2D Locus Configurations and the Charged Trigonometric Calogero-Moser System

TL;DR

This paper links hyperplane arrangements in complex space to a particle system on the circle, providing explicit examples of real configurations and Schrödinger operators with special functions.

Contribution

It establishes a connection between locus configurations and the charged trigonometric Calogero-Moser system, leading to new explicit real configurations and Schrödinger operators.

Findings

Unique equilibrium for particles on S^1 in C^*

Explicit real locus configurations constructed

New Schrödinger operators with Baker-Akhiezer functions

Abstract

A central hyperplane arrangement in C^2 with multiplicity is called a `locus configuration' if it satisfies a series of `locus equations' on each hyperplane. Following Chalykh, Feigin and Veselov [CFV99], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on C^*: the charged trigonometric Calogero-Moser system. When the particles lie on S^1 in C^*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schr\"odinger operators with Baker-Akhiezer functions.