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

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


Introduction
Adinkras are diagrams that encode supersymmetric (SUSY) transformation laws with complete fidelity in one space-time dimension and in two space-time 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 noteably 4D, 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, N = 4 Super Yang-Mills theory. The utility of adinkras in analyzing extended supersymmetric systems was demonstrated in [5] where the adinkra parameter χ 0 was used to classify which 4D, N = 2 offshell supersymmetric systems can be represented with finite numbers of auxilliary field and which can not. Third, adinkras relate supersymmetric systems that exist in different dimensions, possibly providing a holographic path to uncover unknown represenations. Adinkras can be "shadows" of higher dimensional supersymmetry where an adinkra can be drawn which encodes the entire transformation laws when the system is considered to depend only on one or two of the space-time dimensions.
Classifying supersymmetric systems in terms of which adinkras they reduce to is known as supersymmetric genomics [6,7,8] where as 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 have a normalized inner product of one. As shown in [12,16], a set of holoraumy matricesṼ IJ can be constructed from the transformation laws encoded by the adinkra. These holoraumy matrices exist in a space spanned by six basis elements αâ and βâ. The basis elements αâ and βâ 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 identicalṼ IJ 's. Following [19], we define holoraumy-equivalence classes of adinkras whose inner products equal one. We organize our results in terms of BC 3 and BC 4 : the signed permutation groups of three and four elements, respectively. The relationships between the different holoraumyequivalence classes are presented in terms of BC 3 color transformations that map one equivalence class to another. The main results of this paper are 1. the discovery of the BC 3 mappings to allṼ-equivalence classes, 2. the generation of all 36,864 four color, four boson and four fermion adinkras in terms of BC 4 boson × BC 3 color transformations of twoṼ-inequivalent adinkras dubbed the quaternion adinkras 3. the presentation of a formula that encodes all 36,864 four color, four boson and four fermion adinkras in terms of theirṼ-equivalence classes, 4. the explanation of the count of all possible gadget values in terms of equivalence classes, and 5. 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 upon 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 world line 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 σ-models.
The topic of one dimensional N = 4 σ-models has been developed into a substantial literature [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47] since the time of the work in [22]. Parts of this work has 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 WDVV equations and roots, and (g.) Hyper-Kähler, and Clifford Kähler geometries with torsion. The reader should keep in mind 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 use of harmonic superspace techniques [32,33,34], to describe all such models. Such a universal formalism is what we mean by use of the term "model space." We next review the issue of control of the model space in the more familiar context of 4D, N = 1, 2D, N = 2, and 2D, N = (2, 0) σ-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 BC 4 Coxeter Group non-linear σ-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 howṼ-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 96Ṽ-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 sections 5 and 6 we demonstrate connections between holoraumy and dynamics. Specifically, in section 5 we demonstrate how the holoraumy for four different 2D SUSY sigma models are all different, thus have 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 sections 5 and 6 will be particularly important as we continue our quest to develop SUSY holography.

Adinkra Review
In this section we review how adinkras are used to encode supersymmetry entirely and 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 (γ µ ) ab .

Adinkras and One-Dimensional Supersymmetry
Consider four bosonic fields Φ i and four fermionic fields Ψĵ in one dimension, that of time. The Lagrangian for these fields is where a dot above a field indicates a time derivative,Φ i = dΦ i /dτ for example. We now explain how the two adinkras shown in Fig. 1 each encode a set of SUSY transformation laws that leave the Lagrangian (2.1) invariant. The white nodes of the adinkras encode the bosons Φ i , and the black nodes encode the fermions mulitplied by the imaginary number iΨĵ. 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 Fig. 1, we define χ 0 = +1 for the leftmost adinkra (the cis-adinkra) and χ 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: For fermion to boson we have: Adinkras such as those in Fig. 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 with R I and L I inverses and transposes of each other The adinkra matrices L I and R I satisfy the GR(d, N ) algebra also known as the garden algebra: The 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 Fig. 1 Using Eq. (2.7) we concisely write the matrices L I and R I for the transformation laws in Eqs. (2.2) and (2.3) for the cis-and trans-adinkra as A SUSY transformation between two fields is encoded as a nonzero entry in the corresponding row and column of each of the corresponding L and R matrices. A solid (dashed) line in an adinkra encodes a plus (minus) one in the corresponding matrix entry. The only difference between the cisand trans-valise adinkras in fig. 1 are the orange lines: where they are dashed in one adinkra, they are solid in the other, hence matrices L 1 , L 2 , and L 4 are identical but L 3 is opposite for the cis-and trans-valise adinkras. For an arbitrary d = 4, N = 4 valise adinkra, we define χ 0 off the following chromocharacter equation [6] T r( We have for all d = 4, N = 4 valise adinkras χ 0 = ±1. Here we define the χ 0 -equivalence class: Two adinkras are χ 0 -equivalent if they share the same value of χ 0 .

