Geometrical Foundations of Fractional Supersymmetry

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 an nth 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 nontrivial 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.