Group Theoretical Foundations of Fractional Supersymmetry

TL;DR

This paper explores the mathematical structure of fractional supersymmetry using generalized Grassmann variables, revealing its group properties and geometric interpretations of derivatives, thus extending the theoretical framework of supersymmetry.

Contribution

It provides explicit formulas for fractional supersymmetry transformations and demonstrates their group properties, offering new insights into the geometric nature of the theory.

Findings

Derived explicit transformation formulas for fractional supersymmetry.

Showed that derivatives act as generators of the symmetry group.

Clarified the geometric interpretation of generalized derivatives.

Abstract

Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, $\theta = \bar{\theta}, \, \theta^n = 0$, for arbitrary integer $n = 2, 3, ...$. An explicit formula is given in the case of general $n$ for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalised derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as $\theta$.