4D, $\mathcal{N}=1$ Matter Gravitino Genomics

Adinkras are graphs that encode a supersymmetric representation's transformation laws that have been reduced to one dimension, that of time. A goal of the supersymmetry ``genomics'' project is to classify all 4D, $\mathcal{N}=1$ off-shell supermultiplets in terms of their adinkras. In~previous works, the genomics project uncovered two fundamental isomer adinkras, the cis- and trans-adinkras, into which all multiplets investigated to date can be decomposed. The number of cis- and trans-adinkras describing a given multiplet define the isomer-equivalence class to which the multiplet belongs. A further refining classification is that of a supersymmetric multiplet's holoraumy: the commutator of the supercharges acting on the representation. The one-dimensionally reduced, matrix representation of a multiplet's holoraumy defines the multiplet's holoraumy-equivalence class. Together, a multiplet's isomer-equivalence and holoraumy-equivalence classes are two of the main characteristics used to distinguish the adinkras associated with different supersymmetry multiplets in higher dimensions. This paper focuses on two matter gravitino formulations, each with 20 bosonic and 20 fermionic off-shell degrees of freedom, analyzes them in terms of their isomer- and holoraumy-equivalence classes, and compares with non-minimal supergravity which is also a 20x20 multiplet. This analysis fills a missing piece in the supersymmetry genomics project, as now the isomer-equivalence and holoraumy-equivalence for representations up to spin two in component fields have been analyzed for 4D, $\mathcal{N}=1$ supersymmetry. To handle the calculations of this research effort, we have used the Mathematica software package called Adinkra.m. This package is open-source and available for download at a GitHub Repository. Data files associated with this paper are also published open-source at a Data Repository also on GitHub.


Introduction
Generally, there are more representations of supersymmetry (SUSY) in lower dimensions than there are in higher dimensions. Given a particular higher dimensional SUSY representation, one can always reduce the representation to lower dimension simply by considering the fields of the multiplet to depend only on the subset of coordinates required. For instance, in efforts known as "supersymmetric genomics" [1,2,3], a 4D SUSY multiplet is reduced to 1D by considered the fields in the multiplet to depend only on time τ . This is known as reducing to the 0-brane and the transformation laws for the 4D, N = 1 chiral multiplet, for instance, when reduced to the 0-brane can be entirely encoded in a graph known as an adinkra [4,5,6,7,8,9,10,11] as shown in Fig. 1. The precise meaning of the lines in the diagram will be described in section 2. Adinkras have been and continue to be used to discover new representations of supersymmetry: the 4D, N = 2 off-shell relaxed extended tensor mutliplet [12] and a finite representation of the hypermultiplet [13], for instance. Previous supersymmetric genomics works [1,2,3] found adinkra  Table 1 where χ 0 = n c − n t .   Notice in Table 1 that the values of n c and n t only serve to partially differentiate the various multiplets at the adinkra level. Specifically, the CM is seen to be distinct from the VM and TM, but the VM and TM are indistinguishable based solely on their cis/trans adinkra content. Similarly, there is no difference between CLS and mSG at the adinkra-level. Another tool is needed to completely sort this out.
Dimensional enhancement, or SUSY holography, is the effort to build higher dimensional SUSY representations from lower dimensional representations [14,15,16]. To aid in sorting out which of the multitude of lower dimensional systems are candidates for dimensional enhancement, a tool known as holoraumy is being developed [17,18,19,20,21,22,23,24,25,26,27]. On the 0-brane with N SUSY transformations D I , holoraumy is defined as the commutator of two supersymmetric transformations D I acting on the bosons Φ or fermions Ψ of a particular representation (1.1) where a dot above a field indicates a time derivative,Φ = dΦ/dτ for example. Contrast holoraumy with the SUSY algebra which is the anti-commutator of two SUSY transformations D I . A closed 1D SUSY algebra takes the following form on all fields Holoraumy is the tool being developed to split the degeneracy of the cis-and trans-information as shown in Table 1. For instance, holoraumy was first shown to separate the VM and TM at the adinkra level in [17,18]. To do so a dot product-like "gadget" was introduced in [17,18] and studied further in [22,23,24,25,26].
In this paper, we further the supersymmetry genomics efforts by decomposing the two representations of 4D, N = 1 matter gravitino, one as described in [28,29] and the other as described in [30,31] in terms of the cis-and trans-adinkra content as well as their holoraumy. These multiplets have the same degrees of freedom (20 bosons and 20 fermions) as mSG [32] so we compare their cis-and trans-adinkra and holoraumy to this multiplet as well. To the knowledge of the authors, this paper is the first time that the transformation laws for the two matter gravitino multiplets and mSG have each been written in terms of a one parameter family of transformation laws that encodes a field redefinition of the auxiliary fermions that preserves the diagonal character of the Lagrangian. Where the cis-and trans-adinkra content are shown to be independent of this parameter, the holoraumy is not. In a sense, this paper is a sister paper of the recent work [26]: this paper reviews the status of SUSY genomics, focusing on 4D dynamics, where as [26] reviews the status of SUSY holography, focusing on 1D dynamics. It is the view of at least one of the authors of this paper (KS) that building a complete picture of SUSY holography [14,15,16] will require efforts into SUSY genomics [1,2,3], enumeration techniques [33,34], and classification schemes [17,18,19,20,21,22,23,24,25,26,27].
This paper is organized as follows. In section 2 we review adinkras. In section 3 we review supersymmetry genomics. Sections 4, 5, and 6 review the two matter gravitino multiplets and the mSG multiplet, expressed in terms of the one parameter diagonal Lagrangian family of transformations. Section 7 presents the adinkra and 1D holoraumy content of each of the 20 × 20 multiplets and makes comparisons via the gadget. For all gamma matrix conventions, we follow precisely the previous three supersymemtry genomics works [1,2,3]

