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

TL;DR

This paper explores the combinatorial and geometric properties of a five-color hypercube Adinkra, linking it to the Jacobian of a specific Riemann surface, and develops algorithms to analyze height functions and divisors.

Contribution

It introduces a novel combinatorial algorithm for computing height function images on a hypercube Adinkra related to a product of elliptic curves, connecting graph theory with algebraic geometry.

Findings

Height functions on a single elliptic curve are multiples of a specific divisor.

Vertices' raising and lowering correspond to adding or subtracting this divisor.

Derived bounds on coefficients of the generating divisor.

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.