Dubrovin duality for open Hurwitz flat F-manifolds

TL;DR

This paper demonstrates how Dubrovin duality extends to open Hurwitz flat F-manifolds, revealing their bi-flat structure, providing geometric insights into open WDVV solutions, and presenting explicit examples including new solutions.

Contribution

It establishes the extension of Dubrovin duality to open Hurwitz flat F-manifolds and explores their duality, enriching the geometric understanding of open WDVV equations and providing explicit examples.

Findings

Dubrovin duality extends to open Hurwitz flat F-manifolds.

Universal curves acquire bi-flat F-manifold structures.

Explicit examples for types A, D, and E are computed, including new solutions.

Abstract

We prove that the Dubrovin dual of a Hurwitz Frobenius manifold extends naturally to an F-manifold with compatible flat connection on the universal curve, in the sense of the open WDVV equations. A similar result is proven for the Frobenius manifold itself in arXiv:2503.09258 . This equips the universal curve with two F-manifolds with compatible flat structure, and we study their duality. We show that they combine into a bi-flat F-manifold. Conditions on open WDVV solutions imposed in previous work are retrieved in this setting, thus providing them with a geometrical meaning. Finally, explicit examples are computed. For Saito Frobenius manifolds of types $A$ and $D$, the extended prepotentials coincide with open WDVV solutions computed independently, whereas even the existence of the solution in type $E$ had not been previously discussed. On the other hand, new non-homogeneous solutions are constructed by duality.