Adinkras and Four-Dimensional Supersymmetry
In this section we will describe how adinkras can partially encode SUSY transformations in fourdimensions. All valise adinkras belong to either the χ 0 = +1 or χ 0 = −1 equivalence class. In [6], it was shown that the cis-and trans-valise adinkras can encode the 0-brane reduced transformation laws (the transformation laws when only temporal dependence of the fields is considered) of the 4D, N = 1 chiral multiplet (CM ), tensor multiplet (T M ), and vector multiplet (V M ). In terms of χ 0 equivalence classes, we have specifically that χ 0 = +1 for CM and χ 0 = −1 for V M and T M . This first step of simply identifying which type of fields are described by an 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 0-brane as shown in Figs. 2, 3, and 4 for the CM , V M , and T M . 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 0-brane to 4D that exist (namely that time-dependent parts), we will also demonstrate that the χ 0 -equivalence class can be used to partially separate 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 Figs. 2, 3, and 4 encode the transformation laws in Eq. (2.4) with the following node identifications and adinkra matrices L I and R I . The node identifications for each adinkra are The L I for the CM , T M , and V M adinkras are shown below. Note that the R I are the transposes of these as in Eq. (2.5).
The Lagrangian for the CM , V M , and T M are, in terms of their Φ i and Ψ i , all the same: Eq. (2.1). Substituting in the nodal definitions Eqs. (2.13), (2.14), and (2.15), we arrive at their respective Lagrangians and transformation laws where we start to see differences: Again, we are using the gamma matrix conventions of [6]. The corresponding transformation laws for the CM are The corresponding transformation laws for the T M are (2.23) where early latin indices a, b, · · · = 1, 2, 3, 4 and late latin indices m, n, · · · = 1, 2, 3. Also, we define B mn = −B nm . The corresponding transformation laws for the V M are (2.24) 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 0-brane Lagrangians such as (2.19), (2.20), (2.21) and their corresponding transformation laws (2.22), (2.23), and (2.24) to arrive at full 4D, 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 0-brane transformation laws and Lagrangians. We introduce greek indices µ, ν, · · · = 0, 1, 2, 3 where for the CM we have to merely introduce more derivatives on the gauge fields, and more derivatives and gamma matrices for the fermions: For the T M and V M , 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 For the V M we have The preceding illustrates four missing steps in the SUSY holography procedure.
1. Nodal field definitions and which adinkras encode which particle spins.
2. How to augment with spatial derivatives.
3. How to introduce gamma matrices.
4. 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 γ 0 ∂ τ → γ µ ∂ µ . This is related to the third point about γ µ matrices which was addressed in [15,16] where a correspondence between the 0-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 χ 0 -equivalence class. Performing the calculation in Eq. (2.12) on the adinkra L I -matrices for the CM , V M , and T M result in χ 0 = +1 for the CM and χ 0 = −1 for the V M and T M . The CM is therefore in a different χ 0 -equivalence class from the V M and T M . Furthermore, the nodes of the chiral, tensor, and vector valise adinkras in Figs. 2, 3, and 4 can be rearranged to arrive at a different adinkras that equivalently encodes 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 R to a new representation R corresponds to multiplication by matrices X and Y as in: (2.33) As Eq. (2.12) is invariant with respect to such nodal flips and flops, they leave the adinkra in the same χ 0 -equivalence class. Both the tensor and vector adinkras can be node flopped and flipped to the trans-adinkra (χ 0 = −1) in Fig. 1 and the chiral adinkra can be node flopped and flipped to the cis-adinkra (χ 0 = +1) in Fig. 1 [6]. The χ 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.

