Integrable systems in Gromov-Witten and symplectic field theory

This workshop, sponsored by AIM and the NSF, 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:

• compute new SFT examples (for example using the ongoing work by Bourgeois-Ekholm-Eliashberg on Stein(-fillable) manifolds and Cieliebak-Latschev on unit cotangent bundles)
• axiomatize the class of Hamiltonian systems arising in SFT, in view of the new universal relations discovered therein and investigate the possible relation between completeness of a SFT-Hamiltonian system and underlying topology of the contact manifold
• prove the reconstruction scheme for descendants from primaries in SFT by combining the work by Fabert-Rossi with the ongoing work on non-equivariant SFT
• in the case of circle bundles over symplectic manifolds, understand and make explicit the relation between higher genus Gromov-Witten theory (dispersive hierarchy) and SFT (quantum dispersionless hierarchy). As an interesting example, write the quantization of dispersionless Toda hierarchy corresponding to higher genus SFT of circle bundles over P¹
• understand what the topological solution and Witten's conjecture means in the circle bundle SFT context and investigate Hamilton-Jacobi theory in the Frobenius manifold context
• investigate other possibile frameworks for the quantum Hamiltonian systems arising from SFT and the corresponding methods to prove quantum integrability (e.g. the quantum dKdV hierarchy arising from SFT of a circle resembles a continuous limit of the spin-s XXX chain, whose integrability can be proved via Bethe Ansatz, moreover it is related with vertex algebras)

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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