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Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants (2001)
| Citations: | 93 - 2 self |
BibTeX
@MISC{Dubrovin01normalforms,
author = {Boris Dubrovin and Youjin Zhang},
title = {Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants},
year = {2001}
}
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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.
Keyphrases
frobenius manifold normal form gromov-witten invariant integrable pdes certain simple axiom bihamiltonian structure first term main result universal loop equation jet space certain class universal identity perturbative expansion perturbative reconstruction intersection number gromov witten class integrable hierarchy small parameter local bihamiltonian structure semisimple frobenius manifold gromov witten invariant integrable system
