Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into ℓ- and ℓ ˜ -Equivalence Classes

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 $B{C}_3$, the signed permutation group of three elements, and $B{C}_4$, the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via $B{C}_4$ boson × $B{C}_3$ color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the 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 $B{C}_3$ and $B{C}_4$. We also comment on the importance of the gadget as it relates to separating out dynamics in terms of Kähler-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 $B{C}_4$ non-linear $\sigma$-model.

1. Introduction

Adinkras are diagrams that encode supersymmetric (SUSY) transformation laws with complete fidelity in one spacetime dimension and in two spacetime dimensions [1]. Different colored lines in adinkra diagrams encode the action of distinct one-dimensional supercharges on the field variables of a supermultiplet. The lines connect nodes that encode fields related by the supersymmetry transformation. Adinkras are useful theoretical tools for many reasons. First, adinkras are elegant and concise classification tools that encode a plethora of mathematics similar to Dynkin diagrams and Feynman diagrams. Second, adinkras have proven useful in discovering previously-unknown supersymmetric multiplets [2]. We seek to further develop adinkras as a search tool to uncover finite realizations of off-shell supersymmetric representations: most notably 4D, $\mathcal{N}=4$ super Yang–Mills theory and 10D and 11D supergravity. Such representations lie outside the no-go theorem of [3]. A finite representation of the off-shell superconformal hypermultiplet has recently been uncovered [4]. This is promising evidence pointing toward the possibility of a finite realization of 4D, $\mathcal{N}=4$ super Yang–Mills theory. The utility of adinkras in analyzing extended supersymmetric systems was demonstrated in [5] where the adinkra parameter ${\chi }_0$ was used to classify which 4D, $\mathcal{N}=2$ off-shell supersymmetric systems can be represented with finite numbers of auxiliary field and which cannot. Third, adinkras relate supersymmetric systems that exist in different dimensions, possibly providing a holographic path to uncover unknown representations. Adinkras can be “shadows” of higher dimensional supersymmetry where an adinkra can be drawn that encodes the entire transformation laws when the system is considered to depend only on one or two of the spacetime dimensions.

Classifying supersymmetric systems in terms of which adinkras they reduce to is known as supersymmetric genomics [6,7,8], whereas building higher dimensional supersymmetry from lower dimensional supersymmetry (dimensional enhancement) is known as supersymmetric holography [9,10,11,12,13,14,15,16,17]. As there are generally more low-dimensional than high-dimensional supersymmetric systems, a classification scheme is necessary to sort out which lower dimensional systems are related to which higher dimensional systems. Defining equivalence classes is essential to the adinkra sorting process.

In this paper, we report the discovery of 96 equivalence classes of four-color, four-boson, and four-fermion adinkras. These equivalence classes can be thought of as classes of inner products within an orthogonal basis for adinkras: two adinkras can be thought of as equivalent if they decompose the same way in this basis, thus having a normalized inner product of one. As shown in [12,16], a set of holoraumy (the word “holoraumy” was defined in [15] as a combination of the Greek word holos (complete) and the German word raum (space)) matrices ${\tilde{\mathbf{V}}}_{\mathrm{IJ}}$ can be constructed from the transformation laws encoded by the adinkra. These holoraumy matrices exist in a space spanned by six basis elements ${\mathit{\alpha }}^{\widehat{a}}$ and ${\mathit{\beta }}^{\widehat{a}}$. The basis elements ${\mathit{\alpha }}^{\widehat{a}}$ and ${\mathit{\beta }}^{\widehat{a}}$ form mutually-commuting su(2) algebras. An inner product was first defined for this basis in [12]. This inner product has been called the gadget in many subsequent works such as [16,18,19]. The gadget between two adinkra representations equals one if the adinkras have identical ${\tilde{\mathbf{V}}}_{IJ}$’s. Following [19], we define holoraumy-equivalence classes of adinkras whose inner products equal one. We organize our results in terms of $B{C}_3$ and $B{C}_4$: the signed permutation groups of three and four elements, respectively. The relationships between the different holoraumy-equivalence classes are presented in terms of $B{C}_3$ color transformations that map one equivalence class to another. The main results of this paper are

• the discovery of the $B{C}_3$ mappings to all $\tilde{\mathbf{V}}$-equivalence classes,
• the generation of all 36,864 four-color, four-boson, and four-fermion adinkras in terms of $B{C}_4$ boson × $B{C}_3$ color transformations of two $\tilde{\mathbf{V}}$-inequivalent adinkras, dubbed the quaternion adinkras,
• the presentation of a formula that encodes all 36,864 four-color, four-boson, and four-fermion adinkras in terms of their $\tilde{\mathbf{V}}$-equivalence classes,
• the explanation of the count of all possible gadget values in terms of equivalence classes, and
• the connections between the gadget, holoraumy, and dynamics through Kähler-like potentials.

The second and third results will elucidate why there are 36,864 four-color, four-boson, and four-fermion adinkras, as well as encode all such adinkras in two succinct equations. The matrix representations of the quaternion adinkras we define satisfy the quaternion algebra. Though lesser discussed at present, describing supersymmetry in the language of quaternions has been investigated before [20].

The fifth result is the major achievement of this work. For the first time, a universal non-linear σ-model is defined over the entirety of the 36,864 adinkras that provide the basis for all linear representations of 1D, N = 4 supersymmetry.

The vast majority of our previous efforts in the study of adinkras has concentrated on the issue of building a rigorous representation theory. However, there have been two exceptions. In the work of [21], it was shown how to couple one-dimensional SUSY models with arbitrary extensions of the numbers of worldline supersymmetries to external magnetic fields. The existence of a “super Zeeman effect” was noted. The other exception [22] explored the question of the compatibility of the traditional superfield approach of one-dimensional supersymmetrical theories with the approach of using adinkras in these same theories. Compatibility was shown, and this opens the path for explaining how the adinkra approach leads to a uniform and universal formalism for describing the “model space” of 1D, N = 4 supersymmetric $\sigma$-models.

The topic of one-dimensional N = 4 $\sigma$-models, which began in 1991 [23,24,25], developed into a substantial literature [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51], some even prior to the work in [22]. Parts of this work have been empowered by some of the insights (e.g., “root superfields” provide one example) gained from adinkras. The efficacy of these models can be seen by the range of concepts to which they connect such as:

(a)

Hopf maps,

(b)

superconformal mechanics,

(c)

supersymmetric Calogero models,

(d)

supersymmetric CP(n) mechanics,

(e)

superconformal mechanics and black holes,

(f)

supersymmetric WDVVequations and roots, and

(g)

Hyper-Kähler and Clifford–Kähler geometries with torsion.

The reader should keep in mind that the recitation and citations of this paragraph constitute only a very small slice of the literature. If one looks at the cited work of this paragraph, it can be noted there was an effort to establish a universal formalism, via the use of harmonic superspace techniques [36,37,38], to describe all such models. Such a universal formalism is what we mean by the use of the term “model space.” We next review the issue of the control of the model space in the more familiar context of 4D, $\mathcal{N}$ = 1, 2D, $\mathcal{N}$ = 2, and 2D, $\mathcal{N}$ = (2, 0) $\sigma$-models. These are domains in which these issues are well understood and settled. We use this discussion as the basis for the construction of a universal formalism for describing the universal $B{C}_4$ Coxeter group non-linear $\sigma$-model.

This paper is organized as follows. In Section 2, we briefly review adinkras and how they can describe higher dimensional systems, as well as show how $\tilde{\mathbf{V}}$-equivalence classes have already been used to distinguish some of these systems. We illustrate the process of SUSY holography, demonstrating the missing steps and commenting on possible solutions. In Section 3, we introduce the quaternion adinkras, and in Section 4, we report the 96 $\tilde{\mathbf{V}}$-equivalence classes that are generated from the quaternion adinkras and span all 36,864 four-color, four-boson, and four-fermion adinkras. We use these equivalence classes to explain the counts of the four different gadget values calculated in [19] and present the formula that encodes all 36,864 four-color, four-boson, and four-fermion adinkras. In Section 5 and Section 6, we demonstrate connections between holoraumy and dynamics. Specifically, in Section 5, we demonstrate how the holoraumy for each of four different 2D SUSY sigma models is different from the others, thus having gadgets differ from unity. In Section 6, we demonstrate how the holoraumy matrices appear in a set of 1D SUSY actions generated by a Kähler-like potential and demonstrate how a particular choice of Kähler-like potential yields the common action for all 36,864 1D SUSY models investigated throughout the rest of the paper. The connections shown in Section 5 and Section 6 will be particularly important as we continue our quest to develop SUSY holography.

2. Adinkra Review

In this section, we review how adinkras are used to encode supersymmetry entirely in one-dimension and partially in four-dimensions. We demonstrate the (at present incomplete) holographic map from 1D adinkras to 4D SUSY. For a more complete review, we refer the reader to [6,7,8], whose conventions we follow for four-dimensional gamma matrices ${\left({\gamma }^{\mu }\right)}_{ab}$.

2.1. Adinkras and One-Dimensional Supersymmetry

Consider four bosonic fields ${\Phi }_i$ and four fermionic fields ${\Psi }_{\widehat{j}}$ in one dimension, that of time. The Lagrangian for these fields is:

$$\begin{array}{c}\hfill \mathcal{L}=\frac{1}{\phantom{2}2}{\delta }^{ij}{\dot{\Phi }}_i{\dot{\Phi }}_j-i\frac{1}{\phantom{2}2}{\delta }^{\widehat{i}\widehat{j}}{\Psi }_{\widehat{i}}{\dot{\Psi }}_{\widehat{j}}\end{array}$$

(1)

where a dot above a field indicates a time derivative, ${\dot{\Phi }}_i=d{\Phi }_i/d\tau$ for example. We now explain how the two adinkras shown in Figure 1 each encode a set of SUSY transformation laws that leave the Lagrangian (1) invariant.

The white nodes of the adinkras encode the bosons ${\Phi }_i$, and the black nodes encode the fermions multiplied by the imaginary number $i{\Psi }_{\widehat{j}}$. A line connecting two nodes indicates a SUSY transformation law between the corresponding fields. Each of the $N=4$ colors encodes a different SUSY transformation. A solid (dashed) line indicates a plus (minus) sign in SUSY transformations. In transforming from a higher node to a lower node (higher mass dimension field to one-half lower mass dimension field), a time derivative appears on the field of the lower node.

For the two adinkras in Figure 1, we define ${\chi }_0=+1$ for the leftmost adinkra (the cis-adinkra) and ${\chi }_0=-1$ for the rightmost adinkra (the trans-adinkra), which allows us to simultaneously write the transformation laws encoded by each adinkra as follows. For boson to fermion, we have:

$$\begin{array}{c}{\mathrm{D}}_1{\Phi }_1=i{\Psi }_1,{\mathrm{D}}_2{\Phi }_1=i{\Psi }_2,{\mathrm{D}}_3{\Phi }_1={\chi }_{0}i{\Psi }_3,{\mathrm{D}}_4{\Phi }_1=-i{\Psi }_4\hfill \end{array}$$

(2a)

$$\begin{array}{c}{\mathrm{D}}_1{\Phi }_2=i{\Psi }_2,{\mathrm{D}}_2{\Phi }_2=-i{\Psi }_1,{\mathrm{D}}_3{\Phi }_2={\chi }_{0}i{\Psi }_4,{\mathrm{D}}_4{\Phi }_2=i{\Psi }_3\hfill \end{array}$$

(2b)

$$\begin{array}{c}{\mathrm{D}}_1{\Phi }_3=i{\Psi }_3,{\mathrm{D}}_2{\Phi }_3=-i{\Psi }_4,{\mathrm{D}}_3{\Phi }_3=-{\chi }_{0}i{\Psi }_1,{\mathrm{D}}_4{\Phi }_3=-i{\Psi }_2\hfill \end{array}$$

(2c)

$$\begin{array}{c}{\mathrm{D}}_1{\Phi }_4=i{\Psi }_4,{\mathrm{D}}_2{\Phi }_4=i{\Psi }_3,{\mathrm{D}}_3{\Phi }_4=-{\chi }_{0}i{\Psi }_2,{\mathrm{D}}_4{\Phi }_4=i{\Psi }_1\hfill \end{array}$$

(2d)

For fermion to boson, we have:

$$\begin{array}{c}{\mathrm{D}}_1{\Psi }_1={\dot{\Phi }}_1,{\mathrm{D}}_2{\Psi }_1=-{\dot{\Phi }}_2,{\mathrm{D}}_3{\Psi }_1=-{\chi }_0{\dot{\Phi }}_3,{\mathrm{D}}_4{\Psi }_1={\dot{\Phi }}_4\hfill \end{array}$$

(3a)

$$\begin{array}{c}{\mathrm{D}}_1{\Psi }_2={\dot{\Phi }}_2,{\mathrm{D}}_2{\Psi }_2={\dot{\Phi }}_1,{\mathrm{D}}_3{\Psi }_2=-{\chi }_0{\dot{\Phi }}_4,{\mathrm{D}}_4{\Psi }_2=-{\dot{\Phi }}_3\hfill \end{array}$$

(3b)

$$\begin{array}{c}{\mathrm{D}}_1{\Psi }_3={\dot{\Phi }}_3,{\mathrm{D}}_2{\Psi }_3={\dot{\Phi }}_4,{\mathrm{D}}_3{\Psi }_3={\chi }_0{\dot{\Phi }}_1,{\mathrm{D}}_4{\Psi }_3={\dot{\Phi }}_2\hfill \end{array}$$

(3c)

$$\begin{array}{c}{\mathrm{D}}_1{\Psi }_4={\dot{\Phi }}_4,{\mathrm{D}}_2{\Psi }_4=-{\dot{\Phi }}_3,{\mathrm{D}}_3{\Psi }_4={\chi }_0{\dot{\Phi }}_2,{\mathrm{D}}_4{\Psi }_4=-{\dot{\Phi }}_1\hfill \end{array}$$

(3d)

Adinkras such as those in Figure 1 that have all bosons at the same height and all fermions at the same height are known as valise adinkras. The SUSY transformation laws encoded by a valise adinkra can be succinctly written as:

$$\begin{array}{c}\hfill {\mathrm{D}}_{\mathrm{I}}\Phi =i{\mathbf{L}}_{\mathrm{I}}\Psi ,{\mathrm{D}}_{\mathrm{I}}\Psi ={\mathbf{R}}_{\mathrm{I}}\dot{\Phi },\end{array}$$

(4)

with ${\mathbf{R}}_{\mathrm{I}}$ and ${\mathbf{L}}_{\mathrm{I}}$ inverses and transposes of each other:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathrm{I}}={\mathbf{L}}_{\mathrm{I}}^T={\mathbf{L}}_{\mathrm{I}}^{-1}.\end{array}$$

(5)