Adinkra Review
Adinkras are graphs that encode supersymmetry transformation laws in one dimension (that of time τ ) with complete fidelity. Take for example four dynamical bosonic fields Φ i and four dynamical fermionic fields Ψĵ that depend only on time. Two distinct possible sets of supersymmetry transformation laws for Φ i and Ψĵ can be succinctly encoded as and where a dot above a field indicates a time derivative,Φ i = dΦ i /dτ for example. One set of transformation laws is encoded by the choice χ 0 = +1 and another by χ 0 = −1. For Eqs. (2.1) and (2.2) there is no possible set of field redefinitions for which the χ 0 = +1 transformation laws reduce to the transformation laws for which χ 0 = −1. Owing to an analogy to isomers in chemistry, in [1] the χ 0 = +1 transformation laws were dubbed the cis-multiplet and χ 0 = −1 the trans-multiplet. Both the cis-and trans-transformation laws satsify the closure relationship, Eq. (1.2).
The adinkras in Fig. 2 can be seen to encode the transformation laws in Eqs. (2.1) and (2.2) as follows. 5. 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.
The adinkras in Fig. 2 are what are known as valise adinkras: adinkras with a single row of bosons and a single row of fermions. The distinction between the two set of transformation laws encoded in Eqs. (2.1) and (2.2) can be seen easily in the adinkras in Fig. 2: the two adinkras are identical aside from the orange lines which are dashed in one adinkra where they are solid in the other. This is reflected in the D 3 transformation laws in Eqs. (2.1) and (2.2) which differ by a minus sign.
Both of the χ 0 = ±1 supersymmetric transformation laws are symmetries of the Lagrangian The transformation laws in Eqs. (2.1) and (2.2) can succinctly be written as where the adinkra matrices L I are given by and the R I given by In the specific case of the matrices 2.5, we also have the orthogonality relationship where the T denotes transpose. Supersymmetric multiplets whose adinkra matrices satisfy the orthogonality relationship Eq. (2.7) are said to be in an adinkraic representation, that is, these multiplets can be expressed as an adinkra picture as in Fig. 2. Generally, larger multiplets like those investigated in this paper are non-adinkraic when nodes are chosen to be identified with single fields as in Fig. 6. Non-adinkraic multiplets have been investigated previously in [35,36] The closure relation, Eq. (1.2), for an adinkriac system is reflected in the adinkra matrices L I and R I satisfying the GR(d, N ) algebra also known as the garden algebra [37,38] L I R J + L J R I = 2δ IJ I 4 , R I L J + R J L I = 2δ IJ I 4 . (2.8) 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. Specifically, the χ 0 = ±1 adinkras in Fig. 2 each satisfy the GR(4, 4) algebra.
The matrix representation of 1D holoraumy, Eq. (1.1) is [23,24] where V IJ is the matrix representation of the bosonic holoraumy tensor B IJ and V IJ is the matrix representation of the fermionic holoraumy tensor F IJ . Adinkras that share the same value of χ 0 and have the same number of degrees of freedom d are said to be in the same χ 0 -equivalence class [26]. There are two possible χ 0 -equivalence classes for GR (4,4) valise adinkras: the cis-equivalence class defined by χ 0 = +1 and the trans-equivalence class defined by χ 0 = −1. In [25] all possible 36,864 GR(4, 4) adinkras were investigated and tabulated and in [26] all were categorized in terms of χ 0 -equivalence classes and holoraumy-equivalence classes.

