Advances in Theoretical and Mathematical Physics

Volume 19 (2015)

Number 5

Geometrization of $N$-extended 1-dimensional supersymmetry algebras, I

Pages: 1043 – 1113

DOI: http://dx.doi.org/10.4310/ATMP.2015.v19.n5.a4

Authors

Charles Doran (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada)

Kevin Iga (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada)

Jordan Kostiuk (Natural Science Division, Pepperdine University, Malibu, California, U.S.A.)

Greg Landweber (Mathematics Program, Bard College, Annandale-on-Hudson, New York, U.S.A.)

Stefan Méndez-Diez (Department of Mathematics and Statistics, Utah State University, Logan, Ut., U.S.A.)

Abstract

The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincaré algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as Adinkras. In this paper we canonically realize these graphs as Grothendieck “dessins d’enfants,” or Belyi curves uniformized by certain normal torsion-free subgroups of the $(N, N, 2)$-triangle group. We exhibit an explicit algebraic model over $\mathbb{Q}(\zeta_{2N})$, as a complete intersection of quadrics in projective space, and use Galois descent to prove that the curves are, in fact, definable over $\mathbb{Q}$ itself. The stage is thereby set for the geometric interpretation of the remaining Adinkra decorations in Part II.

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Published 31 March 2016