# Dimensional Enhancement via Supersymmetry

Authors: M. G. Faux; K. M. Iga; G. D. Landweber
Journal: Advances in Mathematical Physics
 (2011)

---

## Abstract

We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory
of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can “enhance” to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one-dimensional reasoning exclusively. And we introduce the notion of “phantoms” which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics.


---

## Body

## 1. Introduction

Supersymmetry [1–5] imposes increasingly rigid constraints on the construction of quantum field theories [6–11] as the number of space-time dimensions increases. Thus, there are fewer supersymmetric models in six dimensions than there are in four, and yet  fewer in ten dimensions [12]. In eleven dimensions there seems to be a unique possibility [13], at least on-shell. (Anomaly freedom imposes seemingly distinct algebraic constraints which make this situation even more interesting.) However, the off-shell representation theory for supersymmetry is well understood only for relatively few supersymmetries, and remains a mysterious subject in contexts of special interest, such as N=4 Super Yang Mills theory, and the four ten-dimensional supergravity theories [14].Many lower-dimensional models can be obtained from higher-dimensional models by dimensional reduction [15, 16]. Thus, a subset of lower-dimensional supersymmetric theories derives from the landscape of possible ways that extra dimensions can be removed. But most lower-dimensional theories do not seem to be obtainable from higher-dimensional theories by such a process; they seem to exist only in lower-dimensions. We refer to a lower-dimensional model obtained by dimensional reduction of a higher-dimensional model as the “shadow” of the higher-dimensional model. So we could rephrase our comment above by saying that not all lower dimensional supersymmetric theories may be interpreted as shadows.It is a straightforward process to construct a shadow theory from a given higher-dimensional theory. But it is a more subtle proposition to construct a higher-dimensional supersymmetric model from a lower-dimensional model, or to determine whether a lower dimensional model actually does describe a shadow, especially of a higher-dimensional theory which is also supersymmetric. We have found resident within lower-dimensional supersymmetry an algebraic key which provides access to this information. A primary purpose of this paper is to explain this.It is especially interesting to consider reduction to one time-like dimension, by switching off the dependence of all fields on all of the spatial coordinates. Such a process reduces quantum field theory to quantum mechanics. Upon making such a reduction, information regarding the spin representation content of the component fields is replaced withℛ-charge assignments. But it is not obvious whether the full higher-dimensional field content, or the fact that the one-dimensional model can be obtained in this way, is accessible information given the one-dimensional theory alone. As it turns out, this information lies encoded within the extended one-dimensional supersymmetry transformation rules.We refer to the process of restructuring a one-dimensional theory so that fields depend also on extra dimensions in a way consistent with covariant𝔰𝔭𝔦𝔫(1,D-1) assignments and other structures, such as higher-dimensional supersymmetries, as “dimensional enhancement”. This process describes the reverse of dimensional reduction. We like to envision this in terms of the relationship between a higher-dimensional “ambient” theory, and the restriction to a zero-brane embedded in the larger space. A supersymmetric quantum mechanics then describes the “worldline" physics on the zero-brane. And the question as to whether this worldline physics “enhances” to an ambient space-time field theory is the reverse of viewing the worldline physics as the restriction of a target-space theory to the zero-brane.If the particular supersymmetric quantum mechanics obtained by restriction of a given theory to a zero-brane depended on the particular𝔰𝔭𝔦𝔫(1,D-1)-frame described by that zero-brane, the higher-dimensional theory would not respect 𝔰𝔭𝔦𝔫(1,D-1)-invariance. Thus, if a one-dimensional theory enhances into a 𝔰𝔭𝔦𝔫(1,D-1)-invariant higher-dimensional theory, then the higher-dimensional theory obtained in this way should be agnostic regarding the presence or absence of an actual zero-brane on which such a one-dimensional theory might live. This observation, in conjunction with the requirement of higher-dimensional supersymmetry, provides the requisite constraint needed to resolve the enhancement question. In particular, by imposing 𝔰𝔭𝔦𝔫(1,D-1)-invariance on the enhanced supercharge operator, we are able to complete the ambient field-theoretic supercharge operator given merely the “time-like" restrictions of this operator. We find this interesting and surprising.The proposition that one can systematically delineate those one-dimensional theories which can enhance to higher-dimensions, and also discern how the higher-dimensional spin structures may be switched back on, is empowered by the fact that the representation theory of one-dimensional supersymmetry is relatively tame when compared with the representation theory of higher-dimensional supersymmetry, for a variety of reasons. This enables the prospect of disconnecting the problem of spin assignments from the problem of classifying and enumerating supersymmetry representations, allowing these concerns to be addressed separately, and then merged together afterwards. With this motivation in mind, we have been developing a mathematical context for the representation theory of one-dimensional supersymmetry, also with other collaborators.The primary purpose of this paper is to demonstrate that the landscape of supersymmetry representation theory (in any number of space-time dimensions) resides fully-encoded within the seemingly-restricted regime of one-dimensional worldline supersymmetry representation theory. We find this result remarkable, compelling, and noteworthy, regardless of how complicated it may prove to algorithmically “extract" this information. But we demonstrate below that algorithms to perform such extractions do exist. In fact, we present explicit examples of algorithms which delineate one-dimensional models which are shadows of higher-dimensional models from those which are not. We do not purport that our algorithms are optimized. And we view this paper as a plateau from which more efficient algorithms could be developed. A cursory accounting of the complexity of the general problem is addressed in Section5.In a sequence of papers [17–23], we have explored the connection between representations of supersymmetry and aspects of graph theory. We have shown that elements of a wide and physically relevant class of one-dimensional supermultiplets with vanishing central charge are equivalent to specific bipartite graphs which we call Adinkras; all of the salient algebraic features of the multiplets translate into restrictive and defining features of these objects. A systematic enumeration of those graphs meeting the requisite criteria would thereby supply means for a corresponding enumeration of representations of supersymmetry.In [24, 25], we have developed the paradigm further, explaining how, in the case of N-extended supersymmetry, the topology of all connected Adinkras are specified by quotients of N-dimensional cubes, and how the quotient groups are equivalent to doubly-even linear binary block codes. Thus, the classification of connected Adinkras is related to the classification of such codes. In this way we have discovered an interesting connection between supersymmetry representation theory and coding theory [26–28]. All of this is part of an active endeavor aimed at delineating a mathematically-rigorous representation theory in one-dimension.In this paper we use the language of Adinkras, in a way which does not presuppose a deep familiarity with this topic. We have included AppendixB as a brief and superficial primer, which should enable the reader to appreciate the entirety of this paper self-consistently. Further information can be had by consulting our earlier papers on the subject.In this paper we focus on the special case of enhancement of one-dimensionalN=4 supersymmetric theories into four-dimensional N=1 theories. This is done to keep our discussion concise and concrete. Another motivating reason is because the supersymmetry representation theory for 4D N=1 theories is well known. Thus, part and parcel of our discussion amounts to a consistency check on the very formalism we are developing. From this point of view, this paper provides a first step in what we hope is a continuing process by which yet-unknown aspects of off-shell supersymmetry can be discerned. In the context of 4D theories, we use standard physics nomenclature, and refer to 𝔰𝔭𝔦𝔫(1,3)-invariance as “Lorentz" invariance.We should mention that the prospect that aspects of higher-dimensional supersymmetry might be encoded in one-dimensional theories was suggested years ago in unpublished work [29] by Gates et al. Accordingly, we had used that attractive proposition as a prime motivator for developing the Adinkra technology in our earlier work. This paper represents a tangible realization of that conjecture. Complementary approaches towards resolving a supersymmetry representation theory have been developed in [30–36]. Other ideas concerning the relevance of one-dimensional models to higher-dimensional physics were explored in [37, 38].This paper is structured as follows. In Section2 we describe an algebraic context for discussing supersymmetry tailored to the process of dimensional reduction to zero-branes and, vice-versa, to enhancing one-dimensional theories. We explain how higher-dimensional spin structures can be accommodated into vector spaces spanned by the boson and fermion fields, and how the supercharges can be written as first-order linear differential operators which are also matrices which act on these vector spaces. This is done by codifying the supercharge in terms of diophantine “linkage matrices", which describe the central algebraic entities for analyzing the enhancement question.In Section3 we explain how Lorentz invariance allows one to determine “space-like” linking matrices from the “time-like” linking matrices associated with one-dimensional supermultiplets, and thereby construct a postulate enhancement. We then use this to derive nongauge enhancement conditions, which provide an important sieve which identifies those one-dimensional multiplets which cannot enhance to four-dimensional nongauge matter multiplets.In Section4 we apply our formalism in a methodical and pedestrian manner to the context of minimal one-dimensional N=4 supermultiplets, and show explicitly how the known structure of 4D N=1 nongauge matter may be systematically determined using one-dimensional reasoning coupled only with a choice of 4D spin structure. We explain also how our nongauge enhancement condition provides the algebraic context which properly delineates the chiral multiplet shadow from its 1D “twisted" analog, explaining why the latter cannot enhance.In Section6 we generalize our discussion to include 2-form field-strengths subject to Bianchi identities. This allows access to the question of enhancement to vector multiplets. In the process we introduce the notion of one-dimensional “phantom" fields which prove useful in understanding how gauge invariance manifests on shadow theories. We use the context of the 4D N=1 Abelian vector multiplet as an archetype for future generalizations.We also include five appendices which are an important part of this paper. AppendixA is especially important, as this provides the mathematical proof that imposing Lorentz invariance of the postulate linkage matrices allows one to correlate the entire higher-dimensional supercharge with its time-like restriction. We also derive in this appendix algebraic identities related to the spin structure of enhanced component fields, which should provide for interesting study in the future generalizations of this work.AppendixB is a brief summary of our Adinkra conventions, explaining technicalities, such as sign conventions, appearing in the bulk of the paper. Appendix C explains the dimensional reduction of the 4D N=1 chiral multiplet, complementary to the nongauge enhancement program described in Section 4. Appendix D explains the dimensional reduction of the 4D N=1 Maxwell field-strength multiplet, complementary to Section 5. This shows in detail how phantom sectors correlate with gauge aspects of the higher-dimensional theory. Appendix E is a discussion of four-dimensional spinors useful for understanding details of our calculations.We use below some specialized terminology. Accordingly, we finish this introduction section by providing the following three-term glossary, for reference purposes.Shadow
We refer to the one-dimensional multiplet which results from dimensional reduction of a higher-dimensional multiplet as the “shadow" of that higher-dimensional construction.Adinkra
The term Adinkra refers to 1D supermultiplets represented graphically, as explained in AppendixB. We sometimes use the terms Adinkra, supermultiplet, and multiplet synonymously.Valise
A Valise supermultiplet, or a Valise Adinkra, is one in which the component fields span exactly two distinct engineering dimensions. These multiplets form representative elements of larger “families" of supermultiplets derived from these using vertex-raising operations, as explained below. Thus, larger families of multiplets may be unpacked, as from a suitcase (or a valise), starting from one of these multiplets.


## 2. Ambient versus Shadow Supersymmetry

It is easy to derive a one-dimensional theory by dimensionally reducing any given higher-dimensional supersymmetric theory. Practically, this is done by switching off the dependence of all fields on the spatial coordinates, by setting∂a→0. One way to envision this process is as a compactification, whereby the spatial dimensions are rendered compact and then shrunk to zero size. Alternatively, we may envision this process as describing a restriction of a theory onto a zero-brane, which is a time-like one-dimensional submanifold embedded in a larger, ambient, space-time. Using this latter metaphor, we refer to the restricted theory as the “shadow" of the ambient theory, motivated by the fact that physical shadows are constrained to move upon a wall or a wire upon which the shadow is cast.


