Klein and conformal superspaces, split algebras and spinor orbits

Abstract

We discuss N = 1 Klein and Klein-Conformal superspaces in D = (2, 2) space-time dimensions, realizing them in terms of their functor of points over the split composition algebra C s . We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras; this leads to a natural interpretation of the the sections of the spinor bundle in the critical split dimensions D = 4, 6 and 10 as C 2 s , H 2 s and O 2 s , respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-Conformal superspace structure. IntroductionSupersymmetry (Susy) is a deep and elegant symmetry relating half-integer spin fields (fermions, constituents of matter) to integer-spin fields (bosons, giving rise to interactions). Such a symmetry was originally formulated, as a global symmetry of fields, back in the early 70's in former Soviet Union by physicists Gol'fand and Likhtman [1], Volkov and Akulov [2], and independently in Europe by Wess and Zumino [3].A major advance in the formulation of supersymmetric theories in space-time, which then allowed for the construction of manifestly invariant interactions, was due to Salam and Strathdee, who were the first to introduce the concept of superfield [4,5]. In fact, depending on the number s and t of spacelike resp. timelike dimensions, space-time Susy recasts bosonic and fermionic fields into multiplet structures, each providing a certain representation of such an underlying symmetry. Within the simplest formulation of Susy, in which a unique fermionic generator exists besides the bosonic ones, fields defined in a space M s,t ∼ = R s,t (which in the case s = 3 and t = 1 yields the usual Minkowski space-time) are assembled into a unique object, named superfield, defined into the so-called N = 1, (s + t)-dimensional superspace M s,t|1 , which is characterized by the presence of an anti-commuting Grassmannian coordinate besides the usual commuting bosonic coordinates of M s,t .Such developments eventually led to major advances in Quantum Field Theory, constituting the foundational pillars on which consistent candidates for a unified theory encompassing Quantum Gravity and the Standard Model of particle interactions were constructed. In combination with local gauge invariance, global Susy allowed for the formulation of Supersymmetric Yang-Mills Theories (SYM's) [6]. In such a framework, Susy gives rise to remarkable cancellations between bosons and fermions in their quantum corrections, thus allowing for a study of SYM's beyond perturbation theory. This generally provides a framework for a possible solution of the hierarchy problem, for the search of natural candidates for dark matter, as well as for addressing the conceptual issue of the dark energy.In presence of general diffeomorphisms covariance, Susy becomes a local symmetry. In 1976, Ferrara, Freedman, Van Niewenhuizen [7] and Deser and Zumino [8] succeeded in formulating Su...

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Quantum Klein Space and Superspace

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“…Moreover, 4-dimensional Kleinian signature essentially pertains to twistors [56], which provide a powerful computational tool of scattering amplitudes [65]. In [31] the 4-dimensional Klein space M 2,2 was studied, through its definition inside the related Klein-conformal space, along with its supersymmetric extensions, namely the Klein N = 1 superspace M 2,2|1 and the corresponding Klein-conformal N = 1 superspace. To the best of our knowledge, this, together with our previous work [31], is the first investigation of the superspace with Kleinian (bosonic) signature.…”

“…A direct approach to quantum deformations stems from [30,31], and it exhibits an intrinsic elegance based on split algebras A s 's in particular the split complex numbers C s . This is the subject of the present paper, which, as mentioned, will consider the quantum deformation of both real Klein and Klein-conformal N = 1 spaces.…”

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