An interpretation of some Hitchin Hamiltonians in terms of isomonodromic deformation (Q414520)

The author studies moduli spaces of framed principal bundles with connections having irregular singularities over a compact Riemann surface, which have been constructed by \textit{Boalch} by means of an infinite-dimensional symplectic reduction [Adv. Math. 163, No. 2, 137--205 (2001; Zbl 1001.53059); Int. Math. Res. Not. 2002, No. 22, 1129--1166 (2002; Zbl 1003.58028)].NEWLINENEWLINEIn the first part of the paper (Sections 1--4), a detailled account on the known results is given. Also, it is proved that the symplectic structure induced from the Atiyah-Bott form agrees with the one given in terms of hypercohomology (Section 3). The main results are presented in Section 5. It is proved that Hamiltonian vector fields corresponding to some of the Hitchin Hamiltonians on moduli spaces of irregular connections coincide with vector fields that arise from differences of isomonodromic flows defined in two different ways.NEWLINENEWLINEThe proofs are based on constructions by \textit{I. Krichever} [Mosc. Math. J. 2, No. 4, 717--752 (2002; Zbl 1044.70010)] and \textit{J. Hurtubise} [J. Geom. Phys. 58, No. 10, 1394--1406 (2008; Zbl 1153.37034)], as well as on a realization of open sets in the moduli space of principal bundles as Hecke modifications of a fixed bundle, which was previously studied by author in [``Hecke modifications, wonderful compactifications and moduli of principal bundles'', Ann. Sc. Norm. Super. Pisa, Cl. Sci. (in press); \url{arXiv:1010.0877}].