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Solutions to Open WDVV Equations for the Universal Whitham Hierarchy

Shilin Ma

TL;DR

The paper constructs explicit solutions to the open WDVV equations for the genus-zero universal Whitham hierarchy by introducing two functions $\Omega(\mathbf{u},s)$ and $\hat{\Omega}(\mathbf{u},s)$ on the infinite-dimensional Frobenius manifold $\mathcal{M}$. These functions define flat F‑manifold structures and yield explicit principal hierarchies, with proven compatibility under finite-dimensional reductions to rational superpotentials and $\\mathbb{Z}_2$‑symmetric submanifolds, thus unifying several known open WDVV solutions. In particular, the results recover Basalaev and Buryak’s A‑ and D‑type polynomial solutions in appropriate limits and connect to open Saito theory and open r‑spin theories. The work also elaborates on how the infinite‑dimensional construction specializes to finite‑dimensional Frobenius manifolds and their dispersionless limits, offering a framework to relate infinite‑dimensional flat F‑manifolds, integrable hierarchies, and enumerative geometry. Altogether, the paper extends open WDVV theory to a broad, geometrically rich setting tied to universal Whitham dynamics and its reductions, providing new avenues for both theoretical insight and potential applications in integrable systems and geometry.

Abstract

In this paper, we construct a pair of solutions to the open WDVV equations associated with the infinite-dimensional Frobenius manifolds that underlie the genus-zero universal Whitham hierarchy, and for the resulting flat F-manifolds, we explicitly construct their principal hierarchies. We further demonstrate that this construction is compatible with finite-dimensional reductions, yielding solutions for Frobenius manifolds associated with general rational superpotentials and those subject to a $\mathbb{Z}_{2}$-symmetry reduction. In particular, the polynomial solutions derived by Basalaev and Buryak via open Saito theory for A- and D-type singularities are recovered as special cases.

Solutions to Open WDVV Equations for the Universal Whitham Hierarchy

TL;DR

The paper constructs explicit solutions to the open WDVV equations for the genus-zero universal Whitham hierarchy by introducing two functions and on the infinite-dimensional Frobenius manifold . These functions define flat F‑manifold structures and yield explicit principal hierarchies, with proven compatibility under finite-dimensional reductions to rational superpotentials and ‑symmetric submanifolds, thus unifying several known open WDVV solutions. In particular, the results recover Basalaev and Buryak’s A‑ and D‑type polynomial solutions in appropriate limits and connect to open Saito theory and open r‑spin theories. The work also elaborates on how the infinite‑dimensional construction specializes to finite‑dimensional Frobenius manifolds and their dispersionless limits, offering a framework to relate infinite‑dimensional flat F‑manifolds, integrable hierarchies, and enumerative geometry. Altogether, the paper extends open WDVV theory to a broad, geometrically rich setting tied to universal Whitham dynamics and its reductions, providing new avenues for both theoretical insight and potential applications in integrable systems and geometry.

Abstract

In this paper, we construct a pair of solutions to the open WDVV equations associated with the infinite-dimensional Frobenius manifolds that underlie the genus-zero universal Whitham hierarchy, and for the resulting flat F-manifolds, we explicitly construct their principal hierarchies. We further demonstrate that this construction is compatible with finite-dimensional reductions, yielding solutions for Frobenius manifolds associated with general rational superpotentials and those subject to a -symmetry reduction. In particular, the polynomial solutions derived by Basalaev and Buryak via open Saito theory for A- and D-type singularities are recovered as special cases.

Paper Structure

This paper contains 15 sections, 7 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Frobenius manifold structure on $\mathcal{M}$
  4. Invariant metric
  5. Flat coordinates
  6. Levi-Civita connection
  7. Multiplication structure
  8. Principal hierarchy
  9. Proofs of the Main Theorems
  10. Flat F-manifolds and principal hierarchy
  11. Proof of Theorem \ref{['mainthm1']}
  12. Proof of Theorem \ref{['mainthm2']}
  13. Reductions
  14. Proof of Corollary \ref{['maincor4']}
  15. Proof of Corollaries \ref{['maincor5']} and \ref{['maincor6']}

Key Result

Theorem 1.1

There exists a pair of functions, $\Omega(\mathbf{u},s)$ on $\mathcal{M}\times \mathbf{D}^{\mathrm{ext}}$ and $\hat{\Omega}(\mathbf{u},s)$ on $\mathcal{M}\times \mathbf{D}^{\mathrm{int}}$, that provide solutions to the open WDVV equations for $\mathcal{M}$. These functions are defined by the followi Here, the projection operators $(\cdot)_{\pm}, (\cdot)_{\infty,\ge0}$ and $(\cdot)_{\varphi_{j},\le

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 3.1
  • Lemma 3.2
  • proof