4-dimensional Frobenius manifolds and Painleve' VI

TL;DR

This paper establishes a link between 4-dimensional Frobenius manifolds with special structures and the Painleve VIμ equation, providing a method to construct such manifolds from 3-dimensional cases, with explicit examples on Hurwitz spaces.

Contribution

It demonstrates that semisimple 4D Frobenius manifolds with tri-hamiltonian structure are described by Painleve VIμ and offers a procedure to construct them from 3D Frobenius manifolds.

Findings

Frobenius manifolds of dimension four are linked to Painleve VIμ equations.

Explicit construction method from 3D to 4D Frobenius manifolds.

Examples provided on Hurwitz spaces.

Abstract

A Frobenius manifold has tri-hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We focus on the case of dimension four and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlev\'e VI$\mu$ equation. This yields an explicit procedure associating to any semisimple Frobenius manifold of dimension three a tri-hamiltonian Frobenius manifold of dimension four. We carry out explicit examples for the case of Frobenius structures on Hurwitz spaces.