Quantum Isomonodromic Deformations and the Knizhnik--Zamolodchikov Equations

TL;DR

This paper presents a new formulation connecting the Knizhnik--Zamolodchikov equations with quantum isomonodromic deformations, simplifying previous constructions and revealing their underlying algebraic structure.

Contribution

It introduces a novel quantum Schlesinger equations framework by transforming KZ equations into a Heisenberg representation, linking them to quantum isomonodromic deformations.

Findings

Derivation of quantum Schlesinger equations from KZ equations.

Simplified algebraic formulation of quantum isomonodromic deformations.

Connection established between KZ equations and quantum loop algebra structures.

Abstract

Viewing the Knizhnik--Zamolodchikov equations as multi--time, nonautonomous Shr\"odinger equations, the transformation to the Heisenberg representation is shown to yield the quantum Schlesinger equations. These are the quantum form of the isomonodromic deformations equations for first order operators of the form $\DD_\l = {\di \over \di \l} - \wh{\NN}(\l)$, where $\wh{\NN}(\l)$ is a rational $r\times r$ matrix valued function of $\l$ having simple poles only, and the matrix entries are interpreted as operators on a module of the rational $R$--matrix loop algebra $\wt{\frak{gl}}(r)_R$. This provides a simpler formulation of a construction due to Reshetikhin, relating the KZ equations to quantum isomonodromic deformations.