Higher algebraic structures in Hamiltonian Floer theory I

TL;DR

This paper explores how advanced algebraic structures from symplectic field theory can be applied to symplectic cohomology, revealing new homotopy Lie brackets and conjectural links to mirror symmetry and complex structure deformations.

Contribution

It introduces a homotopy extension of the Lie bracket on symplectic cohomology using SFT of Hamiltonian mapping tori, connecting algebraic formalism to geometric and mirror symmetry applications.

Findings

Defined a homotopy extension of the Lie bracket on symplectic cohomology.

Outlined conjectural relations between $L_{}$-structures and mirror symmetry.

Discussed applications to the existence of closed Reeb orbits.

Abstract

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket on symplectic cohomology. Apart from discussing applications to the existence of closed Reeb orbits, we outline how the $L_{\infty}$-structure is conjecturally related via mirror symmetry to the extended deformation theory of complex structures.