Integrable systems in Gromov-Witten and symplectic field theory

January 30 to February 3, 2012

at the

American Institute of Mathematics, San Jose, California

organized by

Boris Dubrovin, Oliver Fabert, Todor Milanov, and Paolo Rossi

Original Announcement

This workshop will be devoted to the relation between symplectic invariants (namely Gromov-Witten and Symplectic Field Theory) and the theory of infinite dimensional integrable systems.

Starting with the formulation of Witten's conjecture in the 90's, which relates the full Gromov-Witten potential of the point with the commuting integrals of the KdV integrable hierarchy, it has become widely known that there is a deep and subtle relation between Gromov-Witten invariants and the theory of integrable systems of Hamiltonian PDEs.

As outlined by Eliashberg in his ICM2006 plenary talk, the integrable systems of rational Gromov-Witten theory of a symplectic manifold very naturally arise in rational Symplectic Field Theory of circle bundles over such symplectic manifold. More in general, after carefully defining a generalization of gravitational descendants, SFT provides a framework that associates an infinite dimensional Hamiltonian system with infinite symmetries to each contact manifold (or more in general stable Hamiltonian structure). Adding higher genera curves provides a quantization of such system.

Recently some progresses have been made in the investigation of the Hamiltonian systems from SFT (in particular by studying the analogue of tautological relations for psi-classes in Symplectic Field Theory). The goal of this workshop is, hence, to bring together experts from the field of symplectic geometry working on SFT and related topics with experts from the field of integrable systems and their relation to Gromov-Witten theory, in order to push this line of research further in many related directions:

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The zero section of the universal semiabelian variety, and the double ramification cycle
by  Samuel Grushevsky and Dmitry Zakharov,  Duke Math. J. 163 (2014), no. 5, 953-982  MR3189435
The double ramification cycle and the theta divisor
by  Samuel Grushevsky and Dmitry Zakharov,  Proc. Amer. Math. Soc. 142 (2014), no. 12, 4053-4064  MR3266977