Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

TL;DR

This paper introduces asymptotic boundary KZB operators for semisimple Lie groups, proves their properties, and uses them to define new quantum superintegrable systems called quantum Calogero-Moser spin chains, linking conformal field theory and integrable models.

Contribution

It defines and analyzes asymptotic boundary KZB operators for real semisimple Lie groups and introduces quantum Calogero-Moser spin chains as a new class of superintegrable systems.

Findings

Proved algebraic properties and commutativity of asymptotic boundary KZB operators.

Derived explicit expressions for Hamiltonians of quantum Calogero-Moser spin chains.

Connected boundary KZB equations to quantum integrable systems.

Abstract

Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G and an one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first order quantum Hamiltonians. We also explicitly describe the Schr\"odinger operator.