On the quantization of isomonodromic deformations on the torus

Abstract

The quantization of isomonodromic deformation of a meromorphic connection on the torus is shown to lead directly to the Knizhnik-Zamolodchikov-Bernard equations in the same way as the problem on the sphere leads to the system of Knizhnik-Zamolodchikov equations. The Poisson bracket required for a Hamiltonian formulation of isomonodromic deformations is naturally induced by the Poisson structure of Chern-Simons theory in a holomorphic gauge fixing. This turns out to be the origin of the appearance of twisted quantities on the torus.