Flat Structure on the Space of Isomonodromic Deformations

Abstract

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

References in corpus (6)

• Classification of Fuchsian systems and their connection problem
• Some explicit solutions to the Riemann-Hilbert problem
• From Darboux-Egorov system to bi-flat $F$-manifolds
• $F$-manifolds, multi-flat structures and Painlevé transcendents
• Fractional calculus of Weyl algebra and Fuchsian differential equations
• On Algebraic Solutions to Painleve VI

Cited by in corpus (11)

• A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$
• Regular non-semisimple Dubrovin-Frobenius manifolds
• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures
• 3-dimensional F-manifolds
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• Almost duality for Saito structure and complex reflection groups II: the case of Coxeter and Shephard groups
• The Saito determinant for Coxeter discriminant strata
• A Frobenius manifold for $\ell$-Kronecker quiver
• A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups
• Differential relations for almost Belyi maps