New aspects of the Z$_{\textrm 2}$ $\times$ Z$_{\textrm 2}$-graded 1D superspace: induced strings and 2D relativistic models
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 ${\bf S}^1$, 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 $(β$-deformed, where $β\geq 0$ is a real parameter) class of point-particle models than the ones discussed so far in the literature (recovered at $β=0$) is obtained. While the spectrum of the $β=0$ point-particle models is degenerate (due to its relation with an ${\cal N}=2$ supersymmetry), this is no longer the case for the $β> 0$ models.