The adinkra matrices ${\mathbf{L}}_{\mathrm{I}}$ and ${\mathbf{R}}_{\mathrm{I}}$ satisfy the $\mathcal{GR}(d,N)$ algebra, also known as the garden algebra:

$$\begin{array}{c}\hfill {\mathbf{L}}_{\mathrm{I}}{\mathbf{R}}_{\mathrm{J}}+{\mathbf{L}}_{\mathrm{J}}{\mathbf{R}}_{\mathrm{I}}=2{\delta }_{\mathrm{IJ}}{\mathbf{I}}_4,{\mathbf{R}}_{\mathrm{I}}{\mathbf{L}}_{\mathrm{J}}+{\mathbf{R}}_{\mathrm{J}}{\mathbf{L}}_{\mathrm{I}}=2{\delta }_{\mathrm{IJ}}{\mathbf{I}}_4.\end{array}$$

(6)

The $\mathcal{GR}(d,N)$ algebra is the algebra of general, real matrices encoding the supersymmetry transformation laws between d bosons, d fermions, and N supersymmetries. The adinkras in Figure 1 each have $d=4$ and $N=4$. Hitherto, the word adinkra shall refer to $d=4$, $N=4$ valise adinkras, unless otherwise specified. With the bosonic ($i,j,k,\dots$) and fermionic ($\widehat{i},\widehat{j},\widehat{k},\dots$) indices exposed, Equations (4) and (6) are:

$$\begin{array}{cc}\hfill & {\mathrm{D}}_{\mathrm{I}}{\Phi }_i=i{\left({\mathrm{L}}_{\mathrm{I}}\right)}_i^{\widehat{j}}{\Psi }_{\widehat{j}},{\mathrm{D}}_{\mathrm{I}}{\Psi }_{\widehat{j}}={\left({\mathrm{R}}_{\mathrm{I}}\right)}_{\widehat{j}}^i{\dot{\Phi }}_i,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\left({\mathrm{L}}_{\mathrm{I}}\right)}_i^{\widehat{j}}{\left({\mathrm{R}}_{\mathrm{J}}\right)}_{\widehat{j}}^k+{\left({\mathrm{L}}_{\mathrm{J}}\right)}_i^{\widehat{j}}{\left({\mathrm{R}}_{\mathrm{I}}\right)}_{\widehat{j}}^k=2{\delta }_{\mathrm{IJ}}{\delta }_i^k,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\left({\mathrm{R}}_{\mathrm{I}}\right)}_{\widehat{j}}^i{\left({\mathrm{L}}_{\mathrm{J}}\right)}_i^{\widehat{k}}+{\left({\mathrm{R}}_{\mathrm{J}}\right)}_{\widehat{j}}^i{\left({\mathrm{L}}_{\mathrm{I}}\right)}_i^{\widehat{k}}=2{\delta }_{\mathrm{IJ}}{\delta }_{\widehat{j}}^{\widehat{k}}.\hfill \end{array}$$

(9)

Using Equation (7), we concisely write the matrices ${\mathbf{L}}_{\mathrm{I}}$ and ${\mathbf{R}}_{\mathrm{I}}$ for the transformation laws in Equations (2) and (3) for the cis- and trans-adinkra as:

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{L}}_1=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_2=\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{L}}_3={\chi }_0\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_4=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right)\end{array}.\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{R}}_1=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{R}}_2=\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{R}}_3={\chi }_0\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{R}}_4=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ -1& 0& 0& 0\end{array}\right)\end{array}.\hfill \end{array}$$

(11)

A SUSY transformation between two fields is encoded as a nonzero entry in the corresponding row and column of each of the corresponding $\mathbf{L}$ and $\mathbf{R}$ matrices. A solid (dashed) line in an adinkra encodes a plus (minus) one in the corresponding matrix entry. The only difference between the cis- and trans-valise adinkras in Figure 1 is the orange lines, where they are dashed in one adinkra, solid in the other; hence, matrices $\begin{array}{c}\hfill {\mathbf{L}}_1\end{array}$, $\begin{array}{c}\hfill {\mathbf{L}}_2\end{array}$, and $\begin{array}{c}\hfill {\mathbf{L}}_4\end{array}$ are identical, but $\begin{array}{c}\hfill {\mathbf{L}}_3\end{array}$ is opposite for the cis- and trans-valise adinkras. For an arbitrary $d=4$, $N=4$ valise adinkra, we define ${\chi }_0$ from the following chromocharacter equation [6]:

$$\begin{array}{c}\hfill Tr\left({\mathbf{L}}_{\mathrm{I}}{\mathbf{L}}_{\mathrm{J}}^T{\mathbf{L}}_{\mathrm{K}}{\mathbf{L}}_{\mathrm{L}}^T\right)=4({\delta }_{\mathrm{I}\mathrm{J}}{\delta }_{\mathrm{K}\mathrm{L}}-{\delta }_{\mathrm{I}\mathrm{L}}{\delta }_{\mathrm{J}\mathrm{K}}+{\delta }_{\mathrm{I}\mathrm{J}}{\delta }_{\mathrm{K}\mathrm{L}}+{\chi }_0{ϵ}_{\mathrm{I}\mathrm{J}\mathrm{K}\mathrm{L}})\end{array}$$

(12)

We have for all $d=4$, $N=4$ valise adinkras, ${\chi }_0=\pm 1$. Here, we define the ${\chi }_0$-equivalence class:

Two adinkras are χ₀-equivalent if they share the same value of χ₀.

2.2. Adinkras and Four-Dimensional Supersymmetry

In this section, we will describe how adinkras can partially encode SUSY transformations in four-dimensions. All valise adinkras belong to either the ${\chi }_0=+1$ or ${\chi }_0=-1$ equivalence class. In [6], it was shown that the cis- and trans-valise adinkras can encode the zero-brane reduced transformation laws (the transformation laws when only temporal dependence of the fields is considered) of the 4D, $\mathcal{N}=1$ chiral multiplet ($CM$), tensor multiplet ($TM$), and vector multiplet ($VM$). In terms of ${\chi }_0$ equivalence classes, we have specifically that ${\chi }_0=+1$ for $CM$ and ${\chi }_0=-1$ for $VM$ and $TM$. This first step of simply identifying which type of fields are described by adinkras nodes is at present unknown simply given the adinkra.

Given that starting point, however, we now demonstrate how the 4D transformation laws can be seen to arise from the adinkra representations that exist on the zero-brane, as shown in Figure 2, Figure 3 and Figure 4 for $CM$, $VM$, and $TM$. Along with the nodal definitions, the other missing piece of such a full holographic map is how to fill in the spatial dependence. This will be illustrated in the following. While showing the parts of the map from the zero-brane to 4D that exist (namely that time-dependent parts), we will also demonstrate that the ${\chi }_0$-equivalence class can be used to separate partially which adinkras correspond to which higher dimensional multiplets. Holoraumy allows us to separate more completely, which we discuss in the next section.

The adinkras in Figure 2, Figure 3 and Figure 4 encode the transformation laws in Equation (4) with the following node identifications and adinkra matrices ${\mathbf{L}}_{\mathrm{I}}$ and ${\mathbf{R}}_{\mathrm{I}}$. The node identifications for each adinkra are:

$$\begin{array}{cc}\hfill CM:& i{\Psi }_1={\psi }_1,i{\Psi }_2={\psi }_2,i{\Psi }_3={\psi }_3,i{\Psi }_4={\psi }_4\hfill \end{array}$$

(13a)

$$\begin{array}{cc}\hfill & {\Phi }_1=A,{\Phi }_2=B,{\Phi }_3=\int d\tau F,{\Phi }_4=\int d\tau G\hfill \end{array}$$

(13b)

$$\begin{array}{cc}\hfill TM:& i{\Psi }_1={\chi }_1,i{\Psi }_2={\chi }_2,i{\Psi }_3={\chi }_3,i{\Psi }_4={\chi }_4\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill & {\Phi }_1=\phi ,{\Phi }_2=2B_{12},{\Phi }_3=2B_{23},{\Phi }_4=2B_{31}\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill TM:& i{\Psi }_1={\lambda }_1,i{\Psi }_2={\lambda }_2,i{\Psi }_3={\lambda }_3,i{\Psi }_4={\lambda }_4\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & {\Phi }_1=A_1,{\Phi }_2=A_2,{\Phi }_3=A_3,{\Phi }_4=\int d\tau \mathrm{d}\hfill \end{array}$$

(15b)

The ${\mathbf{L}}_{\mathrm{I}}$ for the $CM$, $TM$, and $VM$ adinkras are shown below. Note that the ${\mathbf{R}}_{\mathrm{I}}$ are the transposes of these as in Equation (5).

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{L}}_1^{\left(CM\right)}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 1& 0& 0\\ 0& 0& -1& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_2^{\left(CM\right)}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{L}}_3^{\left(CM\right)}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_4^{\left(CM\right)}=\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right)\end{array}.\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{L}}_1^{\left(TM\right)}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\\ 0& -1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_2^{\left(TM\right)}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\\ 1& 0& 0& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{L}}_3^{\left(TM\right)}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_4^{\left(TM\right)}=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right).\end{array}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{L}}_1^{\left(VM\right)}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 0& -1& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_2^{\left(VM\right)}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& -1\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{L}}_3^{\left(VM\right)}=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_4^{\left(VM\right)}=\left(\begin{array}{cccc}0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 1& 0& 0\end{array}\right).\end{array}\hfill \end{array}$$

(18)

The Lagrangian for the $CM$, $VM$, and $TM$ are, in terms of their ${\Phi }_i$ and ${\Psi }_i$, all the same: Equation (1). Substituting in the nodal definitions Equations (13)–(15), we arrive at their respective Lagrangians and transformation laws, where we start to see differences:

$${\displaystyle {\mathcal{L}}_{CM}^{\left(0\right)}=\frac{1}{\phantom{2}2}{\dot{A}}^2+\frac{1}{\phantom{2}2}{\dot{B}}^2+i\frac{1}{\phantom{2}2}{\left({\gamma }^0\right)}^{ab}{\psi }_a{\dot{\psi }}_b+\frac{1}{\phantom{2}2}{F}^2+\frac{1}{\phantom{2}2}{G}^2}$$

(19)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{TM}}^{\left(0\right)}=& 2\left({\dot{B}}_{12}^2+{\dot{B}}_{23}^2+{\dot{B}}_{31}^2\right)+\frac{1}{\phantom{2}2}{\dot{\phi }}^2+\frac{1}{\phantom{2}2}i{\left({\gamma }^0\right)}^{ab}{\chi }_a{\dot{\chi }}_b\hfill \end{array}$$

(20)

$${\displaystyle {\mathcal{L}}_{VM}^{\left(0\right)}=\frac{1}{\phantom{2}2}({\dot{A}}_1^2+{\dot{A}}_2^2+{\dot{A}}_3^2)+\frac{1}{\phantom{2}2}i{\left({\gamma }^0\right)}^{ab}{\lambda }_a{\dot{\lambda }}_b+\frac{1}{\phantom{2}2}{\mathrm{d}}^2}$$

(21)

Again, we are using the gamma matrix conventions of [6]. The corresponding transformation laws for the CM are:

$${\displaystyle \begin{array}{c}{\mathrm{D}}_{a}A={\psi }_a,{\mathrm{D}}_{a}B=i{\left({\gamma }^5\right)}_a{}^b{\psi }_b,\hfill \\ {\mathrm{D}}_{a}F={\left({\gamma }^0\right)}_a{}^b{\partial }_{\tau }{\psi }_b,{\mathrm{D}}_{a}G=i{\left({\gamma }^5{\gamma }^0\right)}_a{}^b{\partial }_{\tau }{\psi }_b,\hfill \\ {\mathrm{D}}_a{\psi }_b=i{\left({\gamma }^0\right)}_{ab}\left({\partial }_{\tau }A\right)-{\left({\gamma }^5{\gamma }^0\right)}_{ab}\left({\partial }_{\tau }B\right)-i{C}_{ab}F+{\left({\gamma }^5\right)}_{ab}G.\hfill \end{array}}$$

(22)

The corresponding transformation laws for the $TM$ are:

$$\begin{gathered} {\displaystyle \mathrm{D}{}_a\phi ={\chi }_a,\mathrm{D}{}_a{B}_{mn}=-\frac{1}{4}{\left([{\gamma }_m,{\gamma }_n]\right)}_a{}^b{\chi }_b, \\ \mathrm{D}{}_a{\chi }_b=i{\left({\gamma }^0\right)}_{ab}{\partial }_{\tau }\phi -i\frac{1}{2}{\left({\gamma }^0[{\gamma }^m,{\gamma }^n]\right)}_{ab}{\partial }_{\tau }{B}_{mn}.} \end{gathered}$$

(23)

where early Latin indices $a,b,\cdots =1,2,3,4$ and late Latin indices $m,n,\cdots =1,2,3$. Furthermore, we define $B_{mn}=-B_{nm}$. The corresponding transformation laws for the $VM$ are:

$$\begin{gathered} {\displaystyle \mathrm{D}{}_a{A}_m={\left({\gamma }_m\right)}_a{}^b{\lambda }_b,{\mathrm{D}}_a\mathrm{d}=i{\left({\gamma }^5{\gamma }^0\right)}_a{}^b{\partial }_{\tau }{\lambda }_b, \\ \mathrm{D}{}_a{\lambda }_b=-i{\left({\gamma }^0{\gamma }^m\right)}_{ab}\left({\partial }_{\tau }{A}_m\right)+{\left({\gamma }^5\right)}_{ab}\mathrm{d}.} \end{gathered}$$

(24)

So far, the only missing piece in the holographic map has been the nodal definitions. The next step, where we insert the spatial dependence, is the other unknown piece. Said another way, a main open question in SUSY holography is:

What needs to be added to zero-brane Lagrangians such as (19), (20), (21), and their corresponding transformation laws (22), (23), and (24) to arrive at full 4D, $\mathcal{N}=1$ off-shell SUSY representations?

