Dimensional Enhancement via Supersymmetry

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.

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 with ℛ-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 covariant 𝔰𝔭𝔦𝔫(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 particular 𝔰𝔭𝔦𝔫(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 Section 5.

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 Appendix B 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-dimensional 𝑁=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 Section 2 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 Section 3 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 Section 4 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 Section 6 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. Appendix A 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.

Appendix B 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 Appendix B. 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 as 𝜙_𝑖 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 𝛿_𝐿𝜙_𝑖=12𝜃^𝜇𝜈𝑇_𝜇𝜈𝑖^𝑗𝜙_𝑗,𝛿_𝐿𝜓𝚤̂=12𝜃^𝜇𝜈𝑇_𝜇𝜈𝚤̂𝚥̂𝜓𝚥,̂ (2.1) where the label 𝐿 is a mnemonic which specifies these as “Lorentz" transformations. Here, (𝑇_𝜇𝜈)_𝑖^𝑗 represents the spin algebra as realized on the boson fields and (𝑇_𝜇𝜈)𝚤̂𝚥̂ represents the spin algebra as realized on the fermion fields, while 𝜃^0𝑎 parameterizes a boost in the 𝑎th spatial direction and 𝜃^𝑎𝑏 parameterizes a rotation in the 𝑎𝑏-plane. According to the spin-statistics theorem, (𝑇_𝜇𝜈)𝚤̂𝚥̂ should describe a spinor representation and (𝑇_𝜇𝜈)_𝑖^𝑗 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 the 𝑁=1 supersymmetry algebra is generated by a Majorana spinor supercharge with components 𝒬_𝐴 subject to the anticommutator relationship {𝒬_𝐴,𝒬_𝐵}=2𝑖𝐺^𝜇_𝐴𝐵𝜕_𝜇 where 𝐺^𝜇_𝐴𝐵=−(Γ^𝜇𝐶⁻¹)_𝐴𝐵. A parameter-dependent supersymmetry transformation is generated by 𝛿_𝑄(𝜖)=−𝑖𝜖^𝐴𝒬_𝐴, where 𝜖_𝐴 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 𝐺⁰_𝐴𝐵=𝛿_𝐴𝐵. With this choice, we can rewrite our supersymmetry transformation as 𝛿_𝑄(𝜖)=−𝑖𝜖^𝐴𝑄_𝐴, where 𝑄_𝐴=(Γ₀)^𝐵_𝐴𝒬_𝐵. (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 𝑄_𝐴, is then given by 𝑄_𝐴,𝑄_𝐵=2𝑖𝛿_𝐴𝐵𝜕_𝜏−2𝑖𝐺^𝑎_𝐴𝐵𝜕_𝑎, (2.2) where 𝑥⁰∶=𝜏 is the time-like coordinate parameterizing the the zero-brane to which we intend to restrict, and 𝑥^𝑎∶=(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 writing 𝛿_𝑄𝜙_𝑖=−𝑖𝜖^𝐴(𝑄_𝐴)_𝑖𝚤𝜓̂𝚤̂ and 𝛿_𝑄𝜓𝚤̂=−𝑖𝜖^𝐴(𝑄_𝐴)𝚤^𝑖̂𝜙_𝑖, appending a tilde to 𝑄_𝐴 when this describes a fermion transformation rule. The supercharges may be represented as first-order linear differential operators, as 𝑄_𝐴𝑖𝚤̂=𝑢_𝐴𝑖𝚤̂+Δ^𝜇_𝐴𝑖𝚤̂𝜕_𝜇,𝑄_𝐴𝚤^𝑖̂=𝑖𝑢_𝐴𝚤^𝑖̂+𝑖Δ^𝜇_𝐴𝚤^𝑖̂𝜕_𝜇, (2.3) where 𝑢_𝐴, 𝑢_𝐴, Δ^𝜇_𝐴, and Δ^𝜇_𝐴 are real valued “linkage matrices" which play a central role in our discussion below.

The matrices (𝑢_𝐴)_𝑖𝚤̂ describe “links” corresponding to supersymmetry maps from the bosons 𝜙_𝑖 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 (𝑢_𝐴)𝚤^𝑖̂ codify “upward“ maps connecting lower-weight fermions to higher-weight bosons. The matrices (Δ^𝜇_𝐴)_𝑖𝚤̂ and (Δ^𝜇_𝐴)𝚤^𝑖̂ 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 𝑢_(𝐴𝑢_𝐵)𝑖^𝑗=0,𝑢_(𝐴𝑢_𝐵)𝚤̂𝚥̂Δ=0,_(𝜇(𝐴Δ^𝜈)_𝐵)𝑖^𝑗=0,Δ^(𝜇_(𝐴Δ^𝜈)_𝐵)𝚤̂𝚥̂=0, (2.4) which describes a higher-dimensional analog of the Adinkra loop parity rule described in [17] and below, and also implies 𝑢_(𝐴Δ^𝜇_𝐵)+Δ^𝜇_(𝐴𝑢_𝐵)𝑖^𝑗=Λ^𝜇_𝐴𝐵𝛿_𝑖^𝑗,𝑢_(𝐴Δ^𝜇_𝐵)+Δ^𝜇_(𝐴𝑢_𝐵)𝚤̂𝚥̂=Λ^𝜇_𝐴𝐵𝛿𝚤𝚥̂,̂ (2.5) where Λ^𝜇_𝐴𝐵=(Γ₀𝐺^𝜇Γ₀)_𝐴𝐵=(Γ₀Γ^𝜇Γ₀𝐶⁻¹)_𝐴𝐵, whereby Λ⁰_𝐴𝐵=𝐺⁰_𝐴𝐵 and Λ^𝑎_𝐴𝐵=−𝐺^𝑎_𝐴𝐵. 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 (𝑇_𝜇𝜈)_𝑖^𝑗 and (𝑇_𝜇𝜈)𝚤̂𝚥̂.

