Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations. Free fields approach

Abstract

We investigate the classical limit of the Knizhnik-Zamolodchikov-Bernard equations, considered as a system of non-stationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced non-autonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlevé VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the space of flat connections of $G$ bundles with some additional data in the marked points. HID can be derived from some free field theory by the hamiltonian reduction under the action of the gauge symmetries and subsequent factorization with respect to diffeomorphisms of curve. This approach allows to define the Lax equations associated with HID and the linear system whose isomonodromic deformations are provided by HID. In addition, it leads to description of solutions of HID by the projection method. In some special limit HID 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