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 E of its tangent bundle T X . It gives a quantum deformation of the cohomology ring of the exterior algebra of 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 E = T X .

Marginal deformations of heterotic G2 sigma models

“…Similar sectors have been found in the (0, 2) setting [36], and been used to compute exact results in the worldsheet theory [37,38]. The fact that we reproduce expected results on heterotic moduli gives further credence to the original topological twist and BRST proposals in [39,40].…”

“…Although our leading order result is already known from these works, our method can likely be extended to make exact worldsheet statements via topological twisting, in analogy the results of [36][37][38] in the (0, 2) setting. Our arguments borrow from the analysis of the superconformal algebra [39] associated to type II compactifications on manifolds of G 2 holonomy.…”

Supersymmetric localization in GLSMs for supermanifolds

“…(See e.g. [35][36][37] for a discussion of (0, 2) deformations of tangent bundles of products of projective spaces and results in quantum sheaf cohomology.) Then the M 's are given by:…”

Testing the (0,2) mirror map

“…where ζ = e 2πi 5 . The most general polynomial of degree 5 invariant under (3.26) is given by 27) and the unique invariant complex structure coordinate reads κ = α 1 · · · α 5 α 5 0 .…”