Though we do not generally know a holographic procedure, from dimensional reduction, we know the answer [6]. Below are the 4D transformation laws and corresponding Lagrangians that reduce to the zero-brane transformation laws and Lagrangians. We introduce Greek indices $\mu ,\nu ,\cdots =0,1,2,3$ where for the $CM$ we have merely to introduce more derivatives on the gauge fields and more derivatives and gamma matrices for the fermions:

$${\displaystyle {\mathcal{L}}_{\mathrm{CM}}=-\frac{1}{\phantom{2}2}\left({\partial }_{\mu }A\right)\left({\partial }^{\mu }A\right)-\frac{1}{\phantom{2}2}\left({\partial }_{\mu }B\right)\left({\partial }^{\mu }B\right)+i\frac{1}{\phantom{2}2}{\left({\gamma }^{\mu }\right)}^{ab}{\psi }_a{\partial }_{\mu }{\psi }_b+\frac{1}{\phantom{2}2}{F}^2+\frac{1}{\phantom{2}2}{G}^2}$$

(25)

$${\displaystyle \begin{array}{c}{\mathrm{D}}_{a}A={\psi }_a,{\mathrm{D}}_{a}B=i{\left({\gamma }^5\right)}_a{}^b{\psi }_b\hfill \\ {\mathrm{D}}_a{\psi }_b=i{\left({\gamma }^{\mu }\right)}_{ab}\left({\partial }_{\mu }A\right)-{\left({\gamma }^5{\gamma }^{\mu }\right)}_{ab}\left({\partial }_{\mu }B\right)-i{C}_{ab}F+{\left({\gamma }^5\right)}_{ab}G\hfill \\ {\mathrm{D}}_{a}F={\left({\gamma }^{\mu }\right)}_a{}^b{\partial }_{\mu }{\psi }_b,{\mathrm{D}}_{a}G=i{\left({\gamma }^5{\gamma }^{\mu }\right)}_a{}^b{\partial }_{\mu }{\psi }_b\hfill \end{array}}$$

(26)

For the $TM$ and $VM$, in addition to adding more derivatives and more gamma matrices, we must also introduce components that were gauge fixed to zero at the adinkra level: $B_{01}$, $B_{02}$, and $B_{03}$ for the TM and $A_0$ for the VM. For the TM, we have:

$${\displaystyle {\mathcal{L}}_{\mathrm{TM}}=-\frac{1}{\phantom{2}3}{H}_{\mu \nu \alpha }{H}^{\mu \nu \alpha }-\frac{1}{\phantom{2}2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi +\frac{1}{\phantom{2}2}i{\left({\gamma }^{\mu }\right)}^{bc}{\chi }_b{\partial }_{\mu }{\chi }_c}$$

(27)

where:

$$H_{\mu \nu \alpha }\equiv {\partial }_{\mu }{B}_{\nu \alpha }+{\partial }_{\nu }{B}_{\alpha \mu }+{\partial }_{\alpha }{B}_{\mu \nu }$$

(28)

$${\displaystyle \begin{array}{ccc}\hfill {\mathrm{D}}_a\phi & =& {\chi }_a\hfill \\ \hfill {\mathrm{D}}_{a}B{}_{\mu \nu }& =& -\frac{1}{\phantom{2}4}\left([{\gamma }_{\mu },{\gamma }_{\nu }]\right){}_a{}^b{\chi }_b\hfill \\ \hfill {\mathrm{D}}_a{\chi }_b& =& i\left({\gamma }^{\mu }\right){}_{ab}{\partial }_{\mu }\phi -\left({\gamma }^5{\gamma }^{\mu }\right){}_{ab}ϵ{}_{\mu }{}^{\rho \sigma \tau }{\partial }_{\rho }B{}_{\sigma \tau }\hfill \end{array}}$$

(29)

For the $VM$, we have:

$${\displaystyle {\mathcal{L}}_{\mathrm{VM}}=-\frac{1}{\phantom{2}4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{\phantom{2}2}i{\left({\gamma }^{\mu }\right)}^{ab}{\lambda }_a{\partial }_{\mu }{\lambda }_b+\frac{1}{\phantom{2}2}{\mathrm{d}}^2}$$

(30)

$${\displaystyle \begin{array}{ccc}\hfill \mathrm{D}{}_a{A}_{\mu }& =& {\left({\gamma }_{\mu }\right)}_a{}^b{\lambda }_b\hfill \\ \hfill \mathrm{D}{}_a{\lambda }_b& =& -i\frac{1}{\phantom{2}2}{\left({\gamma }^{\mu }{\gamma }^{\nu }\right)}_{ab}F{}_{\mu \nu }+{\left({\gamma }^5\right)}_{ab}\mathrm{d}\hfill \\ \hfill {\mathrm{D}}_a\mathrm{d}& =& i{\left({\gamma }^5{\gamma }^{\mu }\right)}_a{}^b{\partial }_{\mu }{\lambda }_b\hfill \end{array}}$$

(31)

where:

$$\begin{array}{c}\hfill F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }\end{array}$$

(32)

The preceding illustrates four missing steps in the SUSY holography procedure.

• Nodal field definitions and which adinkras encode which particle spins.
• How to augment with spatial derivatives.
• How to introduce gamma matrices.
• How to introduce more degrees of freedom for the gauge fields.

The second has arguably a simple solution as applied to the previous examples: simply replace ${\gamma }^0{\partial }_{\tau }\to {\gamma }^{\mu }{\partial }_{\mu }$. This is related to the third point about ${\gamma }^{\mu }$ matrices, which was addressed in [15,16], where a correspondence between the zero-brane holoraumy matrices and the 4D gamma matrices was shown. The first and fourth are the least understood, but likely will have a common solution: knowing which particle spins are encoded by a given adinkra should lead to the knowledge of which gauge degrees of freedom must be added and vice versa. In the remainder of this paper, we continue to develop holoraumy as tool to be applied to the first and fourth steps in future works.

We conclude this section with some comments on the ${\chi }_0$-equivalence class. Performing the calculation in Equation (12) on the adinkra ${\mathbf{L}}_{\mathrm{I}}$-matrices for the $CM$, $VM$, and $TM$ results in ${\chi }_0=+1$ for the $CM$ and ${\chi }_0=-1$ for the $VM$ and $TM$. The $CM$ is therefore in a different ${\chi }_0$-equivalence class from the $VM$ and $TM$. Furthermore, the nodes of the chiral, tensor, and vector valise adinkras in Figure 2, Figure 3 and Figure 4 can be rearranged to arrive at different adinkras that equivalently encode the SUSY transformations. We define permutations of the nodes as flops and sign flips of the nodes as flips. Flipping and flopping the nodes of a representation $\mathcal{R}$ to a new representation ${\mathcal{R}}^{\prime }$ corresponds to multiplication by matrices $\mathcal{X}$ and $\mathcal{Y}$ as in:

$$\begin{array}{cc}\hfill {\Phi }^{\left({\mathcal{R}}^{\prime }\right)}=\mathcal{X}{\Phi }^{\left(\mathcal{R}\right)},& {\Psi }^{\left({\mathcal{R}}^{\prime }\right)}={\mathcal{Y}}^T{\Psi }^{\left(\mathcal{R}\right)}\hfill \\ \hfill {\mathbf{L}}_{\mathrm{I}}^{\left({\mathcal{R}}^{\prime }\right)}=\mathcal{X}{\mathbf{L}}_{\mathrm{I}}^{\left(\mathcal{R}\right)}\mathcal{Y},& {\mathbf{R}}_{\mathrm{I}}^{\left({\mathcal{R}}^{\prime }\right)}={\mathcal{Y}}^{\mathit{T}}{\mathbf{R}}_{\mathrm{I}}^{\left(\mathcal{R}\right)}{\mathcal{X}}^T.\hfill \end{array}$$

(33)

The matrices $\mathcal{X}$ and $\mathcal{Y}$ can in principle be any non-singular matrix; however, in this paper, they will be elements of $B{C}_4$.

As Equation (12) is invariant with respect to such nodal flips and flops, they leave the adinkra in the same ${\chi }_0$-equivalence class. Both the tensor and vector adinkras can be node flopped and flipped to the trans-adinkra (${\chi }_0=-1$) in Figure 1, and the chiral adinkra can be node flopped and flipped to the cis-adinkra (${\chi }_0=+1$) in Figure 1 [6]. The ${\chi }_0$-equivalence class can therefore be used to distinguish adinkras that are representative of the chiral multiplet from adinkras that are representative of the tensor or vector multiplet.

2.3. $\mathbf{V}$- and $\tilde{\mathbf{V}}$-equivalence Classes

Once ${\chi }_0$-equivalence classes are seen to separate the chiral multiplet from both the tensor and vector multiplets, the natural questions is: Do other equivalence classes exist that separate the vector and tensor multiplets? In [12,16,18,19], the holoraumy matrices ${\mathbf{V}}_{\mathrm{IJ}}$ and ${\tilde{\mathbf{V}}}_{\mathrm{IJ}}$ were seen to do just that (these are the conventions of [19], which we use throughout this paper):

$$\begin{array}{cc}\hfill {\mathbf{V}}_{\mathrm{IJ}}=-i\frac{1}{\phantom{2}2}{\mathbf{L}}_{[\mathrm{I}}{\mathbf{R}}_{\mathrm{J}]}=& \sum _{\widehat{a}=1}^3({\kappa }_{\mathrm{IJ}}^{\widehat{a}}{\mathit{\alpha }}^{\widehat{a}}+{\tilde{\kappa }}_{\mathrm{IJ}}^{\widehat{a}}{\mathit{\beta }}^{\widehat{a}})\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\tilde{\mathbf{V}}}_{\mathrm{IJ}}=-i\frac{1}{\phantom{2}2}{\mathbf{R}}_{[\mathrm{I}}{\mathbf{L}}_{\mathrm{J}]}=& \sum _{\widehat{a}=1}^3({\ell }_{\mathrm{IJ}}^{\widehat{a}}{\mathit{\alpha }}^{\widehat{a}}+{\tilde{\ell }}_{\mathrm{IJ}}^{\widehat{a}}{\mathit{\beta }}^{\widehat{a}}).\hfill \end{array}$$

(35)

The matrices ${\mathit{\alpha }}^{\widehat{a}}$ and ${\mathit{\beta }}^{\widehat{a}}$ are given in Appendix A.

Note that by construction ${\ell }_{IJ}^{\widehat{a}}=-{\ell }_{JI}^{\widehat{a}}$ and similarly for $\tilde{\ell }$, $\kappa$, and $\tilde{\kappa }$. For the $CM$, $VM$, and $TM$ adinkras in Equations (16)–(18), the only independent, non-vanishing ${\tilde{\mathbf{V}}}_{\mathrm{I}\mathrm{J}}$ coefficients are:

$$\begin{array}{c}CM:{\ell }_{12}^2={\ell }_{13}^3={\ell }_{14}^1={\ell }_{23}^1=-{\ell }_{24}^3={\ell }_{34}^2=1\hfill \end{array}$$

(36a)

$$\begin{array}{c}TM:{\tilde{\ell }}_{12}^3={\tilde{\ell }}_{13}^2={\tilde{\ell }}_{14}^1=-{\tilde{\ell }}_{23}^1={\tilde{\ell }}_{24}^2=-{\tilde{\ell }}_{34}^3=1\hfill \end{array}$$

(36b)

$$\begin{array}{c}VM:-{\tilde{\ell }}_{12}^3={\tilde{\ell }}_{13}^2=-{\tilde{\ell }}_{14}^1={\tilde{\ell }}_{23}^1={\tilde{\ell }}_{24}^2={\tilde{\ell }}_{34}^3=1\hfill \end{array}$$

(36c)

Once $\mathbf{V}$ and $\tilde{\mathbf{V}}$ are calculated for two adinkra representations $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$, their gadgets can be computed as (these are the conventions of [12,19], which we use throughout this paper):

$$\begin{array}{c}\hfill {\mathcal{G}}_B[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=\frac{1}{\phantom{2}48}\sum _{\mathrm{IJ}}Tr\left({\mathbf{V}}_{\mathrm{IJ}}^{\left(\mathcal{R}\right)}{\mathbf{V}}_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)}\right)=\frac{1}{\phantom{2}12}\sum _{\widehat{a}\mathrm{IJ}}[{\kappa }_{\mathrm{IJ}}^{\left(\mathcal{R}\right)\widehat{a}}{\kappa }_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)\widehat{a}}+{\tilde{\kappa }}_{\mathrm{IJ}}^{\left(\mathcal{R}\right)\widehat{a}}{\tilde{\kappa }}_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)\widehat{a}}]\end{array}$$

(37)

$$\begin{array}{c}\hfill \mathcal{G}[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=\frac{1}{\phantom{2}48}\sum _{\mathrm{IJ}}Tr\left({\tilde{\mathbf{V}}}_{\mathrm{IJ}}^{\left(\mathcal{R}\right)}{\tilde{\mathbf{V}}}_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)}\right)=\frac{1}{\phantom{2}12}\sum _{\widehat{a}\mathrm{IJ}}[{\ell }_{\mathrm{IJ}}^{\left(\mathcal{R}\right)\widehat{a}}{\ell }_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)\widehat{a}}+{\tilde{\ell }}_{\mathrm{IJ}}^{\left(\mathcal{R}\right)\widehat{a}}{\tilde{\ell }}_{\mathrm{IJ}}^{\left({\mathcal{R}}^{\prime }\right)\widehat{a}}]\end{array}$$

(38)

For the chiral, tensor, and vector multiplets, we have the gadgets [12,16,18,19]:

$$\begin{array}{cc}\hfill \mathcal{G}\left[\right(CM),(CM\left)\right]=& \mathcal{G}\left[\right(VM),(VM\left)\right]=\mathcal{G}\left[\right(TM),(TM\left)\right]=1,\hfill \\ \hfill \mathcal{G}\left[\right(CM),(VM\left)\right]=& \mathcal{G}[\left(CM\right),\left(TM\right)]=0,\mathcal{G}[\left(VM\right),\left(TM\right)]=-\frac{1}{\phantom{2}3}.\hfill \end{array}$$

(39)

Gadgets are thought of as inner products in a $\kappa$, $\tilde{\kappa }$, ℓ, and $\tilde{\ell }$ space that is normalized to one for identical $\mathbf{V}$ and $\tilde{\mathbf{V}}$’s. We conclude that the chiral, vector, and tensor multiplets are all in separate $\tilde{\mathbf{V}}$-equivalence classes.

Generalizing, we define two adinkras to be $\mathbf{V}$-equivalent or $\tilde{\mathbf{V}}$-equivalent if they have identical $\mathbf{V}$’s or $\tilde{\mathbf{V}}$’s, respectively. The set of all adinkras that are $\tilde{\mathbf{V}}$-equivalent ($\mathbf{V}$-equivalent) comprise a $\tilde{\mathbf{V}}$-equivalence class ($\mathbf{V}$-equivalence class). We separate $\mathbf{V}$-equivalence classes into $\kappa$- and $\tilde{\kappa }$-equivalence classes and separate $\tilde{\mathbf{V}}$-equivalence classes into ℓ and $\tilde{\ell }$-equivalence classes. Two adinkras are $\kappa$-equivalent if and only if their $\kappa$ coefficients are identical and their $\tilde{\kappa }$ coefficients vanish. Similarly, we define two adinkras to be $\tilde{\kappa }$-equivalent if and only if their $\tilde{\kappa }$ coefficients are identical, and their $\kappa$ coefficients vanish. Two adinkras are ℓ-equivalent if and only if their ℓ coefficients are identical and their $\tilde{\ell }$ coefficients vanish. Similarly, we define two adinkras to be $\tilde{\ell }$-equivalent if and only if their $\tilde{\ell }$ coefficients are identical and their ℓ coefficients vanish.

