Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras

TL;DR

This paper establishes a geometric framework for classifying N-extended 1D supersymmetry algebras using algebraic curves and dessins d'enfants, linking graph representations to algebraic geometry over rational fields.

Contribution

It provides a canonical algebraic model of Adinkra graphs as Belyi curves over $\\mathbb{Q}$, enabling geometric interpretation of supersymmetry representations.

Findings

Realization of Adinkra graphs as Belyi curves.

Explicit algebraic models over $\mathbb{Q}(\zeta_{2N})$.

Proof of definability over $\mathbb{Q}$.

Abstract

The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincar\'{e} algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as {\em Adinkras}. In this paper we {\em 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.