Extended Gauge Invariance in Geometrical Particle Models and the Geometry of W-Symmetry

TL;DR

This paper demonstrates that certain particle models with actions based on curvature functions exhibit multiple gauge invariances, which are geometrically characterized and linked to W-symmetry, providing new insights into their algebraic structure and invariants.

Contribution

It establishes a geometric framework connecting curvature-based particle models to W-symmetry algebras and introduces a new global invariant for constrained curves.

Findings

Particle models with curvature actions have n+1 gauge invariances.

The gauge algebra corresponds to the W_{n+2} algebra.

A new global invariant for four-dimensional curvature-constrained curves is proposed.

Abstract

We prove that particle models whose action is given by the integrated $n$-th curvature function over the world line possess $n+1$ gauge invariances. A geometrical characterization of these symmetries is obtained via Frenet equations by rephrasing the $n$-th curvature model in $\reals^d$ in terms of a standard relativistic particle in $S^{d-n}$. We ``prove by example'' that the algebra of these infinitesimal gauge invariances is nothing but $\W_{n+2}$, thus providing a geometrical picture of the $\W$-symmetry for these models. As a spin-off of our approach we give a new global invariant for four-dimensional curves subject to a curvature constraint.