Poisson Hierarchy of Discrete Strings

TL;DR

This paper explores the Poisson geometric structure of discrete strings in three-dimensional space, introducing a hierarchy of Poisson algebras via spinor representations and transfer matrices, with connections to the Virasoro algebra.

Contribution

It develops a novel hierarchical framework of Poisson algebras for discrete strings using spinor Frenet equations and transfer matrices, linking to classical algebraic structures.

Findings

Hierarchy of Poisson algebras constructed for discrete strings

Identification of the Virasoro algebra as a sub-algebra within the hierarchy

Extension of continuous string geometry to a discrete, algebraic setting

Abstract

The Poisson geometry of a discrete string in three dimensional Euclidean space is investigated. For this the Frenet frames are converted into a spinorial representation, the discrete spinor Frenet equa- tion is interpreted in terms of a transfer matrix formalism, and Poisson brackets are introduced in terms of the spinor components. The construction is then generalised, in a self-similar manner, into an infinite hierarchy of Poisson algebras. As an example, the classical Virasoro (Witt) algebra that determines reparametrisation diffeomorphism along a continuous string, is identified as a particular sub-algebra, in the hierarchy of the discrete string Poisson algebra.