### 2.1. Ambient Supersymmetry

Supersymmetry transformation rules can be written in terms of off-shell degrees of freedom, by expressing all fields and parameters in terms of individual tensor or spinor components. Thus, without loss of generality, we can write the set of boson components asϕi and the set of fermion components as ψı̂, without being explicitly committal as the the 𝔰𝔭𝔦𝔫(1,D-1)-representation implied by these index structures. Generically, a 𝔰𝔭𝔦𝔫(1,D-1)-transformation acts on these components as  (2.1)δLϕi=12θμν(Tμν)ijϕj,δLψı̂=12θμν(T̃μν)ı̂̂ψ̂,
where the label L is a mnemonic which specifies these as “Lorentz" transformations. Here, (Tμν)ij represents the spin algebra as realized on the boson fields and (T̃μν)ı̂𝚥̂ represents the spin algebra as realized on the fermion fields, while θ0a parameterizes a boost in the ath spatial direction and θab parameterizes a rotation in the ab-plane. According to the spin-statistics theorem, (T̃μν)ı̂𝚥̂ should describe a spinor representation and (Tμν)ij should describe a direct product of tensor representations. The spin representations may also involve constraints. For example, boson components may configure as closed p-forms.In four-dimensions theN=1 supersymmetry algebra is generated by a Majorana spinor supercharge with components 𝒬A subject to the anticommutator relationship {𝒬A,𝒬B}=2iGABμ∂μ where GABμ=-(ΓμC-1)AB. A parameter-dependent supersymmetry transformation is generated by δQ(ϵ)=-iϵ¯A𝒬A, where ϵA describes an infinitesimal Majorana spinor parameter, and ϵ¯A=(ϵ†Γ0)A is the corresponding barred spinor. It proves helpful, for our express purpose of restricting to a zero-brane, to use a Majorana basis where all spinor components, and all four gamma matrices, are real. (See Appendix E for specifics related to this basis.) Furthermore, in this basis, we have the nice result GAB0=δAB. With this choice, we can rewrite our supersymmetry transformation as δQ(ϵ)=-iϵAQA, where QA=(Γ0)AB𝒬B. (This merely technical reorganization facilitates dimensional reduction of 4D multiplets, as done in Appendices C and D.) The four-dimensional N=1 supersymmetry algebra, written in terms of the operators QA, is then given by  (2.2){QA,QB}=2iδAB∂τ-2iGABa∂a,
where x0:=τ is the time-like coordinate parameterizing the the zero-brane to which we intend to restrict, and xa:=(x1,x2,x3) are the three space-like coordinates transverse to the zero-brane. To dimensionally reduce a four-dimensional field theory to a one-dimensional field theory, we set ∂a=0. In this way, the second term on the right-hand side of (2.2) disappears, and we obtain the one-dimensional N=4 supersymmetry algebra.It proves helpful to add a notational distinction, by writingδQϕi=-iϵA(QA)iı̂ψı̂ and δQψı̂=-iϵA(Q̃A)ı̂iϕi, appending a tilde to Q̃A when this describes a fermion transformation rule. The supercharges may be represented as first-order linear differential operators, as  (2.3)(QA)iı̂=(uA)iı̂+(ΔAμ)iı̂∂μ,(Q̃A)ı̂i=i(ũA)ı̂i+i(Δ̃Aμ)ı̂i∂μ,
where uA, ũA, ΔAμ, and Δ̃Aμ are real valued “linkage matrices" which play a central role in our discussion below.The matrices(uA)iı̂ describe “links” corresponding to supersymmetry maps from the bosons ϕi to fermions ψı̂ having engineering dimension one-half unit greater than the bosons. Therefore, these codify “upward” maps connecting lower-weight fermions to higher-weight bosons. (The term “weight” refers to the engineering dimension of the field. We sometimes use the term weight in lieu of dimension, to avoid confusion with space-time dimension. The weight of a field correlates with the vertex “height“ on an Adinkra diagram.) Similarly, the matrices (ũA)ı̂i codify “upward“ maps connecting lower-weight fermions to higher-weight bosons. The matrices (ΔAμ)iı̂ and (Δ̃Aμ)ı̂i codify “downward" maps accompanied by their respective derivatives ∂μ.The component fields may be construed so that the linkage matrices conform to a special structure, known as the Adinkraic structure. This says that there is at most one nonvanishing entry in each column and at most one nonvanishing entry in each row. Moreover, the nonvanishing entries take the values±1. All known higher-dimensional off-shell representations in the standard literature satisfy this condition. (The only counter examples that we know of were contrived by us in [20], as special deformations of one-dimensional Adinkraic representations. And we suspect that these do not enhance. Further scrutiny will be needed to ascertain any relevance of nonAdinkraic multiplets to physics. We find it sensible for now to focus on Adinkraic representations, especially since all known field theoretic multiplets are in this class.)The supersymmetry algebra (2.2) implies  (2.4)(u(AũB))ij=0,(ũ(AuB))ı̂̂=0,(Δ(A(μΔ̃B)ν))ij=0,(Δ̃(A(μΔB)ν))ı̂̂=0,
which describes a higher-dimensional analog of the Adinkra loop parity rule described in [17] and below, and also implies  (2.5)(u(AΔ̃B)μ+Δ(AμũB))ij=ΛABμδij,(ũ(AΔB)μ+Δ̃(AμuB))ı̂̂=ΛABμδı̂̂,
where ΛABμ=(Γ0GμΓ0)AB=(Γ0ΓμΓ0C-1)AB, whereby ΛAB0=GAB0 and ΛABa=-GABa. Equations (2.5) play a central role in this paper.The classification of representations of supersymmetry in diverse dimensions is equivalent to the question of classifying and enumerating the possible sets of real linkage matrices which can satisfy the algebraic requirements in (2.4) and (2.5), and identifying the corresponding spin representation matrices (Tμν)ij and (T̃μν)ı̂𝚥̂.


### 2.2. Shadow Supersymmetry

The one-dimensionalN=4 superalgebra is specified by {QA,QB}=2iδAB∂0, which corresponds to (2.2) in the limit ∂a→0. In this case, the supercharges are represented as  (2.6)(QA)iı̂=(uA)iı̂+(dA)iı̂∂τ,(Q̃A)ı̂i=i(ũA)ı̂i+i(d̃A)ı̂i∂τ.
This is identical to (2.3) except the index μ is restricted to the sole value μ=0, and the down matrices have been renamed by writing ΔA0 as dA and Δ̃A0 as d̃A. As mentioned above, the fields may be configured so that each linkage matrix has not more than one nonvanishing entry in each row and likewise in each column, and the nonvanishing entries are ±1. This specialized structuring enables the faithful translation of 1D supercharges in terms of helpful and interesting graphs known as Adinkras, as mentioned in Section 1. The reader should consult Appendix B for a simple-but-practical overview of this concept.The algebra obeyed by 1D linkage matrices may be obtained from (2.4) and (2.5) by allowing only the value 0 for the space-time indices μ and ν. Thus, the linkage matrices are constrained by  (2.7)(u(AũB))ij=0,(ũ(AuB))ı̂̂=0,(d(Ad̃B))ij=0,(d̃(AdB))ı̂̂=0.
These relationships imply a “loop parity" rule, described in our earlier papers, which says that any closed bicolor loop on an Adinkra diagram must involve an odd number of edges with odd parity. The linkage matrices are further constrained by  (2.8)(u(Ad̃B)+d(AũB))ij=δABδij,(ũ(AdB)+d̃(AuB))ı̂̂=δABδı̂̂.
In this context, the algebra defined by (2.8) was called a “Garden algebra” by Gates and Rana in [39, 40], and the the matrices uA and dA were called Garden matrices. The larger algebra given in (2.4) and (2.5) generalizes this concept to diverse space-time dimensions, and accordingly subsumes these smaller algebras.A one-dimensional supermultiplet is specified by the set of linkage matricesuA, ũA, dA, and d̃A or equivalently by the Adinkra diagram representing these matrices. Given a set of linkage matrices one can construct the equivalent Adinkra. Alternatively, given an Adinkra, one can use this to “read off” the equivalent set of linkage matrices. Given either of these, one can ascertain supersymmetry transformation rules and invariant action functionals from which one can study one-dimensional physics. The linkage matrices associated with any Adinkra satisfy the algebra (2.7) and (2.8) by definition.


## 2.1. Ambient Supersymmetry

Supersymmetry transformation rules can be written in terms of off-shell degrees of freedom, by expressing all fields and parameters in terms of individual tensor or spinor components. Thus, without loss of generality, we can write the set of boson components asϕi and the set of fermion components as ψı̂, without being explicitly committal as the the 𝔰𝔭𝔦𝔫(1,D-1)-representation implied by these index structures. Generically, a 𝔰𝔭𝔦𝔫(1,D-1)-transformation acts on these components as  (2.1)δLϕi=12θμν(Tμν)ijϕj,δLψı̂=12θμν(T̃μν)ı̂̂ψ̂,
where the label L is a mnemonic which specifies these as “Lorentz" transformations. Here, (Tμν)ij represents the spin algebra as realized on the boson fields and (T̃μν)ı̂𝚥̂ represents the spin algebra as realized on the fermion fields, while θ0a parameterizes a boost in the ath spatial direction and θab parameterizes a rotation in the ab-plane. According to the spin-statistics theorem, (T̃μν)ı̂𝚥̂ should describe a spinor representation and (Tμν)ij should describe a direct product of tensor representations. The spin representations may also involve constraints. For example, boson components may configure as closed p-forms.In four-dimensions theN=1 supersymmetry algebra is generated by a Majorana spinor supercharge with components 𝒬A subject to the anticommutator relationship {𝒬A,𝒬B}=2iGABμ∂μ where GABμ=-(ΓμC-1)AB. A parameter-dependent supersymmetry transformation is generated by δQ(ϵ)=-iϵ¯A𝒬A, where ϵA describes an infinitesimal Majorana spinor parameter, and ϵ¯A=(ϵ†Γ0)A is the corresponding barred spinor. It proves helpful, for our express purpose of restricting to a zero-brane, to use a Majorana basis where all spinor components, and all four gamma matrices, are real. (See Appendix E for specifics related to this basis.) Furthermore, in this basis, we have the nice result GAB0=δAB. With this choice, we can rewrite our supersymmetry transformation as δQ(ϵ)=-iϵAQA, where QA=(Γ0)AB𝒬B. (This merely technical reorganization facilitates dimensional reduction of 4D multiplets, as done in Appendices C and D.) The four-dimensional N=1 supersymmetry algebra, written in terms of the operators QA, is then given by  (2.2){QA,QB}=2iδAB∂τ-2iGABa∂a,
where x0:=τ is the time-like coordinate parameterizing the the zero-brane to which we intend to restrict, and xa:=(x1,x2,x3) are the three space-like coordinates transverse to the zero-brane. To dimensionally reduce a four-dimensional field theory to a one-dimensional field theory, we set ∂a=0. In this way, the second term on the right-hand side of (2.2) disappears, and we obtain the one-dimensional N=4 supersymmetry algebra.It proves helpful to add a notational distinction, by writingδQϕi=-iϵA(QA)iı̂ψı̂ and δQψı̂=-iϵA(Q̃A)ı̂iϕi, appending a tilde to Q̃A when this describes a fermion transformation rule. The supercharges may be represented as first-order linear differential operators, as  (2.3)(QA)iı̂=(uA)iı̂+(ΔAμ)iı̂∂μ,(Q̃A)ı̂i=i(ũA)ı̂i+i(Δ̃Aμ)ı̂i∂μ,
where uA, ũA, ΔAμ, and Δ̃Aμ are real valued “linkage matrices" which play a central role in our discussion below.The matrices(uA)iı̂ describe “links” corresponding to supersymmetry maps from the bosons ϕi to fermions ψı̂ having engineering dimension one-half unit greater than the bosons. Therefore, these codify “upward” maps connecting lower-weight fermions to higher-weight bosons. (The term “weight” refers to the engineering dimension of the field. We sometimes use the term weight in lieu of dimension, to avoid confusion with space-time dimension. The weight of a field correlates with the vertex “height“ on an Adinkra diagram.) Similarly, the matrices (ũA)ı̂i codify “upward“ maps connecting lower-weight fermions to higher-weight bosons. The matrices (ΔAμ)iı̂ and (Δ̃Aμ)ı̂i codify “downward" maps accompanied by their respective derivatives ∂μ.The component fields may be construed so that the linkage matrices conform to a special structure, known as the Adinkraic structure. This says that there is at most one nonvanishing entry in each column and at most one nonvanishing entry in each row. Moreover, the nonvanishing entries take the values±1. All known higher-dimensional off-shell representations in the standard literature satisfy this condition. (The only counter examples that we know of were contrived by us in [20], as special deformations of one-dimensional Adinkraic representations. And we suspect that these do not enhance. Further scrutiny will be needed to ascertain any relevance of nonAdinkraic multiplets to physics. We find it sensible for now to focus on Adinkraic representations, especially since all known field theoretic multiplets are in this class.)The supersymmetry algebra (2.2) implies  (2.4)(u(AũB))ij=0,(ũ(AuB))ı̂̂=0,(Δ(A(μΔ̃B)ν))ij=0,(Δ̃(A(μΔB)ν))ı̂̂=0,
which describes a higher-dimensional analog of the Adinkra loop parity rule described in [17] and below, and also implies  (2.5)(u(AΔ̃B)μ+Δ(AμũB))ij=ΛABμδij,(ũ(AΔB)μ+Δ̃(AμuB))ı̂̂=ΛABμδı̂̂,
where ΛABμ=(Γ0GμΓ0)AB=(Γ0ΓμΓ0C-1)AB, whereby ΛAB0=GAB0 and ΛABa=-GABa. Equations (2.5) play a central role in this paper.The classification of representations of supersymmetry in diverse dimensions is equivalent to the question of classifying and enumerating the possible sets of real linkage matrices which can satisfy the algebraic requirements in (2.4) and (2.5), and identifying the corresponding spin representation matrices (Tμν)ij and (T̃μν)ı̂𝚥̂.


## 2.2. Shadow Supersymmetry

The one-dimensionalN=4 superalgebra is specified by {QA,QB}=2iδAB∂0, which corresponds to (2.2) in the limit ∂a→0. In this case, the supercharges are represented as  (2.6)(QA)iı̂=(uA)iı̂+(dA)iı̂∂τ,(Q̃A)ı̂i=i(ũA)ı̂i+i(d̃A)ı̂i∂τ.
This is identical to (2.3) except the index μ is restricted to the sole value μ=0, and the down matrices have been renamed by writing ΔA0 as dA and Δ̃A0 as d̃A. As mentioned above, the fields may be configured so that each linkage matrix has not more than one nonvanishing entry in each row and likewise in each column, and the nonvanishing entries are ±1. This specialized structuring enables the faithful translation of 1D supercharges in terms of helpful and interesting graphs known as Adinkras, as mentioned in Section 1. The reader should consult Appendix B for a simple-but-practical overview of this concept.The algebra obeyed by 1D linkage matrices may be obtained from (2.4) and (2.5) by allowing only the value 0 for the space-time indices μ and ν. Thus, the linkage matrices are constrained by  (2.7)(u(AũB))ij=0,(ũ(AuB))ı̂̂=0,(d(Ad̃B))ij=0,(d̃(AdB))ı̂̂=0.
These relationships imply a “loop parity" rule, described in our earlier papers, which says that any closed bicolor loop on an Adinkra diagram must involve an odd number of edges with odd parity. The linkage matrices are further constrained by  (2.8)(u(Ad̃B)+d(AũB))ij=δABδij,(ũ(AdB)+d̃(AuB))ı̂̂=δABδı̂̂.
In this context, the algebra defined by (2.8) was called a “Garden algebra” by Gates and Rana in [39, 40], and the the matrices uA and dA were called Garden matrices. The larger algebra given in (2.4) and (2.5) generalizes this concept to diverse space-time dimensions, and accordingly subsumes these smaller algebras.A one-dimensional supermultiplet is specified by the set of linkage matricesuA, ũA, dA, and d̃A or equivalently by the Adinkra diagram representing these matrices. Given a set of linkage matrices one can construct the equivalent Adinkra. Alternatively, given an Adinkra, one can use this to “read off” the equivalent set of linkage matrices. Given either of these, one can ascertain supersymmetry transformation rules and invariant action functionals from which one can study one-dimensional physics. The linkage matrices associated with any Adinkra satisfy the algebra (2.7) and (2.8) by definition.