Supersymmetry Genomics Review
In this section, we review the previous SUSY genomics works [1,2,3]. In doing so, we demonstrate how the the cis-adinkra and trans-adinkra shown in Figure 2 encode the 0-brane reduced transformation laws for various 4D, N = 1 off-shell multiplets and comment on their values of χ 0 , holoraumy, and gadgets in the cases where this is known.
The dynamical field content of the CM is a scalar A, psuedoscalar B, and Majorana fermion ψ a . The auxiliary field content of the CM is a scalar F and pseudoscalar G. The 4D component Lagrangian for the CM is given by The transformation laws that are a symmetry of the Lagrangian (3.1) are Reducing to the 0-brane with nodal field definitions reduces the Lagrangian (3.1) to Eq. (2.3) and transformation laws to Eq. (2.4) with the L I matrices given by and R I given by Eq. (2.7) and satsify the GR(4, 4) algebra Eq. (2.8). By the rules explained in section 2, it can be seen that the 0-brane transformation laws for the CM, Eq. (2.4) with L I matrices as in Eq. (3.4) and nodal field definitions (3.3), are entirely described by the adinkra in Fig. 3 which is the same image as Fig. 1 in the introductory material of this paper. The CM is in the cis-equivalence class as can be seen in the following two ways 1. Calculating the trace in Eq. (2.9), which produces the result χ 0 = +1.
The field redefinitions mentioned in the second of these were dubbed flips and flops in the recent work [26]. The dynamical field content of the TM is a scalar ϕ, anti-symmetric rank-two tensor B µν , and a Majorana fermion χ a . There are no auxiliary fields in the TM. The 4D component Lagrangian for the TM is given by The transformation laws that are a symmetry of the Lagrangian (3.5) are Choosing temporal gauge B 0µ = 0 and reducing to the 0-brane with nodal field definitions reduces the Lagrangian (3.5) to Eq. (2.3) and transformation laws to Eq. (2.4) with the L I matrices given by and R I given by Eq. (2.7) and satsify the GR(4, 4) algebra Eq. (2.8). By the rules explained in section 2, it can be seen that the 0-brane transformation laws for the TM, Eq. (2.4) with L I matrices as in Eq. (3.9) and nodal field definitions (3.8), are described by the adinkra in Fig. 4. The TM is in the trans-equivalence class (χ 0 = −1) as can be seen through either Eq. (2.9) or through field redefinitions transforming Fig. 4 into the trans-adinkra in Fig. 2. Figure 4: An adinkra for the 0-brane reduced transformation laws of the 4D, N = 1 tensor multiplet. The dynamical field content of the VM is a U (1) gauge vector field A µ and a Majorana fermion λ a . The only auxiliary field in the VM is a pseudoscalar d. The 4D component Lagrangian for the VM is given by The transformation laws that are a symmetry of the Lagrangian (3.10) are Choosing temporal gauge A 0 = 0 and reducing to the 0-brane with nodal field definitions reduces the Lagrangian (3.10) to Eq. (2.3) and transformation laws to Eq. (2.4) with the L I matrices given by and R I given by Eq. (2.7) and satsify the GR(4, 4) algebra Eq. (2.8). By the rules explained in section 2, it can be seen that the 0-brane transformation laws for the VM, Eq. (2.4) with L I matrices as in Eq. (3.14) and nodal field definitions (3.13), are described the adinkra in Fig. 5. The VM is in the trans-equivalence class (χ 0 = −1) as can be seen through either Eq. (2.9) or through field redefinitions transforming Fig. 5 into the trans-adinkra in Fig. 2. The dynamical field content of the CLS is the same as for the CM: a