A Mathematical Theory of Quantum Sheaf Cohomology

TL;DR

This paper develops a rigorous mathematical framework for quantum sheaf cohomology associated with toric varieties and deformations of tangent bundles, extending physical results and establishing independence from certain deformations.

Contribution

It introduces a mathematical theory of quantum sheaf cohomology for toric varieties and proves the independence of correlation functions from nonlinear deformations.

Findings

Correlation functions are deformation-independent.

Quantum sheaf cohomology relations generalize quantum cohomology.

The theory extends Batyrev's quantum cohomology to deformed tangent bundles.

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 $\sheaf E$ 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 $\sheaf E = T_X$.