## 3. Enhancement Criteria for Shadow Supermultiplets

The requirement that the linkage matrices appearing in the supercharges (2.3) are 𝔰𝔭𝔦𝔫(1,D-1)-invariant has remarkable implications. One of these is the fact that the “space-like” linkage matrices ΔAa are completely determined by the “time-like” linkage matrices ΔA0. The proof of this assertion is given as Appendix A, with the result(3.1)(ΔAa)iı̂=-(Γ0Γa)AB(ΔB0)iı̂,(Δ̃Aa)ı̂i=-(Γ0Γa)AB(Δ̃B0)ı̂i.
It is interesting that the matrix (Γ0Γa)AB is precisely twice a boost operator in the ath spatial direction, in the spinor representation. It is also interesting that the result (3.1) holds irrespective of the 𝔰𝔭𝔦𝔫(1,D-1)-representations described by the component fields. That is, the assignment of the matrices (Tμν)ij and (T̃μν)ı̂𝚥̂ defined in (2.1) does not influence (3.1). These nontrivial consequences are derived explicitly in Appendix A. (The Lorentz invariance of the linkage matrices does imply interesting and interlocking constraints on the allowable choices of (Tμν)ij and (T̃μν)ı̂𝚥̂. These are exhibited in Appendix A. Such correlations are certainly expected, and we suspect that (A.4) and (A.5) have deep and useful implications, which we hope to explore in future work.)It is worth mentioning that the form of (3.1) agrees precisely with the linkage matrices derived from Salam-Strathdee superfields. Also, the appearance of Γa on the right-hand side is tied closely to the appearance of the Γa in the defining supersymmetry algebra.The result (3.1) is the crux ingredient which allows one to determine whether a given one-dimensional supermultiplet describes the shadow of a higher-dimensional supermultiplet. This follows because any one-dimensional multiplet organizes as (2.3) where the index μ assumes only the value 0. To probe whether that multiplet describes a shadow, one creates “provisional” off-brane linkages using the powerful expression (3.1). Since there is no algebraic guarantee that the transformation rules so-extended will properly close the higher-dimensional superalgebra, nor that the boson and fermion vector spaces will properly assemble into representations of the higher-dimensional spin group, the higher-dimensional superalgebra itself, applied to this construction, provides the requisite analytic probe of that possibility: if the one-dimensional multiplet is a shadow then the provisional construction will close the higher-dimensional superalgebra; if it is not possible, then it will not.The supersymmety algebra inD-dimensions closes only if (ΩABμ)ij∂μϕj=0 and (Ω̃ABμ)ı̂𝚥̂∂μψ𝚥̂=0, where we define the following useful matrices:  (3.2)(ΩABμ)ij=(u(AΔ̃B)μ+Δ(AμũB))ij-ΛABμδij,(Ω̃ABμ)ı̂̂=(ũ(AΔB)μ+Δ̃(AμuB))ı̂̂-ΛABμδı̂̂.
This requirement is a minor restructuring of (2.5). In this way, we have written the supersymmetry algebra as a linear algebra problem, cast as matrix equations.Many important supemultiplets exhibit gauge invariances, manifest as physical redundancies inherent in the vector spaces spanned by the component fields. In these cases, the matrices(ΩABμ)ij and (Ω̃ABμ)ı̃𝚥̃, are not unique. Instead these describe classes of matrices interrelated by operations faithful to the gauge structure. We describe this interesting situation, in Section 6. It is useful, however, to begin our discussion with what we call nongauge matter multiplets, which do not exhibit redundancies of this sort. For this smaller but nevertheless interesting and relevant class of supermultiplets, the higher-dimensional supersymmetry algebra is satisfied only if  (3.3)(ΩABμ)ij=0,(Ω̃ABμ)ı̂̂=0.
We refer to these equations as our nongauge enhancement criteria. These enable a practical algorithm for testing whether a given 1D supermultiplet represents the shadow of a 4D nongauge matter multiplet.We use the linkage matrices for a given 1D supermultiplet in conjuction with the 4D gamma matrices to compute all of thed×d matrices ΩABμ and Ω̃ABμ, defined in (3.2). If (3.3) is satisfied, that is, if all of these matrices are identically null, then the 1D multiplet passes an important, nontrivial, and necessary requirement for enhancement to a 4D nongauge matter multiplet. If these matrices do not vanish, then the 1D multiplet cannot enhance to a 4D nongauge matter multiplet. In the latter case, further analysis must be done to probe whether this multiplet can enhance to a gauge multiplets. Equation (3.3) represents a useful “sieve" in the separation of 1D multiplets into groups as shadows versus nonshadows.A second important sieve derives from the spin-statistics theorem. As it turns out, a minority of 1D multiplets actually pass the test (3.3). But those that do come in pairs related in-part by a Klein flip, which is an involution under which the statistics of the fields are reversed—boson fields are replaced with fermion fields and vice versa. Thus, we can organize those multiplets which pass the test (3.3) into such pairs. We then ascertain which elements of each pair satisfy the requirement that fermions assemble as spinors and the bosons as tensors. Those multiplets that do not pass this test describe another class of multiplets which do not describe ordinary shadows. Typically, one multiplet out of each pairing satisfies the spin-statistics test while the other multiplet fails this test. (The important role of the Klein flip in the representation theory of superalgebras was addressed by one of the authors (G. L.) in previous work [41].)In the explicit examples analyzed below in this paper, it is obvious when certain multiplets which pass the first enhancement test (3.3) fail the spin-statistics test. This occurs when the multiplicity of fermions with a common engineering dimension is not a multiple of four, thereby obviating assemblage into 4D spinors. In fact our analysis below is remarkably clean. In more general cases, we suspect that more careful attention to the implications of the Lorentz invariance of the provisional supercharge, codified by equations such as (A.4), will provide the requisite sophistication needed to address enhancement at higher N and higher D. We think this will be a most interesting undertaking.


## 4. Nongauge Matter Multiplets

In this section we impose our enhancement equation (3.3) on the linkage matrices associated with all of the minimal N=4 Adinkras, of which there are 60 in total, to ascertain which of these represent shadows of 4D N=1 nongauge matter multiplets. Since the represention theory for minimal irreducible multiplets in 4D N=1 supersymmetry is well known, this setting provides a natural laboratory for testing our technology. The principal result of this section is that our enhancement equation properly corroborates what is known about nongauge matter in 4D N=1 supersymmetry, thereby passing an important consistency test. Another principal result of this section identifies our enhancement equation as the natural algebraic sieve which distinguishes the chiral multiplet shadow from its “twisted” analog.By the term “nongauge matter multiplets” we refer to 4D supermultiplets which involve component fields neither subject to gauge transformations nor subject to differential constraints, such as Bianchi identities. This excludes the vector and the tensor multiplets, as well as the the corresponding field-strength multiplets. We postpone a discussion of these interesting cases until the next section. In fact, the only nongauge matter multiplet in 4DN=1 supersymmetry is the chiral multiplet. (An antichiral multiplet, which can be formed as the Hermitian conjugate of a chiral multiplet, is not distinct from the latter as representation of the 4D N=1 supersymmetry algebra separate from inherent complex structures; the assignment of possible U(1) charge assignments represents “extra" data not considered overtly in this paper. Ignoring the complex structure, the chiral and the antichiral multiplets have indistinguishable shadows.) As we will see, among the 60 different minimal N=4 Adinkras there are exactly four which satisfy our primary enhancement condition (3.3). For two of these, the fermions configure as a spinor and the bosons configure as Lorentz scalars. For the other two the fermions configure as Lorentz scalars while the bosons configure as a spinor. The latter case fails the spin-statistics test, which says that fermions must assemble as spinors, and bosons must assemble as tensors. Thus, our method identifies the two Adinkras which can provide shadows of 4D minimal nongauge matter multiplets. (Conceivably, the fact that there are two such enhanceable N=4 minimal Adinkras may relate to the fact that there are two complementary choices of complex structure, related to the chiral and antichiral multiplets, as mentioned in the previous footnote.)It is noteworthy that there are two separate minimalN=4 Adinkra families, related by a so-called twist, implemented by toggling the parity of one of the four edge colors. Thus, the shadow of the chiral multiplet has a twisted analog which cannot enhance to 4D. That multiplet, which has been called the twisted chiral multiplet, describes 1D physics which cannot be obtained by restriction from four-dimensions. We have long wondered what algebraic feature distinguishes these two. (We learned about this interesting curiosity from Jim Gates, in the context of a former collaboration.) As it turns out, the linkage matrices for the chiral multiplet shadow satisfy the enhancement equations (3.3) whereas the linkage matrices for the twisted chiral multiplet do not. This answers this long-puzzling question. Details are presented below in this section.In order to ascertain whether a given Adinkra enhances to 4D we need to subject the corresponding linkage matrices to the space-like subset of the equations in (3.3). (The time-like equations are satisfied automatically, since an Adinkra is a representation of 1D supersymmetry by construction.) In the case of testing enhancement to 4D N=1 supersymmetry, each of the two conditions in (3.3) describes 30 matrix equations for each of the 60 Adinkras to be tested, since for each of the three choices for a, the corresponding symmetric matrices GABa=G(AB)a have ten independent components. Thus, according to the crudest counting argument, in order to test both the bosonic and fermionic conditions in (3.3) for all the minimal N=4 Adinkras, we need to check 60×30×2=3600 matrix equations, each involving products of 4×4 matrices. This is a simple matter which we have managed expediently using rudimentary Mathematica programming. (The complexity of the more general problem is addresed in the following section.)The smallest Adinkras which can possibly enhance to describe 4D supersymmetry areN=4 Adinkras describing 4+4 off-shell degrees of freedom. We therefore start by considering N=4 bosonic 4-4 Valise Adinkras. There are exactly two of these not interrelated by cosmetic field redefinitions. These are exhibited in Figure 1.  In this paper we correlate the four edge colors with choices of the index A so that   purple,  blue,  green, and  red correspond respectively to the operators Q1,2,3,4. For purposes of setting a convention for ordering the rows and columns of our linkage matrices, we sequence the boson fields ϕi and the fermion fields ψı̂ using the obvious faithful correspondence with the index choices. Furthermore, in the Adinkras exhibited in this section, the white vertices labeled 1, 2, 3, 4 represent the boson fields ϕi with corresponding index choices, while the black vertices labeled 1, 2, 3, 4 represent the fermion fields ψı̂ with corresponding index choices. This allows us to readily translate each Adinkra into precise linkage matrices, using the technology explained in Appendix B.Figure 1
The twoN=4 Valise Adinkras. The Adinkra on the right is obtained from the Adinkra on the left by implementing a “twist”, toggling the parity of the green edges.The linkage matrices(uA)iı̂ corresponding to the first Adinkra in Figure 1 are exhibited in Table 1. (The diligent reader should verify the correspondence between Figure 1 and Table 1 using the simple technology explained in Appendix B.) These codify the “upward" links connecting the bosons ϕi to the fermions ψı̂ having greater engineering dimension. Since there are no edges linking downward from any of the boson vertices, it follows that (dA)iı̂=0 in this case. Similarly, we have (ũA)ı̂i=0, reflecting the fact that none of the fermions have upward directed edges. Finally, we have (d̃A)ı̂j=δABδjk(uB)kk̂δk̂ı̂ and (dA)i𝚥̂=δABδ𝚥̂k̂(ũB)k̂kδki, schematically d̃A=uAT and dA=ũAT, reflecting the fact that every edge describes a pairing of an upward directed term and a corresponding downward directed term. The relationships d̃A=uAT and dA=ũAT are characteristic of “standard Adinkras". Nonstandard Adinkras, which can include “one-way" upward Adinkra edges, appear in gauge multiplet shadows, as explained below, and in also in other contexts of interest. (Some considerations involving “one-way" Adinkra edges were described in both [21, 23].)Table 1
The boson “up” linkage matrices for the left 4-4 Valise Adinkra shown in Figure1. The up matrices for the right Adinkra in that figure are obtained from these by changing the sign of u3.


### 4.1. TheN=4 Bosonic 4-4 Adinkras

Using the featuresũA=dB=0 and d̃A=uAT, and using the matrices GABa in (E.10), we can begin to analyze the enhancement equations associated with the left Adinkra in Figure 1. Consider the first equation in (3.3) for the index choices (a∣A,B)=(a∣1,1). Since Λ111=-G111=0, that 4×4 matrix equation reads u1Δ̃11=0. We then use (3.1), along with the gamma matrices in (E.3) to determine Δ̃11=-Δ̃30, which is equivalent to Δ̃11=-d̃3 using the nomenclature Δ̃A0≡d̃A. Thus, the first equation in (3.3) reduces for the left Adinkra in Figure 1 and the index selections (a∣A,B)=(1∣1,1) to the simple matrix equation u1u3T=0, where we have also used d̃3=u3T. Using Table 1, it is easy to check that this simple requirement is not satisfied. This tells us that the left Adinkra in Figure 1 cannot enhance to a 4D nongauge matter multiplet. Since the linkage matrices associated with the right Adinkra in Figure 1 are obtained from Table 1 by toggling the overall sign on u3 only, and since the enhancement equation u1u3T=0 is unchanged by such an operation, it follows that neither Adinkra in Figure 1 can enhance to a 4D nongauge matter multiplet.The methodology explained in the last paragraph can be applied systematically for each possible index choice(a∣A,B) for any selected Adinkra. In each case the time-like linkage matrices ΔABa are determined using (3.1), so that the enhancement equation can be translated to a matrix statement involving the linkage matrices specific to the 1D multiplet directly corresponding to the Adinkra. In the following discussion we do not repeat most of these steps. But the reader should be aware that (3.1) is used in each example which we discuss, and the use of this equation is what allows us to cast the enhancement equation in terms of the matrices uA, dA=ΔA0, and their transposes.


