Parafermions for higher order extensions of the Poincar\'e algebra and their associated superspace

TL;DR

This paper explores how parafermions of orders two and three can be used to build superspaces for higher-order extensions of the Poincaré algebra, including superfield construction and covariant derivative operators.

Contribution

It introduces a novel approach using parafermions to extend the Poincaré algebra and constructs related superfields and covariant derivatives.

Findings

Parafermions enable higher-order Poincaré algebra extensions.

Superfields for cubic and quartic extensions are explicitly constructed.

Operators acting as covariant derivatives are identified and formulated.

Abstract

Parafermions of order two and three are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincar\'e algebra. The corresponding superfields are constructed, and some of their main properties analyzed in detail. In this context, the existence problem of operators acting like covariant derivatives is analyzed, and the associated operators are explicitly constructed.