Symplectic Manifolds and Isomonodromic Deformations

Abstract

We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah-Bott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painleve equations and Schlesinger's equations into a uniform framework.

Published in 2001 (based on part of 1999 Oxford thesis), posted here for easier accessibility

Cited by in corpus (12)

• Geometry of Multiplicative Preprojective Algebra
• The Hamiltonian Structure of the Second Painleve Hierarchy
• On the Geometry of Isomonodromic Deformations
• Hamiltonian structure of rational isomonodromic deformation systems
• Counting the fission trees and nonabelian Hodge graphs (untwisted case)
• On the tau function of the hypergeometric equation
• Topology of irregular isomonodromy times on a fixed pointed curve
• Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Interpreting the Ooguri-Vafa symplectic form à la Atiyah-Bott
• Meromorphic Projective Structures, Opers and Monodromy
• Polystability of Stokes representations and differential Galois groups