Properties of HYMNs in Examples of Four-Color, Five-Color, and Six-Color Adinkras

TL;DR

This work extends the Height Yielding Matrix Numbers (HYMNs) framework by including fermion node lifting and analyzes Banchoff matrices derived from adinkras across four-color, minimal five-color, and minimal six-color cases. It demonstrates that the eigenvalues of the Banchoff matrices, together with their traces and determinants, encode intrinsic, dashings-insensitive information about the adinkra shapes and are stable under lifting transformations. The authors provide explicit calculations for GR(4,4) multiplets (Chiral, Vector, Tensor) and several non-minimal and higher-color adinkras, including two five-color and two six-color examples, highlighting systematic patterns in the spectra. These results suggest a route to classifying adinkras into equivalence classes via spectral polynomials and hint at connections to SUSY holography and the geometry of adinkras. Overall, the paper strengthens the link between combinatorial adinkra data and algebraic structures, offering new tools for understanding supersymmetric representations.

Abstract

The mathematical concept of a "Banchoff index" associated with discrete Morse functions for oriented triangular meshes has been shown to correspond to the height assignments of nodes in adinkras. In recent work there has been introduced the concept of "Banchoff matrices" leading to HYMNs - height yielding matrix numbers. HYMNs map the shape of an adinkra to a set of eigenvalues derived from Banchoff matrices. In the context of some examples of four-color, minimal five-color, and minimal six-color adinkras, properties of the HYMNs are explored.

Figures (12)

• Figure 1: Chiral Supermultiplet Adinkra
• Figure 2: Vector Supermultiplet Adinkra
• Figure 3: Tensor Supermultiplet Adinkra
• Figure 4: Adinkra Diagram for SM-I
• Figure 5: Adinkra Diagram for SM-II