Geometrical foundations of fractional supersymmetry

TL;DR

This paper develops a geometric framework for fractional supersymmetry using $q$-calculus, relating it to $q$-deformed bosons, and explores algebraic structures, derivatives, and superspace measures, especially at roots of unity.

Contribution

It introduces a new algebraic structure based on $q$-calculus for fractional supersymmetry, connecting it with $q$-deformed bosons and extending the Hopf algebra framework.

Findings

Identifies derivatives with charge and covariant derivative in fractional supersymmetry.

Establishes a generalized Berezin integral and superspace measure.

Shows the algebra's non-trivial Hopf structure at roots of unity.

Abstract

A deformed $q$-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge $Q$ and covariant derivative $D$ encountered in ordinary/fractional supersymmetry and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When $q$ is a root of unity the algebra is found to have a non-trivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when $q$ is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the $q$-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.