Painlevé—Calogero Correspondence

Abstract

It is proved that the Painlevé VI equation PVI_α,β,γ,δ for the special values of constants $\alpha = {\nu ^2}/4,\,\beta = - {\nu ^2}/4,\,\gamma = {\nu ^2}/4,\,\delta = {1 \over 2} - {\nu ^2}/4$ is a reduced Hamiltonian system. Its phase space is the set of flat SL(2, ℂ) connections over elliptic curves with a marked point, and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite Hamiltonian system. The phase space of the latter is the affine space of smooth connections, and the times are the Beltrami differentials. This approach allows us to define the associate linear problem, whose isomonodromic deformations are provided by the Painlevé equation and the Lax pair. In addition, it leads to a description of solutions by a linear procedure. This scheme can be generalized to G bundles over Riemann curves with marked points, where G is a simple complex Lie group. In some special limit such Hamiltonian systems convert into the Hitchin systems. In particular, for SL bundles over elliptic curves with a marked point, we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik—Zamolodchikov—Bernard equations are discussed.