2.2. Shadow Supersymmetry

The one-dimensional 𝑁=4 superalgebra is specified by {𝑄_𝐴,𝑄_𝐵}=2𝑖𝛿_𝐴𝐵𝜕₀, which corresponds to (2.2) in the limit 𝜕_𝑎→0. In this case, the supercharges are represented as 𝑄_𝐴𝑖𝚤̂=𝑢_𝐴𝑖𝚤̂+𝑑_𝐴𝑖𝚤̂𝜕_𝜏,𝑄_𝐴𝚤^𝑖̂=𝑖𝑢_𝐴𝚤^𝑖̂+𝑖𝑑_𝐴𝚤^𝑖̂𝜕_𝜏. (2.6) 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 Δ⁰_𝐴 as 𝑑_𝐴 and Δ⁰_𝐴 as 𝑑_𝐴. 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 𝑢_(𝐴𝑢_𝐵)𝑖^𝑗=0,𝑢_(𝐴𝑢_𝐵)𝚤̂𝚥̂𝑑=0,_(𝐴𝑑_𝐵)𝑖^𝑗=0,𝑑_(𝐴𝑑_𝐵)𝚤̂𝚥̂=0. (2.7) 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 𝑢_(𝐴𝑑_𝐵)+𝑑_(𝐴𝑢_𝐵)𝑖^𝑗=𝛿_𝐴𝐵𝛿_𝑖^𝑗,𝑢_(𝐴𝑑_𝐵)+𝑑_(𝐴𝑢_𝐵)𝚤̂𝚥̂=𝛿_𝐴𝐵𝛿𝚤𝚥̂.̂ (2.8) 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 𝑢_𝐴 and 𝑑_𝐴 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 matrices 𝑢_𝐴, 𝑢_𝐴, 𝑑_𝐴, and 𝑑_𝐴 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 Δ^𝑎_𝐴 are completely determined by the “time-like” linkage matrices Δ⁰_𝐴. The proof of this assertion is given as Appendix A, with the resultΔ^𝑎_𝐴𝑖𝚤̂=−Γ⁰Γ^𝑎_𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂,Δ^𝑎_𝐴𝚤^𝑖̂=−Γ⁰Γ^𝑎_𝐴^𝐵Δ⁰_𝐵𝚤^𝑖̂. (3.1) It is interesting that the matrix (Γ⁰Γ^𝑎)_𝐴^𝐵 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 (𝑇_𝜇𝜈)_𝑖^𝑗 and (𝑇_𝜇𝜈)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 (𝑇_𝜇𝜈)_𝑖^𝑗 and (𝑇_𝜇𝜈)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 Γ^𝑎 on the right-hand side is tied closely to the appearance of the Γ^𝑎 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 in 𝐷-dimensions closes only if (Ω^𝜇_𝐴𝐵)_𝑖^𝑗𝜕_𝜇𝜙_𝑗=0 and (Ω^𝜇_𝐴𝐵)𝚤̂𝚥̂𝜕_𝜇𝜓𝚥̂=0, where we define the following useful matrices: Ω^𝜇_𝐴𝐵𝑖^𝑗=𝑢_(𝐴Δ^𝜇_𝐵)+Δ^𝜇_(𝐴𝑢_𝐵)𝑖^𝑗−Λ^𝜇_𝐴𝐵𝛿_𝑖^𝑗,Ω^𝜇_𝐴𝐵𝚤̂𝚥̂=𝑢_(𝐴Δ^𝜇_𝐵)+Δ^𝜇_(𝐴𝑢_𝐵)𝚤̂𝚥̂−Λ^𝜇_𝐴𝐵𝛿𝚤𝚥̂.̂ (3.2) 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 (Ω^𝜇_𝐴𝐵)_𝑖^𝑗 and (Ω^𝜇_𝐴𝐵)_̃𝚤𝚥̃, 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 Ω^𝜇_𝐴𝐵𝑖^𝑗Ω=0,_𝜇𝐴𝐵𝚤̂𝚥̂=0. (3.3) 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 the 𝑑×𝑑 matrices Ω^𝜇_𝐴𝐵 and Ω^𝜇_𝐴𝐵, 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 4D 𝑁=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 minimal 𝑁=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 𝐺^𝑎_𝐴𝐵=𝐺^𝑎_(𝐴𝐵) 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 are 𝑁=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 𝑄_1,2,3,4. For purposes of setting a convention for ordering the rows and columns of our linkage matrices, we sequence the boson fields 𝜙_𝑖 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 𝜙_𝑖 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 two 𝑁=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 (𝑢_𝐴)_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 𝜙_𝑖 to the fermions 𝜓𝚤̂ having greater engineering dimension. Since there are no edges linking downward from any of the boson vertices, it follows that (𝑑_𝐴)_𝑖𝚤̂=0 in this case. Similarly, we have (𝑢_𝐴)𝚤^𝑖̂=0, reflecting the fact that none of the fermions have upward directed edges. Finally, we have (𝑑^𝐴)𝚤^𝑗̂=𝛿^𝐴𝐵𝛿^𝑗𝑘(𝑢_𝐵)_𝑘𝑘𝛿𝑘𝚤̂ and (𝑑^𝐴)_𝑖𝚥̂=𝛿^𝐴𝐵𝛿𝚥𝑘̂(𝑢_𝐵)𝑘^𝑘𝛿_𝑘𝑖, schematically 𝑑_𝐴=𝑢^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴, reflecting the fact that every edge describes a pairing of an upward directed term and a corresponding downward directed term. The relationships 𝑑_𝐴=𝑢^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴 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 Figure 1. The up matrices for the right Adinkra in that figure are obtained from these by changing the sign of 𝑢₃.

4.1. The 𝑁=4 Bosonic 4-4 Adinkras

Using the features 𝑢_𝐴=𝑑^𝐵=0 and 𝑑^𝐴=𝑢^𝑇_𝐴, and using the matrices 𝐺^𝑎_𝐴𝐵 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 𝑢₁Δ¹₁=0. We then use (3.1), along with the gamma matrices in (E.3) to determine Δ¹₁=−Δ⁰₃, which is equivalent to Δ¹₁=−𝑑₃ using the nomenclature Δ⁰_𝐴≡𝑑_𝐴. 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 𝑢₁𝑢^𝑇₃=0, where we have also used 𝑑₃=𝑢^𝑇₃. 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 𝑢₁𝑢^𝑇₃=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 Δ^𝑎_𝐴𝐵 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 𝑢_𝐴, 𝑑_𝐴=Δ⁰_𝐴, and their transposes.

4.2. The 𝑁=4 Bosonic 3-4-1 Adinkras

Consider next those Adinkras obtained by raising one vertex starting with each Adinkra in Figure 1. 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 Figure 1 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 bosons 𝜙_𝑖 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 Figure 2, 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 𝑑_𝐴 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 𝑢_𝐴=𝑑^𝑇_𝐴, 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 Figure 2. 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 the 𝑛th boson vertex if and only if the 𝑛th row of each boson down matrix is null, that is, provided (𝑑_𝐴)_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 𝑢_𝐴=𝑑^𝑇_𝐴, 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 𝑢_𝐴 with the 𝑛th row of the respective boson down matrix 𝑑_𝐴. Thus, we implement the matrix reorganizations (𝑢_𝐴)_𝑛𝚤̂↔(𝑑_𝐴)_𝑛𝚤̂. At the same time, we must interchange the 𝑛th column of each fermion up matrix 𝑢_𝐴 with the 𝑛th column of the respective fermionic down matrix 𝑑_𝐴, via (𝑢_𝐴)𝚤^𝑛̂↔(𝑑_𝐴)𝚤^𝑛̂. The latter operation preserves the standard relationships 𝑢_𝐴=𝑑^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴. 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 standard 𝑁=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 𝑢_𝐴=𝑑^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴. Using the 𝐺^𝑎_𝐴𝐵 given in (E.10), the first condition in (3.3) reduces for the choice (𝑎∣𝐴𝐵)=(1∣11) to the matrix equation 𝑢₁𝑢^𝑇₃+𝑑₃𝑑^𝑇₁=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 Figure 2 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, 𝑢₃ and 𝑑₃ each by a multiplicative minus sign. The (𝑎∣𝐴𝐵)=(1∣1,1) enhancement equation, 𝑢₁𝑢^𝑇₃+𝑑₃𝑑^𝑇₁=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. The 𝑁=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 minimal 𝑁=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 Figure 3. The linkage matrices for the right Adinkra in that figure are obtained from these by changing the sign of 𝑢₃ and 𝑑₃.

Figure 3 

The 𝑁=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 𝑢_𝐴=𝑑^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴.

We found above that for each of the two 𝑁=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 𝑢₁𝑢^𝑇₃+𝑑₃𝑑^𝑇₁ 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 𝑢₁𝑢^𝑇₃+𝑑₃𝑑^𝑇₁ 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 𝑢₁𝑢^𝑇₂+𝑑₂𝑑^𝑇₁+𝑢₄𝑢^𝑇₃+𝑑₃𝑑^𝑇₄=0. (4.1) 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 Figure 3, 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 Table 3 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 𝑢₂𝑢^𝑇₂+𝑢₃𝑢^𝑇₃+𝑑₂𝑑^𝑇₂+𝑑₃𝑑^𝑇₃=2, (4.2) 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 Figure 3 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 Figure 3 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-Member 𝑁=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 Figure 1. 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 Figure 1 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 4D 𝑁=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 to 𝐷 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 a 128×128 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 Δ⁰_𝐴=𝑢^𝑇_𝐴 and Δ⁰_𝐴=𝑢^𝑇_𝐴. 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 components 𝜙_𝑖 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 closed 𝑝-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 closed 𝑝-form multiplet components. Since closed 1-forms do not involve gauge invariance, it follows that the simplest case involves closed 2-forms, such as 𝐹_𝜇𝜈 subject to 𝜕_[𝜆𝐹_𝜇𝜈]=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 𝑃_𝐴𝑖𝚤̂∶=𝑢^𝑇_𝐴−Δ⁰_𝐴𝑖𝚤̂, (6.1) where 𝑢^𝑇_𝐴 is the transpose of the 𝐴th fermion “up” matrix. A nonvanishing phantom matrix indicates the presence of one-way upward-directed Adinkra edges. If 𝑃_𝐴 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 Δ^𝑎_𝐴=−Γ⁰Γ^𝑎Δ⁰_𝐴−12Γ⁰Γ^𝑎𝑃_𝐴+𝑇^0𝑎𝑃_𝐴−𝑃_𝐴𝑇^0𝑎. (6.2) 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 4D 𝑁=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 𝐹_𝜇𝜈. It is natural to write 𝐸_𝑎=𝐹_0𝑎 and 𝐵^𝑎=(1/2)𝜀^𝑎𝑏𝑐𝐹_𝑏𝑐. The Bianchi identity 𝜕_[𝜆𝐹_𝜇𝜈]=0 correlates 𝐸_𝑎 and 𝐵^𝑎. Locally, we can solve the Bianchi identity in terms of a vector potential 𝐴_𝜇, so that 𝐸_𝑎=𝜕₀𝐴_𝑎−𝜕_𝑎𝐴₀ and 𝐵^𝑎=𝜀^𝑎𝑏𝑐𝜕_𝑏𝐴_𝑐. Upon restriction to the zero-brane we take 𝜕_𝑎→0, so that 𝐸_𝑎→𝜕₀𝐴_𝑎 and 𝐵_𝑎→0. Since the magnetic fields vanish on the zero-brane, it is natural to think of the 𝐸_𝑎 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 𝐵^𝑎, along with constraints. To do this, we allow for “phantom” bosons on the worldline, which correspond to the 𝐵^𝑎 off of the worldline.

