Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

TL;DR

This paper explores trigonometric solutions to WDVV equations, characterizes associated Veselov systems, and links these to generalized Calogero-Moser-Sutherland operators with factorized eigenfunctions.

Contribution

It introduces geometric conditions for trigonometric WDVV solutions, classifies small Veselov systems, and establishes a bidirectional link between these systems and eigenfunction factorizability.

Findings

Classified all trigonometric Veselov systems with up to five vectors.

Established that factorized eigenfunctions occur if and only if the system is a trigonometric Veselov system.

Connected Veselov systems to generalized Calogero-Moser-Sutherland operators.

Abstract

We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system ($\vee$-system) and we determine all trigonometric $\vee$-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.