Adinkra Height Yielding Matrix Numbers: Eigenvalue Equivalence Classes for Minimal Four-Color Adinkras

TL;DR

This work extends adinkra analysis beyond valise graphs by introducing Height Yielding Matrix Numbers (HYMNs), the eigenvalues of B-matrices constructed from L- and R-matrices that encode node lifting in non-valise adinkras. By defining the nodal raising operator $M(m,w)$ and studying the dependence on the ratio $\rho=m/\mu$, the authors classify the isomorphism classes of lifted adinkras for all $36{,}864$ valise $BC_4$ adinkras (plus raised variants), revealing that HYMN eigenvalues capture the full shape information of adinkras and are invariant under node relabelings and certain color transformations. They provide detailed classifications for GR$(2,2)$ and GR$(4,4)$, including explicit class counts: valise, one- to four-boson lifts, with HYMN distributions forming well-structured equivalence classes, such as $χ_{ m o}$-based splits and EB$_i^{(χ_{ m o})}$ groups. The results establish a concrete, computable framework to map 1D adinkras to higher-dimensional SUSY representations and set the stage for future work linking HYMN structure to holoraumy, gadgets, and broader color-generalizations. A Python-based calculator and a Mathematica notebook are provided for reproducing and extending the HYMN classifications.

Abstract

An adinkra is a graph-theoretic representation of spacetime supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, $\cal N$ = 1 supermultiplets. Valise adinkras, although an important subclass, do not encode all the information present when a 4D supermultiplet is reduced to 1D. Eigenvalue equivalence classes for valise adinkra matrices exist, known as $χ_{\rm o}$ equivalence classes, where valise adinkras within the same $χ_{\rm o}$ equivalence class are isomorphic in the sense that adinkras within a $χ_{\rm o}$-equivalence class can be transformed into each other via field redefinitions of the nodes. We extend this to non-valise adinkras, via Python code, providing a complete eigenvalue classification of "node-lifting" for all 36,864 valise adinkras associated with the Coxeter group $BC{}_4$. We term the eigenvalues associated with these node-lifted adinkras Height Yielding Matrix Numbers (HYMNs) and introduce HYMN equivalence classes. These findings have been summarized in a $Mathematica$ notebook that can found at the HEPTHools Data Repository (https://hepthools.github.io/Data/) on GitHub.

Figures (5)

• Figure 1: Valise Adinkra for Chiral Supermultiplet with $F$ & $G$ Nodes Lowered
• Figure 2: Adinkra for Chiral Supermultiplet without $F$ & $G$ Nodes Lowered
• Figure 3: An example of a $\mathcal{GR}(2,2)$ valise adinkra.
• Figure 4: Valise and raised versions of the ${\cal {GR}}(2,2)$ adinkra of Fig. \ref{['f:GR22L0']}. The tilded bosons are identified as $\tilde{\Phi}_i = m \Phi_i$.
• Figure 5: All 16 ${\cal {GR}}(2,2)$ valise adinkras. The white nodes are bosons, the black fermions, in numerical left to right order as in Fig. \ref{['f:GR22L0']}.