  In this paper, we discuss various aspects of a class of A-twisted heterotic
Landau-Ginzburg models on a Kaehler variety X. We provide a classification of
the R-symmetries in these models which allow the A-twist to be implemented,
focusing on the case in which the gauge bundle is either a deformation of the
tangent bundle of X or a deformation of a sub-bundle of the tangent bundle of
X. Some anomaly-free examples are provided. The curvature constraint imposed by
supersymmetry in these models when the superpotential is not holomorphic is
reviewed. Constraints of this nature have been used to establish properties of
analogues of pullbacks of Mathai-Quillen forms which arise in the correlation
functions of the corresponding A-twisted or B-twisted heterotic Landau-Ginzburg
models. The analogue most relevant to this paper is a deformation of the
pullback of a Mathai-Quillen form. We discuss how this deformation may arise in
the class of models studied in this paper. We then comment on how analogues of
pullbacks of Mathai-Quillen forms not discussed in previous work may be
obtained. Standard Mathai-Quillen formalism is reviewed in an appendix. We also
include an appendix which discusses the deformation of the pullback of a
Mathai-Quillen form.
