[Paper Review] Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

Boris Dubrovin, Youjin Zhang|ArXiv.org|Aug 23, 2001

Nonlinear Waves and Solitons104 references250 citations

TL;DR

This paper establishes a universal loop equation on the jet space of semisimple Frobenius manifolds that enables perturbative reconstruction of integrable hierarchies of 1+1 PDEs. By linking bihamiltonian structures to Frobenius manifolds and using tau-structures and quasitriviality, the authors show that the first few terms of the expansion reproduce universal identities among Gromov-Witten invariants and their descendants, embedding topological recursion and quantum cohomology into integrable systems theory via a classification of normal forms of Hamiltonian structures with a small parameter ǫ.

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.

Motivation & Objective

• To classify a class of bihamiltonian 1+1 PDEs with a small parameter ǫ using normal forms of local Poisson brackets.
• To embed the theory of Gromov-Witten invariants of all genera into the framework of integrable systems via Frobenius manifolds.
• To establish a universal loop equation on the jet space of semisimple Frobenius manifolds for perturbative reconstruction of integrable hierarchies.
• To demonstrate that the perturbative expansion of the hierarchy correctly reproduces universal identities between intersection numbers of Gromov-Witten classes and their descendents.

Proposed method

• Classify (n,0) and (0,n) local Poisson brackets on extended formal loop spaces using Miura group transformations.
• Introduce tau-structures and tau-covers to define canonical coordinates and Hamiltonians for bihamiltonian hierarchies.
• Construct the Principal Hierarchy from a semisimple Frobenius manifold using deformed flat coordinates and the spectrum of the manifold.
• Apply quasitriviality to relate the hierarchy to the Gromov-Witten potential and derive a loop equation on the jet space.
• Use Virasoro symmetries and free field realizations to analyze the structure and solutions of the hierarchy.
• Derive the loop equation in genus 1 and genus 2, showing agreement with known intersection numbers.

Experimental results

Research questions

• RQ1 How can bihamiltonian structures of evolutionary PDEs be classified using normal forms and Miura transformations?
• RQ2 What is the precise relationship between semisimple Frobenius manifolds and integrable hierarchies of PDEs?
• RQ3 Can the universal loop equation on the jet space of a Frobenius manifold reconstruct the perturbative expansion of Gromov-Witten invariants?
• RQ4 How does quasitriviality connect the bihamiltonian hierarchy to the topological recursion and tau-function formalism?
• RQ5 What role do Virasoro symmetries play in the solution space of the Principal Hierarchy?

Key findings

• The universal loop equation on the jet space of a semisimple Frobenius manifold correctly reproduces the first few terms of the perturbative expansion of Gromov-Witten invariants and their descendents.
• The Principal Hierarchy associated with a semisimple Frobenius manifold is completely integrable and admits a tau-function via the bihamiltonian recursion procedure.
• Quasitrivial bihamiltonian structures allow the hierarchy to be reconstructed from the Gromov-Witten potential, with the leading-order term being the dispersionless limit.
• The genus 1 loop equation takes a final form that matches the known anomaly equation in topological gravity and Gromov-Witten theory.
• For n=3, the degenerate Frobenius manifold structure is governed by the classical Euler equations of rigid body rotation, with solutions expressible in terms of Prym theta functions.
• The spectral curve of the system is a plane algebraic curve of degree n with genus (n−1)(n−2)/2, and the Baker-Akhiezer function provides a complete solution in terms of theta functions.