4-Dimensional Frobenius manifolds and Painleve’ VI

Abstract

A Frobenius manifold has tri-hamiltonian structure if it is evendimensional 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 Painlevé VIµ 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.

First integrals of affine connections and Hamiltonian systems of hydrodynamic type

“…BEX 13656/13-9. We thank Robert Bryant for discussions about invariants of ODEs, and Guido Carlet for discussions about Frobenius manifolds, and for bringing reference [17] to our attention.…”

The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁

“…The Hurwitz spaces H 1,0,0 are classified by the group J Ã 1 , hence we increase the knowledge of the WDVV/discrete group correspondence. Recently, the case J Ã 1 attracted the attention of experts, due to its application in integrable systems [5,13,16].…”

Regular Flat Structure and Generalized Okubo System

“…Change the variables z →z = (z 1,0 − z 2,0 )z + z 2,0 and t → z 1,1 = (z 1,0 − z 2,0 )t. Diagonalizing C V by applying a gauge transformation to (45), we obtain from S V a matrix T satisfying (43). If the resulting matrix T satisfies the condition (30) in Theorem 3.12 (which is a generic condition), we obtain a flat coordinate system and a potential vector field satisfying the condition (36).…”

“…Remark 5.1. The correspondence between particular 4-dimensional Frobenius manifolds and generic solutions to a one-parameter family of the Painlevé VI equation was treated by S. Romano [30] (in a somewhat different context).…”