TY - JOUR

T1 - Higher algebraic structures in Hamiltonian Floer theory

AU - Fabert, Oliver

PY - 2019/9/11

Y1 - 2019/9/11

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.

KW - Hamiltonian Floer theory

KW - Reeb orbit

KW - symplectic field theory

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U2 - 10.1515/advgeom-2019-0017

DO - 10.1515/advgeom-2019-0017

M3 - Article

VL - 20

SP - 179

EP - 215

JO - Advances in Geometry

JF - Advances in Geometry

IS - 2

ER -