Degenerate symplectic fixed points and Gromov-Witten invariants

TL;DR

This paper links Gromov-Witten invariants with fixed points of Hamiltonian diffeomorphisms on rational symplectic manifolds, extending known theorems and introducing new cuplength estimates to analyze fixed points.

Contribution

It generalizes Givental's fixed point theorem to broader classes of symplectic manifolds and introduces deformed quantum cuplength for estimating fixed points.

Findings

Established a connection between Gromov-Witten invariants and fixed points.

Generalized fixed point theorem to non-Fano rational symplectic manifolds.

Proved a new cuplength estimate for symplectic fixed points.

Abstract

We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants.