Quantization of Integrable Systems with Spectral Parameter on a Riemann Surface

Abstract

Given an integrable system defined by a Lax representation with spectral parameter on a Riemann surface, we construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields by means of operators of covariant derivatives with respect to the Knizhnik–Zamolodchikov connection. It is a Dirac-type prequantization of the integrable system from a physical point of view. Simultaneously, it establishes a correspondence between integrable systems in question and conformal field theories. In the present paper, we focus on systems whose spectral curves possess a holomorphic involution. Examples are presented by Hitchin systems of the types Bn, $${{C}_{n}}$$ , $${{D}_{n}}$$ , and also of the type An on hyperelliptic curves.