  The purpose of this paper is to present a mathematical theory of the
half-twisted $(0,2)$ gauged linear sigma model and its correlation functions
that agrees with and extends results from physics. The theory is associated to
a smooth projective toric variety $X$ and a deformation $\sheaf E$ of its
tangent bundle $T_X$. It gives a quantum deformation of the cohomology ring of
the exterior algebra of $\sheaf E^*$. We prove that in the general case, the
correlation functions are independent of `nonlinear' deformations. We derive
quantum sheaf cohomology relations that correctly specialize to the ordinary
quantum cohomology relations described by Batyrev in the special case $\sheaf E
= T_X$.