In this paper, we focus on the gadget $\mathcal{G}$, hitherto referred to as simply the gadget, and so focus on ℓ- and $\tilde{\ell }$-equivalence classes. It has been known for some time [13,17] that any adinkra has either vanishing ℓ or vanishing $\tilde{\ell }$ coefficients. Therefore, the adinkras that belong to the ℓ- and $\tilde{\ell }$-equivalence classes constitute all possible adinkras. The main result of this paper is to explain the results of [19], where all gadgets between all adinkras were presented, in terms of their ℓ- and $\tilde{\ell }$-equivalence classes. Note that gadgets between different ℓ-equivalence classes are always less than one: they will either be $+1/3$, 0, or $-1/3$. If a gadget between two adinkras is one, then they necessarily belong to the same ℓ- or $\tilde{\ell }$-equivalence class.

3. Quaternion Adinkras

We denote the group of signed permutations of three elements as $B{C}_3$. Any element of $B{C}_3$ can be expressed as a sign flip element $H^a$ times a permutation element $S_3^{\mu }$. The indices take the values $a,b,c,\cdots =1,\dots 8$ and $\mu ,\nu ,\cdots =1,\dots ,6$.

$$\begin{array}{cc}\hfill H^a=& \{\left(\right),\left(\overline{12}\right),\left(\overline{13}\right),\left(\overline{23}\right),\left(\overline{1}\right),\left(\overline{2}\right),\left(\overline{3}\right),\left(\overline{123}\right)\}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill S_3^{\mu }=& \left\{\right(),(12),(13),(23),(123),(132\left)\right\}\hfill \end{array}$$

(41)

A line over a number indicates a sign flip for that element. The explicit matrix forms for these elements are given in Appendix B. Hitherto, we shall refer to permutation elements as flops and sign flips as simply flips.

A general element of $B{C}_3$ is given by:

$$\begin{array}{cc}\hfill B{C}_3^{a\mu }=& H^a{S}_3^{\mu }\hfill \end{array}$$

(42)

The Vierergruppe ${\mathcal{V}}^A$, also known as the Klein four-group, is a subgroup of the permutation group of four elements $S_4$:

$$\begin{array}{cc}\hfill {\mathcal{V}}^A& =\left\{\right(),(12\left)\right(34),(13\left)\right(24),(14\left)\right(23\left)\right\}\hfill \end{array}$$

(43)

where $A,B,\cdots =1,2,3,4$. A general element of $B{C}_4$, the group of signed permutations of four elements, can be expressed as plus or minus one times an $B{C}_3$ element times an element of the Vierergruppe.

$$\begin{array}{cc}\hfill B{C}_4^{\pm a\mu A}=& \pm H^a{S}_3^{\mu }{\mathcal{V}}^A\hfill \end{array}$$

(44)

Left cosets of the Vierergruppe via $S_3$ generate all elements of $S_4$ as follows [13,52]:

$$\begin{array}{cc}\hfill \mathcal{V}& =\left\{\right(),(12\left)\right(34),(13\left)\right(24),(14\left)\right(23\left)\right\}\hfill \\ \hfill \left(12\right)\mathcal{V}& =\left\{\right(12),(34),(1324),(1423\left)\right\}\hfill \\ \hfill \left(13\right)\mathcal{V}& =\left\{\right(13),(1234),(24),(1432\left)\right\}\hfill \\ \hfill \left(23\right)\mathcal{V}& =\left\{\right(23),(1342),(1243),(14\left)\right\}\hfill \\ \hfill \left(123\right)\mathcal{V}& =\left\{\right(123),(134),(243),(142\left)\right\}\hfill \\ \hfill \left(132\right)\mathcal{V}& =\left\{\right(132),(234),(124),(143\left)\right\}.\hfill \end{array}$$

(45)

Consider the adinkras in Figure 5, dubbed the quaternion adinkras Q and $\tilde{Q}$:

These have the matrix representations:

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\mathbf{L}}_1^{\left(Q\right)}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_2^{\left(Q\right)}=\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\mathbf{L}}_3^{\left(Q\right)}=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\mathbf{L}}_4^{\left(Q\right)}=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right)\end{array},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\tilde{\mathbf{L}}}_1^{\left(\tilde{Q}\right)}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\tilde{\mathbf{L}}}_2^{\left(\tilde{Q}\right)}=\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\tilde{\mathbf{L}}}_3^{\left(\tilde{Q}\right)}=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\tilde{\mathbf{L}}}_4^{\left(\tilde{Q}\right)}=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right)\end{array}.\hfill \end{array}$$

(47)

Notice that $\tilde{Q}$ is the same as the trans adinkra in Figure 1.

We can express any adinkra matrices as elements of $B{C}_4$. For the quaternion adinkras, we have:

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)}=& \{\left(\right),\left(\overline{13}\right)\left(12\right)\left(34\right),-\left(\overline{23}\right)\left(13\right)\left(24\right),\left(\overline{12}\right)\left(14\right)\left(23\right)\}\hfill \\ \hfill =& \{\mathbf{I},-i{\mathit{\alpha }}^2,-i{\mathit{\alpha }}^3,-i{\mathit{\alpha }}^1\}=\{\mathrm{I},\mathit{i},\mathit{j},\mathit{k}\}\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\tilde{\mathbf{L}}}_{\mathrm{I}}^{\left(\tilde{Q}\right)}=& \{\left(\right),\left(\overline{23}\right)\left(12\right)\left(34\right),\left(\overline{12}\right)\left(13\right)\left(24\right),\left(\overline{13}\right)\left(14\right)\left(23\right)\}\hfill \\ \hfill =& \{\mathbf{I},i{\mathit{\beta }}^3,-i{\mathit{\beta }}^2,-i{\mathit{\beta }}^1\}=\{\mathrm{I},\tilde{\mathit{i}},\tilde{\mathit{j}},\tilde{\mathit{k}}\}\hfill \end{array}$$

(49)

with $\mathrm{I}=1,2,3,4$. The matrices $\mathit{\alpha }$ and $\mathit{\beta }$ are given in Appendix A, and $\mathbf{I}$ is the $4\times 4$ identity matrix. Forgetting the bosonic and fermionic nature of the rows and columns of ${\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)}$ and ${\tilde{\mathbf{L}}}_{\mathrm{I}}^{\left(\tilde{Q}\right)}$, they satisfy the quaternion multiplication rules (Technically, two $\mathbf{L}$-matrices can not be multiplied together. Here, it is meant that ${\mathbf{L}}_{\mathrm{I}}^2=-{\mathbf{L}}_{\mathrm{I}}{\mathbf{R}}_{\mathrm{I}}$; no $\mathrm{I}$ sum and for $\mathrm{I}=2,3,\mathrm{or}4$.):

$$\begin{array}{cc}\hfill {\mathbf{L}}_2^2=& {\mathbf{L}}_3^2={\mathbf{L}}_4^2={\mathbf{L}}_2{\mathbf{L}}_3{\mathbf{L}}_4=-\mathbf{I}\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\tilde{\mathbf{L}}}_2^2=& {\tilde{\mathbf{L}}}_3^2={\tilde{\mathbf{L}}}_4^2={\tilde{\mathbf{L}}}_2{\tilde{\mathbf{L}}}_3{\tilde{\mathbf{L}}}_4=-\mathbf{I}.\hfill \end{array}$$

(51)

They are also mutually commuting:

$$\begin{array}{c}\hfill [{\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)},{\tilde{\mathbf{L}}}_{\mathrm{J}}^{\left(\tilde{Q}\right)}]=0\mathrm{for}\mathrm{I}=2,3,4\end{array}$$

The quaternion adinkras Q and $\tilde{Q}$ belong to separate ℓ and $\tilde{\ell }$-equivalence classes:

$$\begin{array}{cc}\hfill Q:& {\ell }_{23}^1={\ell }_{41}^1={\ell }_{21}^2={\ell }_{34}^2={\ell }_{31}^3={\ell }_{42}^3=+1,{\tilde{\ell }}_{\mathrm{I}\mathrm{J}}^{\widehat{a}}=0\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill \tilde{Q}:& {\tilde{\ell }}_{23}^1={\tilde{\ell }}_{41}^1={\tilde{\ell }}_{31}^2={\tilde{\ell }}_{42}^2={\tilde{\ell }}_{12}^3={\tilde{\ell }}_{43}^3=+1,{\ell }_{\mathrm{I}\mathrm{J}}^{\widehat{a}}=0\hfill \end{array}$$

(53)

These values of ℓ and $\tilde{\ell }$ can be succinctly written as $\mathit{\beta }$ matrices:

$$\begin{array}{cc}\hfill Q:i{\ell }_{\mathrm{I}\mathrm{J}}^1& ={\beta }_{\mathrm{I}\mathrm{J}}^1,i{\ell }_{\mathrm{I}\mathrm{J}}^2={\beta }_{\mathrm{I}\mathrm{J}}^3,i{\ell }_{\mathrm{I}\mathrm{J}}^3={\beta }_{\mathrm{I}\mathrm{J}}^2,{\tilde{\ell }}_{\mathrm{I}\mathrm{J}}^{\widehat{a}}=0\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \tilde{Q}:i{\tilde{\ell }}_{\mathrm{I}\mathrm{J}}^1& ={\beta }_{\mathrm{I}\mathrm{J}}^1,i{\tilde{\ell }}_{\mathrm{I}\mathrm{J}}^2={\beta }_{\mathrm{I}\mathrm{J}}^2,i{\tilde{\ell }}_{\mathrm{I}\mathrm{J}}^3=-{\beta }_{\mathrm{I}\mathrm{J}}^3,{\ell }_{\mathrm{I}\mathrm{J}}^{\widehat{a}}=0.\hfill \end{array}$$

(55)

The gadget between the quaternion adinkras is therefore zero: $\mathcal{G}[(Q),(\tilde{Q})]=0$. Both quaternion adinkras belong to the ${\chi }_0$-equivalence class ${\chi }_0=-1$.

4. The 96 $\tilde{\mathbf{V}}$-Equivalence Classes of Adinkras

We define $B{C}_4$ color transformations as $B{C}_4$ elements acting on the color indices $\mathrm{I},\mathrm{J},\dots$ of the adinkras, $B{C}_4$ boson transformations as $B{C}_4$ elements acting on the bosonic indices $i,j,\dots$ of the adinkras, and $B{C}_4$ fermion transformations as $B{C}_4$ elements acting on the fermionic indices $\widehat{i},\widehat{j},\dots$ of the adinkras.

$$\begin{array}{cc}\hfill B{C}_4\mathrm{color}:{(B{C}_4^{\pm a\mu A})}_{\mathrm{I}}^{\mathrm{J}}=& \pm {\left(H^a{S}_3^{\mu }{\mathcal{V}}^A\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill B{C}_4\mathrm{boson}:{(B{C}_4^{\pm a\mu A})}_i^j=& \pm {\left(H^a{S}_3^{\mu }{\mathcal{V}}^A\right)}_i^j\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill B{C}_4\mathrm{fermion}:{(B{C}_4^{\pm a\mu A})}_{\widehat{i}}^{\widehat{j}}=& \pm {\left(H^a{S}_3^{\mu }{\mathcal{V}}^A\right)}_{\widehat{i}}^{\widehat{j}}\hfill \end{array}$$

(58)

The $B{C}_4$ boson (fermion) transformations act on the adinkra matrices as the $\mathcal{X}$ ($\mathcal{Y}$) matrices in Equation (33). To generate adinkras starting from the quaternion adinkras, any combination of these transformation laws can be used. As the gadget is invariant with respect to $B{C}_4$ boson transformations, we will use these transformation laws and exclude the fermion transformation laws so that the boson transformations will take us around the orbit of the equivalence classes. We will also have to use $B{C}_3$ color transformations to be able to generate all adinkras as the $B{C}_4$ boson does not have enough elements to do so alone.

4.1. Examples

We now show how the representations discussed thus far can be generated from the quaternion adinkras. Consider the following set of calculations that generate the $CM$:

$$\begin{array}{cc}\hfill {(B{C}_4^{+362})}_i^j{(-L_1^{\left(Q\right)})}_j^{\widehat{j}}& ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^j{[-\left(\right)]}_j^{\widehat{j}}\hfill \\ & =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)\hfill \\ & =\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 1& 0& 0\\ 0& 0& -1& 0\end{array}\right)={\left[\left(\overline{24}\right)\left(234\right)\right]}_i^{\widehat{j}}\hfill \\ & ={(L_1^{\left(CM\right)})}_i^{\widehat{j}}\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {(B{C}_4^{+362})}_i^j{(L_2^{\left(Q\right)})}_j^{\widehat{j}}& ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^j{(L_2^{\left(Q\right)})}_j^{\widehat{j}}\hfill \\ & =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)\hfill \\ & =\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right)={\left[\left(\overline{34}\right)\left(132\right)\right]}_i^{\widehat{j}}\hfill \\ & ={(L_2^{\left(CM\right)})}_i^{\widehat{j}}\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {(B{C}_4^{+362})}_i^j{(L_3^{\left(Q\right)})}_j^{\widehat{j}}& ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^j{(L_3^{\left(Q\right)})}_j^{\widehat{j}}\hfill \\ & =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right)\hfill \\ & =\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\end{array}\right)={\left[\left(\overline{23}\right)\left(143\right)\right]}_i^{\widehat{j}}\hfill \\ & ={(L_3^{\left(CM\right)})}_i^{\widehat{j}}\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {(B{C}_4^{+362})}_i^j{(L_4^{\left(Q\right)})}_j^{\widehat{j}}& ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^j{(L_4^{\left(Q\right)})}_j^{\widehat{j}}\hfill \\ & =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right)\hfill \\ & =\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right)={\left(124\right)}_i^{\widehat{j}}\hfill \\ & ={(L_4^{\left(CM\right)})}_i^{\widehat{j}}\hfill \end{array}$$

(62)

Putting this all together, we find that $CM$ is generated from Q via the ${(B{C}_4^{+362})}_i^j={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^j$ boson flip and flop and the ${(B{C}_3^{51})}_{\mathrm{I}}^{\mathrm{J}}={\left(\overline{1}\right)}_{\mathrm{I}}^{\mathrm{J}}$ color flip:

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\left(CM\right)})}_i^{\widehat{j}}=& {(B{C}_4^{+362})}_i^j{(B{C}_3^{51})}_{\mathrm{I}}^{\mathrm{J}}{({\mathrm{L}}_{\mathrm{J}}^{\left(Q\right)})}_j^{\widehat{j}}.\hfill \end{array}$$

(63)

A more succinct derivation of Equation (63) is shown below in permutation notation, where colored text/numbers encode the corresponding free color index. For instance, a row of green in a matrix indicates that that row multiplying a column vector yields the $\mathrm{I}=1$ element, violet $\mathrm{I}=2$, orange $\mathrm{I}=3$, and red $\mathrm{I}=4$.