To accommodate the phantom bosons, we consider an enlarged bosonic vector space, 𝜙_𝑖=(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 𝜙_𝑖=(𝐸_𝑎,𝐷∣𝐵^𝑎), where 𝐵^𝑎 is the magnetic phantom associated with 𝐸_𝑎. Thus, 𝑎 and 𝑎 each assume the values 1,2,3, and we have 𝜙_1,2,3=𝐸_1,2,3 and 𝜙_5,6,7=𝐵^1,2,3, 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 𝜙_1,2,3,4=(𝐸₁,𝐸₂,𝐸₃,𝐷). Matrices with two boson indices then divide into four sectors, 𝑋_𝑖^𝑗, 𝑋_𝑖^𝑎, 𝑋_𝑎^𝑗, and 𝑋_𝑎^𝑏.

The shadow transformation rules associated with the Maxwell multiplet can be written as (2.3), but the linkage matrices are not square! Instead, (𝑢_𝐴)𝚤^𝑗̂ is 7×4 and (Δ^𝜇_𝐴)_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, 𝜕_[𝜆𝐹_𝜇𝜈]=0, usefully rewritten as 𝜕₀𝐵^𝑎=𝜀^𝑎𝑏𝑐𝜕_𝑏𝐸_𝑐,𝜕_𝑎𝐵^𝑎=0, (6.3) 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 Ω⁰_𝐴𝐵𝑖^𝑎Ω⟶0,_{𝑎𝐴𝐵𝑖𝑏}⟶Ω^𝑎_𝐴𝐵𝑖^𝑏+𝜀^𝑎𝑏_𝑐Ω⁰_𝐴𝐵𝑖^𝑐, (6.4) whereby we exchange each appearance of 𝜕₀𝐵^𝑎 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 Ω¹_𝐴𝐵𝑖¹Ω⟶0,_{2𝐴𝐵𝑖2}⟶Ω²_𝐴𝐵𝑖²−Ω¹_𝐴𝐵𝑖¹,Ω³_𝐴𝐵𝑖³⟶Ω³_𝐴𝐵𝑖³−Ω¹_𝐴𝐵𝑖¹. (6.5) In this way, we define a canonical structure of the matrices (Ω^𝜇_𝐴𝐵)_𝑖^𝑗, ensured by the transformations (6.4) and (6.5), enabled by the Bianchi identity (6.3), whereby (Ω⁰_𝐴𝐵)_𝑖^𝑎=0 and (Ω¹_𝐴𝐵)_𝑖¹=0. The first equation in (3.3) may now be interpreted as saying that (Ω^𝜇_𝐴𝐵)_𝑖^𝑗→0 under these transformations.

We remark that that each of the 40 7×7 matrices (Ω^𝜇_𝐴𝐵)_𝑖^𝑗 defined by (3.2), using the linkage matrices exhibited in Appendix D, do satisfy (Ω^𝜇_𝐴𝐵)_𝑖^𝑗→0 using the transformations (6.4) and (6.5). The diligent reader is encouraged to check this assertion.

6.4. The 𝑝=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 rule 𝛿_𝑄𝜆 given by (1/2)𝜀_𝑎𝑏𝑐𝐵^𝑎Γ^𝑏𝑐𝜖 or by (1/2)𝜀_𝑎𝑏𝑐𝐵^𝑎Γ^𝑏𝑐Γ₅𝜖. 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 𝛿_𝑄𝜆=⋯+(1/2)𝜀_𝑎𝑏𝑐𝐵^𝑎Γ^𝑏𝑐𝜖. By rearranging this term into the form 𝛿_𝑄𝜆_𝑖=⋯+𝜖^𝐴(𝑢_𝐴)^𝑎_𝑖𝐵_𝑎, 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.) 𝑢_𝐴𝚤^𝑎̂=12𝜀^𝑎𝑏𝑐Γ_𝑏𝑐𝚤_𝐴̂,Δ⁰_𝐴𝑎𝚤̂=0, (6.6) whereby using (6.1) we determine the nonvanishing part of the phantom matrix as 𝑃_𝐴𝑎𝚤̂=12𝜀_𝑎𝑏𝑐Γ^𝑏𝑐_𝐴𝚤̂. (6.7) The entire phantom matrix has nonvanishing entries only in its final three rows.

We can resolve the Δ^𝑎_𝐴 matrices in two parts, using different methods. First we resolve the phantom part (Δ^𝑎_𝐴)_𝑎𝚤̂. Then we resolve the nonphantom part (Δ^𝑎_𝐴)_𝑎𝚤̂.

We determine the phantom part of the space-like boson down matrices using the fact that (Δ⁰_𝐴)_𝑎𝚤̂=0, which says that there are no connections linking downward from the phantoms. Thus, equation (A.4) tells us (Δ^𝑎_𝐴)_𝑎𝚤̂=−(𝑇^0𝑎)_𝑎^𝑖(Δ⁰_𝐴)_𝑖𝚤̂. Next, we use the fact that a boost shuffles magnetic fields into electric fields, via (𝑇_0𝑎)_𝑏^𝑐𝐵_𝑐=𝜀^𝑐_𝑎𝑏𝐸_𝑐, to determine Δ^𝑎_𝐴𝑏𝚤̂=𝜀_𝑏^𝑎𝑐Δ⁰_𝐴𝑐𝚤̂. (6.8) This determines the 𝑏th 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 (𝑃_𝐴)_𝑎𝚤̂=0. The third term on the right-hand side can be resolved by noting that a boost shuffles electric fields into magnetic fields, via (𝑇^0𝑎)_𝑏^𝑐𝐸_𝑐=𝜀_𝑏^𝑎𝑐𝐵_𝑐, whereby (𝑇^0𝑎)_𝑏^𝑐=𝜀_𝑏^𝑎𝑐. Substituting this result, along with (6.7), we derive Δ^𝑎_𝐴𝑏𝚤̂=−Γ⁰Γ^𝑎_𝐴^𝐵Δ⁰_𝐴𝑏𝚤̂+Γ^𝑎_𝑏𝐴𝚤̂. (6.9) 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 −2𝜀_𝑏^𝑎𝑐(ℛ_𝑐)_𝐴𝚤̂, where ℛ_𝑐 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 minimal 𝑁=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 Figure 1. (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 𝑢_𝐴 and the time-like down matrices Δ⁰_𝐴 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 Δ^𝑎_𝐴, 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 (Ω^𝜇_𝐴𝐵)_𝑖^𝑗 and (Ω^𝜇_𝐴𝐵)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 of Γ₅. 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 basis 𝜙_𝑖. 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 that 𝜙_1,2,3 correspond to these.(2)Compute time-like linkage matrices 𝑢_𝐴, 𝑢_𝐴, Δ⁰_𝐴, and Δ⁰_𝐴 from the Adinkra.(3)Augment the boson vector space by adding on a phantom sector consisting of three new fields 𝜙_1,2,3 with the same engineering dimension as 𝜙_1,2,3.(4)Add phantom sectors to the up matrices 𝑢_𝐴, by adding three extra columns, and add phantom sectors to the time-like down matrices Δ⁰_𝐴 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 (Ω^𝑎_𝐴𝐵)_𝑖^𝑗 and (Ω^𝑎_𝐴𝐵)𝚤̂𝚥̂ using (3.2).(8)Reshuffle the boson matrix (Ω^𝑎_𝐴𝐵)_𝑖^𝑗 using the prescription (6.4) and (6.5), to obtain a new matrix, in canonical form, Ω^𝑎_𝐴𝐵𝑖^𝑗⟶Ω^𝑎_𝐴𝐵𝑖^𝑗. (6.10) The presence of the hat indicates canonical form.(9)The 𝑝=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 Ω^𝜇_𝐴𝐵𝑖^𝑗Ω=0,_𝜇𝐴𝐵𝚤̂𝚥̂=0. (6.11)

This is our 𝑝=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 Ω 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 higher 𝑁, 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-dimensional 𝑁=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.