### 4.2. TheN=4 Bosonic 3-4-1 Adinkras

Consider next those Adinkras obtained by raising one vertex starting with each Adinkra in Figure1. There are four possibilities starting from each of the two Valises, namely one possibility associated with raising any one of the four bosons. For example, if we raise the boson vertex labeled “4” starting from each Valise, what results are the two Adinkras in Figure 2. In these cases, we end up with three bosons at the lowest level, four fermions at the next level, and a single boson at the next level. We refer to Adinkras with this distribution of vertex multiplicities as bosonic 3-4-1 Adinkras, where the sequence of numerals faithfully enumerates the sequence of vertex multiplicities at successively higher levels. (These alternate between boson and fermion multiplicities, naturally.) It is easy to see that there are exactly eight bosonic 3-4-1 N=4 Adinkras, and that these split evenly into two groups interrelated by a twist operation.Figure 2
The two 3-4-1 Adinkras obtained from the Valise Adinkras in Figure1 by raising one vertex. Here we have raised the boson vertex labeled 4.We should point out that two Adinkras are equivalent if they are mapped into each other by cosmetic renaming of vertices, equivalent to linear automorphisms on the vector spaces spanned by the bosonsϕi or fermions ψı̂, in cases where these maps preserve all vertex height assignments. Such transformations have been called “inner automorphisms”. The simplest examples correspond to rescaling any component field by a factor of −1. This manifests on an Adinkra by simultaneously toggling the parity of every edge connected to the vertex representing that field, that is, by changing dashed edges into solid edges and vice-versa. (This is referred to as “flipping the vertex”, and was described already in [17].) Our observation that there are two distinct families of minimal N=4 Adinkras interrelated by a twist operation refers to the readily-verifiable fact that one cannot “undo” a twist by any inner automorphism. (The curious reader might find it amusing to draw Adinkra diagrams, and investigate this statement for his or her self.) It is also true that there are only two twist classes of minimal N=4 Adinkras, despite the fact that there are four different colors which can be used to implement a twist. This is so because a given twist applied using any chosen edge color can be equivalently implemented as a twist applied using any other edge color augmented by a suitable inner automorphism.When we raise an Adinkra vertex, the up and down linkage matrices accordingly modify. For example, consider the the 3-4-1 Adinkra on the left in Figure2, obtained from the Adinkra on the left in Figure 1 by raising the ϕ4 vertex. The corresponding boson up and down matrices, which are straightforward to read off of the Adinkra, are shown in Table 2. Note that in this case the boson down matrices dA no longer vanish as they did in the case of the Valise. This is because the ϕ4 vertex obtains downward links after being raised. The fermion up matrices, which are determined for this standard Adinkra using ũA=dAT, are also nonvanishing after this vertex raise, since each of the fermions obtains an upward link to the boson ϕ4.Table 2
Linkage matrices for the left 3-4-1 Adinkra shown in Figure2. The linkage matrices for the right Adinkra in that figure are obtained from these by changing the sign of u3 and d3.Given a standard Adinkra, it is possible to raise thenth boson vertex if and only if the nth row of each boson down matrix is null, that is, provided (dA)nı̂=0 for all values of ı̂. This criterion ensures that the nth boson vertex does not have any downward links which would preclude the vertex from being raised. (Since for standard Adinkras we have ũA=dAT, this criterion also implies that there are no lower fermions which link upward to the boson in question.) Absent such a tethering, the boson is free to be raised. This operation is implemented algebraically by interchanging the nth row of each boson up matrix uA with the nth row of the respective boson down matrix dA. Thus, we implement the matrix reorganizations (uA)nı̂↔(dA)nı̂. At the same time, we must interchange the nth column of each fermion up matrix ũA with the nth column of the respective fermionic down matrix d̃A, via (ũA)ı̂n↔(d̃A)ı̂n. The latter operation preserves the standard relationships ũA=dAT and d̃A=uAT. It is easy to check that the linkage matrices in Table 2 are obtained from the linkage matrices in Table 1 by appropriately interchanging the fourth rows of the boson up and down matrices according to the above discussion.We now use the enhancement equation to analyze the eight standardN=4 bosonic 3-4-1 Adinkras to ascertain if any of these can enhance to a 4D N=1 nongauge matter multiplet. To begin, we start with the left Adinkra in Figure 2, by using the boson linkage matrices in Table 2 and the fermion linkage matrices determined by ũA=dAT and d̃A=uAT. Using the GABa given in (E.10), the first condition in (3.3) reduces for the choice (a∣AB)=(1∣11) to the matrix equation u1u3T+d3d1T=0. Using the explicit matrices in Table 2, it is easy to see that this requirement is not satisfied. This tells us that the left Adinkra in Figure 2 cannot enhance to a 4D nongauge matter multiplet.Since the right Adinkra in Figure2 is obtained from the left Adinkra in that figure by twisting the green edges, corresponding to replacing Q3→-Q3, the linkage matrices for that second 3-4-1 Adinkra are obtained from those in Table 2 by scaling the matrices u3, d3, ũ3 and d̃3 each by a multiplicative minus sign. The (a∣AB)=(1∣1,1) enhancement equation, u1u3T+d3d1T=0, is unchanged by this operation. So we conclude that neither Adinkra in Figure 2 can enhance to a nongauge 4D matter multiplet. It is straightforward to repeat this analysis for all cases associated with raising any possible single boson vertex starting with either of the Valise Adinkras in Figure 1. It follows, after careful analysis of each case, that the nongauge enhancement equation (3.3) is not satisfied for any of the eight bosonic 3-4-1 Adinkras.


### 4.3. TheN=4 Bosonic 2-4-2 Adinkras

Things become more interesting when we raise one of the lower bosons in 3-4-1 Adinkras to obtain 2-4-2 Adinkras. In the end there are twelve minimalN=4 bosonic 2-4-2 Adinkras—six obtained by two vertex raises starting from the left Adinkra in Figure 1 and six obtainable by two vertex raises starting from the right Adinkra in Figure 1. The six possibilities in each class correspond to the six different ways to select pairs from four choices. For example, if we raise ϕ3 and ϕ4 in either case then what results are the two 2-4-2 Adinkras shown in Figure 3. For the left Adinkra in Figure 3, the boson linkage matrices are shown in Table 3. (It is straightforward to read these matrices off of the Adinkra. It is also straightforward to obtain these matrices algebraically, as explained above, by interchanging the third rows of the 3-4-1 up and down matrices shown in Table 2.)Table 3
Linkage matrices for the left 2-4-2 Adinkra shown in Figure3. The linkage matrices for the right Adinkra in that figure are obtained from these by changing the sign of u3 and d3.Figure 3
TheN=4 2-4-2 Adinkras may be obtained from 3-4-1 Adinkras by raising one vertex. Here we have raised the third boson vertex starting with the two Adinkras shown in Figure 2.We now use the enhancement equation (3.3) to analyze the twelve standard N=4 2-4-2 Adinkras to ascertain if any of these can enhance to a 4D N=1 nongauge matter multiplet. To begin, we start with the left Adinkra in Figure 3, equivalently described by the boson linkage matrices specified in Table 3 and by the fermion linkage matrices determined from these by ũA=dAT and d̃A=uAT.We found above that for each of the twoN=4 Valise Adinkras and for each of the eight 3-4-1 Adinkras the enhancement equation corresponding to (a∣A,B)=(1∣1,1) is not satisfied. This is equivalent to the statement that the 4×4 matrix determined by u1u3T+d3d1T does not vanish in these cases. The reader should compute this combination in those cases, and then also compute this combination using the linkage matrices in Table 3. It is noteworthy that in this latter case, that is, using the matrices in Table 3, the computation of u1u3T+d3d1T does indeed produce the 4×4 null matrix. So the particular obstruction which we identified in the 4-4 and 3-4-1 Adinkras is notably absent for the specific 2-4-2 Adinkra shown on the left in Figure 1.Having satisfied the(a∣A,B)=(1∣1,1) equation, it remains to analyze all of the other possible choices for (a∣A,B) and check the enhancement equations (3.3) in each case. It is interesting to comment on the case (a∣A,B)=(1∣1,4), for example. In this case the enhancement equation reads  (4.1)u1u2T+d2d1T+u4u3T+d3d4T=0.
Note that this is satisfied using the matrices in Table 3. So the left 2-4-2 Adinkra in Figure 3 passes this second enhancement test. (Thus, this Adinkra passes two out of 60 different tests, counting the both the bosonic and the fermionic enhancement conditions for each of the 30 index choices (a∣A,B).)It is interesting that, unlike the left Adinkra in Figure3, the right Adinkra in Figure 3 fails the test (4.1). This can be seen by noting that (4.1) is sensitive to the parity on any one of the four edge colors since the overall signs on the first two terms flip upon toggling the sign on Q1 or Q2 while the sign on the third and fourth terms flip upon toggling the sign on Q3 or Q4. More specifically, the linkage matrices for the second Adinkra in Figure 3 are obtained from Table 3 by toggling the sign on Q3, which toggles the overall sign on the matrices u3 and d3. If we substitute the linkage matrices for the right Adinkra in Figure 3, obtained in this way, into (4.1) we find that this equation is no longer satisfied. Thus, we conclude that the right Adinkra in Figure 3 cannot enhance to a 4D nongauge matter multiplet. Again, the reader should check these assertions by doing a few simple matrix calculations.Further analysis of all of the remaining 58 enhancement conditions shows that the matrices in Table3 pass every one of these tests. This is a nontrivial accomplishment, which indicates that the left Adinkra in Figure 3 does represent the shadow of a 4D N=1 nongauge matter multiplet. As a representative example, consider the bosonic enhancement condition for the choice (a∣A,B)=(2∣3,2). This equation reads  (4.2)u2u2T+u3u3T+d2d2T+d3d3T=2,
where the factor of 2 on the right-hand side means twice the 4×4 unit matrix. The reader should verify that (4.2) is the bosonic enhancement equation described by (3.3) in this case. The reader should also verify that (4.2) is satisfied by the linkage matrices in Table 3.The fact that the first Adinkra in Figure3 describes the shadow of the 4D N=1 chiral multiplet is easy to check by performing a direct dimensional reduction of the chiral multiplet. This is done explicitly in Appendix C. In that appendix we derive the shadow Adinkra, shown in Figure 4, by direct translation of the 4D chiral multiplet transformation rules. The left Adinkra in Figure 3 is obtained from the Adinkra in Figure (4) by reorganizing fields according to the following four permutation operations: ϕ1↔ϕ2, ϕ3↔ϕ4, ψ1↔ψ4, and ψ2↔ψ3. This describes a cosmetic inner automorphism, indicating that the two Adinkras are equivalent.Figure 4
The shadow of the chiral multiplet, expressed as an Adinkra equivalent to the transformation rules (C.3).The fact that the left Adinkra in Figure3 passes the enhancement criteria while the right Adinkra does not identifies the left Adinkra as the chiral multiplet shadow and identifies the right Adinkra as the so-called twisted chiral multiplet. We also find that the 2-4-2 Adinkra obtained by raising the vertices ϕ1 and ϕ2 starting from the left 4-4 Valise in Figure 1 also passes all of the enhancement criteria whereas the twisted analog of this does not. Scrutiny of all twelve bosonic 2-4-2 Valises confirms that only those two cases in the nontwisted family, associated with the left Valise in Figure 1, obtained by raising either the pair (ϕ1,ϕ2) or the pair (ϕ3,ϕ4) can enhance to nongauge matter multiplets in 4D.


### 4.4. The Two 30-MemberN=4 Adinkra Families

It is straighforward to systematically check the enhancement conditions for all 30 Adinkras in each of the two families—one family associated with each of the two Valises in Figure1. In each case, the 30-member family consists of the bosonic 4-4 Adinkra (the Valise), four bosonic 3-4-1 Adinkras, six bosonic 2-4-2 Adinkras, four bosonic 1-4-3 Adinkras, and the Klein flip of each one of these 15 representatives. (The Klein flipped Adinkras are the fermionic 4-4 Valise and the 14 fermionic Adinkras obtainable from this by various vertex raises.)In total there are exactly four out of the 60 minimal Adinkras which pass our nongauge enhancement criteria (3.3). The first two are the bosonic 2-4-2 chiral multiplet shadows obtained by raising either ϕ1 and ϕ2 or by raising ϕ3 and ϕ4 starting from the left Adinkra in Figure 1. The other two Adinkras reside in the other (relatively twisted) family, and are obtained from the right Adinkra in Figure 1 by first raising all four bosonic vertices, then raising either the fermionic vertices ψ1 and ψ2 or by raising the fermionic vertices ψ3 and ψ4. These operations produce fermionic 2-4-2 Adinkras corresponding to twisted Klein flips of the two enhanceable bosonic 2-4-2 Adinkras. Since in these cases there are at most two fermions at any given height assignment, it is clear that these cannot assemble as 4D spinors. As we explained above, the Adinkras which pass the enhancement criteria come in pairs, one element of which passes the spin-statistics test and one which does not. In this way we conclude that of the 60 minimal Adinkras specified above only the two bosonic 2-4-2 cases can describe shadows of nongauge 4D matter multiplets.It might appear curious that the four bosonic 2-4-2 Adinkras obtained from the right Adinkra in Figure1 by raising (ϕ1,ϕ3) or (ϕ1,ϕ4) or (ϕ2,ϕ3) or (ϕ2,ϕ4) do not pass the enhancement criteria whereas the two Adinkras obtained by raising (ϕ1,ϕ2) or (ϕ3,ϕ4) do pass this test. The reason why certain combinations of component fields appear favored relates to the fact that we have made a choice of spin structure when we selected the particular gamma matrices in (E.9). It is interesting that we lose no generality in making such a choice, however, since the freedom to choose a 4D spin basis is replaced by a corresponding freedom to perform inner automorphisms on the vector space spanned by the 1D component fields.Upon selecting a higher-dimensional spin basis, the enhancement equations (3.3) place restrictions on the component fields which are legitimately meaningful; the result that exactly two out of the 60 minimal N=4 Adinkras enhance to nongauge 4D supersymmetric matter, along with the observation that those 1D multiplets which do enhance have 2-4-2 component multiplicities says something salient about 4D supersymmetry representation theory. Specifically it says that any 4D N=1 nongauge matter multiplet must have two physical bosons, four fermions, and two auxiliary bosons. This corroborates what has long been known about the minimal representations of 4D N=1 supersymmetry. What is remarkable is that we have hereby shown that this information is fully extractable using merely 1D supersymmetry and a choice of 4D spin structure—that this information lies fully encoded in the 1D supersymmetry representation theory codified by the families of Adinkras, and that the key to unlocking this information is contained in our enhancement equation (3.3). (We believe that the extra structures, namely that the bosons complexify and that the fermions assemble into chiral spinors, is also encoded in our formalism, using the equations (A.5). We also believe that deeper scrutiny of those equations should provide an algebraic context for broadly resolving natural organizations of supermultiplets, including complex structures, and quaternionic structures in diverse dimensions. But this lies beyond the scope of this introductory paper on this topic.)


