Infinite planar string: cusps, braids and soliton exitations

TL;DR

This paper explores infinite strings in (2+1)D space-time, introducing a novel Hamiltonian framework that reveals complex excitations like cusps and braids, with implications for understanding soliton dynamics.

Contribution

It develops a new Hamiltonian description for infinite strings, separating internal and external variables, and analyzes N-soliton excitations leading to cuspidal points and braids.

Findings

Reconstructed strings exhibit cuspidal points.

String excitations form braids of various topologies.

Phase space involves entangled variables from constraints.

Abstract

We investigate infinite strings in $(2+1)D$ space-time, which may be considered as excitations of straight lines on the spatial plane. We also propose the hamiltonian description of such objects that differs from the standard hamiltonian description of the string. The hamiltonian variables are separated into two independent groups: the "internal" and "external" variables. The first ones are invariant under space-time transformations and are connected with the second form of the world-sheet. The "external" variables define the embedding of the world-sheet into space-time. The constructed phase space is nontrivial because the finite number of constraints entangles the variables from these groups. First group of the variables constitute the coefficients for the pair of first-order spectral problems; the solution of these problems is necessary for the reconstruction of the string world-sheet. We consider the excitations, which correspond to "N- soliton" solution of the spectral problem, and demonstrate that the reconstructed string has cuspidal points. World lines of such points form braids of various topologies.