Hamiltonian structure of rational isomonodromic deformation systems

TL;DR

This paper extends the Hamiltonian framework for isomonodromic deformations to rational connections with irregular singularities, revealing a rich Poisson structure and integrable deformation dynamics on the Riemann sphere.

Contribution

It introduces a Hamiltonian approach for rational isomonodromic systems with irregular singularities, linking deformation parameters to Casimir functions and Birkhoff invariants.

Findings

Established a Poisson structure on the space of rational connections.

Identified deformation parameters as Casimir functions.

Derived commuting Hamiltonian vector fields for spectral invariants.

Abstract

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincar\'e rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space $L^*gl(r)$ of the loop algebra $Lgl(r)$. Nonautonomous isomonodromic counterparts of the isospectral systems generated by spectral invariants are obtained by identifying the deformation parameters as Casimir functions on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, extended by the exponents of formal monodromy, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves, locally, may be identified as the orbits of a free abelian group. The projection of the infinitesimal isomonodromic deformations vector fields to the quotient manifold under this action gives the commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.