Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

摘要

We present a project of classification of a certain class of bihamiltonian1+1 PDEs depending on a small parameter. Our aim is to embed the theory ofGromov - Witten invariants of all genera into the theory of integrable systems.The project is focused at describing normal forms of the PDEs and their localbihamiltonian structures satisfying certain simple axioms. A Frobenius manifoldor its degeneration is associated to every bihamiltonian structure of our type.The main result is a universal loop equation on the jet space of a semisimpleFrobenius manifold that can be used for perturbative reconstruction of theintegrable hierarchy. We show that first few terms of the perturbativeexpansion correctly reproduce the universal identities between intersectionnumbers of Gromov - Witten classes and their descendents.