Coxeter's frieze patterns and discretization of the Virasoro orbit

TL;DR

This paper demonstrates that Coxeter's frieze patterns serve as a discrete analogue of Virasoro coadjoint orbits, linking discrete geometry with continuous conformal structures and symplectic forms.

Contribution

It establishes a novel connection between Coxeter's frieze patterns and the Virasoro algebra's coadjoint orbits, providing a discretization of the Kirillov symplectic form.

Findings

Frieze patterns form a discrete model of Virasoro coadjoint orbits.

The symplectic form on frieze patterns discretizes the Kirillov form.

Continuous frieze patterns relate to 2D conformal metrics.

Abstract

We show that the space of classical Coxeter's frieze patterns can be viewed as a discrete version of a coadjoint orbit of the Virasoro algebra. The canonical (cluster) (pre)symplectic form on the space of frieze patterns is a discretization of the Kirillov symplectic form. We relate a continuous version of frieze patterns to conformal metrics of constant curvature in dimension 2.