## 4.1. TheN=4 Bosonic 4-4 Adinkras

Using the featuresũA=dB=0 and d̃A=uAT, and using the matrices GABa in (E.10), we can begin to analyze the enhancement equations associated with the left Adinkra in Figure 1. Consider the first equation in (3.3) for the index choices (a∣A,B)=(a∣1,1). Since Λ111=-G111=0, that 4×4 matrix equation reads u1Δ̃11=0. We then use (3.1), along with the gamma matrices in (E.3) to determine Δ̃11=-Δ̃30, which is equivalent to Δ̃11=-d̃3 using the nomenclature Δ̃A0≡d̃A. Thus, the first equation in (3.3) reduces for the left Adinkra in Figure 1 and the index selections (a∣A,B)=(1∣1,1) to the simple matrix equation u1u3T=0, where we have also used d̃3=u3T. Using Table 1, it is easy to check that this simple requirement is not satisfied. This tells us that the left Adinkra in Figure 1 cannot enhance to a 4D nongauge matter multiplet. Since the linkage matrices associated with the right Adinkra in Figure 1 are obtained from Table 1 by toggling the overall sign on u3 only, and since the enhancement equation u1u3T=0 is unchanged by such an operation, it follows that neither Adinkra in Figure 1 can enhance to a 4D nongauge matter multiplet.The methodology explained in the last paragraph can be applied systematically for each possible index choice(a∣A,B) for any selected Adinkra. In each case the time-like linkage matrices ΔABa are determined using (3.1), so that the enhancement equation can be translated to a matrix statement involving the linkage matrices specific to the 1D multiplet directly corresponding to the Adinkra. In the following discussion we do not repeat most of these steps. But the reader should be aware that (3.1) is used in each example which we discuss, and the use of this equation is what allows us to cast the enhancement equation in terms of the matrices uA, dA=ΔA0, and their transposes.


## 4.2. TheN=4 Bosonic 3-4-1 Adinkras

Consider next those Adinkras obtained by raising one vertex starting with each Adinkra in Figure1. There are four possibilities starting from each of the two Valises, namely one possibility associated with raising any one of the four bosons. For example, if we raise the boson vertex labeled “4” starting from each Valise, what results are the two Adinkras in Figure 2. In these cases, we end up with three bosons at the lowest level, four fermions at the next level, and a single boson at the next level. We refer to Adinkras with this distribution of vertex multiplicities as bosonic 3-4-1 Adinkras, where the sequence of numerals faithfully enumerates the sequence of vertex multiplicities at successively higher levels. (These alternate between boson and fermion multiplicities, naturally.) It is easy to see that there are exactly eight bosonic 3-4-1 N=4 Adinkras, and that these split evenly into two groups interrelated by a twist operation.Figure 2
The two 3-4-1 Adinkras obtained from the Valise Adinkras in Figure1 by raising one vertex. Here we have raised the boson vertex labeled 4.We should point out that two Adinkras are equivalent if they are mapped into each other by cosmetic renaming of vertices, equivalent to linear automorphisms on the vector spaces spanned by the bosonsϕi or fermions ψı̂, in cases where these maps preserve all vertex height assignments. Such transformations have been called “inner automorphisms”. The simplest examples correspond to rescaling any component field by a factor of −1. This manifests on an Adinkra by simultaneously toggling the parity of every edge connected to the vertex representing that field, that is, by changing dashed edges into solid edges and vice-versa. (This is referred to as “flipping the vertex”, and was described already in [17].) Our observation that there are two distinct families of minimal N=4 Adinkras interrelated by a twist operation refers to the readily-verifiable fact that one cannot “undo” a twist by any inner automorphism. (The curious reader might find it amusing to draw Adinkra diagrams, and investigate this statement for his or her self.) It is also true that there are only two twist classes of minimal N=4 Adinkras, despite the fact that there are four different colors which can be used to implement a twist. This is so because a given twist applied using any chosen edge color can be equivalently implemented as a twist applied using any other edge color augmented by a suitable inner automorphism.When we raise an Adinkra vertex, the up and down linkage matrices accordingly modify. For example, consider the the 3-4-1 Adinkra on the left in Figure2, obtained from the Adinkra on the left in Figure 1 by raising the ϕ4 vertex. The corresponding boson up and down matrices, which are straightforward to read off of the Adinkra, are shown in Table 2. Note that in this case the boson down matrices dA no longer vanish as they did in the case of the Valise. This is because the ϕ4 vertex obtains downward links after being raised. The fermion up matrices, which are determined for this standard Adinkra using ũA=dAT, are also nonvanishing after this vertex raise, since each of the fermions obtains an upward link to the boson ϕ4.Table 2
Linkage matrices for the left 3-4-1 Adinkra shown in Figure2. The linkage matrices for the right Adinkra in that figure are obtained from these by changing the sign of u3 and d3.Given a standard Adinkra, it is possible to raise thenth boson vertex if and only if the nth row of each boson down matrix is null, that is, provided (dA)nı̂=0 for all values of ı̂. This criterion ensures that the nth boson vertex does not have any downward links which would preclude the vertex from being raised. (Since for standard Adinkras we have ũA=dAT, this criterion also implies that there are no lower fermions which link upward to the boson in question.) Absent such a tethering, the boson is free to be raised. This operation is implemented algebraically by interchanging the nth row of each boson up matrix uA with the nth row of the respective boson down matrix dA. Thus, we implement the matrix reorganizations (uA)nı̂↔(dA)nı̂. At the same time, we must interchange the nth column of each fermion up matrix ũA with the nth column of the respective fermionic down matrix d̃A, via (ũA)ı̂n↔(d̃A)ı̂n. The latter operation preserves the standard relationships ũA=dAT and d̃A=uAT. It is easy to check that the linkage matrices in Table 2 are obtained from the linkage matrices in Table 1 by appropriately interchanging the fourth rows of the boson up and down matrices according to the above discussion.We now use the enhancement equation to analyze the eight standardN=4 bosonic 3-4-1 Adinkras to ascertain if any of these can enhance to a 4D N=1 nongauge matter multiplet. To begin, we start with the left Adinkra in Figure 2, by using the boson linkage matrices in Table 2 and the fermion linkage matrices determined by ũA=dAT and d̃A=uAT. Using the GABa given in (E.10), the first condition in (3.3) reduces for the choice (a∣AB)=(1∣11) to the matrix equation u1u3T+d3d1T=0. Using the explicit matrices in Table 2, it is easy to see that this requirement is not satisfied. This tells us that the left Adinkra in Figure 2 cannot enhance to a 4D nongauge matter multiplet.Since the right Adinkra in Figure2 is obtained from the left Adinkra in that figure by twisting the green edges, corresponding to replacing Q3→-Q3, the linkage matrices for that second 3-4-1 Adinkra are obtained from those in Table 2 by scaling the matrices u3, d3, ũ3 and d̃3 each by a multiplicative minus sign. The (a∣AB)=(1∣1,1) enhancement equation, u1u3T+d3d1T=0, is unchanged by this operation. So we conclude that neither Adinkra in Figure 2 can enhance to a nongauge 4D matter multiplet. It is straightforward to repeat this analysis for all cases associated with raising any possible single boson vertex starting with either of the Valise Adinkras in Figure 1. It follows, after careful analysis of each case, that the nongauge enhancement equation (3.3) is not satisfied for any of the eight bosonic 3-4-1 Adinkras.


## 4.3. TheN=4 Bosonic 2-4-2 Adinkras

Things become more interesting when we raise one of the lower bosons in 3-4-1 Adinkras to obtain 2-4-2 Adinkras. In the end there are twelve minimalN=4 bosonic 2-4-2 Adinkras—six obtained by two vertex raises starting from the left Adinkra in Figure 1 and six obtainable by two vertex raises starting from the right Adinkra in Figure 1. The six possibilities in each class correspond to the six different ways to select pairs from four choices. For example, if we raise ϕ3 and ϕ4 in either case then what results are the two 2-4-2 Adinkras shown in Figure 3. For the left Adinkra in Figure 3, the boson linkage matrices are shown in Table 3. (It is straightforward to read these matrices off of the Adinkra. It is also straightforward to obtain these matrices algebraically, as explained above, by interchanging the third rows of the 3-4-1 up and down matrices shown in Table 2.)Table 3
Linkage matrices for the left 2-4-2 Adinkra shown in Figure3. The linkage matrices for the right Adinkra in that figure are obtained from these by changing the sign of u3 and d3.Figure 3
TheN=4 2-4-2 Adinkras may be obtained from 3-4-1 Adinkras by raising one vertex. Here we have raised the third boson vertex starting with the two Adinkras shown in Figure 2.We now use the enhancement equation (3.3) to analyze the twelve standard N=4 2-4-2 Adinkras to ascertain if any of these can enhance to a 4D N=1 nongauge matter multiplet. To begin, we start with the left Adinkra in Figure 3, equivalently described by the boson linkage matrices specified in Table 3 and by the fermion linkage matrices determined from these by ũA=dAT and d̃A=uAT.We found above that for each of the twoN=4 Valise Adinkras and for each of the eight 3-4-1 Adinkras the enhancement equation corresponding to (a∣A,B)=(1∣1,1) is not satisfied. This is equivalent to the statement that the 4×4 matrix determined by u1u3T+d3d1T does not vanish in these cases. The reader should compute this combination in those cases, and then also compute this combination using the linkage matrices in Table 3. It is noteworthy that in this latter case, that is, using the matrices in Table 3, the computation of u1u3T+d3d1T does indeed produce the 4×4 null matrix. So the particular obstruction which we identified in the 4-4 and 3-4-1 Adinkras is notably absent for the specific 2-4-2 Adinkra shown on the left in Figure 1.Having satisfied the(a∣A,B)=(1∣1,1) equation, it remains to analyze all of the other possible choices for (a∣A,B) and check the enhancement equations (3.3) in each case. It is interesting to comment on the case (a∣A,B)=(1∣1,4), for example. In this case the enhancement equation reads  (4.1)u1u2T+d2d1T+u4u3T+d3d4T=0.
Note that this is satisfied using the matrices in Table 3. So the left 2-4-2 Adinkra in Figure 3 passes this second enhancement test. (Thus, this Adinkra passes two out of 60 different tests, counting the both the bosonic and the fermionic enhancement conditions for each of the 30 index choices (a∣A,B).)It is interesting that, unlike the left Adinkra in Figure3, the right Adinkra in Figure 3 fails the test (4.1). This can be seen by noting that (4.1) is sensitive to the parity on any one of the four edge colors since the overall signs on the first two terms flip upon toggling the sign on Q1 or Q2 while the sign on the third and fourth terms flip upon toggling the sign on Q3 or Q4. More specifically, the linkage matrices for the second Adinkra in Figure 3 are obtained from Table 3 by toggling the sign on Q3, which toggles the overall sign on the matrices u3 and d3. If we substitute the linkage matrices for the right Adinkra in Figure 3, obtained in this way, into (4.1) we find that this equation is no longer satisfied. Thus, we conclude that the right Adinkra in Figure 3 cannot enhance to a 4D nongauge matter multiplet. Again, the reader should check these assertions by doing a few simple matrix calculations.Further analysis of all of the remaining 58 enhancement conditions shows that the matrices in Table3 pass every one of these tests. This is a nontrivial accomplishment, which indicates that the left Adinkra in Figure 3 does represent the shadow of a 4D N=1 nongauge matter multiplet. As a representative example, consider the bosonic enhancement condition for the choice (a∣A,B)=(2∣3,2). This equation reads  (4.2)u2u2T+u3u3T+d2d2T+d3d3T=2,
where the factor of 2 on the right-hand side means twice the 4×4 unit matrix. The reader should verify that (4.2) is the bosonic enhancement equation described by (3.3) in this case. The reader should also verify that (4.2) is satisfied by the linkage matrices in Table 3.The fact that the first Adinkra in Figure3 describes the shadow of the 4D N=1 chiral multiplet is easy to check by performing a direct dimensional reduction of the chiral multiplet. This is done explicitly in Appendix C. In that appendix we derive the shadow Adinkra, shown in Figure 4, by direct translation of the 4D chiral multiplet transformation rules. The left Adinkra in Figure 3 is obtained from the Adinkra in Figure (4) by reorganizing fields according to the following four permutation operations: ϕ1↔ϕ2, ϕ3↔ϕ4, ψ1↔ψ4, and ψ2↔ψ3. This describes a cosmetic inner automorphism, indicating that the two Adinkras are equivalent.Figure 4
The shadow of the chiral multiplet, expressed as an Adinkra equivalent to the transformation rules (C.3).The fact that the left Adinkra in Figure3 passes the enhancement criteria while the right Adinkra does not identifies the left Adinkra as the chiral multiplet shadow and identifies the right Adinkra as the so-called twisted chiral multiplet. We also find that the 2-4-2 Adinkra obtained by raising the vertices ϕ1 and ϕ2 starting from the left 4-4 Valise in Figure 1 also passes all of the enhancement criteria whereas the twisted analog of this does not. Scrutiny of all twelve bosonic 2-4-2 Valises confirms that only those two cases in the nontwisted family, associated with the left Valise in Figure 1, obtained by raising either the pair (ϕ1,ϕ2) or the pair (ϕ3,ϕ4) can enhance to nongauge matter multiplets in 4D.


