Commuting symplectomorphisms and Dehn twists in divisors

TL;DR

This paper establishes an elliptic relation between commuting symplectomorphisms affecting Floer cohomologies, and applies it to show certain Dehn twists have infinite order in the symplectic mapping class group.

Contribution

It proves the elliptic relation for commuting symplectomorphisms and uses it to analyze the order of Dehn twists in symplectic mapping class groups.

Findings

Supertraces of actions of commuting symplectomorphisms are equal.

Lower bounds on Floer cohomology dimensions are derived from Lefschetz numbers.

Dehn twists around vanishing Lagrangian spheres have infinite order in most hypersurfaces in Grassmannians.

Abstract

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of $HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.