scalar field and pseudoscalar field, here called K and L respectively, and a Majorana fermion, here called ζ a . A key difference from CLS and CM is the auxiliary field content: a scalar M , pseudoscalar N , vector V µ , pseudovector U µ , a low-dimensional fermion ρ a with [ρ a ] = 3/2 and a high-dimensional fermion β a with [β a ] = 5/2. The 4D component Lagrangian for CLS is The transformation laws that are a symmetry of the Lagrangian (3.15) are Unlike the minimal CM, TM, and VM cases, the 0-brane reduction for CLS requires nodal definitions that are linear combinations of the 0-brane reduced fields for the resulting L I matrices to be adinkraic (Eq. (2.7)). A particular choice of nodal field definitions for CLS that are adinkraic arė Here we have defined Φ and Ψ in terms of their derivativesΦ andΨ for notational convenience 3 . With these nodal field definitions, the Lagrangian (3.15) reduces to Eq. (2.3) and transformation laws to Eq. (2.4) with the L I matrices given by with the α and β matrices as in appendix C and R I given by Eq. (2.7).
By the rules explained in section 2, it can be seen that the 0-brane transformation laws for the CLS, Eq. (2.4) with L I matrices as in Eq. (3.18) and nodal field definitions (3.17), are described by the adinkra in Fig. 6. The CLS has χ 0 = −1 as can be seen through either Eq. (2.9) or through the fact that comparing  The dynamical field content of mSG is the symmetric, rank two graviton h µν and gravitino ψ µa . The auxiliary fields of mSG are a scalar S, pseudoscalar P , and axial vector A µ . The 4D component Lagrangian for mSG is The transformation laws that are a symmetry of the Lagrangian (3.19) are In the valise adinkra in Fig. 13, we have put the third piece below the first two even though all bosons have the same dimension and all the fermions have the same dimension. From here on, we will generally split up these larger valise adinkras into more than one row with the understanding that all fermions have the same dimension and all bosons have the same dimension. Comparing Fig. 13 with Fig. 2, we clearly see that the valise adinkra for the CLS splits into n c = 1 cis-adinkra and n t = 2 trans-adinkras.
The splitting is easily seen in the block diagonal matrix representation of Eq. (2.42) 21 Figure 6: An adinkra for the 0-brane reduced transformation laws of the 4D, N = 1 complex linear superfield multiplet (CLS) and old-minimal supergravity multiplet (mSG).
Similar to the CLS, the 0-brane reduction for mSG requires nodal definitions that are linear combinations of the 0-brane reduced fields for the resulting L I matrices to be adinkraic (Eq. (2.7)). As shown in [3], the most general choice of nodal field definitions for mSG that are adinkraic and match with the cis-and/or trans-adinkra Fig. 2 are Here we have defined Φ and Ψ in terms of their derivativesΦ andΨ for notational convenience as in the CLS case 4 . With these nodal field definitions, the Lagrangian (3.19) reduces to Eq. (2.3) and transformation laws to Eq. (2.4) with the L I matrices given by with the α and β matrices as in appendix C and R I given by Eq. (2.7).
By the rules explained in section 2, it can be seen that the 0-brane transformation laws for the mSG, Eq. (2.4) with L I matrices as in Eq. (3.23) and nodal field definitions (3.21), are described by the adinkra in Fig. 6: the same adinkra as for CLS. Thus as in the CLS case, the mSG has χ 0 = −1 and can be decomposed as one cis-adinkra (n c = 1) and two trans-adinkras (n t = 2) with χ 0 = n c − n t = −1.

