Quantum Steenrod operations and Fukaya categories

Abstract

The recent introduction of mod p equivariant operations to symplectic Gromov-Witten theory has fueled exciting developments in the field. In this thesis, we develop new tools for understanding these operations and explore an application to the quantum connection. In one direction, we construct certain operations on the equivariant Hochschild (co)homology of a general A∞-category. We show that when applied to the Fukaya category of a nondegenerated closed monotone symplectic manifold, this construction can be identified with the quantum Steenrod operations via Ganatra’s cyclic open-closed maps. A key ingredient in this identification is a new homotopy theoretic framework for studying various equivariant open-closed maps at once, using a combination of cyclic categories, edgewise subdivision and Abouzaid-Groman-Varolgunes’ operadic Floer theory. In another direction, we utilize quantum Steenrod operations, and Lee’s observation that it is related to the p-curvature of the quantum connection, to study singularities of the quantum connection in characteristic 0, and prove the exponential type conjecture for all closed monotone symplectic manifolds.