4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension n = 4 and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the PainlevéVIµ equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension n = 3, this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces