of 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 $X$. 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, where the obstruction vanishes for
$\it{any}$ smooth manifold $X$, we obtain a purely mathematical description of
the half-twisted variant of the topological A-model and (if $c_1(X) =0$) its
elliptic genus. By studying the half-twisted $(2,2)$ model on $X=\mathbb
{CP}^1$, one can show that a subset of the infinite-dimensional space of
physical operators generates an underlying super-affine Lie algebra.
Furthermore, on a non-K\"ahler, parallelised, group manifold with torsion, we
uncover a direct relationship between the modulus of the corresponding sheaves
of chiral de Rham complex, and the level of the underlying WZW theory.
