• Corpus ID: 118640886

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

@article{Dubrovin2001NormalFO,
  title={Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants},
  author={Boris Dubrovin and You-jin Zhang},
  journal={arXiv: Differential Geometry},
  year={2001}
}

• B. Dubrovin, You-jin Zhang
• Published 23 August 2001
• Mathematics
• arXiv: Differential Geometry

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.