V-andṼ-equivalence Classes
Once χ 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 V IJ andṼ IJ were seen to do just that 6 The matrices αâ and βâ are given in appendix A.
Note that by construction â IJ = − â JI and similarly for˜ , κ,andκ. For the CM , V M , and T M adinkras in Eqs. (2.16), (2.17), and (2.18) the only independent, non-vanishingṼ IJ coefficients are: Once V andṼ are calculated for two adinkra representations R and R , their gadgets can be computed as 7 For the chiral, tensor, and vector multiplets, we have the gadgets [12,16,18,19] Gadgets are thought of as inner products in a κ,κ, , and˜ space that is normalized to one for identical V andṼ's. We conclude that the chiral, vector, and tensor multiplets are all in separatẽ V-equivalence classes.
Generalizing, we define two adinkras to be V-equivalent orṼ-equivalent if they have identical V's orṼ's, respectively. The set of all adinkras that areṼ-equivalent (V-equivalent) comprise aṼequivalence class (V-equivalence class). We separate V-equivalence classes into κ-andκ-equivalence classes and separateṼ-equivalence classes into and˜ -equivalence classes. Two adinkras are κequivalent if and only if their κ coefficients are identical, and theirκ coefficients vanish. Similarly, we define two adinkras to beκ-equivalent if and only if theirκ coefficients are identical, and their κ coefficients vanish. Two adinkras are -equivalent if and only their coefficients are identical, and their˜ coefficients vanish. Similarly, we define two adinkras to be˜ -equivalent if and only if their˜ coefficients are identical, and their coefficients vanish.
In this paper, we focus on the gadget G, hitherto refers to as simply the gadget, and so focus on -and˜ -equivalence classes. It has been known for some time [13,17] that any adinkra has either vanishing or vanishing˜ coefficients. Therefore, the adinkras that belong to the -and -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 -and -equivalence classes. Note that gadgets between different -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 -or˜ -equivalence class.

Quaternion Adinkras
We denote the group of signed permuations of three elements as BC 3 . Any element of BC 3 can be expressed as a sign flip element H a times a permutation element S µ 3 . The indices take the values a, b, c, · · · = 1, . . . 8 and µ, ν, · · · = 1, . . . , 6. 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 BC 3 is given by The Viererrgruppe is where A, B, · · · = 1, 2, 3, 4. A general element of BC 4 , the group of signed permuattions of four elements, can be expressed as plus or minus one times an BC 3 element times an element of the Viererrgruppe.
Left cosets of the Vierergrupe via S 3 generate all elements of S 4 as follows [13,48]: These have the matrix representations Notice thatQ is the same as the trans adinkra in Fig 1. We can express any adinkra matrices as elements of BC 4 . For the quaternion adinkras, we have:  The quaternion adinkras Q andQ belong to separate and˜ -equivalence classes: These values of and˜ can be succinctly written as β matrices: The gadget between the quaternion adinkras is therefore zero: G[(Q), (Q)] = 0. Both quaternion adinkras belong to the χ 0 -equivalence class χ 0 = −1.

The 96Ṽ-Equivalence Classes of Adinkras
We define BC 4 color transformations as BC 4 elements acting on the color indices I, J, . . . of the adinkras, BC 4 boson transformations as BC 4 elements acting on the bosonic indices i, j, . . . of the adinkras, and BC 4 fermion transformations as BC 4 elements acting on the fermionic indicesî,ĵ, . . . of the adinkras.
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 BC 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 BC 3 color transformations to be able to generate all adinkras as BC 4 boson does not have enough elements to do so alone.

