Lax equations and Knizhnik-Zamolodchikov connection

TL;DR

This paper constructs a projective unitary representation of Hamiltonian vector fields from a Lax system with spectral parameter on a Riemann surface, providing a prequantization and analyzing operator commutation properties.

Contribution

It introduces a novel method to prequantize Lax systems via Knizhnik-Zamolodchikov operators, linking integrable systems with representation theory.

Findings

Representation operators of Poisson commuting Hamiltonians projectively commute.

Operators commute when Hamiltonians depend only on action variables.

Provides a framework connecting Lax systems with quantum representations.

Abstract

Given a Lax system of equations with the spectral parameter on a Riemann surface we construct a projective unitary representation of the Lie algebra of Hamiltonian vector fields by Knizhnik-Zamolodchikov operators. This provides a prequantization of the Lax system. The representation operators of Poisson commuting Hamiltonians of the Lax system projectively commute. If Hamiltonians depend only on action variables then the corresponding operators commute.