Quantum Steenrod operations and Fukaya categories

Abstract

This paper is concerned with quantum cohomology and Fukaya categories of a closed monotone symplectic manifold X, where we use coefficients in a field k of characteristic p > 0. The main result of this paper is that the quantum Steenrod operations Q\Sigma admit an interpretation in terms of certain operations on the (equivariant) Hochschild invariants of the Fukaya category of X, via suitable (equivariant) versions of the open-closed maps. As an application, we demonstrate how the categorical perspective provides new tools for computing Q\Sigma beyond the reach of known technology. We also explore potential connections of our work to arithmetic homological mirror symmetry.