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 : Putting this all together, we find that CM is generated from Q via the A more succinct derivation of Eq .(4.8) 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 I = 1 element, violet I = 2, orange I = 3 , and red I = 4.
Similarly, we can calculate how other representations are generated from Q orQ: Notice that the representations in Eqs. that is generated as: The details of this calculation are shown below: The explicit matrices for this representation arẽ The adinkra forL −73426 I is as in Fig. 6. (4.14) There is a two fold degenerate list of eight such isometry transformations for each quaternion adinkra. The isometries of Q are (4.16d) Notice the two fold degeneracy in BC 4 boson × BC 4 color that stems from the fact that for instance (13) = − (24). Also, the isometry transformations for both Q andQ are all signed elements of the Vierergruppe. This has consequences for the equivalence classes as explained in the next section. Color flipping and flopping Q andQ via BC 3 color leaves us with the following set of 96 adinkras:

Flipping and
Each is a member of a distinct -or˜ -equivalence class as summarized in tables 1,2 3, and 4. Each table below describes 24 -or˜ -equivalence classes of the same χ 0 -equivalence class. We define the 24 classes in each -equivalence class table to comprise an isomer-equivalence class (either a transor cis-equivalence class) and we define the 24 classes in each˜ -equivalence class table to comprise an isometer-tilde-equivalence class (either a trans-tilde-or cis-tilde-equivalence class). We label these isomer-and isomer-tilde-equivalence classes as A, B,Ã, andB for l ∼ α, l ∼ β,l ∼ α, orl ∼ β, respectively.
Class bν (BC bν Class bν (BC bν Class bν (BC bν Class bν (BC bν We define color-parity of a given equivalence class as being odd (even) if there are an odd (even) number of total flips plus flops as listed in the (BC bν 3 ) I J column of Tables 1,2 3, and 4. Compare for instance the equivalence classesB 6 andB 16 in Table 4. The equivalence classB 6 has odd colorparity: two flips encoded by (12) and three flops encoded by (123) for a total of five flips plus flops. In constrast, the equivalence classB 16 has even color-parity: three flips encode by (123) and one flop encoded by (12) for a total of four flips plus flops. Notice the equivalence classes in each of the Tables 1,2 3, and 4 are numbered such that classes one through 12 have odd color-parity and classes 13 through 24 have even color-parity. We also point out that the â IJ and˜ â IJ for each equivalence class satisfy This is due to the fact that classes A andÃ with χ 0 = +1 have l ∼ α andl ∼ α, classes B andB with χ 0 = −1 have l ∼ β andl ∼ β, and the fact that the α and β matrices satisfy The above property of the α and β matrices can be seen from inspection of their explicit forms in appendix A.

The One Billion, Three Hundred Fifty Eight Million, Nine Hundred Fifty Four Thousand, Four Hundred Ninety Six Gadgets, Revisited
As first computed in [19], 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 : The matrix above therefore encodes the possible gadget values within each isomer-and isomertilde-equivalence class and shows some interesting color-parity features within each of these classes. Recall from Tables 1,2 3, and 4 that class numbers one through 12 (13 through 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 are shown in Table 5. Notice these counts add up to the total number of gadgets 1,358,954,496. Also shown in Table 5 are the number of times each value appears in each of the matrices in Eqs. (4.22) and (4.23).
Noting that there are 12 = 2 × 6 matrices described by Eqs. (4.22) (twice the number due to symmetry of the Gadget), four matrices described by Eqs. (4.23), and that each matrix entry has a degeneracy of 147, 456 = 384 × 384 (owing to there being 384 adinkras within each -and -equivalence class), we see that the count of each gadget value in Eqs. (4.22)   that of [19]:

Models
The previous chapters 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 chapters' foundation to engage the application of this foundation to the construction of 1D, N = 4 non-linear sigma-models over the BC 4 Coxeter Group.