## 4.4. The Two 30-MemberN=4 Adinkra Families

It is straighforward to systematically check the enhancement conditions for all 30 Adinkras in each of the two families—one family associated with each of the two Valises in Figure1. In each case, the 30-member family consists of the bosonic 4-4 Adinkra (the Valise), four bosonic 3-4-1 Adinkras, six bosonic 2-4-2 Adinkras, four bosonic 1-4-3 Adinkras, and the Klein flip of each one of these 15 representatives. (The Klein flipped Adinkras are the fermionic 4-4 Valise and the 14 fermionic Adinkras obtainable from this by various vertex raises.)In total there are exactly four out of the 60 minimal Adinkras which pass our nongauge enhancement criteria (3.3). The first two are the bosonic 2-4-2 chiral multiplet shadows obtained by raising either ϕ1 and ϕ2 or by raising ϕ3 and ϕ4 starting from the left Adinkra in Figure 1. The other two Adinkras reside in the other (relatively twisted) family, and are obtained from the right Adinkra in Figure 1 by first raising all four bosonic vertices, then raising either the fermionic vertices ψ1 and ψ2 or by raising the fermionic vertices ψ3 and ψ4. These operations produce fermionic 2-4-2 Adinkras corresponding to twisted Klein flips of the two enhanceable bosonic 2-4-2 Adinkras. Since in these cases there are at most two fermions at any given height assignment, it is clear that these cannot assemble as 4D spinors. As we explained above, the Adinkras which pass the enhancement criteria come in pairs, one element of which passes the spin-statistics test and one which does not. In this way we conclude that of the 60 minimal Adinkras specified above only the two bosonic 2-4-2 cases can describe shadows of nongauge 4D matter multiplets.It might appear curious that the four bosonic 2-4-2 Adinkras obtained from the right Adinkra in Figure1 by raising (ϕ1,ϕ3) or (ϕ1,ϕ4) or (ϕ2,ϕ3) or (ϕ2,ϕ4) do not pass the enhancement criteria whereas the two Adinkras obtained by raising (ϕ1,ϕ2) or (ϕ3,ϕ4) do pass this test. The reason why certain combinations of component fields appear favored relates to the fact that we have made a choice of spin structure when we selected the particular gamma matrices in (E.9). It is interesting that we lose no generality in making such a choice, however, since the freedom to choose a 4D spin basis is replaced by a corresponding freedom to perform inner automorphisms on the vector space spanned by the 1D component fields.Upon selecting a higher-dimensional spin basis, the enhancement equations (3.3) place restrictions on the component fields which are legitimately meaningful; the result that exactly two out of the 60 minimal N=4 Adinkras enhance to nongauge 4D supersymmetric matter, along with the observation that those 1D multiplets which do enhance have 2-4-2 component multiplicities says something salient about 4D supersymmetry representation theory. Specifically it says that any 4D N=1 nongauge matter multiplet must have two physical bosons, four fermions, and two auxiliary bosons. This corroborates what has long been known about the minimal representations of 4D N=1 supersymmetry. What is remarkable is that we have hereby shown that this information is fully extractable using merely 1D supersymmetry and a choice of 4D spin structure—that this information lies fully encoded in the 1D supersymmetry representation theory codified by the families of Adinkras, and that the key to unlocking this information is contained in our enhancement equation (3.3). (We believe that the extra structures, namely that the bosons complexify and that the fermions assemble into chiral spinors, is also encoded in our formalism, using the equations (A.5). We also believe that deeper scrutiny of those equations should provide an algebraic context for broadly resolving natural organizations of supermultiplets, including complex structures, and quaternionic structures in diverse dimensions. But this lies beyond the scope of this introductory paper on this topic.)


## 5. Algorithmic Complexity

Much of the analysis needed to fully delineate 4DN=1 supersymmetry representation theory using our techniques succumbs to by-hand matrix manipulations, as explained above. The entirety of the computation of all Adinkras which are size-appropriate to enhance to 4D N=1 supersymmetry may be analyzed, using our nonoptimized algorithm, in a matter of seconds using a simple Mathematica routine. But the reader is likely more interested in the richer (unsolved) problem implied by the unknown elements of supersymmetry representation theory, or how realistically our equations might be used to address this larger problem, with an aim to discover yet-unknown representations which might drive novel supersymmetric field theories for use in physical model building.Using our criteria, testing enhancement toD dimensions would involve at most (1/2)(D-1)d(d+1) independent d×d matrix equations per Adinkra, where d is the number of fermions or bosons in the Adinkra. The minimal-size Adinkra in the case D=1, N=16 is 128+128, whereby d=128. To ascertain whether one of these enhances to D=10, N=1 supersymmetry would involve (1/2)(9)(128)(129)=74,304 equations, each involving products of 128×128 matrices. This number is not prohibitively large given contemporary computer resources. But we should quantify this assertion.To test enhancement of a generic 128+128 vertex Adinkra to a ten-dimensional space-time would require 74,304 equations, each involving a128×128 matrix. This translates to a computation on the order of a trillion floating point operations (1 teraflop). This result derives from the fact that solving a general matrix equation of size N is an 𝒪(N3) computation. However, with good choices of bases or a locality property, solving a matrix equation of size N can often be improved substantially into a mere O(N) computation. On the basis of this, we envision that some interesting algorithmic work not addressed in this paper should enable the ready analysis of more general supersymmetry representations using the analytic base described in this paper.To be more precise, an eight-processor Mac Pro is capable of about 50 Gigaflops, if all processors are used with highly optimized code. A teraflop would take only 20 seconds on such a machine. Even without highly optimized code (but using compiled code), using just one processor, this would take about five minutes. The full computation checking the enhance-ability of a size-appropriate Adinkra to ten space-time dimensions is on the order of a teraflop, which means it would take several minutes, on an eight-processor Mac Pro computer.Thus, the rote enhancement of a given Adinkra is never difficult computationally. However, the sheer number of possibilities provides a separate concern. Certainly the number of ostensibly distinct 128+128 Adinkras is astronomical. A more precise accounting of the combinatorics is described in [42]. However, there are interesting considerations which strongly constrain the search for the subset of these with physical relevance. One of these is the feature that enhancement proceeds as a filtratation as the dimensionality of the ambient space is increased incrementally. So an algorithm based on the feature of enhancement from one-dimension to two-dimensions, and from there to three-dimensions, and so-forth would add great refinement to the brute-force approach we have used above.Indeed, this is the approach advocated in a recent paper [42], where the enhancement problem (called the extension problem in that paper) is engaged first from one to two dimensions. The authors of that paper have already identified theorems (their Theorems 2.2 and 2.3 in that reference) which remove the great majority of Adinkras from consideration from the outset. This shows that there are considerations which can strongly influence the accounting for the difficulties inherent in implementing analyses based on our enhancement equations.Thus, until efficient algorithms beyond the scope of this paper are developed, the problem of making systematic work of resolving interesting unknown representations of supersymmetry using our methods remains daunting in its complexity. But there are indications, partially motivated by the ruminations in this section, which give a reason to optimistically hope that this combinatoric problem is not an insurmountable impasse. The development of the relevant algorithmic approach is work-in-progress.


## 6. Gauge Multiplets

The nongauge enhancement condition (3.3) relies on the result (3.1), which is derived in Appendix A. An important part of that derivation uses the assumptions ΔA0=ũAT and Δ̃A0=uAT. These translate into the statement that every Adinkra edge codifies both an upward-directed term and a downward-directed term in the multiplet transformation rules. (In other words, this result applies to “standard” Adinkras.) But the presence of gauge degrees of freedom or Bianchi identities obviates this assumption. This is demonstrated explicitly by dimensionally reducing the 4D N=1 Maxwell field-strength multiplet, as described in detail in Appendix D.


### 6.1. Introducing Phantoms

In field-strength multiplets, the vector space spanned by the boson componentsϕi is larger than the vector space spanned by the fermion components ψı̂. The physical degrees of freedom balance, however, owing to redundancies in the space of bosons, related to the constraints. This feature manifests in nonsquare linkage matrices, including sectors which decouple on the shadow. We call these “phantom sectors”.The Maxwell multiplet is characteristic of generic multiplets involving closedp-form field-strengths, when p≥2. In these cases, the field-strength divides into an “electric” sector, including components with a time-like index, and a “magnetic” sector involving components which have only space-like indices. The electric sector and the magnetic sector are correlated by the differential constraints implied by the Bianchi identity. Upon reduction to one-dimension, the magnetic sector decouples. The reason for this is that locally the magnetic fields are pure space derivatives, which vanish upon restriction to a zero-brane.Thus, in order to enhance a one-dimensional gauge multiplet to a higher-dimensional analog, not only do we have to resurrect the spatial derivatives,∂a, but we also have to resurrect the gauge sector. In the case of field-strength multiplets, this means reinstating the magnetic fields. Since these are physically decoupled on the shadow, they are reintroduced in an interesting way, at the top of one-way upward-directed Adinkra edges. These nascent magnetic vertices play no role in the one-dimensional supersymmetry algebra respected by the other fields on the shadow. (But they play an important role in closing the algebra in ambient higher dimensions.) We therefore call these vertices “phantoms”.Phantoms may be introduced into one-dimensional supermultiplets to enable possible enhancement involving closedp-form multiplet components. Since closed 1-forms do not involve gauge invariance, it follows that the simplest case involves closed 2-forms, such as Fμν subject to ∂[λFμν]=0. This allows access to the important cases involving vector multiplets. The higher-p cases may be treated similarly, but these involve additional subtlety. In order to keep our presentation relatively concise, we will not address cases p≥3 nor will we address cases involving gauge fermions. Our discussion will remain focussed on the ability to include 4D Abelian field-strengths. We also avoid other subtle technicalities by allowing 4D fermions to assemble only as spin 1/2 fields; that is we will not address the case of spin 3/2, or Rarita-Schwinger fields, in this paper. It is straightforward to generalize our technology to allow for these possibilities. But we defer discussions of these cases to future work, for reasons of bounding complexity.The structure of a phantom sector is usefully codified by phantom link matrices, defined as(6.1)(PA)iı̂:=(ũAT-ΔA0)iı̂,
where ũAT is the transpose of the Ath fermion “up” matrix. A nonvanishing phantom matrix indicates the presence of one-way upward-directed Adinkra edges. If PA is nonvanishing then this modifies the analysis in Appendix A precisely at the point where (A.9) is introduced as the transpose of (A.8). If the phantom matrices are included and the analysis is repeated, it is easy to show that (3.1) generalizes to  (6.2)ΔAa=-(Γ0ΓaΔ0)A-12(Γ0ΓaP)A+T0aPA-PAT̃0a.
Note that the final three terms will contribute nontrivially to this equation only in the gauge sector.It is helpful to briefly review the particular phantom sector associated with the shadow of the Maxwell field-strength multiplet. This provides the archetype for generalizations, and motivates what follows.


### 6.2. Maxwell's Shadow

The 4DN=1 super Maxwell multiplet involves four boson degrees of freedom off-shell. These organize as the auxiliary scalar D plus the three off-shell “electromagnetic” degrees of freedom described by Fμν. It is natural to write Ea=F0a and Ba=(1/2)ɛabcFbc. The Bianchi identity ∂[λFμν]=0 correlates Ea and Ba. Locally, we can solve the Bianchi identity in terms of a vector potential Aμ, so that Ea=∂0Aa-∂aA0 and Ba=ɛabc∂bAc. Upon restriction to the zero-brane we take ∂a→0, so that Ea→∂0Aa and Ba→0. Since the magnetic fields vanish on the zero-brane, it is natural to think of the Ea as more fundamental for our purposes. The shadow is described by a fermionic 4-4 Adinkra where the bosons are (E1,E2,E3,D) and the fermions are (λ1,λ2,λ3,λ4). To enhance this multiplet we must reintroduce ∂a and also reintroduce the fields Ba, along with constraints. To do this, we allow for “phantom” bosons on the worldline, which correspond to the Ba off of the worldline.To accommodate the phantom bosons, we consider an enlarged bosonic vector space,ϕi=(E1,E2,E3,D∣B1,B2,B3) in conjunction with the fermionic vector space ψı̂=(λ1,λ2,λ3,λ4). As a useful index notation, we write these as ϕi=(Ea,D∣Ba¯), where Ba¯ is the magnetic phantom associated with Ea. Thus, a and a¯ each assume the values 1,2,3, and we have ϕ1,2,3=E1,2,3 and ϕ5,6,7=B1¯,2¯,3¯, respectively. In this way, phantom fields are designated by an over-bar on the relevant index. Boson fields not in the phantom sector are indicated by underlined indices, so that ϕ1̲,2̲,3̲,4̲=(E1,E2,E3,D). Matrices with two boson indices then divide into four sectors, Xi̲j̲, Xi̲a¯, Xa¯j̲, and Xa¯b¯.The shadow transformation rules associated with the Maxwell multiplet can be written as (2.3), but the linkage matrices are not square! Instead, (ũA)ı̂j is 7×4 and (ΔAμ)iı̂ is 4×7. We exhibit the precise linkage matrices associated with the Maxwell multiplet in Appendix D. For the Super Maxwell case, the first enhancement equation in (3.3) is a 7×7 matrix equation, whereas the second is a 4×4 matrix equation. The first equation has phantom sectors which can be shuffled away canonically via use of the Bianchi identity, as explained below.


### 6.3. Canonical Reshuffling

Owing to the derivatives in the enhancement condition (3.3), we may use the Bianchi identity, ∂[λFμν]=0, usefully rewritten as  (6.3)∂0Ba¯=ɛa¯bc∂bEc,∂a¯Ba¯=0,
to define “canonical reorganizations” of the matrices in (3.2) under which (3.3) remains unchanged. Specifically, the first equation in (6.3) allows us to redefine  (6.4)(ΩAB0)ia¯⟶0,(ΩABa)ib⟶(ΩABa)ib+ɛc¯ab(ΩAB0)ic¯,
whereby we exchange each appearance of ∂0Ba¯ in a supersymmetry commutator with an equivalent expression involving spatial derivatives on the electric field components. Similarly, the second equation in (6.3) allows us to redefine  (6.5)(ΩAB1)i1¯⟶0,(ΩAB2)i2¯⟶(ΩAB2)i2¯-(ΩAB1)i1¯,(ΩAB3)i3¯⟶(ΩAB3)i3¯-(ΩAB1)i1¯.
In this way, we define a canonical structure of the matrices (ΩABμ)ij, ensured by the transformations (6.4) and (6.5), enabled by the Bianchi identity (6.3), whereby (ΩAB0)ia¯=0 and (ΩAB1)i1=0. The first equation in (3.3) may now be interpreted as saying that (ΩABμ)ij→0 under these transformations.We remark that that each of the 407×7 matrices (ΩABμ)ij defined by (3.2), using the linkage matrices exhibited in Appendix D, do satisfy (ΩABμ)ij→0 using the transformations (6.4) and (6.5). The diligent reader is encouraged to check this assertion.


