Limits of Gaudin Systems: Classical and Quantum Cases

TL;DR

This paper explores the classical and quantum limits of Gaudin systems, introducing new integrals and algebras through pole collision limits, and connecting to Bending flows and multi-Poisson geometry.

Contribution

It presents a novel limiting procedure for Gaudin systems that yields new integrals and algebraic structures, expanding understanding of classical and quantum integrability.

Findings

New families of Liouville integrals in classical Gaudin systems.

Introduction of new Gaudin algebras in the quantum case.

Connection of total pole collisions to Bending flows and multi-Poisson geometry.

Abstract

We consider the XXX homogeneous Gaudin system with $N$ sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new "Gaudin" algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of "Manin matrices" to provide explicit generators of the Gaudin Algebras in the quantum case.