The Superparticle and the Lorentz Group

TL;DR

This paper develops a unified group-theoretical framework for superparticle theories, explaining the origin of twistor-like variables and their relation to Lorentz and supersymmetries, with implications for covariant formulations.

Contribution

It introduces a covariant group-theoretical approach that clarifies the role of twistor-like variables and the coset space structure in superparticle theories.

Findings

Twistor-like variables parametrize the coset space ${ m SO}^(1,d-1)/{ m SO}(d-1)

The framework provides covariantization of light-cone frames

Clarifies the relation between target-space and worldline supersymmetries

Abstract

We present a unified group-theoretical framework for superparticle theories. This explains the origin of the ``twistor-like'' variables that have been used in trading the superparticle's $\kappa$-symmetry for worldline supersymmetry. We show that these twistor-like variables naturally parametrise the coset space ${\cal G}/{\cal H}$, where $\cal G$ is the Lorentz group $SO^\uparrow(1,d-1)$ and $\cal H$ is its maximal subgroup. This space is a compact manifold, the sphere $S^{d-2}$. Our group-theoretical construction gives the proper covariantisation of a fixed light-cone frame and clarifies the relation between target-space and worldline supersymmetries.