### 6.4. Thep=1 Gauge Enhancement Conditions

Based on the above, a means becomes apparent under which we can ascertain which one-dimensional multiplets may enhance to 4D gauge field-strength multiplets, based only on a knowledge of the one-dimensional transformation rules, or equivalently given an Adinkra.For physical gauge fields, the bosonic field-strength tensor has greater engineering dimension than the corresponding gaugino fermions. Therefore, the ambient fermions transform into the magnetic fields via terms in the fermion transformation ruleδQλ given by (1/2)ɛabcBaΓbcϵ or by (1/2)ɛabcBaΓbcΓ5ϵ. These are the only Lorentz covariant possibilities. The former case involves a vector potential and the latter case involves an axial vector potential. We focus first on the former case, and comment on axial vectors afterwards. It is straightforward to determine the phantom “up" links and the time-like fermion “down" links using δQλ=⋯+(1/2)ɛabcBaΓbcϵ. By rearranging this term into the form δQλi=⋯+ϵA(ũA)ia¯Ba¯, we derive (Consistency of (6.6) with (6.1) implies usefully extractable information about the spin representation assignments. We will not explore this arena in this paper.)  (6.6)(ũA)ı̂a¯=12ɛa¯bc(Γbc)ı̂A,(ΔA0)a¯ı̂=0,
whereby using (6.1) we determine the nonvanishing part of the phantom matrix as  (6.7)(PA)a¯ı̂=12ɛa¯bc(Γbc)Aı̂.
The entire phantom matrix has nonvanishing entries only in its final three rows.We can resolve theΔAa matrices in two parts, using different methods. First we resolve the phantom part (ΔAa)a¯ı̂. Then we resolve the nonphantom part (ΔAa)a̲ı̂.We determine the phantom part of the space-like boson down matrices using the fact that(ΔA0)a¯ı̂=0, which says that there are no connections linking downward from the phantoms. Thus, equation (A.4) tells us (ΔAa)a¯ı̂=-(T0a)a¯i(ΔA0)iı̂. Next, we use the fact that a boost shuffles magnetic fields into electric fields, via (T0a)bc¯Bc¯=ɛabcEc, to determine  (6.8)(ΔAa)b¯ı̂=ɛb¯ac̲(ΔA0)c̲ı̂.
This determines the b¯th row in the phantom sector of the Ath space-like down matrices in terms of “electric” rows in the time-like down matrices.We determine the nonphantom part of the space-like down matrices by considering the nonphantom sector in (6.2). Thus, we allow only nonphantom values for the suppressed boson index. In this case, the second and the fourth terms on the right-hand side vanish because (PA)a̲ı̂=0. The third term on the right-hand side can be resolved by noting that a boost shuffles electric fields into magnetic fields, via (T0a)bcEc=ɛbac¯Bc¯, whereby (T0a)b̲c=ɛb̲ac¯. Substituting this result, along with (6.7), we derive  (6.9)(ΔAa)b̲ı̂=-(Γ0Γa)AB(ΔA0)b̲ı̂+(Γb̲a)Aı̂.
This is the same as our nongauge result (3.1), modified by the second term. (Note that the second term in (6.9) can also be written as -2ɛb̲ac(ℛc)Aı̂, where ℛc generates a rotation in the cth spatial dimension.) Taken together, (6.8) and (6.9) generate for us the entire space-like phantom-modified down matrices, generalizing our earlier nongauge result (3.1) to the case in which fields can assemble into closed 1-forms.Note that the Maxwell field-strength shadow, including its phantom sectors, may be reproduced from the minimalN=4 Adinkras using the methods described in this section. Presently, we explain a means to produce a representative in one of the two minimal N=4 Adinkra families which demonstrably enhances to a 4D Maxwell multiplet. Importantly, we can verify that this Adinkra enhances to such a 4D multiplet using only one-dimensional reasoning.This is done by starting with right Adinkra in Figure1. (Note that this Adinkra is in the family twisted relative to the family which includes the chiral multiplet shadow.) From this starting Adinkra, we raise all four boson vertices, to obtain a fermionic 4-4 Adinkra with the four bosons at the higher level. We then perform a permutation of the first and the fourth boson vertices, and a permutation of the the second and the fourth boson vertices. We then flip both the first and the fourth boson vertices. (To flip a vertex means to scale this by an overall minus sign.) Finally we flip the third fermion vertex. We compute the time-like up matrices ũA and the time-like down matrices ΔA0 using the resultant fermionic 4-4 Adinkra. We designate the first three boson vertices in this final orientation as our designated “electric” components. We then apply (6.6), (6.8), and (6.9) to append phantom sectors to these time-like linkage matrices, and to generate provisional space-like down links ΔAa, including phantom sectors. What results from these operations are precisely the matrices shown in Appendix D. We next compute the Ω matrices using (3.2). Finally, we apply a canonical reshuffling of these Ω-matrices, using (6.4) and (6.5). Happily, we find that under this operation all of the matrices (ΩABμ)ij and (Ω̃ABμ)ı̂𝚥̂ vanish. This illustrates that this representative set of operations produces an Adinkra which passes our nontrivial gauge enhancement test.If we repeat the above search allowing for axial vector potential, we would modify all equations in this subsection with an additional factor ofΓ5. What we find is that every multiplet which enhances to provide a vector potential also enhances to provide an axial vector potential. This is loosely similar to the situation involving chiral versus antichiral multiplets, which have identical shadows. It follows that the both the vector multiplet and the axial vector multiplet shadow lie in the family of Adinkras relatively twisted relative to the chiral multiplet. We will not exhibit separate equations for the axial vector case, leaving that as an exercise for the interested reader.


### 6.5. Algorithm

We have already explained that it is possible to systematically generate the linkage matrices for each member of the family associated with a given Valise. For each representative, we can sequentially select triplets of vertices as potential electric field components, and use (6.6), (6.8), and (6.9) to generate postulate phantom sectors. For each of these, we compute the relevant Ω-matrices using (3.2), then shuffle these into canonical form using (6.4) and (6.5). By selecting those Adinkras for which all the canonical Ω-matrices obtained in this way vanish, we thereby obtain an algorithmic search for all multiplets in which vertices can assemble into closed 1-forms. This search is guaranteed to locate those multiplets which do properly enhance. (N.B. we have explained in the previous paragraph an example which we know works.)In the case of closed 1-form multiplets, an enhanceable Adinkra exhibits a synergy between the postulated electric field components and the designated magnetic phantom sector, vis-a-vis the assignment of the component basisϕi. This is because the enhancement criteria are sensitive to the basis choice on the component boson vector via the structure of our imposed phantom sector. (Note that designating the phantom sector using (6.8) and (6.9) does not remove generality from the search, much as choosing 4D gamma matrices does not remove generality, for reasons described above. Instead, this removes redundancies from the answer set.) Practically, this requires, for a comprehensive algorithmic search for enhanceable Adinkras, that in addition to sifting through all possible vertex raises and all possible selections of vertex triplets, we also have to sift through vertex permutations and vertex flips. Thus, inner automorphisms would seem to enlarge the effective search family. Regardless, our discussion does show that the portion of 4D supersymmetry representation theory involving closed 1-form multiplets is accessible and understandable using only 1D supersymmetry. We find this interesting.In summary, following is a method to test an Adinkra to see if it enhances to give a 4D multiplet with a closed 1-form gauge field-strength.(1)
Select three boson vertices with a common engineering dimension as the presumed electric components, and arrange the boson vector space so thatϕ1,2,3 correspond to these.(2)
Compute time-like linkage matricesuA, ũA, ΔA0, and Δ̃A0 from the Adinkra.(3)
Augment the boson vector space by adding on a phantom sector consisting of three new fieldsϕ1¯,2¯,3¯ with the same engineering dimension as ϕ1,2,3.(4)
Add phantom sectors to the up matricesũA, by adding three extra columns, and add phantom sectors to the time-like down matrices ΔA0 by adding three extra rows.(5)
Populate the phantom sector of the up matrices using (6.6).(6)
Generate space-like down matrices, including phantom sectors using (6.8) and (6.9).(7)
Use the complete set of phantom-augmented linkage matrices to determine the matrices(ΩABa)ij and (Ω̃ABa)ı̂𝚥̂ using (3.2).(8)
Reshuffle the boson matrix(ΩABa)ij using the prescription (6.4) and (6.5), to obtain a new matrix, in canonical form,
(6.10)(ΩABa)ij⟶(Ω̂ABa)ij.
The presence of the hat indicates canonical form.(9)
Thep=1 gauged enhancement conditions now correspond to the original enhancement conditions (3.3) augmented by the addition of phantom sectors and a canonical reshuffle. We conclude that a necessary requirement for an Adinkra to enhance to a p=1 field-strength multiplet is
(6.11)(Ω̂ABμ)ij=0,(Ω̃ABμ)ı̂̂=0.This is ourp=1 gauge enhancement condition. The key difference as compared to the nongauge case is that the linkage matrices are not square in the gauge case, owing to the presence of the phantom boson sectors. Furthermore, we must implement the canonical reshuffling maneuver to generate the hatted Ω̂ matrices which describe the nongauge enhancement condition.The way we have designed our formalism is tailored toward implementation via computer-searches. This may require supercomputers for cases with higherN, which will involve large matrix computations. We hope to enlarge our algorithms so that sifting through one-dimensional multiplets is controlled by the relevant lists of doubly-even error-correcting codes which correspond to these, as explained in the introduction. But this lies beyond the scope of the presentation in this paper. We find it sufficiently noteworthy that such algorithms exist, even in principle. Our hope is that this will shed light on unknown aspects of supersymmetry which have defied attack using previous conventions.


## 6.1. Introducing Phantoms

In field-strength multiplets, the vector space spanned by the boson componentsϕi is larger than the vector space spanned by the fermion components ψı̂. The physical degrees of freedom balance, however, owing to redundancies in the space of bosons, related to the constraints. This feature manifests in nonsquare linkage matrices, including sectors which decouple on the shadow. We call these “phantom sectors”.The Maxwell multiplet is characteristic of generic multiplets involving closedp-form field-strengths, when p≥2. In these cases, the field-strength divides into an “electric” sector, including components with a time-like index, and a “magnetic” sector involving components which have only space-like indices. The electric sector and the magnetic sector are correlated by the differential constraints implied by the Bianchi identity. Upon reduction to one-dimension, the magnetic sector decouples. The reason for this is that locally the magnetic fields are pure space derivatives, which vanish upon restriction to a zero-brane.Thus, in order to enhance a one-dimensional gauge multiplet to a higher-dimensional analog, not only do we have to resurrect the spatial derivatives,∂a, but we also have to resurrect the gauge sector. In the case of field-strength multiplets, this means reinstating the magnetic fields. Since these are physically decoupled on the shadow, they are reintroduced in an interesting way, at the top of one-way upward-directed Adinkra edges. These nascent magnetic vertices play no role in the one-dimensional supersymmetry algebra respected by the other fields on the shadow. (But they play an important role in closing the algebra in ambient higher dimensions.) We therefore call these vertices “phantoms”.Phantoms may be introduced into one-dimensional supermultiplets to enable possible enhancement involving closedp-form multiplet components. Since closed 1-forms do not involve gauge invariance, it follows that the simplest case involves closed 2-forms, such as Fμν subject to ∂[λFμν]=0. This allows access to the important cases involving vector multiplets. The higher-p cases may be treated similarly, but these involve additional subtlety. In order to keep our presentation relatively concise, we will not address cases p≥3 nor will we address cases involving gauge fermions. Our discussion will remain focussed on the ability to include 4D Abelian field-strengths. We also avoid other subtle technicalities by allowing 4D fermions to assemble only as spin 1/2 fields; that is we will not address the case of spin 3/2, or Rarita-Schwinger fields, in this paper. It is straightforward to generalize our technology to allow for these possibilities. But we defer discussions of these cases to future work, for reasons of bounding complexity.The structure of a phantom sector is usefully codified by phantom link matrices, defined as(6.1)(PA)iı̂:=(ũAT-ΔA0)iı̂,
where ũAT is the transpose of the Ath fermion “up” matrix. A nonvanishing phantom matrix indicates the presence of one-way upward-directed Adinkra edges. If PA is nonvanishing then this modifies the analysis in Appendix A precisely at the point where (A.9) is introduced as the transpose of (A.8). If the phantom matrices are included and the analysis is repeated, it is easy to show that (3.1) generalizes to  (6.2)ΔAa=-(Γ0ΓaΔ0)A-12(Γ0ΓaP)A+T0aPA-PAT̃0a.
Note that the final three terms will contribute nontrivially to this equation only in the gauge sector.It is helpful to briefly review the particular phantom sector associated with the shadow of the Maxwell field-strength multiplet. This provides the archetype for generalizations, and motivates what follows.


