Contents Online

# 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

#### 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.