Fractional Superspace Formulation of Generalized Super-Virasoro Algebras

We present a fractional superspace formulation of the centerless parasuper-Virasoro and fractional super-Virasoro algebras. These are two different generalizations of the ordinary super-Virasoro algebra generated by the infinitesimal diffeomorphisms of the superline. We work on the fractional superline parametrized by t and θ, with t a real coordinate and θ a paragrassmann variable of order M and canonical dimension 1/F . We further describe a more general structure labelled by M and F with M ≥ F . The case F = 2 corresponds to the parasuper-Virasoro algebra of orderM , while the case F = M leads to the fractional super-Virasoro algebra of order F . The ordinary super-Virasoro algebra is recovered at F = M = 2. The connection with q-oscillator algebras is discussed. ∗ E-mail address: [email protected] 1 Symmetries play a fundamental role in physics. Therefore, all new kinds of symmetry are worth studying. For instance, supersymmetries could be a key ingredient of unifying theories. Up to now, at least two generalizations of supersymmetry are known: the para-supersymmetries [1,2] which already possess interesting applications in quantum mechanics , and the fractional supersymmetries [5,6,7] which appear, in particular, in the context of the chiral Potts model . In this letter, we shall present a new algebraic stucture that allows the unification of the concepts of para-superalgebra and fractional superalgebra, and that furthermore describes new types of symmetries. More precisely, we present a novel generalization of the centerless super-Virasoro algebra which we will call a generalized super-Virasoro algebra. This construction has the following features: • It is parametrized by 2 positive integers M and F with M ≥ F . • It possesses a ZF grading. • It describes operators of fractional spin s = 1 + 1 F . • It is realized within the framework of a fractional superspace formalism using paragrassmann variables of order M and canonical dimension 1/F . • It contains as particular cases, the parasuper-Virasoro algebras of order M (for F = 2) and the fractional super-Virasoro algebras of order F (for M = F ). The ordinary super-Virasoro algebra is recovered at F = M = 2. Thus, it also provides a new unified realization of the “old” fractional and para super-Virasoro algebras. (In Ref. [2] the latter was realized in terms of the Green representation [9] while in Refs. [5, 6, 7] the former was strictly realized in matrix form). Let us first recall some standard results. The well-known centerless superVirasoro algebra is given by [Lm, Ln] = (m− n)Lm+n (1a) [Ln, Gr] = ( 1 2 n− r)Gn+r (1b) {Gr, Gs} = 2Lr+s (1c) with m,n ∈ Z and r, s ∈ Z + 1 2 . When centrally extended, it yields the NeveuSchwarz algebra which plays a central role in superstring theories. The algebraic relations (1), referred to as the two-dimensional superconformal algebra, can also be thought of as realized by the generators of the infinitesimal diffeomorphisms 2 (Ln) and superdiffeomorphisms (Gr) of the superline: Ln = t ∂t − 1 2(n− 1)t θ∂θ, n ∈ Z (2a) Gr = t 1 2 (∂θ + θ∂t), r ∈ Z + 1 2 (2b) where t is the real line parameter (∂t ≡ ∂/∂t) and θ a paragrassmann variable satifying (∂θ ≡ ∂/∂θ): θ = 0, {∂θ, θ} = 1. (3) The generators Ln and Gr respectively have spin 2 and 3/2. Generally speaking, one says that φn has conformal spin s if [Lm, φn] = (