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

TL;DR

This paper aims to classify a specific class of bihamiltonian PDEs and connect them to Gromov-Witten invariants through Frobenius manifolds, providing a framework for reconstructing integrable hierarchies and understanding intersection numbers.

Contribution

It introduces a universal loop equation on jet spaces of Frobenius manifolds, linking integrable PDEs with Gromov-Witten invariants and offering a method for hierarchy reconstruction.

Findings

Universal loop equation for semisimple Frobenius manifolds

Perturbative expansion reproduces Gromov-Witten identities

Normal forms of PDEs linked to Frobenius structures

Abstract

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or 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 semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov - Witten classes and their descendents.