Hamiltonian structure of isomonodromic deformations of rational connections on the Riemann sphere

          @misc{ pirsa_PIRSA:09050039,
            doi = {10.48660/09050039},
            url = {https://pirsa.org/09050039},
            author = {Harnad, John},
            keywords = {},
            language = {en},
            title = {Hamiltonian structure of isomonodromic deformations of rational connections on the Riemann sphere},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {may},
            note = {PIRSA:09050039 see, \url{https://pirsa.org}}
          }
          

Abstract

The classical "split" rational R-matrix Poisson bracket structure on the space of rational connections over the Riemann sphere provides a natural setting for studying deformations. It can be shown that a natural set of Poisson commuting spectral invariant Hamiltonians, which are dual to the Casimir invariants of the Poisson structure, generate all deformations which, when viewed as nonautonomous Hamiltonian systems, preserve the generalized monodromy of the connections, in the sense of Birkhoff (i.e., the monodromy representation, the Stokes parameters and connection matrices). These spectral invariants may be expressed as residues of the trace invariants of the connection over the spectral curve. Applications include the deformation equations for orthogonal polynomials having "semi-classical" measures. The $\tau$ function for such isomonodromic deformations coincides with the Hankel determinant formed from the moments, and is interpretable as a generalized matrix model integral. They are also related to Seiberg-Witten invariants. (This talk is based in part on joint work with: Marco Bertola, Gabor Pusztai and Jacques Hurtubise)