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Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds

Alexey Basalaev

TL;DR

The paper establishes a precise link between genus-zero Hurwitz–Frobenius manifolds and the genus-zero Whitham hierarchy. It proves that the stabilized Frobenius potentials $F^{K}$ define an infinite family of commuting PDEs that are equivalent to the Whitham hierarchy, and provides dual descriptions in Fay form and coordinate-free Lax form. It further connects to the dispersionless multi-KP framework via an $\hbar$-deformation of the differential operators, thereby bridging Hurwitz geometry with both Fay-type and Lax approaches to integrable hierarchies. Together these results illuminate how Hurwitz–Frobenius geometry underpins universal integrable structures and offers a route to dispersionful hierarchies through $\bar{\hbar}$-deformations.

Abstract

B. Dubrovin introduced the structure of a Dubrovin--Frobenius manifold on a space of ramified coverings of a sphere by a genus $g$ Riemann surface with the prescribed ramification profile. This is now known as a genus $g$ Hurwitz--Frobenius manifold. We investigate the genus zero Hurwitz--Frobenius manifolds and their connection to the integrable hierarchies. In particular, we prove that the Frobenius potentials of the genus zero Hurwitz--Frobenius manifolds stabilize and therefore define an infinite system of commuting PDEs. We show that this system of PDEs is equivalent to the genus zero Whitham hierarchy of I. Krichever. Our result shows that this system of PDEs has both Fay form, depending heavily on the flat structure fo the Hurwitz--Frobenius manifold and coordinatefree Lax form. We also show how to extend this system of PDEs to the multicomponent KP hierarchy via the $\hbar$--deformation of the differential operators.

Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds

TL;DR

The paper establishes a precise link between genus-zero Hurwitz–Frobenius manifolds and the genus-zero Whitham hierarchy. It proves that the stabilized Frobenius potentials define an infinite family of commuting PDEs that are equivalent to the Whitham hierarchy, and provides dual descriptions in Fay form and coordinate-free Lax form. It further connects to the dispersionless multi-KP framework via an -deformation of the differential operators, thereby bridging Hurwitz geometry with both Fay-type and Lax approaches to integrable hierarchies. Together these results illuminate how Hurwitz–Frobenius geometry underpins universal integrable structures and offers a route to dispersionful hierarchies through -deformations.

Abstract

B. Dubrovin introduced the structure of a Dubrovin--Frobenius manifold on a space of ramified coverings of a sphere by a genus Riemann surface with the prescribed ramification profile. This is now known as a genus Hurwitz--Frobenius manifold. We investigate the genus zero Hurwitz--Frobenius manifolds and their connection to the integrable hierarchies. In particular, we prove that the Frobenius potentials of the genus zero Hurwitz--Frobenius manifolds stabilize and therefore define an infinite system of commuting PDEs. We show that this system of PDEs is equivalent to the genus zero Whitham hierarchy of I. Krichever. Our result shows that this system of PDEs has both Fay form, depending heavily on the flat structure fo the Hurwitz--Frobenius manifold and coordinatefree Lax form. We also show how to extend this system of PDEs to the multicomponent KP hierarchy via the --deformation of the differential operators.

Paper Structure

This paper contains 11 sections, 19 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Hurwitz--Frobenius manifold
  4. Dubrovin--Frobenius manifolds
  5. Hurwitz--Frobenius manifold
  6. Flat structure
  7. Supplimentary facts
  8. Stabilization
  9. Whitham and multi-KP hierarchies
  10. h--deformation
  11. Proof of Theorem \ref{['theorem: Fay']}

Key Result

Theorem 1

Let $L \ge K$, denote by $F^L$ and $F^K$ the potentials of ${\mathcal{H}}_{0,L}$ and ${\mathcal{H}}_{0,K}$ respectively. For any $0 \le i,j \le N$ let $\alpha \le k_i$, $\beta \le k_j$ if $i \neq j$ and $\alpha+\beta \le k_i-2$ if $i=j$. Then assumed as the equality of function in $s_{r,\bullet}$ and $q_1,\dots,q_N$.

Theorems & Definitions (37)

  • Theorem : Theorem \ref{['theorem: stabilization']} below
  • Corollary
  • Theorem : Theorem \ref{['theorem: Fay']} below
  • Corollary
  • Remark 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Theorem 2.4: Theorem 5.1 of D2
  • ...and 27 more