Dimensional Enhancement via Supersymmetry
JJuly 2009 SUNY-O/701
Dimensional Enhancement via Supersymmetry
M.G. Faux a , K.M. Iga b , and G.D. Landweber ca Department of Physics, State University of New York, Oneonta, NY 13820 [email protected] b Natural Science Division, Pepperdine University, Malibu, CA 90263
[email protected] c Department of Mathematics, Bard College, Annandale-on-Hudson, NY 12504-5000 [email protected]
ABSTRACT
We explain how the representation theory associated with supersym-metry in diverse dimensions is encoded within the representation theoryof supersymmetry in one time-like dimension. This is enabled by alge-braic criteria, derived, exhibited, and utilized in this paper, which in-dicate which subset of one-dimensional supersymmetric models describe“shadows” of higher-dimensional models. This formalism delineates thatminority of one-dimensional supersymmetric models which can “enhance”to accommodate extra dimensions. As a consistency test, we use our for-malism to reproduce well-known conclusions about supersymmetric fieldtheories using one-dimensional reasoning exclusively. And we introducethe notion of “phantoms” which usefully accommodate higher-dimensionalgauge invariance in the context of shadow multiplets in supersymmetricquantum mechanics.PACS: 04.65.+e a r X i v : . [ h e p - t h ] J u l Introduction
Supersymmetry [ ] imposes increasingly rigid constraints on the construction of quan-tum field theories [ ] as the number of spacetime dimensions increases. Thus, there arefewer supersymmetric models in six dimensions than there are in four, and fewer yetin ten dimensions [ ] . In eleven dimensions there seems to be a unique possibility [ ] ,at least on-shell. However, the off-shell representation theory for supersymmetry iswell understood only for relatively few supersymmetries, and remains a mysterioussubject in contexts of special interest, such as N = 4 Super Yang Mills theory, andthe four ten-dimensional supergravity theories [ ] .Many lower-dimensional models can be obtained from higher-dimensional modelsby dimensional reduction [ ] . Thus, a subset of lower-dimensional supersymmetrictheories derives from the landscape of possible ways that extra dimensions can beremoved. But most lower-dimensional theories do not seem to be obtainable fromhigher dimensional theories by such a process; they seem to exist only in lower di-mensions. We refer to a lower-dimensional model obtained by dimensional reductionof a higher-dimensional model as the “shadow” of the higher-dimensional model. Sowe could re-phrase our comment above by saying that not all lower dimensional su-persymmetric theories may be interpreted as shadows.It is a straightforward process to construct a shadow theory from a given higherdimensional theory. But it is a more subtle proposition to construct a higher-dimensional supersymmetric model from a lower-dimensional model, or to determinewhether a lower dimensional model actually does describe a shadow, especially ofa higher dimensional theory which is also supersymmetric. We have found residentwithin lower-dimensional supersymmetry an algebraic key which provides access tothis information. A primary purpose of this paper is to explain this.It is especially interesting to consider reduction to one time-like dimension, byswitching off the dependence of all fields on all of the spatial coordinates. Such aprocess reduces quantum field theory to quantum mechanics. Upon making such areduction, information regarding the spin representation content of the componentfields is replaced with R -charge assignments. But it is not obvious whether the fullhigher-dimensional field content, or the fact that the one-dimensional model can beobtained in this way, is accessible information given the one-dimensional theory alone.As it turns out, this information lies encoded within the extended one-dimensionalsupersymmetry transformation rules. Anomaly freedom imposes seemingly distinct algebraic constraints which make this situationeven more interesting.
