New aspects of the Z2_{\textrm 2} ×\times Z2_{\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-Poincar\'e 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-Poincar\'e algebra which differs from the previous two ${\mathbb Z}_2\times {\mathbb Z}_2$-graded $2D$ versions of super-Poincar\'e 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 $(\beta$-deformed, where $\beta\geq 0$ 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 ${\cal N}=2$ supersymmetry), this is no longer the case for the $\beta> 0$ models.Comment: 28 page

One-dimensional sigma-models with N=5,6,7,8 off-shell supersymmetries

We computed the actions for the 1D N=5 sigma-models with respect to the two inequivalent (2,8,6) multiplets. 4 supersymmetry generators are manifest, while the constraint originated by imposing the 5-th supersymmetry automatically induces a full N=8 off-shell invariance. The resulting action coincides in the two cases and corresponds to a conformally flat 2D target satisfying a special geometry of rigid type. To obtain these results we developed a computational method (for Maple 11) which does not require the notion of superfields and is instead based on the nowadays available list of the inequivalent representations of the 1D N-extended supersymmetry. Its application to systematically analyze the sigma-models off-shell invariant actions for the remaining N=5,6,7,8 (k,8,8-k) multiplets, as well as for the N>8 representations,only requires more cumbersome computations.Comment: 10 pages; one reference adde

On the Construction and the Structure of Off-Shell Supermultiplet Quotients

Recent efforts to classify representations of supersymmetry with no central charge have focused on supermultiplets that are aptly depicted by Adinkras, wherein every supersymmetry generator transforms each component field into precisely one other component field or its derivative. Herein, we study gauge-quotients of direct sums of Adinkras by a supersymmetric image of another Adinkra and thus solve a puzzle from Ref.[2]: The so-defined supermultiplets do not produce Adinkras but more general types of supermultiplets, each depicted as a connected network of Adinkras. Iterating this gauge-quotient construction then yields an indefinite sequence of ever larger supermultiplets, reminiscent of Weyl's construction that is known to produce all finite-dimensional unitary representations in Lie algebras.Comment: 20 pages, revised to clarify the problem addressed and solve

Supersymmetric Extension of Hopf Maps: N=4 sigma-models and the S^3 -> S^2 Fibration

We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension of R^4), II) the (3,4,1)_{lin} linear supermultiplet (supersymmetric extension of R^3), III) the (3,4,1)_{nl} non-linear supermultiplet living on S^3 and IV) the (2,4,2)_{nl} non-linear supermultiplet living on S^2. The I -> II map is the supersymmetric extension of the R^4 -> R^3 bilinear map, while the II -> IV map is the supersymmetric extension of the S^3 -> S^2 first Hopf fibration. The restrictions on the S^3, S^2 spheres are expressed in terms of the stereographic projections. The non-linear supermultiplets, whose supertransformations are local differential polynomials, are not equivalent to the linear supermultiplets with the same field content. The sigma-models are determined in terms of an unconstrained prepotential of the target coordinates. The Uniformization Problem requires solving an inverse problem for the prepotential. The basic features of the supersymmetric extension of the second and third Hopf maps are briefly sketched. Finally, the Schur's lemma (i.e. the real, complex or quaternionic property) is extended to all minimal linear supermultiplets up to N<=8.Comment: 24 page