Appendices

A. A Proof

In this appendix we prove that demanding Lorentz invariance of the linkage matrices defined in (2.3) completely determines all of the space-like linkage matrices Δ^𝑎_𝐴 in terms of the time-like linkage matrices Δ⁰_𝐴, and does so in precisely the manner specified above as equation (3.1). We also show how this same requirements provides constraints on the spin representation content of supermultiplet component fields.

The linkage matrices (Δ^𝜇_𝐴)_𝑖𝚤̂ transform under 𝔰𝔭𝔦𝔫(1,𝐷−1), manifestly, as 𝛿_𝐿Δ^𝜇_𝐴𝑖𝚤̂=𝜃^𝜇_𝜈Δ^𝜈_𝐴𝑖𝚤̂+14𝜃^𝜆𝜎Γ_𝜆𝜎𝐴^𝐵Δ^𝜇_𝐵𝑖𝚤̂+12𝜃^𝜆𝜎𝑇_𝜆𝜎𝑖^𝑗Δ^𝜇_𝐴𝑗𝚤̂−12𝜃^𝜆𝜎Δ^𝜇_𝐴𝑖𝚥̂𝑇_𝜆𝜎𝚥̂𝚤̂. (A.1) In (A.1), the first term indicates that on (Δ^𝜇_𝐴)𝚥_𝑖̂ the 𝜇 index is a vector index, the second term indicates that the 𝐴 index is a spinor index, and the last line accommodates the representation content of the supermultiplet component fields.

Using (A.1), we obtain the following boost and rotation transformations for the “time-like" linkage matrices (Δ⁰_𝐴)_𝑖𝚤̂: 𝛿_boostΔ⁰_𝐴𝑖𝚤̂=𝜃⁰_𝑎Δ^𝑎_𝐴𝑖𝚤̂+12𝜃^0𝑎Γ_0𝑎𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂+𝜃^0𝑎𝑇_0𝑎Δ⁰_𝐴𝑖𝚤̂−𝜃^0𝑎Δ⁰_𝐴𝑇_0𝑎𝑖𝚤̂,𝛿_rotationΔ⁰_𝐴𝑖𝚤̂1=+2𝜃^𝑎𝑏Γ_𝑎𝑏𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂+𝜃^𝑎𝑏𝑇_𝑎𝑏Δ⁰_𝐴𝑖𝚤̂−𝜃^𝑎𝑏Δ⁰_𝐴𝑇_𝑎𝑏𝑖𝚤̂, (A.2) and the following boost and rotation transformations for the “space-like" linkage matrices: 𝛿_boostΔ^𝑎_𝐴𝑖𝚤̂=𝜃^𝑎₀Δ⁰_𝐴𝑖𝚤̂+12𝜃^0𝑏Γ₀Γ_𝑏𝐴^𝐵Δ^𝑎_𝐵𝑖𝚤̂+𝜃^0𝑏𝑇_0𝑏Δ^𝑎_𝐴𝑖𝚤̂−𝜃^0𝑏Δ^𝑎_𝐴𝑇_0𝑏𝑖𝚤̂,𝛿_rotationΔ^𝑎_𝐴𝑖𝚤̂=𝜃^𝑎_𝑏Δ^𝑏_𝐴𝑖𝚤̂+14𝜃^𝑏𝑐Γ_𝑏𝑐𝐴^𝐵Δ^𝑎_𝐵𝑖𝚤̂+12𝜃^𝑏𝑐𝑇_𝑏𝑐Δ^𝑎_𝐴𝑖𝚤̂−12𝜃^𝑏𝑐Δ^𝑎_𝐴𝑇_𝑏𝑐𝑖𝚤̂. (A.3) We demand that the linkage matrices are Lorentz invariant. This imposes that each of the transformations in (A.2) and (A.3) must vanish. (This is similar to “demanding" that the gamma matrices appearing in a Salam-Strathdee superfield be Lorentz invariant—in that case they are, automatically, as a consequence of the Clifford algebra.) Requiring 𝛿_boostΔ⁰=0 imposes Δ^𝑎_𝐴𝑖𝚤̂1=−2Γ⁰Γ^𝑎_𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂−𝑇^0𝑎Δ⁰_𝐴𝑖𝚤̂+Δ⁰_𝐴𝑇^0𝑎_𝑖𝚤̂. (A.4) This determines (Δ^𝑎_𝐴)_𝑖𝚤̂ in terms of (Δ⁰_𝐴)_𝑖𝚤̂ and in terms of the representation assignments of the supermultiplet component fields.

The remaining consequences of imposing Lorentz invariance on the linkage matrices (Δ^𝜇_𝐴)_𝑖𝚤̂ are the following: 12Γ_𝑎𝑏𝐴^𝐵Δ⁰_𝐵=Δ⁰_𝐴𝑇_𝑎𝑏−𝑇_𝑎𝑏Δ⁰_𝐴,𝛿_𝑏^𝑎Δ⁰_𝐴=12Γ₀Γ_𝑏𝐴^𝐵Δ^𝑎_𝐵+𝑇_0𝑏Δ^𝑎_𝐴−Δ^𝑎_𝐴𝑇_0𝑏,𝜂^𝑎[𝑏Δ_𝐴^𝑐]+14Γ^𝑏𝑐_𝐴^𝐵Δ^𝑎_𝐵=12Δ^𝑎_𝐴𝑇^𝑏𝑐−12𝑇^𝑏𝑐Δ^𝑎_𝐴, (A.5) where the (⋅)𝚤_𝑖̂ index structure has been suppressed on each term. These correspond, respectively, to 𝛿_rotationΔ⁰=0, 𝛿_boostΔ^𝑎=0, and 𝛿_rotationΔ^𝑎=0, for arbitrary transformation parameters 𝜃^0𝑎 and 𝜃^𝑎𝑏. Equations (A.5) place significant restrictions on the spin representation content of the component fields. As explained above, we suspect that these equations encode useful and extractable information regarding allowable complements of spin structures in supermultiplets in diverse dimensions.

The linkage matrices (𝑢_𝐴)_𝑖𝚤̂ transform under 𝔰𝔭𝔦𝔫(1,𝐷−1), manifestly, as 𝛿𝑢_𝐴𝑖𝚤̂=14𝜃^𝜇𝜈Γ_𝜇𝜈𝐴^𝐵𝑢_𝐵𝑖𝚤̂+12𝜃^𝜇𝜈𝑇_𝜇𝜈𝑢_𝐴𝑖𝚤̂−12𝜃^𝜇𝜈𝑢_𝐴𝑇_𝜇𝜈𝑖𝚤̂. (A.6) Requiring that these transformations vanish imposes 12Γ_𝜇𝜈𝐴^𝐵𝑢_𝐵𝑖𝚤̂=𝑢_𝐴𝑇_𝜇𝜈𝑖𝚤̂−𝑇_𝜇𝜈𝑢_𝐴𝑖𝚤̂. (A.7) This indicates correlations between the up linkage matrices and the representation content of the component fields.

Similar conditions result from demanding invariance of (𝑢_𝐴)𝚤^𝑖̂ and (Δ^𝜇_𝐴)𝚤^𝑖̂. These are obtained from the above constraints by placing tildes on all matrices which do not have tildes and removing tildes from those that do. For example, invariance of (𝑢_𝐴)𝚤^𝑖̂ imposes 12Γ_𝜇𝜈𝐴^𝐵𝑢_𝐵𝚤^𝑖̂=𝑢_𝐴𝑇_𝜇𝜈𝚤^𝑖̂−𝑇_𝜇𝜈𝑢_𝐴𝚤^𝑖̂. (A.8) Note that for standard Adinkras we have Δ⁰_𝐴=𝑢^𝑇_𝐴. (For nonstandard Adinkras, such as those which accommodate gauge invariances, this relationship does not hold. Instead, we define 𝑢_𝐴=Δ⁰_𝐴+𝑃_𝐴, where 𝑃_𝐴 is a so-called phantom matrix, which encodes the nexus of one-way upward-directed Adinkra edges. This generalization is addressed in Section 6.) Thus, using (A.8) we determine 12Γ_𝜇𝜈𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂=𝑇^𝑇_𝜇𝜈Δ⁰_𝐴𝑖𝚤̂−Δ⁰_𝐴𝑇^𝑇_𝜇𝜈𝑖𝚤̂. (A.9) This equation can be used in conjunction with (A.4) to replace that equation with an analog in which the representation matrices are not included.

