The symplectic and twistor geometry of the general isomonodromic deformation problem

Abstract

Hitchin's twistor treatment of Schlesinger's equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n,C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n,C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by deforming X in Z. This is related to a description of the deformations in terms of Hamiltonian flows on a symplectic manifold constructed from affine orbits in the dual Lie algebra of a loop group.

35 pages, LATEX

References in corpus (1)

• Symplectic Manifolds and Isomonodromic Deformations

Cited by in corpus (5)

• Quasi-Hamiltonian Geometry of Meromorphic Connections
• The Hamiltonian Structure of the Second Painleve Hierarchy
• On the Geometry of Isomonodromic Deformations
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Hamiltonian structure of rational isomonodromic deformation systems