The five-color hypercube Adinkra and the Jacobian of a generalized Fermat curve

TL;DR

This work links Adinkra combinatorics to arithmetic geometry by mapping height functions to divisors on Jacobians of associated curves. For the $N=5$ hypercube, the authors realize the curve as a generalized Fermat curve $X_{ m alg}^5$ and decompose its Jacobian as a product of five elliptic curves with common $j$-invariant $2048$, enabling componentwise analysis via the Mordell–Weil group. They develop a purely combinatorial algorithm using $j$-color splittings to compute height images, prove that these images on each elliptic factor are multiples of a single generator determined by a base height, and bound the possible coefficients to $-8$ and $8$; they also compute explicit Jacobian divisors for key height functions and discuss fields of definition necessary to access infinite-order points. The results illuminate how Adinkra height data encode arithmetic information on high-genus curves and establish precise relationships between combinatorial operations (raising/lowering vertices) and linear changes in the Jacobian, with potential implications for understanding 1D supersymmetry algebras via algebraic geometry.

Abstract

Adinkras are highly structured graphs developed to study 1-dimensional supersymmetry algebras. A cyclic ordering of the edge colors of an Adinkra, or rainbow, determines a Riemann surface and a height function on the vertices of the Adinkra determines a divisor on this surface. We study the induced map from height functions to divisors on the Jacobian of the Riemann surface. In the first nontrivial case, a 5-dimensional hypercube corresponding to a Jacobian given by a product of 5 elliptic curves each with $j$-invariant 2048, we develop and characterize a purely combinatorial algorithm to compute height function images. We show that when restricted to a single elliptic curve, every height function is a multiple of a specified generating divisor, and raising and lowering vertices corresponds to adding or subtracting this generator. We also give strict bounds on the coefficients of this generator that appear in the collection of all divisors of height functions.

Figures (9)

• Figure 1: An $N=3$ Adinkra $A$. The dashing function $d$ on $A$ satisfies $d(e)=1$ if and only if the edge is dashed in the visualization above. The height function values correspond to the $y$-coordinates of the vertices.
• Figure 2: Two extremal examples of heights on $H^5$, the valise height and the fully extended height.
• Figure 3: A triangulated diamond 4-cycle and bow-tie 4-cycle
• Figure 4: Possible saddle point configurations in $X_{\rm alg}^5$. Vertices are labeled by their heights, relative to the height $h_0$ of the central vertex. Edges are labeled by their colors.
• Figure 5: An $N=3$ hypercube Adinkra with an orange color-splitting

Theorems & Definitions (39)

• Remark 2.1
• Proposition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Example 2.6
• Definition 2.8
• Proposition 2.9
• proof
• Definition 2.10