Review of the Discovery of Twisted Reps In Sigma-Models
For 4D, N = 1 σ-models [49] there are two ingredients, chiral superfields Φ I and a Kähler potential K(Φ I ,Φ I ) which come together to define a dynamical system via the supersymmetrical action formula 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 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, N = 2 superspace that does not exist in 4D, N = 1 superspace.
As was first shown in the works of [50,51] a distinct 2D, N = 2 supersymmetric scalar multiplet, the so-called "twisted chiral supermultiplet" (denoted by χÎ and where the range of the index I may be different from that of theÎ index) exists in the lower dimension. Thus, modifying the action to the form 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 N = 1 σ-model, this geometry describes a Riemannian Kähler manifold. In the case of the complete two dimensional N = 2 σ-model, this geometry is a non-Riemannian bi-hermitian manifold with torsion.
The distinction between two dimensional N = 2 σ-models constructed solely from chiral supermultiplets or solely from twisted chiral supermultiplets in comparison to two dimensional N = 2 σ-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" [52].
When two dimensional N = 2 supermultiplets are reduced to one dimensional N = 4 supermultiplets, the distinction between the chiral supermultiplet (CM ) and twisted chiral supermultiplet (T CM ) can be seen in their gadget values [19]: As first discovered in [50,51] and later related to adinkras in [15,16,19], the 2D N = 2 twisted chiral multiplet is the dimensionally reduced 4D, N = 1 vector multiplet. We see in comparing Eq. (5.4) to Eq. (2.39) that the gadget keeps track of this relationship between the twisted chiral and vector supermultiplets.
The two dimensional σ-model actions can also be reduced to one dimensional σ-model actions, dτ d 2 θ d 2θ K(Φ I ,Φ I ; χÎ,χÎ) , (5.5) and the works of [32,33,34] in principle capture all of these. However, here it is useful to recall the experience of the reduction from 4D, N = 1 σ-models to 2D, N = 2 σ-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, N = 2 supersymmetry to first consider the intermediate step where we consider 2D, 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, N = (4,0) supersymmetry must inherit these supermultiplets from 2D, N = (4,0) supersymmetry.

2D, N = (4,0) Supersymmetry Considerations
We introduce the bosonic coordinates for the worldsheet τ and σ assembled into lightcone coordinates σ and σ such that (5.6) For (4,0) superspace, four Grassmann coordinates correspond to the + component of spinor helicity with regard to the world sheet Lorentz group. So we have 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" together with the lighcone derivatives ∂ and ∂ to describe the tangent space to the supermanifold. These definitions ensure the equations

Reviewing The Known 2D, N = (4,0) Minimal Scalar Valise Supermultiplets
Many years ago [53,54], it was demonstrated that there are a minimal of four distinct (4,0) valise supermultiplets each containing four bosons and four fermions. Thus, one can introduce a "representation label" (R) 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 A, B, φ, φ i j , A i and B i and of these only φ and φ 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, N = 1 "chiral supermultiplet" Quantity su(2) − spin J value Table 7: The spin content of the 2D, N = (4,0) supermultiplets.

Multiplet Field Content
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 respectlvely 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 lightcone coordinates σ and σ . 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.

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 were the comparison process can be made in the simplest possible manner. In particular, the first step is 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 an appendix of the work in [55], "L-matrices" and "R-matrices" can be identified in dimensions greater than one. In this particular example of the "L-matrices" and "R-matrices" where shown for superspaces with three bosonic coordinates.
To begin, we note for the operator D +i we can write where the four "supercovariant derivatives" are defined with respect to the four real (Majorana) spinor coordinates for the (4, 0) superspace ζ +Â , ζ +B , ζ +Ĉ , and ζ + D . It is important to note that the labelsÂ,B,Ĉ, andD are fixed values, not indices that take on different values.
Taking the complex conjugate of the results in (5.14), we find 15) which imply also the validity of the complex conjugate version of (5.9). Together (5.14) and (5.15) imply which in turns allows the definition of a real (4, 0) superspace covariant derivative through the equation Let us also highlight that in (5.17) there is a notational device introduced. The quartet Majorana supercovariant derivative operator is denoted by D + I whereas the complex SU(2)-doublet supercovariant derivative operator is denoted by the pair (D +i ,D + i ). Thus, the equation (5.17) 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 .   Thus, by choosing to work in a real basis, the disparate forms of the field content and transformation laws seen in (5.10) -(5.13) have been subjected to a "uniformization."

