Fractional Supersymmetry As a Matrix Model

TL;DR

This paper develops a parafermionic approach to 2D fractional supersymmetry, overcoming previous difficulties, and demonstrates that fractional supersymmetric algebras can be naturally realized as matrix models, with detailed analysis for the case K=3.

Contribution

It introduces a parafermionic method for fractional supersymmetry, linking it to matrix models and providing explicit models for specific fractional supersymmetries.

Findings

Fractional supersymmetry algebras are realized as matrix models.

Links established between fractional and standard supersymmetries.

Explicit field theoretical models for matter multiplets are constructed.

Abstract

Using parafermionic field theoretical methods, the fundamentals of 2d fractional supersymmetry ${\bf Q}^{K} =P$ are set up. Known difficulties induced by methods based on the $U_{q}(sl(2))$ quantum group representations and non commutative geometry are overpassed in the parafermionic approach. Moreover we find that fractional supersymmetric algebras are naturally realized as matrix models. The K=3 case is studied in details. Links between 2d $({1\over 3},0)$ and $(({1\over 3}^{2}),0)$ fractional supersymmetries and N=2 U(1) and N=4 su(2) standard supersymmetries respectively are exhibited. Field theoretical models describing the self couplings of the matter multiplets $(0^{2},({1\over 3})^{2},({2\over 3})^{2})$ and $(0^{4},({1\over 3})^{4},({2\over 3})^{4})$ are given.