On continuous 2-frieze patterns

TL;DR

This paper introduces and explores continuous 2-frieze patterns, linking them to moduli spaces of projective curves and symplectic structures, expanding the understanding of frieze patterns in a continuous setting.

Contribution

It defines continuous 2-frieze patterns and establishes their connections with moduli spaces and symplectic structures, providing new insights into their geometric and algebraic properties.

Findings

Continuous 2-frieze patterns are related to moduli spaces of projective curves.

The symplectic structure on 2-frieze space connects with the Adler-Gelfand-Dikii bracket.

The study bridges combinatorial frieze patterns with differential operators.

Abstract

We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves and relate the (pre)symplectic structure on the space of closed 2-friezes, considered as a cluster variety, with the Adler-Gelfand-Dikii bracket on the space of 3rd order differential operators.