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# Hamiltonian structure of rational isomonodromic deformation systems

Dec 2022

The Hamiltonian approach to isomonodromic deformation systems is extended toinclude generic rational covariant derivative operators on the Riemann spherewith irregular singularities of arbitrary Poincar\'e rank. The space ofrational connections with given pole degrees carries a natural Poissonstructure corresponding to the standard classical rational R-matrix structureon the dual space $L^*{\mathfrak g}(r)$ of the loop algebra $L{\mathfrakg}(r)$. Nonautonomous isomonodromic counterparts of the isospectral systemsgenerated by spectral invariants are obtained by identifying the deformationparameters as Casimir invariants on the phase space. These are shown tocoincide with the higher {\it Birkhoff invariants} determining the localasymptotics near to irregular singular points, together with the pole loci. Infinitesimal isomonodromic deformations are shown to be generated by the sumof the Hamiltonian vector field and an {\em explicit derivative} vector fieldthat is transversal to the symplectic foliation. The Casimir invariants serveas coordinates complementing those along the symplecticeaves, extended by the{\it exponents of formal monodromy}, defining a local symplectomorphism betweenthem. The explicit derivative vector fields preserve the Poisson structure anddefine a flat {\it transversal connection}, spanning an integrable distributionwhose leaves, locally, may be identified as the orbits of a free abelian group.The projection of the infinitesimal isomonodromic deformations vector fields tothe quotient manifold under this action gives the commuting Hamiltonian vectorfields corresponding to the spectral invariants dual to the Birkhoff invariantsand the pole loci.

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