Gadgets
There is not a tremendous amount of diversity in the χ 0 values reviewed thus far, as summarized in Table 1: all reviewed in the previous five sections have χ 0 = −1 aside from the CM which has χ 0 = +1. Clearly, another tool is needed to further separate out at the adinkra level which adinkras relate to which higher dimensional systems.
The gadget G(R, R ) between two different adinkra representations R and R of the GR(d, N ) algebra, defined below, is used to separate multiplets at the adinkra level that holographically correspond to different multiplets in higher dimensions [17,18,21,22,25,26].
In the above, the function d min (4) = 4 is the minimal size of an N = 4 adinkra as proved for general N in [7,10] and used in the subsequent works [36,25]. Two representations that have the same holoraumy V IJ will have a gadget value of n c + n t . Representations that have gadget value different from n c + n t are said to be V IJ -inequivalent. For 4D, N = 1 supersymmetry, the number of colors in the adinkraic representation is N = 4. The notion of holoraumy equivalence and the connection to SUSY holography has recently been exhaustively studied for GR (4,4) in [26].
Gadgets can only be compared between systems of the same size d and number of colors N . The CM, TM, and VM all have d = 4, N = 4, thus the gadget may be calculated amongst them. As first discovered in [17,18], the gadgets between the representations ordered R = R = (CM, T M, V M ) take the following form This demonstrates that the gadget separates the TM and VM at the adinkra level, and so is a further distinguishing calculation that can be done in addition to χ 0 . In the adinkraic basis chosen for mSG and CLS, however, the gadget can not be used to distinguish these 12 × 12 multiplets as their L I and R I matrices, Eqs. (3.18) and (3.23), are identical. Learning this lesson from the 12 × 12 case, for the 20 × 20 multiplets investigated in this paper, we choose nodal field definitions that are not linear combinations of the 0-brane reduced fields which put the L I and R I matrices in a non-adinkraic basis, i.e. matrices that satisfy Eq. (2.6) rather than the full orthogonal relationship Eq. (2.7). The value of χ 0 is insensitive to this, owing to the trace over which it is defined and the fact that it is defined only over a single representation as shown in Eq. (2.9).

