Note on Floer theory and integrable hierarchies

TL;DR

This paper explores how Dubrovin's integrable hierarchies, originally based on Gromov-Witten theory, can be extended to Hamiltonian Floer theory using symplectic field theory, revealing a new connection between these frameworks.

Contribution

It introduces a generalization of Dubrovin's integrable hierarchies to Hamiltonian Floer theory via a novel construction of the PSS isomorphism within symplectic field theory.

Findings

Extension of integrable hierarchies to Floer theory

Construction of PSS isomorphism in symplectic field theory

Bridging Gromov-Witten and Floer theories

Abstract

In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, can be constructed in the framework of Eliashberg-Givental-Hofer's symplectic field theory.