W-symmetry and the rigid particle

TL;DR

This paper demonstrates that the gauge symmetry of the scale-invariant rigid particle is described by the W_3 algebra, linking particle dynamics with integrable systems and extending the analysis to supersymmetric cases.

Contribution

It establishes the W_3 symmetry of the rigid particle and connects its equations of motion to integrable operators like Boussinesq and KdV, including supersymmetric extensions.

Findings

W_3 symmetry is the gauge symmetry of the rigid particle.

Equations of motion relate to Boussinesq and KdV operators.

Supersymmetric extension constructed for the rigid particle.

Abstract

We prove that $\W_3$ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The $\W_3$ generators $T$ and $W$ then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the ``zero-curvature gauge'', and how to rewrite them in terms of the $\KdV$ operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the $\SKdV$ operator. We also partially extend our analysis in the supersymmetric domain to the scale invariant rigid particle by explicitly constructing a supercovariant version of its action. Comment: This is an expanded version of hep-th/9406072 (to be published in the Proceedings of the Workshop on the Geometry of Constrained Dynamical Systems, held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, June 14-18, 1994.).