The de Wit-van Holten Formulation
We refer to the matter gravitino multiplet as described in Ref. [28,29] as the "de Wit van Holten" (dWvH) formulation. 5 The dWvH multiplet consists of a spin one-half superfield with compensators of a vector multiplet and chiral multiplet [40]. The components of this multiplet are as follows. The matter fields are that of a spin three-halves Rarita Schwinger field ψ µb and a spin one vector B µ . The bosonic auxiliary fields in the multiplet (all with dimension-two) are a scalar K, pseudoscalars L and P , rank-two tensor t µν , vector V µ , and axial vector U µ . The fermionic auxiliary fields are a dimension three-halves spinor λ a and dimension five-halves spinor χ a . The transformation laws, Lagrangian, algebra, and adinkras are described in the following subsections in a real Majorana notation.
Using Fierz identities, the term including F αβ within W αβ of the transformation law for the gravitino can be expressed as follows: As the first term encodes the gravitino's gauge transformation, it can be ignored. This is true certainly at the adinkraic level, where this term only shows up in the transformation laws for ψ 0b = 0 in temporal gauge.

Anti-Commutators
Direct calculations of the anti commutators of the D-operators on all the fields yield the results shown in (4.12) -(4.14).
The non-closure terms Λ ab and abc indicate gauge transformations of the B µ and ψ µa fields. As such, the algebra closes on the field strengths F µν and R a :

Lagrangian
The Lagrangian that is invariant (up to a surface term) with respect to these transformation laws takes the form This Lagrangian is also invariant with respect to the following gauge transformations that were indicated by (4.17) (4.20b)

The Ogievetsky-Sokatchev Formulation
The matter gravitino multiplet as described in Ref. [30,31], to which we refer as the OS multiplet for the authors names, consists of a spin one-half superfield with compensators of a vector multiplet and tensor multiplet [40]. The components of this multiplet are as follows. The matter fields are that of a spin three-halves Rarita Schwinger field ψ µb and a spin one vector B µ . The bosonic auxiliary fields (all with dimension-two) are a pseudoscalar P , rank-two tensor t µν , vector V µ , axial gauge vector A µ , and divergenceless axial vector G µ that is actually the field strength of a gauge two form E αβ such that G µ = 1 2 µναβ ∂ ν E αβ . The fermionic auxiliary fields are a dimension three-halves spinor ξ a and dimension five-halves spinor χ a . The transformation laws, Lagrangian, algebra, and adinkras are described in the following subsections in a real Majorana notation.

Anti-Commutators
with Λ ab and abc as in Eq. (4.17) with c 0 → s 0 and the new gauge term The algebra closes on the field strengths

Lagrangian
The Lagrangian that is invariant with respect to the OS transformation laws is The OS Lagrangian is also invariant with respect to the same gauge transformations as the dWvH case, Eq. (4.20).

The non-minimal Supergravity Formulation
The dynamical field content of mSG is that of the graviton h µν , which is symmetric but not traceless in our formulation, and the gravitino ψ µa which is likewise not traceless. The auxiliary field content for mSG consists of a scalar field S, pseudoscalar field P , two pseudovector fields A µ and W µ , a vector field V µ , and two spinors λ a and β a , the former being a leading order fermion in the superfield expansion. The transformation laws, algebra, and Lagrangian for mSG are given in the following subsections.

Anti-Commutators
The algebra closes on the auxiliary fields X = (S, P, A µ , V µ , W µ , λ a , β a ) as The algebras for the physical fields ψ µa and h µν are The gauge terms ζ νab and ε abc are ζ νab =i(γ α ) ab h να (6.5) The algebra closes on the field strengths where the weak field Riemann tensor R αβµν is

Lagrangian
The Lagrangian for mSGis This Lagrangian is invariant with respect to the following gauge transformations that were indicated by Eq. (6.4) The fermionic part of the mSG Lagrangian is identical to those of OS and dWvH under the identification β a = χ a .

