A construction of the quantum Steenrod squares and their algebraic relations

TL;DR

This paper develops a quantum deformation of Steenrod squares for symplectic manifolds, establishing algebraic relations and computing examples, thus advancing the understanding of quantum cohomology operations.

Contribution

It introduces a quantum Steenrod square construction, proves quantum Cartan and Adem relations, and computes these squares for projective spaces and toric varieties.

Findings

Quantum Steenrod squares satisfy quantum Cartan and Adem relations.

Explicit computations of quantum Steenrod squares for CP^n.

Quantum corrections for blowups are determined by classical Steenrod squares.

Abstract

We construct a quantum deformation of the Steenrod square construction on closed monotone symplectic manifolds, based on the work of Fukaya, Betz and Cohen. We prove quantum versions of the Cartan and Adem relations. We compute the quantum Steenrod squares for all CP n and give the means of computation for all toric varieties. As an application, we also describe two examples of blowups along a subvariety, in which a quantum correction of the Steenrod square on the blowup is determined by the classical Steenrod square on the subvariety.