Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology

TL;DR

This paper connects quantum Steenrod squares with equivariant pair-of-pants products in symplectic cohomology, establishing new relations, nonvanishing results, and explicit calculations for certain symplectic manifolds.

Contribution

It introduces an equivariant framework relating quantum Steenrod squares to pair-of-pants products and proves a symplectic Cartan relation, with explicit computations for line bundles.

Findings

Established a relation between quantum Steenrod squares and equivariant pair-of-pants products.

Proved a symplectic Cartan relation in the equivariant setting.

Computed the symplectic square for negative line bundles, extending Ritter's results.

Abstract

We relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We proceed similarly for $\mathbb{Z}/2$-equivariant symplectic cohomology, using an equivariant version of the continuation and $c^*$-maps. We prove a symplectic Cartan relation, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of $SH^*(T^* S^n)$. We finish by calculating the symplectic square for the negative line bundles $M = \text{Tot}(\mathcal{O}(-1) \rightarrow \mathbb{CP}^m)$, proving an equivariant version of a result due to Ritter.