WEEK 3: INTEGRABLE SYSTEMS METHODS IN ENUMERATIVE GEOMETRY
In the early 90’s Witten, motivated by a matrix model description of topological gravity, conjectured that the intersection numbers of tautological classes on moduli spaces of curves are governed by the KdV integrable hierarchy. Witten’s conjecture was first proven by Kontsevich, and his proofs as well as later different proofs by Okounkov-Pandharipande, by Mirzakhani, have all led to a rich theory connecting moduli spaces of stable curves, and more generally Gromov-Witten theory, with 1+1 dimensional integrable systems.
More recently, symplectic field theory has provided another link with integrable systems, including quantum ones. As it was observed by Dubrovin the genus $0$ Gromov-Witten theories are related with the so-called dispersionless integrable hierarchies, while adding the dispersion requires full genus GW theories. On the other hand, while the genus $0$ SFT leads to the same dispersionless hierarchies, the full genus SFT yields quantization of dispersionless hierarchies, rather than introducing a dispersion parameter.
An area of active current research, also motivated by symplectic field theory, is the study of the
double ramification (DR) cycle. The DR viewpoint allows a new interpretation and construction of (quantum) integrable systems from cohomological field theories, and also yields a 2-parametric deformation of dispersionless hierarchies, where one parameter introduces a dispersion, while the second leads to a deformation quantization. A lot of current research revolves around relations of DR cycles with other geometric constructions on moduli of curves and differentials, and in field theories.
This week will include series of lectures by Danilo Lewanski and Dimitri Zvonkine on this circle of ideas, and some individual talks by experts.