$$\begin{array}{cc}\hfill {(B{C}_4^{+362})}_i^k{(B{C}_3^{51})}_{\mathrm{I}}^{\mathrm{J}}{({\mathrm{L}}_{\mathrm{J}}^{\left(Q\right)})}_k^{\widehat{j}}& ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^k{\left(\overline{1}\right)}_{\mathrm{I}}^{\mathrm{J}}{(L_{\mathrm{J}}^{\left(Q\right)})}_k^{\widehat{j}}\hfill \\ & ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^k\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -[\left(\overline{23}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & ={\left[\left(\overline{13}\right)\left(234\right)\right]}_i^k\left(\begin{array}{}-{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -[\left(\overline{23}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{}-[\left(\overline{13}\right)\left(234\right){]}_i^{\widehat{j}}\\ [\left(\overline{13}\right)\left(234\right)\left(\overline{13}\right)\left(12\right)\left(34\right){]}_i^{\widehat{j}}\\ -[\left(\overline{13}\right)\left(234\right)\left(\overline{23}\right)\left(13\right)\left(24\right){]}_i^{\widehat{j}}\\ [\left(\overline{13}\right)\left(234\right)\left(\overline{12}\right)\left(14\right)\left(23\right){]}_i^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{}[\left(\overline{24}\right)\left(234\right){]}_i^{\widehat{j}}\\ [\left(\overline{34}\right)\left(132\right){]}_i^{\widehat{j}}\\ [\left(\overline{23}\right)\left(143\right){]}_i^{\widehat{j}}\\ {\left(124\right)}_i^{\widehat{j}}\end{array}\right)\hfill \\ & ={({\mathrm{L}}_{\mathrm{I}}^{\left(CM\right)})}_i^{\widehat{j}}.\hfill \end{array}$$

(64)

Similarly, we can calculate how other representations are generated from Q or $\tilde{Q}$:

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\left(TM\right)})}_i^{\widehat{j}}& ={(B{C}_4^{+553})}_i^j{(B{C}_3^{21})}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_J^{\left(\tilde{Q}\right)})}_j^{\widehat{j}},\hfill \end{array}$$

(65a)

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\left(VM\right)})}_i^{\widehat{j}}& ={(B{C}_4^{-342})}_i^j{(B{C}_3^{41})}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_J^{\left(\tilde{Q}\right)})}_j^{\widehat{j}},\hfill \end{array}$$

(65b)

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\left(cis\right)})}_i^{\widehat{j}}& ={\left(B{C}_4^{+111}\right)}_i^j{\left(B{C}_3^{71}\right)}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_J^{\left(\tilde{Q}\right)})}_j^{\widehat{j}},\hfill \end{array}$$

(65c)

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\left(trans\right)})}_i^{\widehat{j}}& ={\left(B{C}_4^{+111}\right)}_i^j{\left(B{C}_3^{11}\right)}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_J^{\left(\tilde{Q}\right)})}_j^{\widehat{j}}.\hfill \end{array}$$

(65d)

Notice that the representations in Equations (63) and (65) all utilize color flips, but not flops. For an example of color flips and flops and boson flips and flops, we define the representation ${({\tilde{\mathrm{L}}}_{\mathrm{I}}^{-73426})}_i^{\widehat{j}}$ that is generated as:

$$\begin{array}{cc}\hfill {({\tilde{\mathrm{L}}}_{\mathrm{I}}^{-73426})}_i^{\widehat{j}}\equiv & {\left(B{C}_4^{-734}\right)}_i^j{\left(B{C}_3^{26}\right)}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_{\mathrm{J}}^{\left(\tilde{Q}\right)})}_j^{\widehat{j}}.\hfill \end{array}$$

(66)

The details of this calculation are shown below:

$$\begin{array}{cc}\hfill {(B{C}_4^{-734})}_i^k{(B{C}_3^{26})}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_{\mathrm{J}}^{\left(\tilde{Q}\right)})}_k^{\widehat{j}}& =-{\left[\left(\overline{3}\right)\left(1432\right)\right]}_i^k{\left[\left(\overline{12}\right)\left(132\right)\right]}_{\mathrm{I}}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_{\mathrm{J}}^{\left(\tilde{Q}\right)})}_k^{\widehat{j}}\hfill \\ & =-{\left[\left(\overline{3}\right)\left(1432\right)\right]}_i^k\left(\begin{array}{cccc}0& -1& 0& 0\\ 0& 0& -1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{23}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & =-{\left[\left(\overline{3}\right)\left(1432\right)\right]}_i^k\left(\begin{array}{c}-[\left(\overline{23}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -[\left(\overline{12}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ {\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{c}[\left(\overline{3}\right)\left(1432\right)\left(\overline{23}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ [\left(\overline{3}\right)\left(1432\right)\left(\overline{12}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ -{\left[\left(\overline{3}\right)\left(1432\right)\right]}_k^{\widehat{j}}\\ -[\left(\overline{3}\right)\left(1432\right)\left(\overline{13}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{c}[\left(\overline{123}\right)\left(24\right){]}_i^{\widehat{j}}\\ [\left(\overline{134}\right)\left(1234\right){]}_i^{\widehat{j}}\\ [\left(\overline{124}\right)\left(1432\right){]}_i^{\widehat{j}}\\ [\left(\overline{1}\right)\left(13\right){]}_i^{\widehat{j}}\end{array}\right)\hfill \\ & ={({\tilde{\mathrm{L}}}_{\mathrm{I}}^{-73426})}_i^{\widehat{j}}.\hfill \end{array}$$

(67)

The explicit matrices for this representation are

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\tilde{\mathbf{L}}}_1^{-73426}=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\tilde{\mathbf{L}}}_2^{-73426}=\left(\begin{array}{cccc}0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\end{array}\right)\end{array},\hfill \\ \hfill \begin{array}{c}\hfill {\tilde{\mathbf{L}}}_3^{-73426}=\left(\begin{array}{cccc}0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\\ -1& 0& 0& 0\end{array}\right)\end{array}& ,\begin{array}{c}\hfill {\tilde{\mathbf{L}}}_4^{-73426}=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}.\hfill \end{array}$$

(68)

The adinkra for ${\tilde{\mathbf{L}}}_{\mathrm{I}}^{-73426}$ is as in Figure 6.

Now that we have seen how $B{C}_4$ boson × $B{C}_3$ color can generate different adinkra representations from the quaternion adinkras, we will next discuss which transformations leave ${\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)}$ and ${\tilde{\mathbf{L}}}_{\mathrm{I}}^{\left(\tilde{Q}\right)}$ invariant. Said another way, we will discuss the isometries of ${\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)}$ and ${\tilde{\mathbf{L}}}_{\mathrm{I}}^{\left(\tilde{Q}\right)}$. One isometry is demonstrated in the following example.

$$\begin{array}{cc}\hfill {(B{C}_4^{-312})}_i^k{(B{C}_4^{-312})}_{\mathrm{I}}^{\mathrm{J}}{({\mathrm{L}}_{\mathrm{J}}^{\left(Q\right)})}_k^{\widehat{j}}& ={\left[\left(\overline{24}\right)\left(12\right)\left(34\right)\right]}_i^k{\left[\left(\overline{24}\right)\left(12\right)\left(34\right)\right]}_{\mathrm{I}}^{\mathrm{J}}{\left(L_{\mathrm{J}}^{\left(Q\right)}\right)}_k^{\widehat{j}}\hfill \\ & ={\left[\left(\overline{24}\right)\left(12\right)\left(34\right)\right]}_i^k\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)\left(\begin{array}{c}{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -[\left(\overline{23}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & ={\left[\left(\overline{24}\right)\left(12\right)\left(34\right)\right]}_i^k\left(\begin{array}{c}[\left(\overline{13}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\\ [\left(\overline{23}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{c}[\left(\overline{24}\right)\left(12\right)\left(34\right)\left(\overline{13}\right)\left(12\right)\left(34\right){]}_i^{\widehat{j}}\\ -[\left(\overline{24}\right)\left(12\right)\left(34\right){]}_i^{\widehat{j}}\\ [\left(\overline{24}\right)\left(12\right)\left(34\right)\left(\overline{12}\right)\left(14\right)\left(23\right){]}_i^{\widehat{j}}\\ [\left(\overline{24}\right)\left(12\right)\left(34\right)\left(\overline{23}\right)\left(13\right)\left(24\right){]}_i^{\widehat{j}}\end{array}\right)\hfill \\ & =\left(\begin{array}{c}{\left(\right)}_k^{\widehat{j}}\\ [\left(\overline{13}\right)\left(12\right)\left(34\right){]}_k^{\widehat{j}}\\ -[\left(\overline{23}\right)\left(13\right)\left(24\right){]}_k^{\widehat{j}}\\ [\left(\overline{12}\right)\left(14\right)\left(23\right){]}_k^{\widehat{j}}\end{array}\right)\hfill \\ & ={({\mathrm{L}}_{\mathrm{I}}^{\left(Q\right)})}_i^{\widehat{j}}.\hfill \end{array}$$

(69)

There is a two-fold degenerate list of eight such isometry transformations for each quaternion adinkra. The isometries of Q are:

$$\begin{array}{c}{\left(B{C}_4^{+312}\right)}_i^j{\left(B{C}_4^{+312}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-312}\right)}_i^j{\left(B{C}_4^{-312}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{13}{\left)\left(12\right)\left(34\right)\right)}_i{}^j\left(\right(\overline{13}{\left)\left(12\right)\left(34\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{24}{\left)\left(12\right)\left(34\right)\right)}_i{}^j\left(\right(\overline{24}{\left)\left(12\right)\left(34\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(70a)

$$\begin{array}{c}{\left(B{C}_4^{+413}\right)}_i^j{\left(B{C}_4^{+413}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-413}\right)}_i^j{\left(B{C}_4^{-413}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{23}{\left)\left(13\right)\left(24\right)\right)}_i{}^j\left(\right(\overline{23}{\left)\left(13\right)\left(24\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{14}{\left)\left(13\right)\left(24\right)\right)}_i{}^j\left(\right(\overline{14}{\left)\left(13\right)\left(24\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(70b)

$$\begin{array}{c}{\left(B{C}_4^{+214}\right)}_i^j{\left(B{C}_4^{+214}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-214}\right)}_i^j{\left(B{C}_4^{-214}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{12}{\left)\left(14\right)\left(23\right)\right)}_i{}^j\left(\right(\overline{12}{\left)\left(14\right)\left(23\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{34}{\left)\left(14\right)\left(23\right)\right)}_i{}^j\left(\right(\overline{34}{\left)\left(14\right)\left(23\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(70c)

$$\begin{array}{c}{\left(B{C}_4^{+111}\right)}_i^j{\left(B{C}_4^{+111}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-111}\right)}_i^j{\left(B{C}_4^{-111}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ ={\left(\right)}_i{}^j{\left(\right)}_{\mathrm{I}}{}^{\mathrm{J}}={\left(\overline{1234}\right)}_i{}^j{\left(\overline{1234}\right)}_{\mathrm{I}}{}^{\mathrm{J}}.\hfill \end{array}$$

(70d)

The isometries of $\tilde{Q}$ are:

$$\begin{array}{c}{\left(B{C}_4^{+412}\right)}_i^j{\left(B{C}_4^{+312}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-412}\right)}_i^j{\left(B{C}_4^{-312}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{23}{\left)\left(12\right)\left(34\right)\right)}_i{}^j\left(\right(\overline{13}{\left)\left(12\right)\left(34\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{14}{\left)\left(12\right)\left(34\right)\right)}_i{}^j\left(\right(\overline{24}{\left)\left(12\right)\left(34\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(71a)

$$\begin{array}{c}{\left(B{C}_4^{+213}\right)}_i^j{\left(B{C}_4^{-413}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-213}\right)}_i^j{\left(B{C}_4^{+413}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{12}{\left)\left(13\right)\left(24\right)\right)}_i{}^j\left(\right(\overline{14}{\left)\left(13\right)\left(24\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{34}{\left)\left(13\right)\left(24\right)\right)}_i{}^j\left(\right(\overline{23}{\left)\left(13\right)\left(24\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(71b)

$$\begin{array}{c}{\left(B{C}_4^{+314}\right)}_i^j{\left(B{C}_4^{+214}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-314}\right)}_i^j{\left(B{C}_4^{-214}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ =\left(\right(\overline{13}{\left)\left(14\right)\left(23\right)\right)}_i{}^j\left(\right(\overline{12}{\left)\left(14\right)\left(23\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}=\left(\right(\overline{24}{\left)\left(14\right)\left(23\right)\right)}_i{}^j\left(\right(\overline{34}{\left)\left(14\right)\left(23\right)\right)}_{\mathrm{I}}{}^{\mathrm{J}}\hfill \end{array}$$

(71c)

$$\begin{array}{c}{\left(B{C}_4^{+111}\right)}_i^j{\left(B{C}_4^{+111}\right)}_{\mathrm{I}}^{\mathrm{J}}={\left(B{C}_4^{-111}\right)}_i^j{\left(B{C}_4^{-111}\right)}_{\mathrm{I}}^{\mathrm{J}}\hfill \\ ={\left(\right)}_i{}^j{\left(\right)}_{\mathrm{I}}{}^{\mathrm{J}}={\left(\overline{1234}\right)}_i{}^j{\left(\overline{1234}\right)}_{\mathrm{I}}{}^{\mathrm{J}}.\hfill \end{array}$$

(71d)

Notice the two-fold degeneracy in $B{C}_4\mathrm{boson}\times B{C}_4\mathrm{color}$ that stems from the fact that for instance $\left(\overline{13}\right)=-\left(\overline{24}\right)$. Furthermore, the isometry transformations for both Q and $\tilde{Q}$ are all signed elements of the Vierergruppe. This has consequences for the equivalence classes as explained in the next section.

4.2. Flipping and Flopping via $B{C}_4$ Boson × $B{C}_3$ Color: The Group Theory Reason for 36,864 Adinkras

As ${\tilde{\mathbf{V}}}_{\mathrm{I}\mathrm{J}}$ has only color indices and fermionic indices, explicitly ${({\tilde{\mathrm{V}}}_{\mathrm{IJ}})}_{\widehat{i}}{}^{\widehat{j}}$, it is invariant with respect to $B{C}_4$ boson transformations. Owing to its color indices $\mathrm{I},\mathrm{J}$, ${({\tilde{\mathrm{V}}}_{\mathrm{IJ}})}_{\widehat{i}}{}^{\widehat{j}}$ is not invariant with respect to all $B{C}_4$ color transformations, but it will be invariant with respect to some $B{C}_4$ color transformations. For a given adinkra, ${({\tilde{\mathrm{V}}}_{\mathrm{IJ}})}_{\widehat{i}}{}^{\widehat{j}}$ will have an eight-fold set of $B{C}_4$ color isometries stemming from the isometries of the quaternion adinkras, Equations (70) and (71). Recall from the previous section that the isometry transformations for Q and $\tilde{Q}$ were all elements of the signed Vierergruppe. Since the 48-element group $B{C}_3$ is $B{C}_4$ with ±Vierergruppe removed, Equations (42) and (44), there is a total of 96 $\tilde{\mathbf{V}}$-equivalence classes: 48 $B{C}_3$ color transformations of Q and 48 $B{C}_3$ color transformations of $\tilde{Q}$. The 384 $B{C}_4$ boson transformations then fill each $\tilde{\mathbf{V}}$-equivalence classes with its distinct adinkras. This explains the total number of distinct adinkras as $B{C}_4$ boson × $B{C}_3$ color × 2 quaternions: 36,864 $=384\times 48\times 2$.

