On the quantization of isomonodromic deformations on the torus

TL;DR

This paper demonstrates that quantizing isomonodromic deformations on the torus naturally yields the Knizhnik-Zamolodchikov-Bernard equations, linking geometric deformation theory with quantum field theory structures.

Contribution

It establishes a direct connection between the quantization of isomonodromic deformations on the torus and the derivation of KZB equations, highlighting the role of Chern-Simons Poisson structures.

Findings

Quantization leads to KZB equations on the torus.

Poisson structure from Chern-Simons theory underpins the Hamiltonian formulation.

Twisted quantities emerge naturally due to the torus topology.

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.