Topics in quantum sheaf cohomology

Quantum sheaf cohomology generalizes the theory quantum cohomology, in the sense that it deals with a class of more general sheaves rather than the tangent bundle. In this thesis we study quantum sheaf cohomology of bundles on smooth projective toric varieties. The basic case is when the bundle is a deformation of the tangent bundle. We study the quantum correlators defined by the quantum sheaf cohomology. We give a mathematical proof of a formula that computes the quantum correlators in this case, confirming the conjecture in the physics literature. The next important case is when the bundle is of higher rank than the tangent bundle. We study bundles being deformations of T ⊕ O where T is the tangent bundle and O is the trivial bundle. We give a rudimentary description of the classical and quantum sheaf cohomology ring in this case. We also discuss other interesting cases and the demanding from physics, as well as the connections between them and the previously mentioned ones.