Higher algebraic structures in Hamiltonian Floer theory

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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 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 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.
Original languageEnglish
Pages (from-to)179-215
Number of pages37
JournalAdvances in Geometry
Volume20
Issue number2
Early online date11 Sep 2019
DOIs
Publication statusPublished - Apr 2020

Keywords

  • Hamiltonian Floer theory
  • Reeb orbit
  • symplectic field theory

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