Higher algebraic structures in Hamiltonian Floer theory. (English) Zbl 1437.53070
Summary: 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.
MSC:
| 53D42 | Symplectic field theory; contact homology |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
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