Quantum Steenrod operations and Fukaya categories

TL;DR

This paper connects quantum Steenrod operations with Hochschild invariants of Fukaya categories in symplectic geometry, offering new computational tools and exploring links to mirror symmetry.

Contribution

It introduces an interpretation of quantum Steenrod operations via Hochschild invariants and open-closed maps in Fukaya categories, advancing the categorical understanding of quantum cohomology.

Findings

Quantum Steenrod operations relate to Hochschild invariants of Fukaya categories.

New methods for computing quantum Steenrod operations are developed.

Potential links to arithmetic homological mirror symmetry are discussed.

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.