Adinkranization of the 20 × 20 multiplets
Here we summarize the adinkranization process. More details can be found in the appendices. Considering the fields in the dWvH, OS, and mSG multiplets to be only time dependent, we gauge fix to temporal gauge Next we define the fields for the nodes of the adinkra to be as follows. For the dWvH formulation of the ( 3 2 ,1) supermultiplet, we order the bosons according to for the OS formulation we order the bosons according to where the ordering for t µν is as follows for both the dWvH and OS multiplets: {t 01 , t 02 , t 03 , t 12 , t 23 , t 31 } (7.6) Note that Finally for the non-minimal SG bosons, while for fermions of the non-minimal SG supermultiplet fermions we use, With these definitions, the transformation laws for each multiplet can be succinctly written as As it is not terribly instructive to display all L I and R I matrices for all of these multiplets we have published them along with all of the adinkra data described below for these three multiplets in three Mathematica data files dWvH.m, OS.m, and nmSG.m at the Data repository on GitHub. A master fileCompare20x20Reps.nb is located at the same repository which demonstrates how to display the data and perform the various calculations summarized in the remainder of the paper. The tutorial file Compare20x20Reps.nb utilizes the Mathematica package Adinkra.m, which is available at a different GitHub Repository. Also located at the Adinkra.m repository is a general tutorial AdinkraTutorial.nb that demonstrates the various features of the Adinkra.m package.
In Appendix B, we display the explicit L I and R J matrices for the c 0 = 0 representation of the dWvH multiplet. For all three multiplets, the L I and R J matrices satisfy the GR(d, N ) algebra, the algebra of general, real matrices of size d × d that encode N supersymmetries [1]: As d = 20 and N = 4 for the dWvH, OS, and mSG multiplets, their L I andR J matrices satisfy more specifically the GR(20, 4) algebra.
Recall, the parameter χ 0 is defined through the relationship 14) The parameters n c and n t are referred to as the isomer parameters. They encode the number n c cisisomer adinkras and the number n t trans-isomer adinkras into which a multiplet can be decomposed. The parameter χ 0 = n c − n t . For the dWvH, OS, and mSGmultiplets we find Recall the matrix representations for fermionic and bosonic holoraumy are defined as For any set of matrices L I and R J that satisfy the GR(d, N ) algebra, Eq. (2.8), setting either V IJ = 2t IJ or V IJ = 2t IJ will satisfy the so(N) algebra A proof is given in Appendix A. For the special case of so(4), we define where Einstein summation convention is assumed on the repeated indices K and L. It is straightforward to show that both t IJ = 1/2V ± IJ and t IJ = 1/2 V ± IJ satisfy the so(4) algebra, Eq. (7.17). At the same time, V ± IJ and V ± IJ each only have three independent elements. We display the independent elements of V ± IJ below, those of V ± IJ satisfy similar relations: Furthermore, all V + IJ commute with all V − IJ . In this way, V ± IJ are actually two separate, commuting representations of su (2): 7.2 V IJ , Eigenvalues, and Gadgets for the dWvH, OS, and mSG multiplets The explicit matrix forms of V IJ and V IJ are too large to display and be instructive in this paper. We have published them open-source in the files dWvH.m, OS.m, and nmSG.m at the previously mention GitHub data repository. As an example, in Appendix C we show the explicit form for the V IJ for the c 0 = 0 representation of the dWvH multiplet. Unlike the fundamental CM , T M , and V M representations [17,18,20,21,23,24], the V IJ and V IJ for the dWvH, OS, and mSG representations are each true so(4) representations composed of six linearly independent elements: In contrast, the V IJ and V IJ for the CM , V M , and T M each form a single, non-trivial su(2) representation, with only three linearly independent algebra elements [17,18,20,21,23,24]. That is either the V + IJ or the V − IJ vanish and either the either the V + IJ or the V − IJ vanish for the CM , V M , and T M . This is not the case for the dWvH, OS, and mSG representations: the V ± IJ for these are all nontrivial. We see then for the dWvH, OS, and mSG representations, the V IJ and V IJ all form true so(4) representations, each which separate into two commuting su (2) representations, V ± IJ and V ± IJ , respectively, as shown in the previous section. The eigenvalues for V IJ and V IJ for the dWvH, OS, and mSG multiplets are all ±1.

