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The Stability Limit of Prepotentials for Hurwitz-Frobenius Manifolds: An Infinite-Dimensional Approach

Shilin Ma

TL;DR

The work provides a direct geometric realization of the stabilization phenomenon for prepotential derivatives of Hurwitz-Frobenius manifolds by embedding the problem into an infinite-dimensional Frobenius-manifold framework that underlies the genus-zero universal Whitham hierarchy. It shows that the stable limits of these derivatives coincide with the τ-structure of a sector of the principal hierarchy on this infinite-dimensional manifold, yielding an intrinsic identification with the Whitham hierarchy and extending to open WDVV equations. Reductions to A- and D-type cases, including D-type even/parity reductions, are treated, recovering known results and supplying a unified Lax-like interpretation within a single geometric framework. This establishes a geometric bridge between stabilization, Whitham theory, and open WDVV structures, with potential implications for dispersionless hierarchies and their open-extended generalizations.

Abstract

The stability of prepotential derivatives for Frobenius manifolds associated with A_N and D_N singularities has been utilized to construct (2+1)-dimensional dispersionless integrable hierarchies. Although the generalization of this construction to genus-zero Hurwitz-Frobenius manifolds was shown to yield the genus-zero Whitham hierarchy, a direct geometric explanation of this correspondence has been lacking. In this note, we provide a direct proof of this identification within the framework of infinite-dimensional Frobenius manifolds. We demonstrate that the stability of prepotentials is an intrinsic property of the tau-structure of the Whitham hierarchy. Furthermore, we extend this identification to the hierarchies arising from the stability of solutions to the open WDVV equations with the extensions of the Whitham hierarchy.

The Stability Limit of Prepotentials for Hurwitz-Frobenius Manifolds: An Infinite-Dimensional Approach

TL;DR

The work provides a direct geometric realization of the stabilization phenomenon for prepotential derivatives of Hurwitz-Frobenius manifolds by embedding the problem into an infinite-dimensional Frobenius-manifold framework that underlies the genus-zero universal Whitham hierarchy. It shows that the stable limits of these derivatives coincide with the τ-structure of a sector of the principal hierarchy on this infinite-dimensional manifold, yielding an intrinsic identification with the Whitham hierarchy and extending to open WDVV equations. Reductions to A- and D-type cases, including D-type even/parity reductions, are treated, recovering known results and supplying a unified Lax-like interpretation within a single geometric framework. This establishes a geometric bridge between stabilization, Whitham theory, and open WDVV structures, with potential implications for dispersionless hierarchies and their open-extended generalizations.

Abstract

The stability of prepotential derivatives for Frobenius manifolds associated with A_N and D_N singularities has been utilized to construct (2+1)-dimensional dispersionless integrable hierarchies. Although the generalization of this construction to genus-zero Hurwitz-Frobenius manifolds was shown to yield the genus-zero Whitham hierarchy, a direct geometric explanation of this correspondence has been lacking. In this note, we provide a direct proof of this identification within the framework of infinite-dimensional Frobenius manifolds. We demonstrate that the stability of prepotentials is an intrinsic property of the tau-structure of the Whitham hierarchy. Furthermore, we extend this identification to the hierarchies arising from the stability of solutions to the open WDVV equations with the extensions of the Whitham hierarchy.
Paper Structure (13 sections, 7 theorems, 187 equations)

This paper contains 13 sections, 7 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frobenius manifolds and their principal hierarchies
  4. Open WDVV equations and flat $F$-manifolds
  5. $\tau$-structure of $\mathcal{M}$
  6. Definitions of $\mathcal{M}$
  7. Principal hierarchy and $\tau$-structure
  8. Properties of the $\tau$-structure
  9. Stabilization of $H_{0; \mathbf{n}}$ and Related Hierarchies
  10. Explicit formulas for the prepotential derivatives of $H_{0;\mathbf{n}}$
  11. Stabilization of the Prepotential
  12. Open extension and stable limits
  13. Stabilization of D-type prepotentials

Key Result

Theorem 3.1

The functions $\Omega_{\alpha, p; \beta, q}$ defined by the following contour integrals constitute the $\tau$-structure for the principal hierarchy over the sector $\{h_{k, 0}\}_{k=1}^m \cup \{e\}:$ where and $\gamma = \bigcup_{k=1}^m \gamma_k$ denotes the union of the boundaries of the disks $D_k$.

Theorems & Definitions (9)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Corollary 5.1
  • Corollary 5.2