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 .

We compute instanton corrections to correlators in the genus-zero topological subsector of a (0, 2) supersymmetric gauged linear sigma model with target space P 1 × P 1 , whose left-moving fermions couple to a deformation of the tangent bundle. We then deduce the theory's chiral ring from these correlators, which reduces in the limit of zero deformation to the (2, 2) ring. Finally, we compare our results with the computations carried out by Adams et al.[ABS04] and Katz and Sharpe [KS06]. We find immediate agreement with the latter and an interesting puzzle in completely matching the chiral ring of the former. Date

In this paper, we will outline computations of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, for those deformations describable as deformations of toric Euler sequences. Quantum sheaf cohomology is a heterotic analogue of quantum cohomology, a e-print archive: http://lanl.arXiv.org/abs/1110.3752v2 RON DONAGI ET AL.quantum deformation of the classical product on sheaf cohomology groups, that computes nonperturbative corrections to analogues of 27 3 couplings in heterotic string compactifications. Previous computations have relied on either physics-based gauged linear sigma model (GLSM) techniques or computation-intensive brute-force Cech cohomology techniques. This paper describes methods for greatly simplifying mathematical computations, and derives more general results than previously obtainable with GLSM techniques. We will outline recent results (rigorous proofs will appear elsewhere).

a b s t r a c tIn this paper, we apply the methods developed in recent work for constructing A-twisted (2, 2) Landau-Ginzburg models to analogous (0, 2) models. In particular, we study (0, 2) Landau-Ginzburg models on topologically non-trivial spaces away from large-radius limits, where one expects to find correlation function contributions akin to (2, 2) curve corrections. Such heterotic theories admit A-and B-model twists, and exhibit a duality that simultaneously exchanges the twists and dualizes the gauge bundle. We explore how this duality operates in heterotic Landau-Ginzburg models, as well as other properties of these theories, using examples which renormalization-group flow to heterotic nonlinear sigma models as checks on our methods.

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