4.2.1. Flipping and Flopping between Different $\tilde{\mathbf{V}}$-Equivalence Classes via $B{C}_3$ Color

Color flipping and flopping Q and $\tilde{Q}$ via $B{C}_3$ color leaves us with the following set of 96 adinkras:

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{b\nu })}_i{}^{\widehat{j}}& ={(B{C}_3^{b\nu })}_{\mathrm{I}}{}^{\mathrm{J}}{({\mathrm{L}}_{\mathrm{J}}^{\left(Q\right)})}_i{}^{\widehat{j}}\hfill \end{array}$$

(72a)

$$\begin{array}{cc}\hfill {({\tilde{\mathrm{L}}}_{\mathrm{I}}^{b\nu })}_i{}^{\widehat{j}}& ={(B{C}_3^{b\nu })}_{\mathrm{I}}{}^{\mathrm{J}}{({\tilde{\mathrm{L}}}_{\mathrm{J}}^{\left(\tilde{Q}\right)})}_i{}^{\widehat{j}}\hfill \end{array}$$

(72b)

Each is a member of a distinct ℓ- or $\tilde{\ell }$-equivalence class as summarized in Table 1, Table 2, Table 3 and Table 4. Each table below describes 24 ℓ- or $\tilde{\ell }$-equivalence classes of the same ${\chi }_0$-equivalence class. We define the 24 classes in each ℓ-equivalence class table to comprise an isomer-equivalence class (either a trans- or cis-equivalence class), and we define the 24 classes in each $\tilde{\ell }$-equivalence class table to comprise an isometry-tilde-equivalence class (either a trans-tilde- or cis-tilde-equivalence class). We label these isomer- and isomer-tilde-equivalence classes as A, B, $\tilde{A}$, and $\tilde{B}$ for $\mathit{l}\sim \mathit{\alpha }$, $\mathit{l}\sim \mathit{\beta }$, $\tilde{\mathit{l}}\sim \mathit{\alpha }$, or $\tilde{\mathit{l}}\sim \mathit{\beta }$, respectively.

We define the color-parity of a given equivalence class as being odd (even) if there is an odd (even) number of total flips plus flops as listed in the ${\left(B{C}_3^{b\nu }\right)}_{\mathrm{I}}{}^{\mathrm{J}}$ column of Table 1, Table 2, Table 3 and Table 4. Compare for instance the equivalence classes ${\tilde{B}}_6$ and ${\tilde{B}}_{16}$ in Table 4. The equivalence class ${\tilde{B}}_6$ has odd color-parity: two flips encoded by $\left(\overline{12}\right)$ and three flops encoded by $\left(123\right)$ for a total of five flips plus flops. In contrast, the equivalence class ${\tilde{B}}_{16}$ has even color-parity: three flips encode by ($\overline{123}$) and one flop encoded by $\left(12\right)$ for a total of four flips plus flops. Notice that the equivalence classes in each of the Table 1, Table 2, Table 3 and Table 4 are numbered such that Classes 1–12 have odd color-parity and Classes 13–24 have even color-parity. We also point out that the ${\ell }_{\mathrm{IJ}}^{\widehat{a}}$ and ${\tilde{\ell }}_{\mathrm{IJ}}^{\widehat{a}}$ for each equivalence class satisfy:

$$\begin{array}{c}\hfill {\ell }_{\mathrm{IJ}}^{\widehat{a}}=\frac{1}{\phantom{2}2}{\chi }_0{ϵ}_{\mathrm{IJ}}{}^{\mathrm{KL}}{\ell }_{\mathrm{KL}}^{\widehat{a}},{\tilde{\ell }}_{\mathrm{IJ}}^{\widehat{a}}=\frac{1}{\phantom{2}2}{\chi }_0{ϵ}_{\mathrm{IJ}}{}^{\mathrm{KL}}{\tilde{\ell }}_{\mathrm{KL}}^{\widehat{a}}.\end{array}$$

(73)

This is due to the fact that classes A and $\tilde{A}$ with ${\chi }_0=+1$ have $\mathit{l}\sim \mathit{\alpha }$ and $\tilde{\mathit{l}}\sim \mathit{\alpha }$, classes B and $\tilde{B}$ with ${\chi }_0=-1$ have $\mathit{l}\sim \mathit{\beta }$ and $\tilde{\mathit{l}}\sim \mathit{\beta }$, and the fact that the $\mathit{\alpha }$ and $\mathit{\beta }$ matrices satisfy:

$$\begin{array}{c}\hfill {\alpha }_{\mathrm{IJ}}^{\widehat{a}}=+\frac{1}{\phantom{2}2}{ϵ}_{\mathrm{IJ}}{}^{\mathrm{KL}}{\alpha }_{\mathrm{KL}}^{\widehat{a}},{\beta }_{\mathrm{IJ}}^{\widehat{a}}=-\frac{1}{\phantom{2}2}{ϵ}_{\mathrm{IJ}}{}^{\mathrm{KL}}{\beta }_{\mathrm{KL}}^{\widehat{a}}.\end{array}$$

(74)

The above property of the $\mathit{\alpha }$ and $\mathit{\beta }$ matrices can be seen from inspection of their explicit forms in Appendix A.

On the 96 adinkras ${({\mathrm{L}}_{\mathrm{I}}^{b\nu })}_i{}^{\widehat{j}}$ and ${({\tilde{\mathrm{L}}}_{\mathrm{I}}^{b\nu })}_i{}^{\widehat{j}}$ defined in Equations (72), we perform $B{C}_4$ boson transformations generating 384 distinct adinkras within each ℓ- and $\tilde{\ell }$-equivalence class:

$$\begin{array}{cc}\hfill {({\mathrm{L}}_{\mathrm{I}}^{\pm a\mu Ab\nu })}_i{}^{\widehat{j}}=& {(B{C}_4^{\pm a\mu A})}_i{}^j{({\mathrm{L}}_{\mathrm{I}}^{b\nu })}_j{}^{\widehat{j}},{({\tilde{\mathrm{L}}}_{\mathrm{I}}^{\pm a\mu Ab\nu })}_i{}^{\widehat{j}}={(B{C}_4^{\pm a\mu A})}_i{}^j{({\tilde{\mathrm{L}}}_{\mathrm{I}}^{b\nu })}_j{}^{\widehat{j}}.\hfill \end{array}$$

(75)

Equation (75) therefore encodes all 36,864 = $384\times 96$ adinkras. The counting is summarized below:

$$\begin{array}{ccccccccc}\mathrm{index}& \pm & a& \mu & A& b& \nu & \sim & \mathrm{product}\\ \mathrm{count}& 2& 8& 6& 4& 8& 6& 2& \mathrm{36,864}\end{array}$$

The representations we have used as examples throughout this paper, including their ℓ- or $\tilde{\ell }$-equivalence classes, are:

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(Q\right)}& ={\mathbf{L}}_{\mathrm{I}}^{+11111}\in B_1,\hfill \end{array}$$

(76a)

$$\begin{array}{cc}\hfill {\tilde{\mathbf{L}}}_{\mathrm{I}}^{(\tilde{Q})}& ={\mathbf{L}}_{\mathrm{I}}^{\left(trans\right)}={\tilde{\mathbf{L}}}_{\mathrm{I}}^{+11111}\in {\tilde{B}}_1,\hfill \end{array}$$

(76b)

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(cis\right)}& ={\tilde{\mathbf{L}}}_{\mathrm{I}}^{+11171}\in {\tilde{A}}_3,\hfill \end{array}$$

(76c)

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(CM\right)}& ={\mathbf{L}}_{\mathrm{I}}^{+36251}\in A_1,\hfill \end{array}$$

(76d)

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(TM\right)}& ={\tilde{\mathbf{L}}}_{\mathrm{I}}^{+55321}\in {\tilde{B}}_2,\hfill \end{array}$$

(76e)

$$\begin{array}{cc}\hfill {\mathbf{L}}_{\mathrm{I}}^{\left(VM\right)}& ={\tilde{\mathbf{L}}}_{\mathrm{I}}^{-34241}\in {\tilde{B}}_4.\hfill \end{array}$$

(76f)

4.2.2. The One Billion, Three Hundred Fifty Eight Million, Nine Hundred Fifty Four Thousand, Four Hundred Ninety Six Gadgets, Revisited

As first computed in [19], there are 1,358,954,496 = 36,864 × 36,864 possible gadget values amongst the 36,864 adinkras. The majority of these vanish, and both the non-vanishing and vanishing gadgets can now be easily understood in terms of $B{C}_4$ boson × $B{C}_3$ color. For instance, the 384 adinkras that constitute each ℓ- or $\tilde{\ell }$-equivalence class have gadgets with others within their equivalence classes equal to one. Gadgets between different isomer- or isomer-tilde equivalence classes all vanish.

$$\begin{array}{cc}\hfill \mathcal{G}[\left(B_n\right),\left(A_m\right)]=& \mathcal{G}[\left(B_n\right),({\tilde{B}}_m)]=\mathcal{G}[\left(B_n\right),({\tilde{A}}_m)]=0\hfill \end{array}$$

(77a)

$$\begin{array}{cc}\hfill \mathcal{G}[(A_n),({\tilde{A}}_m)]=& \mathcal{G}[(A_n),({\tilde{B}}_m)]=\mathcal{G}[({\tilde{B}}_n),({\tilde{A}}_m)]=0\hfill \end{array}$$

(77b)

The only non-vanishing gadgets are those within the same isomer or isomer-tilde equivalence class, and each takes the same matrix form where for instance the rows go over $A_m$ and the columns go over $A_n$:

$$\begin{array}{cc}\hfill 3\mathcal{G}[\left(A_m\right),\left(A_n\right)]=& \left(\begin{array}{cccccccccccccccccccccccc}3& -& -& -& 0& 0& 0& 0& 0& 0& 0& 0& +& +& -& -& +& -& +& -& -& +& +& -\\ -& 3& -& -& 0& 0& 0& 0& 0& 0& 0& 0& +& +& -& -& -& +& -& +& +& -& -& +\\ -& -& 3& -& 0& 0& 0& 0& 0& 0& 0& 0& -& -& +& +& +& -& +& -& +& -& -& +\\ -& -& -& 3& 0& 0& 0& 0& 0& 0& 0& 0& -& -& +& +& -& +& -& +& -& +& +& -\\ 0& 0& 0& 0& 3& -& -& -& 0& 0& 0& 0& +& -& +& -& -& +& +& -& +& +& -& -\\ 0& 0& 0& 0& -& 3& -& -& 0& 0& 0& 0& -& +& -& +& +& -& -& +& +& +& -& -\\ 0& 0& 0& 0& -& -& 3& -& 0& 0& 0& 0& +& -& +& -& +& -& -& +& -& -& +& +\\ 0& 0& 0& 0& -& -& -& 3& 0& 0& 0& 0& -& +& -& +& -& +& +& -& -& -& +& +\\ 0& 0& 0& 0& 0& 0& 0& 0& 3& -& -& -& -& +& +& -& +& +& -& -& +& -& +& -\\ 0& 0& 0& 0& 0& 0& 0& 0& -& 3& -& -& +& -& -& +& +& +& -& -& -& +& -& +\\ 0& 0& 0& 0& 0& 0& 0& 0& -& -& 3& -& +& -& -& +& -& -& +& +& +& -& +& -\\ 0& 0& 0& 0& 0& 0& 0& 0& -& -& -& 3& -& +& +& -& -& -& +& +& -& +& -& +\\ +& +& -& -& +& -& +& -& -& +& +& -& 3& -& -& -& 0& 0& 0& 0& 0& 0& 0& 0\\ +& +& -& -& -& +& -& +& +& -& -& +& -& 3& -& -& 0& 0& 0& 0& 0& 0& 0& 0\\ -& -& +& +& +& -& +& -& +& -& -& +& -& -& 3& -& 0& 0& 0& 0& 0& 0& 0& 0\\ -& -& +& +& -& +& -& +& -& +& +& -& -& -& -& 3& 0& 0& 0& 0& 0& 0& 0& 0\\ +& -& +& -& -& +& +& -& +& +& -& -& 0& 0& 0& 0& 3& -& -& -& 0& 0& 0& 0\\ -& +& -& +& +& -& -& +& +& +& -& -& 0& 0& 0& 0& -& 3& -& -& 0& 0& 0& 0\\ +& -& +& -& +& -& -& +& -& -& +& +& 0& 0& 0& 0& -& -& 3& -& 0& 0& 0& 0\\ -& +& -& +& -& +& +& -& -& -& +& +& 0& 0& 0& 0& -& -& -& 3& 0& 0& 0& 0\\ -& +& +& -& +& +& -& -& +& -& +& -& 0& 0& 0& 0& 0& 0& 0& 0& 3& -& -& -\\ +& -& -& +& +& +& -& -& -& +& -& +& 0& 0& 0& 0& 0& 0& 0& 0& -& 3& -& -\\ +& -& -& +& -& -& +& +& +& -& +& -& 0& 0& 0& 0& 0& 0& 0& 0& -& -& 3& -\\ -& +& +& -& -& -& +& +& -& +& -& +& 0& 0& 0& 0& 0& 0& 0& 0& -& -& -& 3\end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}[(A_m),(A_n)]=& \mathcal{G}[(B_m),(B_n)]=\mathcal{G}[({\tilde{A}}_m),({\tilde{A}}_n)]=\mathcal{G}[({\tilde{B}}_m),({\tilde{B}}_n)]\hfill \end{array}$$

(78)

The matrix above therefore encodes the possible gadget values within each isomer- and isomer-tilde-equivalence class and shows some interesting color-parity features within each of these classes. Recall from Table 1, Table 2, Table 3 and Table 4 that Class Numbers 1–12 (13–24) have odd (even) color-parity. Keeping this in mind while inspecting the block off-diagonal elements of the gadget matrix above, we notice that even and odd color-parity classes are never orthogonal: they always have an inner product equal to either plus or minus one-third. In fact, the plus one-third gadget only arises between classes of different color-parity. Inspecting instead the block diagonal elements, where the classes have the same color-parity, notice that different even (odd) flops are always orthogonal and different flips paired with the same flop always have an inner product of minus one-third.

In [19], the 1,358,954,496 = 36,864 × 36,864 gadgets between all possible pairings of the 36,864 adinkras were exhaustively calculated. The counting of each is shown in Table 5. Notice that these counts add up to the total number of gadgets: 1,358,954,496. Also shown in Table 5 is the number of times each value appears in each of the matrices in Equations (77) and (78).

