4-Dimensional Frobenius manifolds and Painleve’ VI

Abstract

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 Painleve $$\hbox {VI}\mu $$ 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.