The symplectic and twistor geometry of the general isomonodromic deformation problem

Details The symplectic and twistor geometry of the general isomonodromic deformation problem The symplectic and twistor geometry of the general isomonodromic deformation problem

2000-07-18

arxiv.org/abs/nlin/0007024v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

35 pages, LATEX

Scientific paper

10.1016/S0393-0440(01)00003-1

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.

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