Using the D=2 (1/k,1/k) superalgebra (i.e. the two-dimensional generalized supersymmetry generated by spin s=1/k and -1/k; , charge operators Q and satisfying among other conditions and where P and are the light components of the energy--momentum operator), we build a superspace representation of this exotic symmetry. This realization generalizing the usual D=2 (1/2,1/2) supersymmetric one is based on the use of parafermionic variables and of spin s=-1/k and 1/k obeying as well as generalized commutation relation rules. Two-dimensional (1/3,0) and (1/3,1/3) invariant scalar superfield models extending the well-known D=2 (1/2,0) and (1/2,1/2) supersymmetric models are given. The link between this exotic symmetry and the periodic representation of with is worked out. Other features are also discussed.
Denoting by D = 2(1/3, 1/3) superalgebra the off critical symmetry of the φ 5/7,5/7 perturbation of the c = 6/7 conformal theory, we build a new superspace solution of the (1/3, 1/3) − subalgebra generated by spin ±1/3 charge operators extending the usual (1/2, 1/2) supersymmetry generated by spin ±1/2 charges (Saidi et al). This solution is based on the use of two Grassmann variables instead of one parafermionic variable θ ±1/3 satisfying the cubic nilpotency condition (θ ±1/3 ) 3 = 0. Known results on the c = 6/7 tricritical Potts model are recovered as special features. The relation with N = 2 Landau-Ginzburg models is also discussed.
We develop the basis of the two dimensional generalized quantum statistical systems by using results on r-generalized Fibonacci sequences. According to the spin value s of the 2d-quasiparticles, we distinguish four classes of quantum statistical systems indexed by s = 0, 1/2 : mod(1), s = 1/M : mod(1), s = n/M : mod(1) and 0 ≤ s ≤ 1 : mod(1). For quantum gases of quasiparticles with s = 1/M : mod(1), M ≥ 2,, we show that the statistical weights densities ρ M are given by the integer hierarchies of Fibonacci sequences. This is a remarkable result which envelopes naturally the Fermi and Bose statistics and may be thought of as an alternative way to the Haldane interpolating statistical method.
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