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.

中文翻译：

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哈密​​顿弗洛尔理论中的高等代数结构

摘要 在本文中，我们展示了如何使用 Eliashberg-Givental-Hofer 辛场论 (SFT) 丰富的代数形式来定义哈密顿弗洛尔理论中的高级代数结构。使用哈密顿映射环面的 SFT，我们定义了著名李括号的同伦扩展，并讨论如何使用它来证明多个闭合 Reeb 轨道的存在。此外，我们在开辛流形的哈密顿 Floer 理论中定义了有理 Gromov-Witten 理论的类似物。更准确地说，我们在哈密顿弗洛尔理论中引入了所谓的上同调 F-流形结构，并证明它推广了有理格罗莫夫-维滕理论中著名的 Frobenius 流形结构。