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arXiv · hep-th/9610087

Geometrical foundations of fractional supersymmetry

R. S. Dunne·A. J. Macfarlane·J. A. de Azcárraga·J. C. Pérez Bueno
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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.

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R. S. Dunne, A. J. Macfarlane, J. A. de Azcárraga, J. C. Pérez Bueno. 1996-10-11. Geometrical foundations of fractional supersymmetry. https://doi.org/10.1142/s0217751x97001729

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