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.

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

Cited by in corpus (40)

• Classical Conformal Blocks and Painleve VI
• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Cluster varieties from Legendrian knots
• Stability conditions and Stokes factors
• Geometry of Multiplicative Preprojective Algebra
• The dependence on the monodromy data of the isomonodromic tau function
• T-Branes at the Limits of Geometry
• Moduli of unramified irregular singular parabolic connections on a smooth projective curve
• The Hamiltonian Structure of the Second Painleve Hierarchy
• Global Weyl groups and a new theory of multiplicative quiver varieties
• Gravitational instantons with faster than quadratic curvature decay (III)
• On some Hamiltonian properties of the isomonodromic tau functions
• Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues
• On the Geometry of Isomonodromic Deformations
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Linear differential equations on the Riemann sphere and representations of quivers
• Duality for the general isomonodromy problem
• The symplectic and twistor geometry of the general isomonodromic deformation problem
• A theory of minimal K-types for flat G-bundles
• Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System
• Middle Convolution and Harnad Duality
• Hamiltonian structure of rational isomonodromic deformation systems
• A Lie-theoretic description of the solution space of the tt*-Toda equations
• Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
• Isomonodromic deformations of connections with singularities of parahoric formal type
• Flat parabolic vector bundles on elliptic curves
• Moduli spaces of irregular singular connections
• An Interpretation of Some Hitchin Hamiltonians In Terms of Isomonodromic Deformation
• Confluent KZ equations for sl_N with Poincare rank 2 at infinity
• The Deligne-Simpson problem for connections on ${\mathbb{G}}_m$ with a maximally ramified singularity
• On the tau function of the hypergeometric equation
• Counting the fission trees and nonabelian Hodge graphs (untwisted case)
• Euler continuants in noncommutative quasi-Poisson geometry
• Topology of irregular isomonodromy times on a fixed pointed curve
• Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
• Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Meromorphic Projective Structures, Opers and Monodromy
• Interpreting the Ooguri-Vafa symplectic form à la Atiyah-Bott
• Polystability of Stokes representations and differential Galois groups