The boost matrices (𝑇_0𝑎)_𝑖^𝑗 and (𝑇_0𝑎)𝚤̂𝚥̂ are generically symmetric (For example, if the fermions assemble as spinors then 𝑇_0𝑎=(1/2)Γ₀Γ_𝑎. In the Majorana basis Γ₀ is antisymmetric and real while Γ_𝑎 are symmetric and real, and since Γ₀ and Γ_𝑎 anticommute, it follows that 𝑇_0𝑎 is symmetric in that case. For vectors 𝑉_𝑎 we have (𝑇_0𝑎)₀^𝑏=𝛿_𝑎^𝑏 and (𝑇_0𝑎)_𝑏⁰=−𝜂_𝑎𝑏; for our metric choice 𝜂_𝑎𝑏=−1, so that these 𝑇_0𝑎 are symmetric. This reasoning generalizes to higher-rank tensors and to all products of tensor and spinor representations. Note too, that if the boost matrices were antisymmetric, then (A.9) and (A.4) could be used together to prove the inconsistent result that Δ^𝑎=0.) Therefore, (A.9) can be rewritten as 𝑇_0𝑎Δ⁰_𝐴𝑖𝚤̂−Δ⁰_𝐴𝑇_0𝑎𝑖𝚤̂=12Γ₀Γ_𝑎𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂. (A.10) Substituting this result for the last two terms in (A.4), we determine Δ^𝑎_𝐴𝑖𝚤̂=−Γ⁰Γ^𝑎_𝐴^𝐵Δ⁰_𝐵𝑖𝚤̂. (A.11) Remarkably, this relationship is completely independent of the representation content of the component fields. This is an interesting result, which says that the space-like linkage matrices Δ^𝑎 are determined from the time-like linkage matrices Δ⁰.

B. Adinkra Conventions

In this appendix we give a very concise overview of the graphical technology of Adinkra diagrams. These were introduced in [17], and have formed the basis of a multidisciplinary research endeavor, aimed at resolving a mathematically rigorous basis for supersymmetry [18, 19, 21, 23–25]. Some of the conventions, notably as regards sign choices, have varied in these references, in part because some of these papers aim at a physics audience and some at a mathematics audience. Thus, one reason for this appendix is to clarify our conventions, as used above, so that this paper can be appreciated without undue confusion. Another is to allow this paper to be functionally self-contained.

A representation of 𝑁-extended supersymmetry in one time-like dimension consists of 𝑑 boson fields 𝜙_𝑖 and 𝑑 fermion fields 𝜓𝚤̂ endowed with a set of transformation rules, generated by 𝛿_𝑄(𝜖), where 𝜖^𝐴 are a set of 𝑁 anticommuting parameters, which respect the 𝑁-extended supersymmetry algebra specified by the commutator [𝛿_𝑄(𝜖₁),𝛿_𝑄(𝜖₂)]=2𝑖𝛿_𝐴𝐵𝜖^𝐴₁𝜖^𝐵₂𝜕_𝜏. The transformation rules can be written for boson fields as 𝛿_𝑄𝜙_𝑖=−𝑖𝜖^𝐴(𝑄_𝐴)_𝑖𝚤𝜓̂𝚤̂ and for fermion fields as 𝛿_𝑄𝜓𝚤̂=−𝑖𝜖^𝐴(𝑄_𝐴)𝚤^𝑖̂𝜙_𝑖, where (𝑄_𝐴)𝚤_𝑖̂ and (𝑄_𝐴)𝚤^𝑖̂ are two sets of 𝑁 abstract 𝑑×𝑑 matrix generators of supersymmetry. By definition these represent 𝑄_(𝐴𝑄_𝐵)𝑖^𝑗=𝑖𝛿_𝑖^𝑗𝜕_𝜏,𝑄_(𝐴𝑄_𝐵)𝚤̂𝚥̂𝚤̂=𝑖𝛿𝚥𝜕̂_𝜏, (B.1) where the symmetrization brackets are defined with “weight-one”, whereby 𝑋_(𝐴𝑌_𝐵)=(1/2)(𝑋_𝐴𝑌_𝐵+𝑋_𝐵𝑌_𝐵).

It is possible to use cosmetic field redefinitions to redefine the component fields 𝜙_𝑖 and 𝜓𝚤̂ into a “frame" where the generators (𝑄_𝐴)_𝑖𝚥̂ and (𝑄_𝐴)𝚤̂𝚥̂ are first-order differential operators with a specialized matrix structure. Specifically, it is possible to write 𝑄_𝐴𝑖𝚤̂=𝑢_𝐴𝑖𝚤̂+𝑑_𝐴𝑖𝚤̂𝜕_𝜏,𝑄_𝐴𝚤^𝑖̂=𝑖𝑢_𝐴𝚤^𝑖̂+𝑖𝑑_𝐴𝚤^𝑖̂𝜕_𝜏, (B.2) where (𝑢_𝐴)_𝑖𝚤̂, (𝑑_𝐴)_𝑖𝚤̂, (𝑢_𝐴)𝚤^𝑗̂, and (𝑑_𝐴)𝚤^𝑗̂ are four sets of 𝑁 real 𝑑×𝑑 “linkage matrices" with the features that every entry of each of these matrices takes only one of three values, 0, 1, or −1, and such that there is at most one nonvanishing entry in every row and at-most one nonvanishing entry in every column of each of these matrices. Remarkably, we lose no generality by specializing to generators of the sort (B.2) with these particular properties. A mathematical proof that any one-dimensional supermultiplet can be written in this manner is provided in [18].

A simple example in the context of 𝑁=2 supersymmetry is given by the following transformation rules, 𝛿_𝑄𝜙₁=−𝑖𝜖¹𝜓₁−𝑖𝜖²𝜓₂,𝛿_𝑄𝜙₂=−𝑖𝜖¹𝜕_𝜏𝜓₂+𝑖𝜖²𝜕_𝜏𝜓₁,𝛿_𝑄𝜓₁=𝜖¹𝜕_𝜏𝜙₁−𝜖²𝜕_𝜏𝜙₂,𝛿_𝑄𝜓₂=𝜖¹𝜙₁+𝜖²𝜕_𝜏𝜙₂. (B.3) It is straightforward to verify that these satisfy the commutator relationship specified above.

The operator 𝜕_𝜏 carries unit engineering dimension, while supersymmetry parameters 𝜖^𝐴 carry engineering dimension one-half. (In a system where ℏ=𝑐=1, a field with engineering dimension 𝑞 carries units of (Mass)^𝑞.) Thus, in order to balance units in the transformation rules (B.3) it follows that the two fermions 𝜓_1,2 have a common engineering dimension one-half greater than 𝜙₁, and that 𝜙₂ has an engineering dimension one-half greater than the fermions, and one unit greater than 𝜙₁.

The transformation rules (B.3) can be expressed equivalently, in terms of linkage matrices, as 𝑢_1𝑖𝚥̂=10,𝑢_2𝑖𝚤̂=10,𝑑_1𝑖𝚤̂=01,𝑑_2𝑖𝚤̂=0,𝑢−1₁𝚤^𝑗̂=01,𝑢₂𝚤^𝑗̂=0,𝑑−1₁𝚤^𝑗̂=10,𝑑₂𝚤^𝑗̂=01, (B.4) where blank matrix entries represent zeros. This set of eight matrices is completely equivalent to the transformation rules (B.3). It is straightforward to verify, using (B.2), that the algebra (B.1) is properly represented using these matrices.

As an example, to illustrate what these matrices mean, consider the matrix 𝑢₂ defined in (B.4). This is the “second fermion up matrix", where the qualifier “second" refers to the subscript on 𝑢₂ and indicates that this matrix encodes a mapping under the second supersymmetry, while the qualifier “fermionic" refers to the tilde, and indicates that this matrix encodes transformations of the fermions. The single nonvanishing term in this matrix is in the first row, second column, which indicates that, of the two fermions, only the first fermion 𝜓₁ transforms under the second supersymmetry, into the second boson 𝜙₂. The fact that this matrix entry is −1 indicates a minus sign in the transformation rule 𝛿_𝑄𝜓₁ on the term proportional to 𝜙₂, that is, 𝛿_𝑄𝜓₁=⋯−𝜙₂𝜖², as seen in (B.3). The reason why this is called an “up" matrix is that it encodes a mapping “upward" from from a field with lower engineering dimension—𝜓₁ in this case–- to a field with higher engineering dimension—𝜙₂ in this case.

The matrices in (B.4) exhibit the properties 𝑢_𝐴=𝑑^𝑇_𝐴 and 𝑑_𝐴=𝑢^𝑇_𝐴. It is easy to see that this indicates a symmetric feature in the transformation rules (B.3), whereby a fermion appearing in a boson transformation rule is correlated with that boson appearing in the transformation rule for that fermion. In other words, terms in these transformation rules come paired. This feature is satisfied by a wide and important class of supermultiplets, which we call “standard". (These are also called “Adinkraic" in the literture.)

