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Infinite-dimensional Frobenius Manifolds and Extensions of Genus-Zero Whitham Hierarchies

Shilin Ma, Chao-Zhong Wu, Dafeng Zuo

TL;DR

The paper develops a comprehensive framework for infinite-dimensional Frobenius manifolds on spaces of pairs of meromorphic functions defined around $m$ circles on the Riemann sphere, establishing a flat metric, cotangent/tangent structures, a Frobenius product, a symmetric 3-tensor, a potential, an Euler vector field, and an intersection form. It then derives the principal hierarchy on the loop space, identifies a bi-Hamiltonian structure, and constructs explicit Hamiltonian densities that satisfy the required flatness and compatibility conditions. A key result is that the genus-zero Whitham hierarchy is embedded as a subhierarchy of the principal hierarchy, with clear reductions to finite-dimensional Frobenius manifolds $M^{cKP}$ in the rational superpotential limit. The work provides a robust algebraic-geometric route to extend dispersionless Whitham dynamics to infinite dimensions and suggests a route toward dispersive Whitham hierarchies, with potential connections to 2+1D integrable systems and topological-field-theory-type structures. Overall, it broadens the scope of Frobenius-manifold techniques to new geometric settings and lays groundwork for dispersive extensions of genus-zero Whitham hierarchies.$

Abstract

In this paper we construct a class of infinite-dimensional Frobenius manifolds in the spaces of pairs of meromorphic functions defined on certain regions of the Riemann sphere. For such Frobenius manifolds, we obtain their principal hierarchies and show them to be extensions of the genus-zero Whitham hierarchies.

Infinite-dimensional Frobenius Manifolds and Extensions of Genus-Zero Whitham Hierarchies

TL;DR

The paper develops a comprehensive framework for infinite-dimensional Frobenius manifolds on spaces of pairs of meromorphic functions defined around circles on the Riemann sphere, establishing a flat metric, cotangent/tangent structures, a Frobenius product, a symmetric 3-tensor, a potential, an Euler vector field, and an intersection form. It then derives the principal hierarchy on the loop space, identifies a bi-Hamiltonian structure, and constructs explicit Hamiltonian densities that satisfy the required flatness and compatibility conditions. A key result is that the genus-zero Whitham hierarchy is embedded as a subhierarchy of the principal hierarchy, with clear reductions to finite-dimensional Frobenius manifolds in the rational superpotential limit. The work provides a robust algebraic-geometric route to extend dispersionless Whitham dynamics to infinite dimensions and suggests a route toward dispersive Whitham hierarchies, with potential connections to 2+1D integrable systems and topological-field-theory-type structures. Overall, it broadens the scope of Frobenius-manifold techniques to new geometric settings and lays groundwork for dispersive extensions of genus-zero Whitham hierarchies.$

Abstract

In this paper we construct a class of infinite-dimensional Frobenius manifolds in the spaces of pairs of meromorphic functions defined on certain regions of the Riemann sphere. For such Frobenius manifolds, we obtain their principal hierarchies and show them to be extensions of the genus-zero Whitham hierarchies.
Paper Structure (18 sections, 24 theorems, 148 equations)

This paper contains 18 sections, 24 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Infinite-dimensional Frobenius manifold structure on $\mathcal{M}$
  5. Flat metric
  6. Cotangent spaces
  7. Frobenius algebra of the tangent space
  8. Symmetric $3$-tensor
  9. Potential
  10. Euler vector field
  11. Intersection form
  12. Principal hierarchy for $\mathcal{M}$
  13. General theory of principal hierarchy
  14. Bi-Hamiltonian structure on the loop space of $\mathcal{M}$
  15. Densities of Hamiltonian functionals
  16. ...and 3 more sections

Key Result

Theorem 1.1

On $\mathcal{M}$ there exists an infinite-dimensional Frobenius manifold structure $(\mathcal{M}, \circ, e, \eta, E)$ of charge $d=1-\frac{2}{n_{0}}$, where the flat metric $\eta$, the invariant multiplication $\circ$ of vector fields with respect to $\eta$, the unity vector field $e$ and the Euler

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 35 more