Noting that there are $12=2\times 6$ matrices described by Equation (77) (twice the number due to the symmetry of the gadget), four matrices described by Equation (78), and that each matrix entry has a degeneracy of 147,456 $=384\times 384$ (owing to there being 384 adinkras within each ℓ- and $\tilde{\ell }$-equivalence class), we see that the count of each gadget value in Equations (77) and (78) matches that of [19]:

$$\begin{array}{cc}\hfill \mathrm{count}=& (\#\mathrm{OF}\mathrm{entries}\mathrm{in}\mathrm{each}\mathrm{matrix}(77)\times 12\mathrm{matrices}\hfill \\ & +\#\mathrm{OF}\mathrm{entries}\mathrm{in}\mathrm{each}\mathrm{matrix}\left(78\right)\times 4\mathrm{matrices})\times \mathrm{entry}\mathrm{degeneracy}\hfill \end{array}$$

(79a)

$$\begin{array}{cc}\hfill \mathcal{G}[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=0:& 1,132,462,080=(576\times 12+192\times 4)\times 147,456\hfill \end{array}$$

(79b)

$$\begin{array}{cc}\hfill \mathcal{G}[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=-1/3:& 127,401,984=(0\times 12+216\times 4)\times 147,456\hfill \end{array}$$

(79c)

$$\begin{array}{cc}\hfill \mathcal{G}[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=+1/3:& 84,934,656=(0\times 12+144\times 4)\times 147,456\hfill \end{array}$$

(79d)

$$\begin{array}{cc}\hfill \mathcal{G}[\left(\mathcal{R}\right),\left({\mathcal{R}}^{\prime }\right)]=1:& 14,155,776=(0\times 12+24\times 4)\times 147,456\hfill \end{array}$$

(79e)

5. Moving Toward 1D, $\mathit{N}$ = 4 Minimal Valises AND ${\mathit{BC}}_{\mathbf{4}}$ Sigma-Models

The previous sections have been devoted to constructing a streamlined mathematical approach to sorting among the 36,864 adinkras that possess four colors, four closed nodes, and four open nodes. This is a problem in representation theory. At this point of our discussion, we will build on the previous sections’ foundation to engage the application of this foundation to the construction of 1D, N = 4 non-linear sigma-models over the $B{C}_4$ Coxeter group.

5.1. Review of the Discovery of Twisted Reps in Sigma-Models

For 4D, $\mathcal{N}$ = 1 $\sigma$-models [53], there are two ingredients, chiral superfields $\Phi {}^I$ and a Kähler potential $K(\Phi {}^I,\overline{\Phi }{}^I)$, which come together to define a dynamical system via the supersymmetrical action formula:

$$S{}_{\sigma }^{\left(4D\right)}=\int d^{4}x{d}^2\theta d^2\overline{\theta }K(\Phi {}^I,\overline{\Phi }{}^I).$$

(80)

This defines the most general possible 4D, $\mathcal{N}$ = 1 $\sigma$-model. For each choice of $K(\Phi {}^I,\overline{\Phi }{}^I)$ and choice of the range of the superscript I on $\Phi {}^I$, there is a model that is well defined. It is also the case that no other 4D, $\mathcal{N}$ = 1 $\sigma$-models exist. Therefore, there is a type of completeness description implicit in the fact that there is only one 4D, N = 1 minimal superfield representation, which implies that a specification of K completely describes the space of 4D, N = 1 $\sigma$-models. This situation is what we refer to as “control of the model space.”

It is simple to reduce the action above to one where only two of the four spacetime manifold coordinates are retained. One is led to write:

$$S{}_{\sigma }^{\left(2D\right)}=\int d^{2}x{d}^2\theta d^2\overline{\theta }K(\Phi {}^I,\overline{\Phi }{}^I),$$

(81)

but control of the model space is lost. There is nothing wrong with the action above. However, what changes is there exists another scalar supermultiplet in 2D, $\mathcal{N}$ = 2 superspace that does not exist in 4D, $\mathcal{N}$ = 1 superspace.

As was first shown in the works of [54,55], a distinct 2D, $\mathcal{N}$ = 2 supersymmetric scalar multiplet, the so-called “twisted chiral supermultiplet” (denoted by $\chi {}^{\widehat{I}}$ and where the range of the index I may be different from that of the $\widehat{I}$ index) exists in the lower dimension. Thus, modifying the action to the form:

$$S{}_{\sigma }^{\left(2D\right)}=\int d^{2}x{d}^2\theta d^2\overline{\theta }K(\Phi {}^I,\overline{\Phi }{}^I;\chi {}^{\widehat{I}},\overline{\chi }{}^{\widehat{I}}),$$

(82)

with the inclusion of the twisted chiral supermultiplet restores control of the model space for minimal off-shell representations. It can be seen that a Kähler-like potential K still controls the geometry. In the case of the four-dimensional $\mathcal{N}$ = 1 $\sigma$-model, this geometry describes a Riemannian Kähler manifold. In the case of the complete two-dimensional $\mathcal{N}$ = 2 $\sigma$-model, this geometry is a non-Riemannian bi-Hermitian manifold with torsion.

The distinction between two-dimensional $\mathcal{N}$ = 2 $\sigma$-models constructed solely from chiral supermultiplets or solely from twisted chiral supermultiplets in comparison to two-dimensional $\mathcal{N}$ = 2 $\sigma$-models constructed from both chiral supermultiplets and twisted chiral supermultiplets arise from the representation theory fact that the two types of supermultiplets are “usefully inequivalent” [56].

When two-dimensional $\mathcal{N}$ = 2 supermultiplets are reduced to one-dimensional $N=4$ supermultiplets, the distinction between the chiral supermultiplet ($CM$) and twisted chiral supermultiplet ($TCM$) can be seen in their gadget values [19]:

$$\begin{gathered} {\displaystyle \mathcal{G}\left[\left(CM\right),\left(CM\right)\right]=\mathcal{G}\left[\left(TCM\right),\left(TCM\right)\right]=1, \\ \mathcal{G}\left[\left(CM\right),\left(TCM\right)\right]=0.} \end{gathered}$$

(83)

As first discovered in [54,55] and later related to adinkras in [15,16,19], the 2D $\mathcal{N}=2$ twisted chiral multiplet is the dimensionally-reduced 4D, $\mathcal{N}=1$ vector multiplet. We see in comparing Equation (83) to Equation (39) that the gadget keeps track of this relationship between the twisted chiral and vector supermultiplets.

The two-dimensional $\sigma$-model actions can also be reduced to one-dimensional $\sigma$-model actions,

$$S{}_{\sigma }^{\left(1D\right)}=\int d\tau d^2\theta d^2\overline{\theta }K(\Phi {}^I,\overline{\Phi }{}^I;\chi {}^{\widehat{I}},\overline{\chi }{}^{\widehat{I}}),$$

(84)

and the works of [36,37,38] in principle capture all of these (as we only consider valise supermultiplets, the chiral and twisted chiral superfields in (84) correspond to starting with their 4D progenitors where both auxiliary fields have been replaced by three-forms). However, here, it is useful to recall the experience of the reduction from 4D, $\mathcal{N}$ = 1 $\sigma$-models to 2D, $\mathcal{N}$ = 2 $\sigma$-models. The loss of control of the model space came about because the actions and supermultiplets that appear in them are “blind” to the appearance of ‘new’ supermultiplets that can result from the reduction process.

In order to demonstrate the loss once more, it is useful to show the reduction from 2D, $\mathcal{N}$ = 2 supersymmetry to first consider the intermediate step where we consider 2D, $\mathcal{N}$ = (4,0) supersymmetry. This will allow the explicit demonstration of the emergence of new supermultiplets in the intermediate step. Thus, any further reduction to 1D, $\mathcal{N}$ = (4,0) supersymmetry must inherit these supermultiplets from 2D, $\mathcal{N}$ = (4,0) supersymmetry.

5.2. 2D, $\mathcal{N}$ = (4,0) Supersymmetry Considerations

We introduce the bosonic coordinates for the worldsheet $\tau$ and $\sigma$ assembled into light cone coordinates ${\sigma }^{⧧}$ and ${\sigma }^{=}$ such that:

$$d{\sigma }^{⧧}d{\sigma }^{=}=\left(d\tau \right){}^2-{\left(d\sigma \right)}^2.$$

(85)

For (4,0) superspace, four Grassmann coordinates correspond to the + component of spinor helicity with regard to the worldsheet Lorentz group. Therefore, we have:

$${\zeta }^{+i}=\left[\begin{array}{c}{\zeta }^{+1}\\ {\zeta }^{+2}\end{array}\right].$$

(86)

As the Grassmann coordinates are complex, the “isospin” indices, denoted by i, j, … etc.) may be regarded as describing an internal su(2) symmetry.

Finally, we introduce the superspace “covariant derivatives”:

$$D_{+i}=\left(\frac{\partial }{\partial {\zeta }^{+i}}+i\frac{1}{\phantom{2}2}\overline{\zeta }{}^{+}{}_i\partial {}_{⧧}\right),{\overline{D}}_{+}^i=\left(\frac{\partial }{\partial \overline{\zeta }{}^{+}{}_i}+i\frac{1}{\phantom{2}2}\zeta {}^{+i}\partial {}_{⧧}\right),$$

(87)

together with the light cone derivatives $\partial {}_{⧧}$ and $\partial {}_{=}$ to describe the tangent space to the supermanifold. These definitions ensure the equations:

$$\{D_{+}{}_i,{\overline{D}}_{+}^j\}=i\delta {}_i{}^j{\partial }_{⧧}.$$

(88)

5.3. Reviewing the Known 2D, $\mathcal{N}$ = (4, 0) Minimal Scalar Valise Supermultiplets

Many years ago [57,58], it was demonstrated that there is a minimum of four distinct (4, 0) valise supermultiplets each containing four bosons and four fermions. Thus, one can introduce a “representation label” $\left(\mathcal{R}\right)$, which takes on four values denoted by SM-I, SM-II, SM-III, and SM-IV. The field content of each is shown in Table 6. All fields with two such indices are traceless. The bosons are $\mathcal{A},\mathcal{B},\varphi ,{\varphi }_i{}^j,{\mathcal{A}}_i$, and ${\mathcal{B}}_i$, and of these, only $\varphi$ and ${\varphi }_i{}^j$ are real (or Hermitian).

Regarding the su(2) symmetry, the supercovariant derivatives, the bosonic fields, and the fermionic fields are distributed among different irreducible representations, as shown in Table 7.

It is noteworthy that the first two of these supermultiplets (i.e., SM-I and SM-II) can be interpreted as arising from a dimensional reduction process applied to the well-known 4D, $\mathcal{N}$ = 1 “chiral supermultiplet” and “vector supermultiplet”, respectively. With respect to the other two supermultiplets, there are no discussions known to us that indicate they can arise solely as the results of such dimensional reductions. The SM-III and SM-IV supermultiplets, however, are related respectively to the SM-I and SM-II supermultiplets by a Klein transformation, where bosonic fields and fermionic fields are exchanged one for the other.

These fields may be interpreted in two ways. In the first interpretation, all these are component fields that are functions solely of light cone coordinates ${\sigma }^{⧧}$ and ${\sigma }^{=}$. In the second interpretation, these are each regarded as the lowest component of a corresponding superfield in the expansion over the basis of the superspace Grassmann coordinates.

The “D-algebra” or “SUSY transformation law” for each supermultiplet is given in Equations (89)–(92), which follow.

$\mathrm{SM}-\mathrm{I}\mathrm{Supermultiplet}:(\mathcal{A},\mathcal{B},{\psi }^{-i})$

$${\displaystyle \begin{array}{c}{D}_{+i}\mathcal{A}=2C_{ij}{\psi }^{-}{}^j,{\overline{D}}_{+}{}^i\mathcal{A}=0,\hfill \\ {\overline{D}}_{+}{}^i\mathcal{B}=i2{\psi }^{-}{}^i,D_{+i}\mathcal{B}=0,\hfill \\ {\overline{D}}_{+}{}^i{\psi }^{-}{}^j=i\frac{1}{\phantom{2}2}{C}^{ij}{\partial }_{⧧}\mathcal{A},D_{+i}{\psi }^{-}{}^j=\frac{1}{\phantom{2}2}{\delta }_i{}^j{\partial }_{⧧}\mathcal{B},\hfill \\ {\overline{D}}_{+}{}^i\overline{\mathcal{A}}=-2C^{ij}\overline{\psi }{}^{-}{}_j,D_{+}{}_i\overline{\mathcal{A}}=0,\hfill \\ D_{+}{}_i\overline{\mathcal{B}}=i2\overline{\psi }{}^{-}{}_i,{\overline{D}}_{+}{}^i\overline{\mathcal{B}}=0,\hfill \\ D_{+}{}_i\overline{\psi }{}^{-}{}_j=-i\frac{1}{\phantom{2}2}{C}_{ij}{\partial }_{⧧}\overline{\mathcal{A}},\overline{D}{}_{+}{}^i{\overline{\psi }}^{-}{}_j={\delta }_i{}^j\frac{1}{\phantom{2}2}{\partial }_{⧧}\overline{\mathcal{B}},\hfill \end{array}}$$

(89)

$\mathrm{SM}-\mathrm{II}\mathrm{Supermultiplet}:(\varphi ,{\varphi }_j{}^k,{\lambda }^{-}{}_j)$

$${\displaystyle \begin{array}{c}{D}_{+i}\varphi =i{\lambda }^{-}{}_i,{\overline{D}}_{+}{}^i\varphi =i{\overline{\lambda }}^{-}{}^i,\hfill \\ D_{+i}{\varphi }_j{}^k=2{\delta }_i{}^k{\lambda }^{-}{}_j-{\delta }_j{}^k{\lambda }^{-}{}_i,\overline{D}{}_{+}{}^i{\varphi }_j{}^k=-2{\delta }_j{}^i{\overline{\lambda }}^{-}{}^k+{\delta }_j{}^k\overline{\lambda }{}^{-}{}^i,\hfill \\ D_{+}{}_i{\overline{\lambda }}^{-}{}^j=\frac{1}{\phantom{2}2}{\delta }_j{}^i{\partial }_{⧧}\varphi -i\frac{1}{\phantom{2}2}{\partial }_{⧧}{\varphi }_j{}^i,D_{+i}{\lambda }^{-}{}_j=0,\hfill \\ {\overline{D}}_{+}{}^i{\lambda }^{-}{}_j=\frac{1}{\phantom{2}2}{\delta }_j{}^i{\partial }_{⧧}\varphi +i\frac{1}{\phantom{2}2}{\partial }_{⧧}{\varphi }_j{}^i,{\overline{D}}_{+}{}^i{\overline{\lambda }}^{-}{}^j=0,\hfill \\ \varphi ={\varphi }^{*},{\varphi }_i{}^j={\left({\varphi }_j{}^i\right)}^{*},{\varphi }_i{}^i=0,\hfill \end{array}}$$

(90)

