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Klein and conformal superspaces, split algebras and spinor orbits

Rita Fioresi

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy

E-mail Address: rita.fioresi@UniBo.it

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Emanuele Latini

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy

E-mail Address: emanuele.latini@UniBo.it

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, and

Alessio Marrani

Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89A, I-00184, Roma, Italy

Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Via 8 Febbraio 1848, 2, 35122 Padova PD, Italy

INFN, Sez. di Padova, Via Marzolo 8, I-35131 Padova, Italy

E-mail Address: Alessio.Marrani@pd.infn.it

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https://doi.org/10.1142/S0129055X17500118Cited by:2 (Source: Crossref)

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 sections of the spinor bundle in the critical split dimensions $D = 4, 6$ and $10$ as $ℂ_{s}^{2}$ , $ℍ_{s}^{2}$ and $𝕆_{s}^{2}$ , 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.

Keywords:

AMSC: 17A35, 17C60, 32C11, 15A66

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