Equivariant quantum connections in positive characteristic

Abstract

In this thesis, we apply techniques from symplectic Gromov--Witten theory to study the equivariant quantum connections in positive characteristic. The main examples of interest arise from symplectic resolutions. We introduce equivariant generalizations of the quantum Steenrod operations of Fukaya, provide nontrivial computations in the example of the cotangent bundle of the projective line, and explore the relationship with Varchenko's construction of mod p solutions to the quantum differential equation. We then prove the compatibility of the equivariant quantum Steenrod operations with the quantum differential and difference connections. As a consequence, we obtain an identification of our operations for divisor classes with the p-curvature of the quantum connection in a wide range of examples.