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

Boris Dubrovin, Youjin Zhang

TL;DR

The paper develops a comprehensive framework to classify bihamiltonian 1+1 PDEs with a small dispersion parameter by linking their dispersionless limits to semisimple Frobenius manifolds and formulating four axioms that govern perturbative reconstruction. It introduces an extended formal loop space and the Miura group to analyze local Poisson brackets, proving key normal-form results for (n,0) and (0,n) brackets and connecting leading-order data to flat pencils of metrics and quasi-Frobenius structures. Through bihamiltonian geometry on extended loop spaces, the authors construct commuting hierarchies via Magri recursion, establish tau-structures and tau-covers, and show that Virasoro symmetries act linearly on tau-functions, yielding loop equations that reproduce Genus-0 to Genus-2 Gromov–Witten identities. The work provides a universal procedure to reconstruct integrable hierarchies from dispersionless data tied to Frobenius manifolds and clarifies the role of tau-functions and Virasoro constraints in encoding higher-genus information. Overall, it forges a principled bridge between the geometry of Frobenius manifolds, Poisson/bihamiltonian formalism, and the GW/invariant framework across genera.

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.

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

TL;DR

The paper develops a comprehensive framework to classify bihamiltonian 1+1 PDEs with a small dispersion parameter by linking their dispersionless limits to semisimple Frobenius manifolds and formulating four axioms that govern perturbative reconstruction. It introduces an extended formal loop space and the Miura group to analyze local Poisson brackets, proving key normal-form results for (n,0) and (0,n) brackets and connecting leading-order data to flat pencils of metrics and quasi-Frobenius structures. Through bihamiltonian geometry on extended loop spaces, the authors construct commuting hierarchies via Magri recursion, establish tau-structures and tau-covers, and show that Virasoro symmetries act linearly on tau-functions, yielding loop equations that reproduce Genus-0 to Genus-2 Gromov–Witten identities. The work provides a universal procedure to reconstruct integrable hierarchies from dispersionless data tied to Frobenius manifolds and clarifies the role of tau-functions and Virasoro constraints in encoding higher-genus information. Overall, it forges a principled bridge between the geometry of Frobenius manifolds, Poisson/bihamiltonian formalism, and the GW/invariant framework across genera.

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.

Paper Structure

This paper contains 38 sections, 110 theorems, 1219 equations.

Table of Contents

  1. Introduction
  2. Normal forms of Hamiltonian structures of evolutionary systems
  3. Brief summary of finite-dimensional Poisson geometry
  4. Formal loop spaces
  5. Local multivectors and local Poisson brackets
  6. Extended formal loop space
  7. Miura group
  8. $(p,q)$-brackets on the extended formal loop space
  9. Classification of $(n,0)$-brackets
  10. Classification of $(0,n)$-brackets
  11. Bihamiltonian geometry of loop spaces
  12. Bihamiltonian structures and hierarchies of commuting flows
  13. Poisson pencils and bihamiltonian recursion procedure: summary of the finite-dimensional case
  14. Construction of bihamiltonian hierarchies on the extended loop spaces
  15. The leading order of $(0,n)$ bihamiltonian structures
  16. ...and 23 more sections

Key Result

Lemma 2.1.3

Let $h=(h^{ij}(x))$ be a Poisson structure of a constant rank $2n<N$ on a sufficiently small ball $U$. 1). A one-cocycle $v=(v^i(x))\in H^1(U,h)$ is trivial iff the vector field $v$ is tangent to the leaves of the symplectic foliation (foliation). 2). A 2-cocycle $f=(f^{ij}(x))\in H^2(U,h)$ is trivi for arbitrary two Casimirs of $h$.

Theorems & Definitions (172)

  • Example 2.1.1
  • Example 2.1.2
  • Lemma 2.1.3
  • Lemma 2.2.1
  • Example 2.2.2
  • Example 2.2.3
  • Theorem 2.2.4: dt
  • Example 2.2.5
  • Lemma 2.2.6
  • Corollary 2.2.7
  • ...and 162 more