"Quantizations" of isomonodromic Hamiltonian Garnier system with two degrees of freedom

TL;DR

This paper constructs explicit solutions for a quantum analogue of the Garnier system, linking isomonodromic Hamiltonian dynamics with solutions to linear differential equations and conformal field theory equations.

Contribution

It introduces a method to obtain solutions of the quantum Garnier system from linear ODEs, connecting integrable systems with conformal field theory.

Findings

Solutions satisfy Belavin-Polyakov-Zamolodchikov equations

Explicit transforms relate solutions to linear ODEs

Bridges between isomonodromic systems and conformal field theory

Abstract

We construct solutions of analogues of the nonstationary Schr\"odinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables.