New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models

Abstract

A novel feature of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry which finds no counterpart in ordinary supersymmetry is the presence of $11$-graded exotic bosons (implied by the existence of two classes of parafermions). Their interpretation, both physical and mathematical, presents a challenge. The role of the "exotic bosonic coordinate" was not considered by previous works on the one-dimensional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superspace (which was restricted to produce point-particle models). By treating this coordinate at par with the other graded superspace coordinates new consequences are obtained. The graded superspace calculus of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded worldline super-Poincaré algebra induces two-dimensional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded relativistic models; they are invariant under a new ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded $2D$ super-Poincaré algebra which differs from the previous two ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded $2D$ versions of super-Poincaré introduced in the literature. In this new superalgebra the second translation generator and the Lorentz boost are $11$-graded. Furthermore, if the exotic coordinate is compactified on a circle ${\mathbf{S}}^1$, a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded closed string with periodic boundary conditions is derived. The analysis of the irreducibility conditions of the $2D$ supermultiplet implies that a larger $(\beta$-deformed, where $\beta \ge 0$ is a real parameter) class of point-particle models than the ones discussed so far in the literature (recovered at $\beta =0$) is obtained. While the spectrum of the $\beta =0$ point-particle models is degenerate (due to its relation with an $\mathcal{N}=2$ supersymmetry), this is no longer the case for the $\beta >0$ models.

28 pages

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