Higher algebraic structures in Hamiltonian Floer theory

Research output: Contribution to JournalArticleAcademicpeer-review

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
JournalAdvances in Geometry
DOIs
Publication statusPublished - 11 Sep 2019

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Algebraic Structure
Field Theory
Frobenius Manifolds
Lie Brackets
Symplectic Manifold
Homotopy
Cohomology
Torus
Orbit
Analogue
Closed
Generalise

Keywords

  • Hamiltonian Floer theory
  • symplectic field theory
  • Reeb orbit

Cite this

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title = "Higher algebraic structures in Hamiltonian Floer theory",
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.",
keywords = "Hamiltonian Floer theory, symplectic field theory, Reeb orbit",
author = "Oliver Fabert",
year = "2019",
month = "9",
day = "11",
doi = "10.1515/advgeom-2019-0017",
language = "English",
journal = "Advances in Geometry",
publisher = "De Gruyter",

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Higher algebraic structures in Hamiltonian Floer theory. / Fabert, Oliver.

In: Advances in Geometry, 11.09.2019.

Research output: Contribution to JournalArticleAcademicpeer-review

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N2 - 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.

AB - 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.

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KW - symplectic field theory

KW - Reeb orbit

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