Abstract

Floer theory (of symplectomorphisms) is a well-established tool in symplectic topology and can be viewed as a generalization of Gromov-Witten theory. On the other hand, it has been well-known for more than 20 years that there is a deep connection between Gromov-Witten theory and the theory of integrable PDE . Generalizing this relation, in this talk I will show that from the Floer theory of a general symplectomorphism one still obtains a (generally infinite-dimensional) Hamiltonian system with infinitely many symmetries. Instead of following the classical path using Frobenius manifolds and bihamiltonian structures, the main idea is to use symplectic field theory. The computation of the sequences of commuting descendant Hamiltonians involves a generating function (with weights) of all higher Floer products (corresponding to Gromov-Witten invariants with more than 3 points), which is an invariant of the symplectomorphism after passing to SFT homology.