There is a third equivalent way to represent the supersymmetry transformations given by (B.3) and by (B.4). This method uses the observation that the generic properties of linkage matrices facilitate a concise system under which the entire collection of linkage matrices for a given multiplet can be faithfully represented by a graph. Such a graph, called an Adinkra, consists of 𝑑 white vertices (one for each boson) and 𝑑 black vertices (one for each fermion). Two vertices are connected by an 𝐴th colored edge if the two fields corresponding to those vertices are interrelated by the 𝐴th supersymmetry. The edge is rendered solid if the corresponding 𝑄_𝐴 matrix entries are +1 and are rendered dashed if the corresponding 𝑄_𝐴 matrix entries are −1. Finally, the vertices are arranged so that their heights on the graph correlate faithfully with the respective engineering dimension.

Thus, if we designate 𝑄₁ using purple edges and 𝑄₂ using blue edges, then the example multiplet described by (B.4), equivalent to (B.3), would have the following Adinkra, (B.5) where the numerals on the vertices specify the fields, for example, the black vertex labeled 2 represents the fermion field 𝜓₂. As an easy exercise, the reader should confirm that (B.4) can be recovered from (B.5) using the rules described above. There is a striking economy exhibited by this graphical method, empowered by the fact that these graphs completely encode every aspect of the transformation rules, in a way which allows for ready translation from any Adinkra into linkage matrices or into parameter-dependent transformation rules.

As another example, consider the following Adinkra, (B.6) This describes a supermultiplet distinct from the previous example, as evidenced by the fact that (B.6) spans only two different engineering dimensions, whereas (B.5) spans three.

We can readily extract the linkage matrices equivalent to (B.6). For example, the boson down matrices 𝑑₁ and 𝑑₂ obviously vanish because the two bosons do not connect “downward" to any lower fermion vertices. Similarly, the two fermion up matrices 𝑢₁ and 𝑢₂ also obviously vanish, since there are no links “upward" from the black vertices. We can determine the nonvanishing linkage matrices by “reading" the diagram. For example, the boson up matrix 𝑢₂ encodes blue edges connecting upward from boson vertices. Thus, since the boson 𝜙₁ links upward via blue edge only to the fermion 𝜓₂, and does so with a solid edge, this tells us that the matrix entry (𝑢₂)²₁=+1. In this way, we can translate the Adinkra (B.6) into the linkage matrices described by 𝑑_𝐴=0, 𝑢_𝐴=0, 𝑢_1𝑖𝚥̂=11,𝑢_2𝑖𝚤̂=1−1, (B.7) and 𝑑_𝐴=𝑢^𝑇_𝐴.

Note that the Adinkra (B.5) can be obtained from (B.6) by an interesting operation: by moving the vertex 𝜙₂ to a new position located one level above the fermions, while continuously maintaining all intervertex edge connections, so that the edges swivel upward during this process. This macrame-like move encodes a transformation which maps one supermultiplet into another, and is called a vertex raising operation.

One of our results concerning Adinkras is a mathematical proof that any standard supermultiplet can be obtained by a sequence of vertex raising operations starting from an Adinkras with vertices which span only two different heights, for example, (B.6). Accordingly, the representation theory of 1D standard multiplets breaks naturally into two parts; first to classify all of the possible two-height Adinkras for a given value of 𝑁, and then to systematize the possible sequences of vertex raises using each of these as a starting point.

Owing to the special role played by the two-height Adinkras, we have given these a special name. Standard Adinkras which span only two height assignments are called Valise Adinkras, or Valises for short. The reason for this nomenclature is based on the observation that a large number of multiplets can be “unpacked", as from a suitcase (or a valise), by judicious choices of vertex raises. (We credit Tristan Hübsch for inventing this catchy and useful term.)

Using the information above, the reader ought to be able to verify the relationships between the Adinkras shown in Figures 1, 2, and 3, with the corresponding linkage matrices exhibited in the respective Tables 1, 2, and 3, and should appreciate our use of the terms Adinkra, Valise, and the concept of vertex raising.

C. The Shadow of the Chiral Multiplet

In this appendix we explain how to dimensionally-reduce the 4D chiral multiplet to extract its shadow.

The 4D 𝑁=1 chiral multiplet has the following transformation rules, ^{𝛿𝑄𝜙=2𝑖𝜖𝐿𝜒𝑅,𝛿}_𝑄^𝜒_𝑅/^{=𝜕𝜙𝜖}_𝐿^+𝐹𝜖_𝑅^,_{𝛿𝑄𝐹=2𝑖𝜖𝑅}/^𝜕𝜒_𝑅^, (C.1) where 𝜙 is a complex scalar, 𝐹 is a complex auxiliary scalar, and 𝜒_𝑅 is a right-chiral Weyl spinor field. The parameter 𝜖_𝑅 is also a right-chiral spinor, while 𝜖_𝐿 describes the corresponding Majorana conjugate, that is, 𝜖_𝐿=𝐶⁻¹𝜖^𝑇_𝑅. The transformation rules (C.1) satisfy ^[𝛿_𝑄^(𝜖₁^),𝛿_𝑄^(𝜖₂^)]=4𝑖𝜖_[2𝐿/^𝜕𝜖_1]𝐿 on all component fields 𝜙, 𝐹, and 𝜒_𝑅. Note that we can define a Majorana spinor parameter via 𝜖=𝜖_𝑅+𝜖_𝐿, so that 𝜖_𝑅,𝐿=(1/2)(1±Γ₅)𝜖 are the corresponding right- and left-chiral projections. In terms of the Majorana spinor, the algebra is ^[𝛿_𝑄^(𝜖₁^),𝛿_𝑄^(𝜖₂^)]=2𝑖𝜖₂/^𝜕𝜖₁.

We express spinors in the Majorana basis described in Appendix E. (The choice of basis is immaterial for the computing the dimensional reduction; we obtain identical results using any other basis. We use the Majorana basis here in order to maintain consistency with other derivations in this paper.) Accordingly, we write the spinor field and the spinor supersymmetry parameter as 𝜒_𝑅=12𝜒₁+𝑖𝜒₂𝜒₂−𝑖𝜒₁𝜒₃+𝑖𝜒₄𝜒₄−𝑖𝜒₃,𝜖_𝑅=12𝜖₁+𝑖𝜖₂𝜖₂−𝑖𝜖₁𝜖₃+𝑖𝜖₄𝜖₄−𝑖𝜖₃, (C.2) where 𝜒_1,2,3,4 are each real anti-commuting fields and 𝜖_1,2,3,4 are each real anti-commuting constant parameters. We also write the complex boson fields as 𝜙=𝜙₁+𝑖𝜙₂, where 𝜙_1,2 are real bosons, and 𝐹=𝐹₁+𝑖𝐹₂, where 𝐹_1,2 are real auxiliary bosons.

Using these definitions, setting 𝜕_𝑎=0, and using the spinor identities in Appendix E, the transformation rules (C.1) become 𝛿_𝑄𝜙₁=−𝑖𝜖₁𝜒₃+𝑖𝜖₂𝜒₄+𝑖𝜖₃𝜒₁−𝑖𝜖₄𝜒₂,𝛿_𝑄𝜙₂=−𝑖𝜖₁𝜒₄−𝑖𝜖₂𝜒₃+𝑖𝜖₃𝜒₂+𝑖𝜖₄𝜒₁,𝛿_𝑄𝜒₁=𝐹₁𝜖₁−𝐹₂𝜖₂−𝜙̇₁𝜖₃−𝜙̇₂𝜖₄,𝛿_𝑄𝜒₂=𝐹₂𝜖₁+𝐹₁𝜖₂−𝜙̇₂𝜖₃+𝜙̇₁𝜖₄,𝛿_𝑄𝜒₃=𝜙₁𝜖₁+𝜙₂𝜖₂+𝐹₁𝜖₃−𝐹₂𝜖₄,𝛿_𝑄𝜒₄=𝜙₂𝜖₁−𝜙₁𝜖₂+𝐹₂𝜖₃+𝐹₁𝜖₄,𝛿_𝑄𝐹₁=−𝑖𝜖₁𝜒̇₁−𝑖𝜖₂𝜒̇₂−𝑖𝜖₃𝜒̇₃−𝑖𝜖₄𝜒̇₄,𝛿_𝑄𝐹₂=−𝑖𝜖₁𝜒̇₂+𝑖𝜖₂𝜒̇₁−𝑖𝜖₃𝜒̇₄+𝑖𝜖₄𝜒̇₃. (C.3) These rules describe the shadow of (C.1). We organize the boson fields so that 𝜙_𝑖=(𝜙₁,𝜙₂,𝐹₁,𝐹₃) and the fermion fields so that 𝜓𝚤̂=(𝜒₁,𝜒₂,𝜒₃,𝜒₄). Using the Adinkra conventions described in Appendix B, along with our edge coloration scheme whereby 𝑄_1,2,3,4 are respectively described by purple, blue, green, and red colored edges, we can unambiguously represent (C.3) as the Adinkra shown in Figure 4.