## 6.2. Maxwell's Shadow

The 4DN=1 super Maxwell multiplet involves four boson degrees of freedom off-shell. These organize as the auxiliary scalar D plus the three off-shell “electromagnetic” degrees of freedom described by Fμν. It is natural to write Ea=F0a and Ba=(1/2)ɛabcFbc. The Bianchi identity ∂[λFμν]=0 correlates Ea and Ba. Locally, we can solve the Bianchi identity in terms of a vector potential Aμ, so that Ea=∂0Aa-∂aA0 and Ba=ɛabc∂bAc. Upon restriction to the zero-brane we take ∂a→0, so that Ea→∂0Aa and Ba→0. Since the magnetic fields vanish on the zero-brane, it is natural to think of the Ea as more fundamental for our purposes. The shadow is described by a fermionic 4-4 Adinkra where the bosons are (E1,E2,E3,D) and the fermions are (λ1,λ2,λ3,λ4). To enhance this multiplet we must reintroduce ∂a and also reintroduce the fields Ba, along with constraints. To do this, we allow for “phantom” bosons on the worldline, which correspond to the Ba off of the worldline.To accommodate the phantom bosons, we consider an enlarged bosonic vector space,ϕi=(E1,E2,E3,D∣B1,B2,B3) in conjunction with the fermionic vector space ψı̂=(λ1,λ2,λ3,λ4). As a useful index notation, we write these as ϕi=(Ea,D∣Ba¯), where Ba¯ is the magnetic phantom associated with Ea. Thus, a and a¯ each assume the values 1,2,3, and we have ϕ1,2,3=E1,2,3 and ϕ5,6,7=B1¯,2¯,3¯, respectively. In this way, phantom fields are designated by an over-bar on the relevant index. Boson fields not in the phantom sector are indicated by underlined indices, so that ϕ1̲,2̲,3̲,4̲=(E1,E2,E3,D). Matrices with two boson indices then divide into four sectors, Xi̲j̲, Xi̲a¯, Xa¯j̲, and Xa¯b¯.The shadow transformation rules associated with the Maxwell multiplet can be written as (2.3), but the linkage matrices are not square! Instead, (ũA)ı̂j is 7×4 and (ΔAμ)iı̂ is 4×7. We exhibit the precise linkage matrices associated with the Maxwell multiplet in Appendix D. For the Super Maxwell case, the first enhancement equation in (3.3) is a 7×7 matrix equation, whereas the second is a 4×4 matrix equation. The first equation has phantom sectors which can be shuffled away canonically via use of the Bianchi identity, as explained below.


## 6.3. Canonical Reshuffling

Owing to the derivatives in the enhancement condition (3.3), we may use the Bianchi identity, ∂[λFμν]=0, usefully rewritten as  (6.3)∂0Ba¯=ɛa¯bc∂bEc,∂a¯Ba¯=0,
to define “canonical reorganizations” of the matrices in (3.2) under which (3.3) remains unchanged. Specifically, the first equation in (6.3) allows us to redefine  (6.4)(ΩAB0)ia¯⟶0,(ΩABa)ib⟶(ΩABa)ib+ɛc¯ab(ΩAB0)ic¯,
whereby we exchange each appearance of ∂0Ba¯ in a supersymmetry commutator with an equivalent expression involving spatial derivatives on the electric field components. Similarly, the second equation in (6.3) allows us to redefine  (6.5)(ΩAB1)i1¯⟶0,(ΩAB2)i2¯⟶(ΩAB2)i2¯-(ΩAB1)i1¯,(ΩAB3)i3¯⟶(ΩAB3)i3¯-(ΩAB1)i1¯.
In this way, we define a canonical structure of the matrices (ΩABμ)ij, ensured by the transformations (6.4) and (6.5), enabled by the Bianchi identity (6.3), whereby (ΩAB0)ia¯=0 and (ΩAB1)i1=0. The first equation in (3.3) may now be interpreted as saying that (ΩABμ)ij→0 under these transformations.We remark that that each of the 407×7 matrices (ΩABμ)ij defined by (3.2), using the linkage matrices exhibited in Appendix D, do satisfy (ΩABμ)ij→0 using the transformations (6.4) and (6.5). The diligent reader is encouraged to check this assertion.


## 6.4. Thep=1 Gauge Enhancement Conditions

Based on the above, a means becomes apparent under which we can ascertain which one-dimensional multiplets may enhance to 4D gauge field-strength multiplets, based only on a knowledge of the one-dimensional transformation rules, or equivalently given an Adinkra.For physical gauge fields, the bosonic field-strength tensor has greater engineering dimension than the corresponding gaugino fermions. Therefore, the ambient fermions transform into the magnetic fields via terms in the fermion transformation ruleδQλ given by (1/2)ɛabcBaΓbcϵ or by (1/2)ɛabcBaΓbcΓ5ϵ. These are the only Lorentz covariant possibilities. The former case involves a vector potential and the latter case involves an axial vector potential. We focus first on the former case, and comment on axial vectors afterwards. It is straightforward to determine the phantom “up" links and the time-like fermion “down" links using δQλ=⋯+(1/2)ɛabcBaΓbcϵ. By rearranging this term into the form δQλi=⋯+ϵA(ũA)ia¯Ba¯, we derive (Consistency of (6.6) with (6.1) implies usefully extractable information about the spin representation assignments. We will not explore this arena in this paper.)  (6.6)(ũA)ı̂a¯=12ɛa¯bc(Γbc)ı̂A,(ΔA0)a¯ı̂=0,
whereby using (6.1) we determine the nonvanishing part of the phantom matrix as  (6.7)(PA)a¯ı̂=12ɛa¯bc(Γbc)Aı̂.
The entire phantom matrix has nonvanishing entries only in its final three rows.We can resolve theΔAa matrices in two parts, using different methods. First we resolve the phantom part (ΔAa)a¯ı̂. Then we resolve the nonphantom part (ΔAa)a̲ı̂.We determine the phantom part of the space-like boson down matrices using the fact that(ΔA0)a¯ı̂=0, which says that there are no connections linking downward from the phantoms. Thus, equation (A.4) tells us (ΔAa)a¯ı̂=-(T0a)a¯i(ΔA0)iı̂. Next, we use the fact that a boost shuffles magnetic fields into electric fields, via (T0a)bc¯Bc¯=ɛabcEc, to determine  (6.8)(ΔAa)b¯ı̂=ɛb¯ac̲(ΔA0)c̲ı̂.
This determines the b¯th row in the phantom sector of the Ath space-like down matrices in terms of “electric” rows in the time-like down matrices.We determine the nonphantom part of the space-like down matrices by considering the nonphantom sector in (6.2). Thus, we allow only nonphantom values for the suppressed boson index. In this case, the second and the fourth terms on the right-hand side vanish because (PA)a̲ı̂=0. The third term on the right-hand side can be resolved by noting that a boost shuffles electric fields into magnetic fields, via (T0a)bcEc=ɛbac¯Bc¯, whereby (T0a)b̲c=ɛb̲ac¯. Substituting this result, along with (6.7), we derive  (6.9)(ΔAa)b̲ı̂=-(Γ0Γa)AB(ΔA0)b̲ı̂+(Γb̲a)Aı̂.
This is the same as our nongauge result (3.1), modified by the second term. (Note that the second term in (6.9) can also be written as -2ɛb̲ac(ℛc)Aı̂, where ℛc generates a rotation in the cth spatial dimension.) Taken together, (6.8) and (6.9) generate for us the entire space-like phantom-modified down matrices, generalizing our earlier nongauge result (3.1) to the case in which fields can assemble into closed 1-forms.Note that the Maxwell field-strength shadow, including its phantom sectors, may be reproduced from the minimalN=4 Adinkras using the methods described in this section. Presently, we explain a means to produce a representative in one of the two minimal N=4 Adinkra families which demonstrably enhances to a 4D Maxwell multiplet. Importantly, we can verify that this Adinkra enhances to such a 4D multiplet using only one-dimensional reasoning.This is done by starting with right Adinkra in Figure1. (Note that this Adinkra is in the family twisted relative to the family which includes the chiral multiplet shadow.) From this starting Adinkra, we raise all four boson vertices, to obtain a fermionic 4-4 Adinkra with the four bosons at the higher level. We then perform a permutation of the first and the fourth boson vertices, and a permutation of the the second and the fourth boson vertices. We then flip both the first and the fourth boson vertices. (To flip a vertex means to scale this by an overall minus sign.) Finally we flip the third fermion vertex. We compute the time-like up matrices ũA and the time-like down matrices ΔA0 using the resultant fermionic 4-4 Adinkra. We designate the first three boson vertices in this final orientation as our designated “electric” components. We then apply (6.6), (6.8), and (6.9) to append phantom sectors to these time-like linkage matrices, and to generate provisional space-like down links ΔAa, including phantom sectors. What results from these operations are precisely the matrices shown in Appendix D. We next compute the Ω matrices using (3.2). Finally, we apply a canonical reshuffling of these Ω-matrices, using (6.4) and (6.5). Happily, we find that under this operation all of the matrices (ΩABμ)ij and (Ω̃ABμ)ı̂𝚥̂ vanish. This illustrates that this representative set of operations produces an Adinkra which passes our nontrivial gauge enhancement test.If we repeat the above search allowing for axial vector potential, we would modify all equations in this subsection with an additional factor ofΓ5. What we find is that every multiplet which enhances to provide a vector potential also enhances to provide an axial vector potential. This is loosely similar to the situation involving chiral versus antichiral multiplets, which have identical shadows. It follows that the both the vector multiplet and the axial vector multiplet shadow lie in the family of Adinkras relatively twisted relative to the chiral multiplet. We will not exhibit separate equations for the axial vector case, leaving that as an exercise for the interested reader.


## 6.5. Algorithm

We have already explained that it is possible to systematically generate the linkage matrices for each member of the family associated with a given Valise. For each representative, we can sequentially select triplets of vertices as potential electric field components, and use (6.6), (6.8), and (6.9) to generate postulate phantom sectors. For each of these, we compute the relevant Ω-matrices using (3.2), then shuffle these into canonical form using (6.4) and (6.5). By selecting those Adinkras for which all the canonical Ω-matrices obtained in this way vanish, we thereby obtain an algorithmic search for all multiplets in which vertices can assemble into closed 1-forms. This search is guaranteed to locate those multiplets which do properly enhance. (N.B. we have explained in the previous paragraph an example which we know works.)In the case of closed 1-form multiplets, an enhanceable Adinkra exhibits a synergy between the postulated electric field components and the designated magnetic phantom sector, vis-a-vis the assignment of the component basisϕi. This is because the enhancement criteria are sensitive to the basis choice on the component boson vector via the structure of our imposed phantom sector. (Note that designating the phantom sector using (6.8) and (6.9) does not remove generality from the search, much as choosing 4D gamma matrices does not remove generality, for reasons described above. Instead, this removes redundancies from the answer set.) Practically, this requires, for a comprehensive algorithmic search for enhanceable Adinkras, that in addition to sifting through all possible vertex raises and all possible selections of vertex triplets, we also have to sift through vertex permutations and vertex flips. Thus, inner automorphisms would seem to enlarge the effective search family. Regardless, our discussion does show that the portion of 4D supersymmetry representation theory involving closed 1-form multiplets is accessible and understandable using only 1D supersymmetry. We find this interesting.In summary, following is a method to test an Adinkra to see if it enhances to give a 4D multiplet with a closed 1-form gauge field-strength.(1)
Select three boson vertices with a common engineering dimension as the presumed electric components, and arrange the boson vector space so thatϕ1,2,3 correspond to these.(2)
Compute time-like linkage matricesuA, ũA, ΔA0, and Δ̃A0 from the Adinkra.(3)
Augment the boson vector space by adding on a phantom sector consisting of three new fieldsϕ1¯,2¯,3¯ with the same engineering dimension as ϕ1,2,3.(4)
Add phantom sectors to the up matricesũA, by adding three extra columns, and add phantom sectors to the time-like down matrices ΔA0 by adding three extra rows.(5)
Populate the phantom sector of the up matrices using (6.6).(6)
Generate space-like down matrices, including phantom sectors using (6.8) and (6.9).(7)
Use the complete set of phantom-augmented linkage matrices to determine the matrices(ΩABa)ij and (Ω̃ABa)ı̂𝚥̂ using (3.2).(8)
Reshuffle the boson matrix(ΩABa)ij using the prescription (6.4) and (6.5), to obtain a new matrix, in canonical form,
(6.10)(ΩABa)ij⟶(Ω̂ABa)ij.
The presence of the hat indicates canonical form.(9)
Thep=1 gauged enhancement conditions now correspond to the original enhancement conditions (3.3) augmented by the addition of phantom sectors and a canonical reshuffle. We conclude that a necessary requirement for an Adinkra to enhance to a p=1 field-strength multiplet is
(6.11)(Ω̂ABμ)ij=0,(Ω̃ABμ)ı̂̂=0.This is ourp=1 gauge enhancement condition. The key difference as compared to the nongauge case is that the linkage matrices are not square in the gauge case, owing to the presence of the phantom boson sectors. Furthermore, we must implement the canonical reshuffling maneuver to generate the hatted Ω̂ matrices which describe the nongauge enhancement condition.The way we have designed our formalism is tailored toward implementation via computer-searches. This may require supercomputers for cases with higherN, which will involve large matrix computations. We hope to enlarge our algorithms so that sifting through one-dimensional multiplets is controlled by the relevant lists of doubly-even error-correcting codes which correspond to these, as explained in the introduction. But this lies beyond the scope of the presentation in this paper. We find it sufficiently noteworthy that such algorithms exist, even in principle. Our hope is that this will shed light on unknown aspects of supersymmetry which have defied attack using previous conventions.


## 7. Conclusions

We have presented algebraic conditions which allow one to systematically locate those representations of one-dimensional supersymmetry which may enhance to higher dimensions. Equivalently, we have explained how to discern whether a given one-dimensional supermultiplet is a shadow of a higher-dimensional analog. This allows the representation theory of supersymmetry in diverse dimensions to be divided into the simpler representation theory of one-dimensional supersymmetry augmented with separate questions pertaining to the possibility of enhancement into higher dimensions.We have shown through explicit examples how information pertaining to four-dimensionalN=1 supersymmetry may be extracted using only information from one-dimensional supersymmetry. We did this comprehensively for the case of 4D N=1 nongauge matter multiplets. We have also explained how this systematics generalizes to cases involving higher-dimensional gauge invariances, specializing our discussion to the case of 4D N=1 Super-Maxwell theory.We intend to use the formalism and the algorithms developed above to seek inroads towards off-shell aspects of interesting supersymmetric contexts where the off-shell physics remains mysterious but potentially relevant.



---

*Source: 259089-2011-10-03.xml*