2D, N = (4, 0) Adinkra Related Matrices and Gadget Values
With the results of the last subsection in hand, we can build on these and are in position to calculate the fermionic holoraumy matrices defined by The gadget values shown in the equation above explicitly show an important lesson about the reduction process. It can be seen that the value of + 1/3 appears. Recalling that for the 2D, N = 2 supermultiplets, this value never occurs. However, in the case of 2D, N = (4,0) supermultiplets, this gadget value readily appears.
As both the 2D, N = 2 supermultiplets, and the 2D, N = (4,0) supermultiplets can be reduced to 1D, N = 4 supermultiplets, this is an explicit demonstration that the model space of any and all possible 1D, N = 4 non-linear σ-models must be larger than that found by dimensional reduction! 5.6 Reducing to 1D, N = 4 Superspace The process of going from 2D, N = (4, 0) superspace to 1D, N = 4 superspace simply amounts to demanding that all fields depend solely on the temporal dimension of the higher dimension. Additionally all the helicity indices are simply dropped from spinors, partial derivatives, etc. as there is no helicity in 1D. However, in 1D we are still able to maintain the distinction between bosons and fermions since the equations must be valid in order to have well defined component actions.
The main goal of this chapter was to provide a convincing argument that knowing the standard supermultiplets associated with the reduction to two dimension of the chiral and vector supermultiplet, totally misses the appearance of two new supermuliplets that can be constructed on the basis of the use of the Klein transformation. All of these supermultiplets are compactly described by quartets of bosons and fermions that are expressed uniformly in the equation of (5.23).
However, from our study of adinkras with four colors, we know that these four supermultiplets are only four among a total of 36,864 such supermultiplets. This raises questions.
How does one distinguish among the 36,864 such supermultiplets?
How does one describe a sigma-model built, not only on the basis of the four supermultiplets (SM-I), (SM-II), (SM-III) and (SM-IV), but one constructed from any or all of these supermultiplets found by the study of adinkras?
The first of the questions is the one of classification and the previous chapters deal with this issue and provide methods for classifying these 1D, N = 4 valise supermultiplets. This was a primary motivation for the arguments developed.
The second question is equivalent to the question of providing a universal formulation of all possible such sigma-models. A solution to this will be presented in subsequent chapter. This is very much a question of physics as descriptions of dynamical systems follow from such actions. 6 The Universal 1D, N = 4 Minimal Valise BC 4 Sigma-

Model
The enumeration of the 36,864 BC 4 adinkras implies that this constitutes a starting point where every possible linear minimal valise representation of 1D, N = 4 supersymmetry has been made explicitly "visible." Thus, if there is a way to concatenate the standard 1D, N = 4 superspace formalism together with the BC 4 -enumerated adinkra-based formalism, we are guaranteed to have complete control of the model space 1D, N = 4 supersymmetric σ-models.
The work in [22] ensures the concatenation is possible. The key observations: (a.) The links in the adinkra use the D I symbol as their representation and in a traditional superfield these can be interpreted at the superspace supercovariant derivative. (b.) Every bosonic and fermionic node of an adinkra may be regarded as the lowest component of a corresponding superfield. These observations imply that one need only re-interpret the symbols used for adinkra based supermultiplets to obtain traditional superfield equations.
In particular this implies that a σ-model action formula involving superfields obtained from BC 4 adinkras via the prescription above must take the form is a choice of the scalar superfields associated with any representation (R) among the 36,864 BC 4 -based valise adinkras, and the notation | as usual means first perform all the indicated differentiations follow by taking the limit as all superspace Grassmann coordinates are set to zero. Now on these superfields we have the following equations. 3) The definitions of these coefficients follow from comparing Eq. (6.3) to Eq. (6.4).
All repeated indices in these equarions (including the representation indices) are to be summed over all possible values that occur in Kähler-like potential. In these expression we have used the notations to denote the derivatives of the Kähler-like potential.
To show this yields the usual component level free action for the supermultiplets defined in terms of adinkras with component fields at the nodes, Eq. (2.1), one should take any superfields and choose K to be the quadratic function (where R can be any of the shown adinkras) of the bosonic nodal fields for each of the adinkras. For this choice of the Kähler-like potential, the geometry of the σ-model is flat. If additional terms higher order in the fields, including even powers of fermion superfields, are included then the geometry associated with the Kähler-like potential becomes non-trivial. vanish. The most important points to take away from this discussion is the forms of the action and the related equations of motion depend on two distinct types of data. One of these is the form of the Kähler-like potential K. The other is the representation of the valises that are chosen to appear in the σ-model. In the action formula, these representation data are contained in the V-matrices and L-matrices.