It is easy to translate the Adinkra in Figure 4 into equivalent up and down matrices. For example, we determine the boson up matrix 𝑢₁ by looking at the purple colored edges extending upward from boson vertices. There are two such edges: a solid edge connecting 𝜙₁ with 𝜓₃ and solid edge connecting 𝜙₂ with 𝜓₄. Thus, there are two nonvanishing entries in 𝑢₁: one in the first row, third column, and the other in the second row, fourth column. These both take the value +1 because both edges are solid. In this way we determine the up matrices shown in Table 4 and the time-like down matrices shown in Table 5.

Table 4 

The boson up linkage matrices for the chiral multiplet shadow. The fermion time-like down matrices are determined from these via 𝑑_𝐴=𝑢^𝑇_𝐴.

Table 5 

The time-like dowm linkage matrices for the chiral multiplet. The fermion up matrices are determined from these via 𝑢_𝐴=𝑑^𝑇_𝐴.

We can also determine the time-like down linkage matrices directly from (C.1). For example, to determine the femion 1-sector down matrices, Δ¹_𝐴 we isolate those terms in the fermion transformation rule 𝛿_𝑄𝜓_𝑅 involving the derivative 𝜕₁. These are given by 𝛿_𝑄⁽¹⁾𝜒_𝑅=Γ¹𝜖_𝐿𝜕₁𝜙. We then use the explicit matrix Γ¹ specified in (E.9), the spinor components specified in (C.2), and we write 𝜙=𝜙₁+𝑖𝜙₂. This allows us, after a small amount of algebra, to rewrite the 1-sector fermion transformation rule as 𝛿_𝑄𝜓𝚤̂=𝜖^𝐴(Δ¹_𝐴)^𝑗𝚤̂𝜕₁𝜙_𝑗, from which we can read off the four matrices (Δ¹_𝐴)^𝑗𝚤̂. It is straightforward to perform this calculation, and then to verify that the matrices thereby obtained satisfy Δ¹_𝐴=−(Γ⁰Γ¹)^𝐵_𝐴Δ⁰_𝐵, where Δ⁰_𝐴=𝑢^𝑇_𝐴. Similar calculations can be done for all of the time-like down matrices, providing a nice consistency check on our powerful assertion (3.1).

D. The Shadow of the Maxwell Field Strength Multiplet

In this appendix we determine the linkage matrices for the 4D 𝑁=4 Maxwell field-strength multiplet. This provides a means to exhibit precisely how 1D phantom sectors arise upon restriction of a 𝑝=1 gauge multiplet to a zero-brane. This appendix is complementary to Section 6 in the main text, above.

The 4D 𝑁=1 super-Maxwell field-strength multiplet has the following transformation rules, ^{𝛿𝑄1𝜆=2𝐹𝜇𝜈Γ𝜇𝜈𝜖−𝑖𝐷Γ5𝛿𝜖,𝑄𝐹𝜇}/^{𝜈=−2𝑖𝜖Γ[𝜇𝜕𝜈]𝛿𝜆,}_𝑄^𝐷=𝜖Γ₅^𝜕𝜆, (D.1) where 𝜆 is a Majorana spinor gaugino field, 𝐷 is a real auxiliary (pseudo)scalar, and the field-strength tensor 𝐹_𝜇𝜈 is subject to the Bianchi identity 𝜕_[𝜆𝐹_𝜇𝜈]=0. We obtain the linkage matrices equivalent to (D.1) by deconstructing these rules using a specific spinor basis, and rewriting them in terms of individual degrees of freedom as specified in (2.3). It follows simply that that Δ^𝑎_𝐴=0 and 𝑢_𝐴=0, since the fermions 𝜆_𝐴 share a common engineering dimension of 3/2 while the bosons 𝐹_𝜇𝜈 and 𝐷 share a common engineering dimension of 2.

We use the specific Majorana basis defined in Appendix E by the gamma matrices given in (E.9). To determine the “up" linkage matrices, it is helpful to rewrite the fermion transformation rule in (D.1) as 𝛿_𝑄𝜆=2𝐸_𝑎ℬ^𝑎𝜖+2𝐵^𝑎ℛ_𝑎𝜖−𝑖𝐷Γ₅𝜖, (D.2) where we have used the definitions 𝐸_𝑎=𝐹_0𝑎 and 𝐵^𝑎=(1/2)𝜀^𝑎𝑏𝑐𝐹_𝑏𝑐, for the electric and magnetic fields, respectively. We have also used the definitions ℬ_𝑎=(1/2)Γ⁰Γ^𝑎, and ℛ_𝑎=(1/4)𝜀_𝑎𝑏𝑐Γ^𝑏𝑐 for the boost and rotation generators also given in Appendix E. We now use the explicit matrices ℬ_𝑎, ℛ^𝑎, and Γ₅ specified in (E.11) and (E.9), to recast (D.2) in matrix form: the left side as a four-component column matrix 𝜆_𝐴=(𝜆₁,𝜆₂∣𝜆₃,𝜆₄)^𝑇 and the right-hand side as a 4×7 matrix multiplying another four-component column matrix given by 𝜖_𝐴=(𝜖₁,𝜖₂∣𝜖₃,𝜖₄)^𝑇. A small amount of algebra then allows us to rewrite the result in the form 𝛿_𝑄𝜆𝚤̂=𝜖^𝐴(𝑢_𝐴)𝚤^𝑗̂𝜙_𝑗, where 𝜙_𝑖∶=(𝐸₁,𝐸₂∣𝐸₃,𝐷||𝐵₁,𝐵₂,𝐵₃)^𝑇, whereupon we can read off each of the four matrices (𝑢_𝐴)𝚤^𝑗̂. The result is shown in Table 6. The fact that 𝜆 transforms nontrivially into 𝐵^𝑎 manifests in the nontriviality of the rightmost three columns in these results.

Table 6 

The four “up" linkage matrices associated with the Maxwell field-strength multiplet.

We then do a similar thing to the boson fields to determine the down matrices Δ^𝜇_𝐴. We do this separately for each of the four choices for 𝜇, referring to these as the 𝜇-sector down matrices. For example, to extract the 0-sector down matrices, we isolate those terms in the boson transformation rules in (D.1) proportional to the derivative 𝜕₀𝜆. These are given by 𝛿_𝑄⁽⁰⁾𝐸_𝑎=𝑖𝜖Γ_𝑎𝜕₀𝛿𝜆,_𝑄(0)𝐷=𝜖Γ₅Γ⁰𝜕₀𝛿𝜆,_𝑄(0)𝐵^𝑎=0. (D.3) Note that the magnetic fields 𝐵^𝑎=(1/2)𝜀^𝑎𝑏𝑐𝐹_𝑏𝑐 do not transform into time derivatives of the gaugino field. This is not surprising since the magnetic field is expressible locally as 𝐵^𝑎=𝜀^𝑎𝑏𝑐𝜕_𝑏𝐴_𝑐. But it is worth noting that (D.3) follows simply from (D.1). This tells us that upon restriction to a zero-brane, there are no downward Adinkra links connecting the three magnetic field components to any other fields; in the shadow these degrees of freedom sit at the top of one-way upward edges. In this way the magnetic fields decouple from the multiplet upon reduction to one-dimension. By utilizing the specific gamma matrices given in Appendix E we can use the same techniques described above to rewrite the 0-sector transformation rules (D.3) as 𝛿_𝑄𝜙_𝑖=−𝑖𝜖^𝐴(Δ⁰_𝐴)_𝑖𝚤̂𝜕₀𝜆𝚤̂, and then read-off the the matrices (Δ⁰_𝐴)_𝑖𝚤̂. The result of this straightforward process is exhibited in Table 7. (It is easy to see that 𝑢^𝑇_𝐴≠Δ⁰_𝐴, so that in this case the phantom matrix defined in (6.1) is nonvanishing. Although the phantom sector is irrelevant to any one-dimensional physics, it is necessary to resurrect this sector should we wish to enhance the shadow theory to its full ambient analog.)

Table 7 

The time-like down matrices associated with the Maxwell field-strength multiplet.

