Poisson Bracket Invariants and Wrapped Floer Homology

TL;DR

This paper links Poisson bracket invariants to wrapped Floer homology, providing lower bounds for specific configurations in symplectic geometry, especially in cotangent bundles of Riemannian manifolds.

Contribution

It establishes a novel connection between Poisson bracket invariants and wrapped Floer homology, offering new lower bounds for configurations in Liouville domain completions.

Findings

Lower bounds on Poisson bracket invariants are expressed via wrapped Floer homology barcodes.

Results apply to cotangent bundles of closed Riemannian manifolds with specific Lagrangian configurations.

The work extends understanding of Hamiltonian chords and symplectic invariants in complex geometric settings.

Abstract

The Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich and further studied by Entov and Polterovich, serve as invariants for quadruples of closed sets in symplectic manifolds. Their nonvanishing has significant implications for the existence of Hamiltonian chords between pairs of sets within the quadruple, with bounds on the time-length of these chords. In this work, we establish lower bounds on the Poisson bracket invariants for certain configurations arising in the completion of Liouville domains. These bounds are expressed in terms of the barcode of wrapped Floer homology. Our primary examples come from cotangent bundles of closed Riemannian manifolds, where the quadruple consists of two fibers over distinct points and two cosphere bundles of different radii, or a single cosphere bundle and the zero section.