4-dimensional Frobenius manifolds and Painleve' VI

2012/09/18 by Stefano Romano, Romano, Stefano
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AG #math.MP #nlin.SI

paper · pdf · doi:10.48550/arxiv.1209.3959

28 pages. One reference added

arxiv created 2012/09/19 · arxiv updated 2012/09/21

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 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.

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