Dubrovin duality for open Hurwitz flat F-manifolds
Alessandro Proserpio, Ian A. B. Strachan
TL;DR
This work establishes that the Dubrovin dual of a Hurwitz Frobenius manifold extends naturally to an F-manifold with a compatible flat connection on the universal curve, aligning with open WDVV equations. It reveals a bi-flat F-manifold structure on the universal curve formed by the original and dual extended multiplications, linked by an eventual identity that encodes homogeneity and rescalings. The authors develop a general rank-one extension framework via Landau-Ginzburg models, show the twisted periods provide flat coordinates for the dual deformed connection, and interpret open WDVV conditions geometrically. They derive explicit ADE-type open WDVV solutions and present several new dual open WDVV solutions across Hurwitz, Dubrovin-Zhang, Ma-Zuo, and Jacobi-group settings, illustrating a unifying approach to open associativity equations with geometric meaning and potential implications for open Gromov-Witten theory.
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.