Symplectic field theory and integrable systems (beyond the circle bundle case)

Using the link between the SFT of a prequantization space and the GW potential of its symplectic base, it was pointed out by Eliashberg in his ICM 2006 talk that the rich algebraic formalism of SFT sheds new light on the deep relations between Gromov-Witten theory and integrable systems. In this talk I will generalize the discussion from circle bundles to general contact manifolds, where the key point is the invariance of the algebraic structures under choices of contact form, cylindrical almost complex structure and abstract perturbations. As a concrete example beyond the circle bundle case I will introduce the SFT of a closed geodesic, which turns out to be nontrivial enough to serve as a toy model for problems and questions.