Poisson hierarchy of discrete strings

• Witt (classical Virasoro) algebra is derived in the case of discrete string.
• Infinite dimensional hierarchy of Poisson bracket algebras is constructed for discrete strings.
• Spinor representation of discrete Frenet equations is developed.

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 equation 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.