By isolating the terms in (D.1) respectively proportional to 𝜕_1,2,3𝜆, writing these explicitly using the Majorana basis gamma matrices shown in (E.9), and then reconfiguring the rules as 𝛿_𝑄𝜙_𝑖=−𝑖𝜖^𝐴(Δ^𝑎_𝐴)_𝑖𝚤̂𝜕_𝑎𝜆𝚤̂, allows us to read off the remaining space-like linkage matrices. The results of this straightforward process are exhibited in Tables 8, 9, and 10.

Table 8 

The 1-sector space-like down matrices for the Maxwell field-strength multiplet.

Table 9 

The 2-sector space-like down matrices for the Maxwell field-strength multiplet.

Table 10 

The 3-sector space-like down matrices for the Maxwell field-strength multiplet.

E. 4D Spinor Bases

gamma matrices satisfy the Clifford relationship {Γ_𝜇,Γ_𝜈}^𝐵_𝐴=−2𝜂_𝜇𝜈𝛿^𝐵_𝐴, where 𝜂_𝜇𝜈=diag(+−−−). These act from the left on spinors 𝜓_𝐴 and from the right on barred spinors 𝜓^𝐴∶=(𝜓^†Γ₀)^𝐴. In four-dimensions, the minimal solution involves 4×4 matrices, so the spinor index 𝐴 takes on four values. The 4D charge conjugation matrix 𝐶 is defined by 𝐶Γ^𝑎𝐶⁻¹=−(Γ^𝑎)^𝑇. In addition, the matrix 𝐶 is real, antisymmetric, and has unit determinant. A chirality operator is defined by Γ₅∶=𝑖Γ₀Γ₁Γ₂Γ₃.

We can change bases by replacing Ψ_𝐴⟶𝑀_𝐴^𝐵𝜓_𝐵,Γ^𝜇_𝐴^𝐵⟶𝑀Γ^𝜇𝑀⁻¹_𝐴^𝐵,𝐶⁻¹_𝐴𝐵⟶1det(𝑀)𝑀𝐶⁻¹𝑀^𝑇_𝐴𝐵, (E.1) where 𝑀 is any nonsingular 4×4 matrix. Two especially useful bases are the Weyl basis and the Majorana basis. These are explained below. Note that 𝐺^𝜇=−Γ^𝜇𝐶⁻¹ transforms as 𝐺^𝜇⟶1det(𝑀)𝑀𝐺^𝜇𝑀^𝑇. (E.2) Given any basis, this allows us to find a similarity transformation to render all spinor components real (a Majorana basis), and moreover one for which 𝐺⁰_𝐴𝐵=𝛿_𝐴𝐵. The resultant basis is then specially tailored for dimensional reduction to 1D, for the simple reason that the four real components of the Majorana supercharge operator 𝑄_𝐴 supply natural real 1D shadow supercharges, which satisfy the algebra {𝑄_𝐴,𝑄_𝐵}=𝑖𝛿_𝐴𝐵𝜕_𝜏.

E.1. The Weyl Basis

In the Weyl basis we choose 4×4 matrices using the following convention: Γ₀=−11,Γ_𝑎=𝜎_𝑎𝜎_𝑎,𝜀𝐶=𝜀,Γ₅=1,−1 (E.3) where 1 is the 2×2 unit matrix, 𝑎=1,2,3, and 𝜎_𝑎 are the Pauli matrices and 𝜀=𝑖𝜎₂.

Right- and left-handed Weyl spinors satisfy the respective constraints Γ₅𝜓_𝑅,𝐿=±𝜓_𝑅,𝐿. In terms of the Weyl basis (E.3), this means that right- and left-handed spinors are respectively configured as 𝜒_𝑅=𝜒₁𝜒₂00,𝜑_𝐿=00𝜑₁𝜑₂, (E.4) where 𝜒₁, 𝜒₂, 𝜑₁, and 𝜑₂ are complex anticommuting fields. Note that Weyl spinors take an especially tidy form in the Weyl basis, since half of the four complex spinor components vanish.

A Majorana spinor satisfies 𝜓=𝐶⁻¹𝜓^𝑇. In terms of the Weyl basis (E.3), this means 𝜓𝜓=¹𝜓₂𝜓^∗₂−𝜓^∗₁, (E.5) where 𝜓₁ and 𝜓₂ are complex anticommuting fields. Note that Majorana spinors are relatively awkward in the Weyl basis.

E.2. The Majorana Basis

Change bases from the Weyl basis to the Majorana basis, using (E.1), by choosing 1𝑀=21−1−𝑖−𝑖11−𝑖𝑖. (E.6) Using the transformation (E.1), right- and left-handed Weyl spinors in the Weyl basis transform into right- and left-handed Weyl spinors in the Majorana basis, as specified respectively by 𝜒_𝑅,𝐿=𝜒₁∓𝑖𝜒₁𝜒₂∓𝑖𝜒₂, (E.7) where 𝜒₁ and 𝜒₂ are complex fields. Note that the difference between left- and right-handedness in this basis manifests in the relative phases appearing in (E.7). Note that Weyl spinors are relatively awkward in the Majorana basis.

Using the transformation (E.1), a Majorana spinor in the Weyl basis, (E.5), transforms into a Majorana spinor in the Majorana basis, as given by 𝜓_𝐴=Re𝜓₁Im𝜓₁Re𝜓₂Im𝜓₂. (E.8) Note that Majorana spinors take an especially tidy form in the Majorana basis, since all four components are independent and real. In this basis, the gamma matrices and the charge conjugation matrices are Γ₀=−111−1,Γ₁=−111,Γ−1₂=1111,Γ₃=1−11−1,𝐶=1−1−11,Γ₅=𝑖−𝑖𝑖,−𝑖 (E.9) as obtained by transforming (E.3) using (E.1). The corresponding 𝐺-Matrices 𝐺^𝑎_𝐴𝐵=−𝜂^𝑎𝑏(Γ_𝑏𝐶⁻¹)_𝐴𝐵 are 𝐺⁰=1111,𝐺¹=1111,𝐺²=1−1−11,𝐺³=11−1.−1 (E.10) Note that 𝐺-matrices are symmetric, real, and traceless.

Also useful are the “boost" operators ℬ^𝑎=(1/2)Γ⁰Γ^𝑎 and the “rotation" operators ℛ_𝑎=(1/4)𝜀_𝑎𝑏𝑐Γ^𝑏𝑐. In the Majorana basis (E.9) these are ℬ¹=121111,ℛ₁=12−11−11,ℬ²=121−1−11,ℛ₂=1211−1−1,ℬ³=1211−1−1,ℛ₃=12−111.−1 (E.11) Note that the boost operators are symmetric while the rotation operators are antisymmetric. Note too that 𝐺^𝑎=2ℬ^𝑎. The operators in (E.11) satisfy the Lorentz algebra ℛ_𝑎,ℛ_𝑏=−𝜀^𝑐_𝑎𝑏ℛ_𝑐,ℬ^𝑎,ℬ^𝑏=𝜀^𝑎𝑏𝑐ℛ_𝑐,ℬ^𝑎,ℛ_𝑏=−𝜀^𝑎_𝑏𝑐ℬ^𝑐. (E.12) The Lorentz algebra (E.12) can be written concisely, and in a manner which is manifestly covariant, as 𝑀_𝜇𝜈,𝑀^𝜆𝜎=𝛿^𝜆_𝜇𝑀^𝜎_𝜈−𝛿^𝜎_𝜇𝑀^𝜆_𝜈+𝛿^𝜎_𝜈𝑀^𝜆_𝜇−𝛿^𝜆_𝜈𝑀^𝜎_𝜇, (E.13) where 𝑀^0𝑎=ℬ^𝑎 and 𝑀_𝑎𝑏=𝜀_𝑎𝑏𝑐ℛ^𝑐. (Note that using our conventions, 𝑀_0𝑎=𝜂₀₀𝜂_𝑎𝑏𝑀^0𝑏=−𝛿_𝑎𝑏𝑀^0𝑏, whereas ℬ_𝑎=𝛿_𝑎𝑏ℬ^𝑏.)

Acknowledgments

The authors are thankful to their long-time collaborators Tristan Hübsch, Charles Doran, and S. J. Gates, Jr., for many relevant discussions which spawned this analysis. M. Faux is thankful also to his muse, Adriana, and to the the Slovak Institute for Basic Research (SIBR), in Podvažie Slovakia, where much of this work was completed, for gracious hospitality. The authors also thank the President's office at the SUNY College at Oneonta, for providing hospitable use of a campus guest house, where the ideas for this paper congealed.