## Asian Journal of Mathematics

- Asian J. Math.
- Volume 18, Number 3 (2014), 387-418.

### A mathematical theory of quantum sheaf cohomology

Ron Donagi, Josh Guffin, Sheldon Katz, and Eric Sharpe

#### 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 $\mathcal{E} = T_X$.

#### Article information

**Source**

Asian J. Math., Volume 18, Number 3 (2014), 387-418.

**Dates**

First available in Project Euclid: 8 September 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1410186663

**Mathematical Reviews number (MathSciNet)**

MR3257832

**Zentralblatt MATH identifier**

1300.32022

**Subjects**

Primary: 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 81T20: Quantum field theory on curved space backgrounds

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

**Keywords**

Quantum cohomology quantum shear cohomology toric varieties primitive collection gauged linear sigma model

#### Citation

Donagi, Ron; Guffin, Josh; Katz, Sheldon; Sharpe, Eric. A mathematical theory of quantum sheaf cohomology. Asian J. Math. 18 (2014), no. 3, 387--418. https://projecteuclid.org/euclid.ajm/1410186663