Asian Journal of Mathematics

Volume 18 (2014)

Number 3

A mathematical theory of quantum sheaf cohomology

Pages: 387 – 418



Ron Donagi (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Josh Guffin (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Sheldon Katz (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Eric Sharpe (Department of Physics, Virginia Tech, Blacksburg, Va., U.S.A.)


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 $\mathcal{E} = T_X$.


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

2010 Mathematics Subject Classification

Primary 32L10, 81T20. Secondary 14N35.

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