Adinkras as Origami? (Edray Goins, Pomona College)

Around 20 years ago, physicists Michael Faux and Jim Gates invented Adinkras as a way to better understand Supersymmetry.  These are bipartite graphs whose vertices represent bosons and fermions and whose edges represent operators which relate the particles.  Recently, Charles Doran, Kevin Iga, Jordan Kostiuk, Greg Landweber and Stefan M\'{e}ndez-Diez determined that Adinkras are a type of Dessin d’Enfant; they showed this by explicitly exhibiting a Belyi map as a composition $\beta: S \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$.  They computed the first arrow as a map from a certain compact connected Riemann surface $S$ to the Riemann sphere $\mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$, and the second as a map which keeps track of the “coloring” of the edges.

Adinkras naturally have square faces.  This keeps track of the non-commutative nature of the supersymmetric operators.  While Dessin d’Enfants correspond to triangular tilings of Riemann surfaces, there is a similar construction — called “origami” — which correspond to square tilings.  In this project, we attempt to discover how to express the construction of Doran, et al. as a composition $\beta: S \to E(\mathbb C) \to \mathbb P^1(\mathbb C)$ for some elliptic curve elliptic curve $E$ such that the map corresponds to an “origami”, that is, a map which is branched over just one point.  This work is conducted as part of the Pomona Research in Mathematics Experience (DMS-2113782).