New aspects of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-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 is presence $11$-graded exotic bosons (implied by existence two classes parafermions). Their interpretation, both physical and mathematical, presents a challenge. The role "exotic bosonic coordinate" was not considered previous works on one-dimensional superspace (which restricted to produce point-particle models). By treating this coordinate at par with other graded coordinates new consequences are obtained. calculus worldline super-Poincar\'e algebra induces two-dimensional relativistic models; they invariant under $2D$ differs from versions introduced literature. In superalgebra second translation generator Lorentz boost $11$-graded. Furthermore, if compactified circle ${\bf S}^1$, closed string periodic boundary conditions derived. analysis irreducibility supermultiplet implies that larger $(\beta$-deformed, where $\beta\geq 0$ real parameter) class models than ones discussed so far literature (recovered $\beta=0$) While spectrum $\beta=0$ degenerate (due its relation an ${\cal N}=2$ supersymmetry), longer case for $\beta> models.