Division algebra representations of <font>SO</font>(4, 2)

Kincaid, Joshua; Dray, Tevian

doi:10.1142/s0217732314501284

Cited by 8 publications

(8 citation statements)

References 17 publications

(42 reference statements)

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“…In [26], the authors constructed the compactified Minkowsky 3d superspace can be reinterpreted by noting the following isomorphism sp 4 (K) = so(n + 2, 2) withsp 4 (K) being the Sudbery symplectic algebra, where, with respect to the traditional definition, the transpose is replaced by hermitian conjugation. Recently, in [27], it was also proposed a Lie group version of the half split 2 × 2 magic square (see also [28,29,30] for further details and the relation with exceptional Lie algebras and groups). Inspired by these observations we then study in details a symplectic characterization of the 4 dimensional (compactified and real) Minkowski space and superspace respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In [26], the authors constructed the compactified Minkowsky 3d superspace with sp 4 (K) being the Sudbery symplectic algebra, where, with respect to the traditional definition, the transpose is replaced by hermitian conjugation. Recently, in [27], it was also proposed a Lie group version of the half split 2 × 2 magic square (see also [28,29,30] for further details and the relation with exceptional Lie algebras and groups).…”

Section: Introductionmentioning

confidence: 99%

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The symplectic origin of conformal and Minkowski superspaces

Fioresi

Latini

2016

Journal of Mathematical Physics

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Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.Comment: 15 pages, references added, minor change

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The symplectic origin of conformal and Minkowski superspaces

Fioresi

Latini

2016

Journal of Mathematical Physics

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“…The Lie algebraic isomorphisms (1.2)-(1.5) have been recently extended to the Lie group level (considering the spin covering of the Lorentz and conformal groups, namely Spin(q + 1, 1) resp. Spin(q + 2, 2)), by explicit constructions worked out in a series of paper [41,42,43,44] by Dray, Manogue and collaborators. In particular, in [41] a Lie group version of the aforementioned order-2 split magic square L 2 (A s , B) was constructed and studied.…”

Section: R C H Omentioning

confidence: 99%

“…It is also worth anticipating here that the symmetry of the order-2 doubly-split magic square L 2 (A s , B s ) (as opposed to the order-2 split magic square L 2 (A s , B), which is not symmetric) -promoted to the Lie group level by relying on the work of Dray, Manogue and collaborators [41,42,43,44] -will play an important role in our treatment. Indeed, the Klein-conformal group Spin(3, 3) in 4 dimensions, besides occurring in the entry L 2 (H s , C s ) and thus being characterized as Spin(3, 3) ∼ = Sp(4, C s ), also appears in the entry L 2 (C s , H s ), and as such it enjoys the isomorphism Spin(3, 3) ∼ = SL(2, H s ), as well.…”

Section: R C H Omentioning

confidence: 99%

Klein and conformal superspaces, split algebras and spinor orbits

2017

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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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“…We will then show that each group in the 2 × 2 magic square can be written in the form SU(2, K ′ ⊗K). Kincaid and Dray [18,19] took the first step in providing a composition algebra description of the third row of the magic squares by showing that SO(4, 2) ≡ SU(2, H ′ ⊗ C). We extend their work by showing that Spin(K ⊕ K ′ ) acts on K ⊕ K ′ just as SU(2, C) acts on the space of 2 × 2 Hermitian matrices.…”

Section: Introductionmentioning

confidence: 99%