A mathematical theory of quantum sheaf cohomology

Abstract

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 $\mathcal{E}$ of its tangent bundle $T_X$. It gives a quantum deformation of the cohomology ring of the exterior algebra of $\mathcal{E}*$. 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 $\mathcal{E} = T_X$.