$\mathrm{SM}-\mathrm{III}\mathrm{Supermultiplet}:({\mathcal{A}}_i,{\pi }^{-},{\rho }^{-})$

$${\displaystyle \begin{array}{c}{D}_{+i}{\mathcal{A}}_j=C_{ij}{\pi }^{-},{\overline{D}}_{+}{}^i{\mathcal{A}}_j={\delta }_j{}^i{\rho }^{-},\hfill \\ {\overline{D}}_{+}{}^i{\overline{\mathcal{A}}}^j=-C^{ij}{\overline{\pi }}^{-},D_{+i}{\overline{\mathcal{A}}}^j=-{\delta }_i{}^j{\overline{\rho }}^{-},\hfill \\ D_{+i}{\rho }^{-}=i{\partial }_{⧧}{\mathcal{A}}_i,{\overline{D}}_{+}{}^i{\rho }^{-}=0,\hfill \\ {\overline{D}}_{+}{}^i{\overline{\rho }}^{-}=-i{\partial }_{⧧}{\overline{\mathcal{A}}}^i,D_{+}{}_i{\overline{\rho }}^{-}=0,\hfill \\ {\overline{D}}_{+}{}^i{\pi }^{-}=i{C}^{ij}{\partial }_{⧧}{\mathcal{A}}_j,D_{+i}{\pi }^{-}=0,\hfill \\ D_{+}{}_i{\overline{\pi }}^{-}=-i{C}_{ij}{\partial }_{⧧}{\overline{\mathcal{A}}}^j,{\overline{D}}_{+}{}^i{\overline{\pi }}^{-}=0,\hfill \end{array}}$$

(91)

$\mathrm{SM}-\mathrm{IV}\mathrm{Supermultiplet}:({\mathcal{B}}_i,{\psi }^{-},{\psi }^{-}{}_i{}^j)$

$${\displaystyle \begin{array}{c}{\overline{D}}_{+}{}^i{\mathcal{B}}_j={\delta }_j{}^i{\psi }^{-}+i2{\psi }^{-}{}_j{}^i,D_{+i}{\mathcal{B}}_j=0,\hfill \\ D_{+}{}_i{\overline{\mathcal{B}}}^j=-{\delta }_i{}^j{\psi }^{-}+i2{\psi }^{-}{}_i{}^j,{\overline{D}}_{+}{}^i{\overline{\mathcal{B}}}^j=0,\hfill \\ D_{+i}{\psi }^{-}=i\frac{1}{\phantom{2}2}{\partial }_{⧧}{\mathcal{B}}_i,D_{+i}{\psi }^{-}{}_j{}^k=\frac{1}{\phantom{2}2}{\delta }_i{}^k{\partial }_{⧧}{\mathcal{B}}_j-\frac{1}{\phantom{2}4}{\delta }_j{}^k{\partial }_{⧧}{\mathcal{B}}_i,\hfill \\ {\overline{D}}_{+}{}^i{\overline{\psi }}^{-}=-i\frac{1}{\phantom{2}2}{\partial }_{⧧}{\overline{\mathcal{B}}}^i,{\overline{D}}_{+}{}^i{\psi }^{-}{}_j{}^k=\frac{1}{\phantom{2}2}{\delta }_j{}^i{\partial }_{⧧}{\overline{\mathcal{B}}}^k-\frac{1}{\phantom{2}4}{\delta }_j{}^k{\partial }_{⧧}{\overline{\mathcal{B}}}^i,\hfill \\ \psi {}^{-}={\left(\psi {}^{-}\right)}^{*},{\psi }^{-}{}_i{}^j={\left({\psi }^{-}{}_j{}^i\right)}^{*},\psi {}^{-}{}_i{}^i=0.\hfill \end{array}}$$

(92)

The fermionic holoraumy for each of these multiplets is given in Appendix C.

5.4. Uniformization via Real Formulations

In order to calculate the values of the first gadget, in the context of these $(4,0)$ supermultiplets described previously, it is necessary to convert all their descriptions into a real basis where the comparison process can be made in the simplest possible manner. In particular, the first step is to obtain the “L-matrices” and “R-matrices” associated with each of the four representations: SM-I, SM-II, SM-III, and SM-IV. As was emphasized in the Appendix of the work in [59], “L-matrices” and “R-matrices” can be identified in dimensions greater than one. In this particular example, the “L-matrices” and “R-matrices” were shown for superspaces with three bosonic coordinates.

To begin, we note that for the operator $D_{+}{}_i$, we can write:

$$D_{+}{}_i=\left[\begin{array}{c}{D}_{+}{}_1\\ D_{+}{}_2\end{array}\right],D_{+}{}_1=({\mathrm{D}}_{+\widehat{A}}-i{\mathrm{D}}_{+\widehat{B}}),D_{+}{}_2=({\mathrm{D}}_{+\widehat{C}}-i{\mathrm{D}}_{+\widehat{D}}),$$

(93)

where the four “supercovariant derivatives” are defined with respect to the four real (Majorana) spinor coordinates for the $(4,0)$ superspace ${\zeta }^{+\widehat{A}}$, ${\zeta }^{+\widehat{B}}$, ${\zeta }^{+\widehat{C}}$, and ${\zeta }^{+D}$. It is important to note that the labels $\widehat{A}$, $\widehat{B}$, $\widehat{C}$, and $\widehat{D}$ are fixed values, not indices that take on different values.

Taking the complex conjugate of the results in (93), we find:

$${\overline{D}}_{+}{}^i=\left[\begin{array}{c}{\overline{D}}_{+}{}^1\\ {\overline{D}}_{+}{}^2\end{array}\right],{\overline{D}}_{+}{}^1=({\mathrm{D}}_{+\widehat{A}}+i{\mathrm{D}}_{+\widehat{B}}),{\overline{D}}_{+}{}^2=({\mathrm{D}}_{+\widehat{C}}+i{\mathrm{D}}_{+\widehat{D}}),$$

(94)

which imply also the validity of the complex conjugate version of (88). Together, (93) and (94) imply:

$${\displaystyle \begin{array}{c}{\mathrm{D}}_{+\widehat{A}}=\left(D_{+}{}_1+{\overline{D}}_{+}{}^1\right),\hfill \\ {\mathrm{D}}_{+\widehat{B}}=i\left(D_{+}{}_1-{\overline{D}}_{+}{}^1\right),\hfill \\ {\mathrm{D}}_{+\widehat{C}}=\left(D_{+}{}_2+{\overline{D}}_{+}{}^2\right),\hfill \\ {\mathrm{D}}_{+\widehat{D}}=i\left(D_{+}{}_2-{\overline{D}}_{+}{}^2\right),\hfill \end{array}}$$

(95)

which in turn allows the definition of a real (4, 0) superspace covariant derivative through the equation:

$${\displaystyle \mathrm{D}{}_{+{}_{\mathrm{I}}}=\left[\begin{array}{c}{\mathrm{D}}_{+\widehat{A}}\\ {\mathrm{D}}_{+\widehat{B}}\\ {\mathrm{D}}_{+\widehat{C}}\\ -{\mathrm{D}}_{+\widehat{D}}\end{array}\right]=\left[\begin{array}{c}\left(D_{+}{}_1+{\overline{D}}_{+}{}^1\right)\\ i\left(D_{+}{}_1-{\overline{D}}_{+}{}^1\right)\\ \left(D_{+}{}_2+{\overline{D}}_{+}{}^2\right)\\ -i\left(D_{+}{}_2-{\overline{D}}_{+}{}^2\right)\end{array}\right],}$$

(96)

where subscript index I takes on the four fixed values $\widehat{A}$, … $\widehat{D}$. This definition implies that the equation:

$$[\mathrm{D}{}_{+{}_{\mathrm{I}}},\mathrm{D}{}_{+{}_{\mathrm{J}}}\}=i2\delta {}_{{}_{\mathrm{I}}{}_{\mathrm{J}}}{\partial }_{⧧},$$

(97)

is satisfied.

Let us also highlight that in (96), there is a notational device introduced. The quartet Majorana supercovariant derivative operator is denoted by $\mathrm{D}{}_{+{}_{\mathrm{I}}}$, whereas the complex SU(2)-doublet supercovariant derivative operator is denoted by the pair $(D_{+}{}_i$, ${\overline{D}}_{+}{}^i)$. Thus, Equation (96) solidifies the definition of the Majorana supercovariant derivative operator. The next step is also do this for all fields.

For the bosonic and fermionic fields in each supermultiplet, we define:

$${\displaystyle {\Phi }_i^{(\mathrm{SM}-\mathrm{I})}=\frac{1}{\phantom{2}2}\left[\begin{array}{c}\left(\mathcal{B}+\overline{\mathcal{B}}\right)\\ +i\left(\mathcal{A}-\overline{\mathcal{A}}\right)\\ -i\left(\mathcal{B}-\overline{\mathcal{B}}\right)\\ \left(\mathcal{A}+\overline{\mathcal{A}}\right)\end{array}\right],{\Psi }_{+\widehat{k}}^{(\mathrm{SM}-\mathrm{I})}=\left[\begin{array}{c}({\overline{\psi }}^{-}{}_1+{\psi }^{-}{}^1)\\ i({\overline{\psi }}^{-}{}_1-{\psi }^{-}{}^1)\\ ({\overline{\psi }}^{-}{}_2+{\psi }^{-}{}^2)\\ -i({\overline{\psi }}^{-}{}_2-{\psi }^{-}{}^2)\end{array}\right],}$$

(98)

$${\displaystyle {\Phi }_i^{(\mathrm{SM}-\mathrm{II})}=\left[\begin{array}{c}\varphi {}_1{}^1\\ \frac{1}{\phantom{2}2}[\varphi {}_1{}^2+{\left(\varphi {}_1{}^2\right)}^{*}]\\ \varphi \\ i\frac{1}{\phantom{2}2}[\varphi {}_1{}^2-{\left(\varphi {}_1{}^2\right)}^{*}]\end{array}\right],{\Psi }_{+\widehat{k}}^{(\mathrm{SM}-\mathrm{II})}=\left[\begin{array}{c}({\lambda }^{-}{}_1+{\overline{\lambda }}^{-}{}^1)\\ -i({\lambda }^{-}{}_1-{\overline{\lambda }}^{-}{}^1)\\ ({\lambda }^{-}{}_2+{\overline{\lambda }}^{-}{}^2)\\ i({\lambda }^{-}{}_2-{\overline{\lambda }}^{-}{}^2)\end{array}\right],}$$

(99)

$${\displaystyle {\Phi }_i^{(\mathrm{SM}-\mathrm{III})}=\left[\begin{array}{c}(\mathcal{A}{}_1+\overline{\mathcal{A}}{}^1)\\ i(\mathcal{A}{}_1-\overline{\mathcal{A}}{}^1)\\ (\mathcal{A}{}_2+\overline{\mathcal{A}}{}^2)\\ i(\mathcal{A}{}_2-\overline{\mathcal{A}}{}^2)\end{array}\right],{\Psi }_{+\widehat{k}}^{(\mathrm{SM}-\mathrm{III})}=\frac{1}{\phantom{2}2}\left[\begin{array}{c}\left({\pi }^{-}+{\overline{\pi }}^{-}\right)\\ i\left({\rho }^{-}-{\overline{\rho }}^{-}\right)\\ i\left({\pi }^{-}-{\overline{\pi }}^{-}\right)\\ \left({\rho }^{-}+{\overline{\rho }}^{-}\right)\end{array}\right],}$$

(100)

$${\displaystyle {\Phi }_i^{(\mathrm{SM}-\mathrm{IV})}=\frac{1}{\phantom{2}2}\left[\begin{array}{c}(\mathcal{B}{}_1+\overline{\mathcal{B}}{}^1)\\ i(\mathcal{B}{}_1-\overline{\mathcal{B}}{}^1)\\ (\mathcal{B}{}_2+\overline{\mathcal{B}}{}^2)\\ i(\mathcal{B}{}_2-\overline{\mathcal{B}}{}^2)\end{array}\right],{\Psi }_{+\widehat{k}}^{(\mathrm{SM}-\mathrm{IV})}=\left[\begin{array}{c}-2{\psi }^{-}{}_1{}^1\\ [{\psi }^{-}{}_1{}^2+{\left({\psi }^{-}{}_1{}^2\right)}^{*}]\\ {\psi }^{-}\\ i[{\psi }^{-}{}_1{}^2-{\left({\psi }^{-}{}_1{}^2\right)}^{*}]\end{array}\right].}$$

(101)

Therefore, when all of the bosons and fermions in Equations (89)–(92) are expressed in terms of real quartets of functions as in (98)–(101) and the Majorana supercovariant derivative in (96) is used, they universally possess SUSY transformation laws in the form:

$${\displaystyle \mathrm{D}{}_{+{}_{\mathrm{I}}}{\Phi }_i^{\left(\mathcal{R}\right)}=i\left(\mathrm{L}{}_{{}_{\mathrm{I}}}^{\left(\mathcal{R}\right)}\right){}_{i\widehat{k}}{\Psi }_{+\widehat{k}}^{\left(\mathcal{R}\right)},\mathrm{D}{}_{+{}_{\mathrm{I}}}{\Psi }_{+\widehat{k}}^{\left(\mathcal{R}\right)}=\left(\mathrm{R}{}_{{}_{\mathrm{I}}}^{\left(\mathcal{R}\right)}\right){}_{\widehat{k}i}{\partial }_{⧧}{\Phi }_i^{\left(\mathcal{R}\right)}.}$$

(102)

In the expression (102), ${\Phi }_i^{\left(\mathcal{R}\right)}$ denotes the i^th boson associated with the $\left(\mathcal{R}\right)$-th (4, 0) supermultiplet and ${\Psi }_{+\widehat{k}}^{\left(\mathcal{R}\right)}$ denotes the $\widehat{k}$-th fermion associated with the $\left(\mathcal{R}\right)$-th supermultiplet, and the explicit forms of the matrices ${\mathbf{L}}_{\mathrm{I}}{}^{\left(\mathcal{R}\right)}$ and ${\mathbf{R}}_{\mathrm{I}}{}^{\left(\mathcal{R}\right)}$ depend on the value of $\left(\mathcal{R}\right)$. For all representations, they satisfy:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathrm{I}}^{\left(\mathcal{R}\right)}={({\mathbf{L}}_{\mathrm{I}}^{\left(\mathcal{R}\right)})}^{-1}={({\mathbf{L}}_{\mathrm{I}}^{\left(\mathcal{R}\right)})}^T.\end{array}$$

(103)

$$\begin{array}{cc}\hfill {\mathbf{L}}_1^{(\mathrm{SM}-\mathrm{I})}=& \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 1& 0& 0\\ 0& 0& -1& 0\end{array}\right),{\mathbf{L}}_2^{(\mathrm{SM}-\mathrm{I})}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right),\hfill \\ \hfill {\mathbf{L}}_3^{(\mathrm{SM}-\mathrm{I})}=& \left(\begin{array}{cccc}0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& -\end{array}\right)\hfill \end{array}$$