Higher algebraic structures in Hamiltonian Floer theory

TL;DR

This paper leverages symplectic field theory to introduce higher algebraic structures in Hamiltonian Floer theory, including homotopy Lie brackets and cohomology F-manifold structures, advancing the understanding of symplectic invariants.

Contribution

It develops new higher algebraic structures in Hamiltonian Floer theory using symplectic field theory, including a homotopy Lie bracket and a cohomology F-manifold, extending Gromov-Witten theory concepts.

Findings

Defined a homotopy extension of the Lie bracket in Floer theory.

Introduced a cohomology F-manifold structure in Hamiltonian Floer theory.

Proved the generalization of Frobenius manifolds in this context.

Abstract

In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existence of multiple closed Reeb orbits. Furthermore we show how to define the analogue of rational Gromov-Witten theory in the Hamiltonian Floer theory of open symplectic manifolds. More precisely, we introduce a so-called cohomology F-manifold structure in Hamiltonian Floer theory and prove that it generalizes the well-known Frobenius manifold structure in rational Gromov-Witten theory.