Conclusions
In this paper, we investigated three different 4D, N = 1 SUSY multiplets with 20 boson × 20 fermion degrees of freedom. Specifically, we investigated two matter gravitino multiplets, dWvH and OS, and non-minimal supergravity ( mSG), each in a one-parameter family of component transformation laws that encoded an auxiliary fermion field redefinition symmetry of the diagonal Lagrangian. We furthered research into SUSY genomics and holography by researching the dimensional reduction, χ 0 values, and adinkralevel fermionic holoraumy of three 20 × 20 multiplets. All three had distinct χ 0 values. Gadgets calculated between the different multiplets indicate some interesting possible connections to holography. The results of section 7.2 demonstrate an elegant choice, Eq. (7.24), for considering the OS and dWvH multiplets to be parallel in terms of the Gadget, that is, have a Gadget value of five. Setting either the dWvH and mSG multiplets parallel or the OS and mSG multiplets parallel required specific values of the supergravity parameter n to be selected. On the other hand, no real solutions exist that set the dWvH and OS multiplets orthogonal, however, at least one elegant solution exists that simultaneously sets both the dWvH and OS multiplets to be orthogonal to the mSG multiplet. These results point to the possibility that holoraumy and the gadget at the adinkra level indicate that the dWvH and OS multiplets are similar in some way, which we know in higher dimensions to be the case as they encode the same dynamical spins of (3/2, 1). Furthermore, fermionic holoraumy and the gadget seem to point to an adinkra-level distinction between the dWvH and mSG multiplets and the OS and mSG multiplets. We know of course that a key difference in 4D, N = 1 is that the dynamical fields of the mSG multiplet have spins (2, 3/2) rather than (3/2, 1) of the matter gravitino multiplets. We pointed out some features of the gadgets for values of the supergravity parameter n = −1, n = −1/3 and n = 0 which correspond to cases where mSG becomes part of a tower of higher spin that extends to N = 2 SUSY [32,41,42,43,44], reduces to old-minimal supergravity [32], and reduces to new-minimal supergravity [45], respectively.
A precise relationship between fermionic holoraumy and the gadget and spin of the higher dimensional system is still unknown. We look to uncover such precise spin-holography relationships not only through more research of these 20 × 20 multiplets, but also into 12 × 12 multiplets of 4D, N = 1 supersymmetry, as well as higher spin mutliplets as in [41,42,43,44]. There are four 12 × 12 representations of 4D, N = 1 offshell supersymmetry, and of these only one (CLS) has the fermionic auxiliary field redefinition symmetry similiar to that presented in this work. This analysis is already being done and we hope to complete it soon. In addition, it would be interesting to see what other gadgets such as those described in [27] and [34] encode for the 12 × 12, 20 × 20, and higher spin multiplets.

Acknowledgments
This work was partially supported by the National Science Foundation grant PHY-1315155. This research was also supported in part by the University of Maryland Center for String & Particle Theory (CSPT). KS would also like to thank Northwest Missouri State University for computing equipment and travel funds and Dartmouth College and the E.E. Just Program for hospitality and travel funds that supported this work. KS thanks Stephen Randall for work done on the dWvH multiplet while at the University of Maryland. The authors would also like to thank Konstantinos Koutrolikos for many helpful discussions throughout this work and S.J. Gates, Jr. for discussions and for providing the conceptual ideas that led to this work.
A Proof That t IJ = 1/2V IJ Satisfies the so(N ) Algebra Swapping L I with R I interchanges V IJ with V IJ , thus proving that t = 1/2V IJ satisfies the so(N ) algebra necessarily means that t = 1/2 V IJ must also satisfy the so(N ) algebra. We therefore prove the latter, the former follows by extension. Substituting t = 1/2V IJ into the so(N ) algebra, Eq. (7.17), results in (A.1) We now prove Eq. (A.1) using repeated use of the garden algebra, Eq. (2.8), rearranged as follows We start by substituting the definition of V IJ , Eq. (2.12) into the left hand side of Eq. (A.1) As an intermediate step, we make repeated use of Eq. (A.2) to modify the last term, momentarily neglecting the antisymmetry between the indices I and J and between L and K Substituting this back into Eq. (A.3) yields The first and fifth terms combine into a single term with V KJ , the second and sixth into V KI and so on:

B Explicit L I and R I matrices
For the choice c 0 = 0, the explicit L I matrices for the dWvH multiplet are The R I matrices are inverses of the L I matrices: R I = L −1 I .