Generating all 36,864 Four-Color Adinkras via Signed Permutations and Organizing into $\ell$- and $\tilde{\ell}$-Equivalence Classes

TL;DR

This paper analyzes the structure of four-color adinkras using signed permutation groups, generating all 36,864 adinkras from quaternion adinkras, and classifies them into equivalence classes based on an inner product called the gadget.

Contribution

It introduces a method to generate all adinkras via signed permutation groups from quaternion adinkras and classifies them into equivalence classes, advancing the understanding of supersymmetrical representations.

Findings

All 36,864 adinkras can be generated from quaternion adinkras using $BC_4$ and $BC_3$ transformations.

96 equivalence classes of adinkras are identified based on the gadget.

The work constructs a universal $BC_4$ non-linear $\sigma$-model.

Abstract

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of $BC_3$, the signed permutation group of three elements, and $BC_4$, the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via $BC_4$ boson $\times$ $BC_3$ color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the \emph{gadget}, which is used to distinguish adinkras. We~show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of $BC_3$ and $BC_4$. We also comment on the importance of the gadget as it relates to separating out dynamics in terms of K\"ahler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal $BC_4$ non-linear $\sigma$-model.