Conclusion
Adinkras are useful and interesting tools with which to study supersymmetry. Perhaps the most useful aspect is the ability to encode information about higher dimensional supersymmetry. To refine holographic procedures, a clear description of equivalent adinkras is necessary. This paper used the gadget to provide a descriptions of 96 such equivalence classes: theṼ-equivalence classes of four color, four boson and four fermion adinkras. These are equivalence classes of 48 BC 3 color transformations (signed three-permutations) of twoṼ-inequivalent quaternion adinkras. Each equivalence class was shown to contain 384 adinkras, hence the 96 equivalence classes contain all 36, 864 = 384 × 96 four color, four boson and four node adinkras.
These equivalence classes serve to elucidate some of the mysteries of the gadget seen in [19]. For instance, we showed that the plus one-third gadget only arises between equivalence classes of different color-parity. We also found correlations between the frequencies of the gadget values and the color-parities of the equivalence classes. The gadget value of zero occurs most frequently because it is the only gadget value that occurs between adinkras of different isomer-and isomertilde-equivalence classes. Within each isomer-and isomer-tilde-equivalence class, color-parity seems to control why the minus-one third gadget appears more frequently than the plus one-third gadget: both appear equally between adinkras of different color-parity but only the minus one-third gadget appears between adinkras of the same color-parity. The gadget value of one is the least frequent as it can only occur between adinkras of the sameṼ-equivalence class.
Another important result of this paper is Eqs. (4.20) which compactly encode all 36,864 four color, four boson and four fermion valise adinkras. Furthermore, the utility of the gadget in distinguishing multiplets related by dimensional reduction was demonstrated. Specifically, the twisted chiral multiplet and the vector multiplet were used as an example. Dimensional reductions of actions were discussed in terms of Kähler-like potentials; technology we plan to use to advance SUSY holography. Future work will focus on utilizing equivalence classes to develop holographic techniques that we plan to extend to adinkras with more supercharges such as 4D, N = 4 Super Yang-Mills theory and 10D and 11D supergravity.
The discussion under (5.38) is a demonstration of the presence of SUSY holography once more. This provides a beautiful physics reason for why the gadget value of +1/3 is so distinctive from the other gadget values of -1/3, 0, 1. The latter occur in the one-dimensional shadows of supermultiplets all the way up to 4D, N = 1 theories, but the former only occurs in the one-dimensional shadows of supermultiplets up to 2D, N =(2,0). The ability of adinkras and their holoraumy to keep track of this level of subtlety should be convincing to any skeptic who doubts the efficacy of this approach to the study of supersymmetric representation theory.
"Equality is not in regarding different things similarly, equality is in regarding different things differently." -Tom Robbins A α and β matrices The α and β matrices used in this paper are: In terms of tensor products of Pauli spin matrices σ i and the 2 × 2 identity matrix I 2 , this can be written as These matrices form two mutually commuting su(2) algebras [αâ, αb] =2i âbĉ αĉ , [βâ, βb] = 2i âbĉ βĉ , [αâ, βb] = 0 .

B Explicit Matrix Forms for Flips and Flops
Since flip matrices with boolean codes greater than seven are simply the negative of a corresponding matrix with code less than seven, it is only necessary to list boolean matrices less than eight:

Valise Supermultiplets
In the text of this work, we calculated the holoraumies associated with the 2D, N = (4,0) supermultiplets by going to a real basis. Of course, this can also be done in the initial su(2) basis used to describe the SM-I, . . . , SM-IV representations. In other words, it is necessary to calculate the effects of the three operators on all of the fermions in each supermultiplet. Upon completion of these calculations we find results shown in the equations of (C.2) -(C.5).