Quantum Isomonodromic Deformations and the Knizhnik--Zamolodchikov Equations

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.