Fractional supersymmetry as a matrix model

Using parafermionic field-theoretical methods, the fundamentals of two-dimensional (2D) fractional supersymmetry 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 avoided in the parafermionic approach. Moreover, we find that fractional supersymmetric algebras are naturally realized as matrix models. The K = 3 case is studied in detail. Links between 2D ((1/3),0) and (((1/3))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 (0²,((1/3))²,((2/3))²) and (0⁴,((1/3))⁴,((2/3))⁴) are given.