2e refer to the process of re-structuring a one-dimensional theory so that fieldsdepend also on extra dimensions in a way consistent with covariant spin (1 , D − spin (1 , D − spin (1 , D − spin (1 , D − spin (1 , D − [ ] , we have explored the connection be-tween representations of supersymmetry and aspects of graph theory. We have shownthat elements of a wide and physically relevant class of one-dimensional supermulti-plets with vanishing central charge are equivalent to specific bipartite graphs which3e call Adinkras; all of the salient algebraic features of the multiplets translate intorestrictive and defining features of these objects. A systematic enumeration of thosegraphs meeting the requisite criteria would thereby supply means for a correspondingenumeration of representations of supersymmetry.In [ ] , we have developed the paradigm further, explaining how, in the caseof N -extended supersymmetry, the topology of all connected Adinkras are specifiedby quotients of N -dimensional cubes, and how the quotient groups are equivalent todoubly-even linear binary block codes. Thus, the classification of connected Adinkrasis related to the classification of such codes. In this way we have discovered an interest-ing connection between supersymmetry representation theory and coding theory [ ] .All of this is part of an active endeavor aimed at delineating a mathematically-rigorousrepresentation theory in one-dimension.In this paper we use the language of Adinkras, in a way which does not presupposea deep familiarity with this topic. We have included Appendix B as a brief andsuperficial primer, which should enable the reader to appreciate the entirety of thispaper self-consistently. Further information can be had by consulting our earlierpapers 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 doneto keep our discussion concise and concrete. Another motivating reason is because thesupersymmetry representation theory for 4D N = 1 theories is well known. Thus, partand parcel of our discussion amounts to a consistency check on the very formalism weare developing. From this point of view, this paper provides a first step in what wehope is a continuing process by which yet-unknown aspects of off-shell supersymmetrycan be discerned. In the context of 4D theories, we use standard physics nomenclature,and refer to spin (1 , [ ] by Gates et al. Accordingly, we had used that attrac-tive proposition as a prime motivator for developing the Adinkra technology in ourearlier work. This paper represents a tangible realization of that conjecture. Comple-mentary approaches towards resolving a supersymmetry representation theory havebeen developed in [ ] . Other ideas concerning the relevance ofone-dimensional models to higher-dimensional physics were explored in [ ] .This paper is structured as follows.In Section 2 we describe an algebraic context for discussing supersymmetry tai-4ored to the process of dimensional reduction to zero-branes and, vice-versa, to en-hancing one-dimensional theories. We explain how higher-dimensional spin structurescan be accommodated into vector spaces spanned by the boson and fermion fields,and how the supercharges can be written as first-order linear differential operatorswhich are also matrices which act on these vector spaces. This is done by codify-ing the supercharge in terms of diophantine “linkage matrices”, which describe thecentral 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. Wethen use this to derive non-gauge enhancement conditions, which provide an impor-tant sieve which identifies those one-dimensional multiplets which cannot enhance tofour-dimensional non-gauge matter multiplets.In Section 4 we apply our formalism in a methodical and pedestrian manner tothe context of minimal one-dimensional N = 4 supermultiplets, and show explicitlyhow the known structure of 4D N = 1 non-gauge matter may be systematicallydetermined using one-dimensional reasoning coupled only with a choice of 4D spinstructure. We explain also how our non-gauge enhancement condition provides thealgebraic context which properly delineates the Chiral multiplet shadow from its 1D“twisted” analog, explaining why the latter cannot enhance.In Section 5 we generalize our discussion to include 2-form field strengths subjectto Bianchi identities. This allows access to the question of enhancement to Vectormultiplets. In the process we introduce the notion of one-dimensional “phantom”fields which prove useful in understanding how gauge invariance manifests on shadowtheories. We use the context of the 4D N = 1 Abelian Vector multiplet as anarchetype for future generalizations.We also include five appendices which are an important part of this paper. Ap-pendix A is especially important, as this provides the mathematical proof that im-posing Lorentz invariance of the postulate linkage matrices allows one to correlate theentire higher-dimensional supercharge with its time-like restriction. We also derivein this Appendix algebraic identities related to the spin structure of enhanced com-ponent fields, which should provide for interesting study in the future generalizationsof this work.Appendix B is a brief summary of our Adinkra conventions, explaining techni-calities, such as sign conventions, appearing in the bulk of the paper. Appendix Cexplains the dimensional reduction of the 4D N = 1 Chiral multiplet, complemen-5ary to the non-gauge enhancement program described in Section 4. Appendix Dexplains the dimensional reduction of the 4D N = 1 Maxwell field strength multiplet,complementary to Section 5. This shows in detail how phantom sectors correlatewith gauge aspects of the higher-dimensional theory. Appendix E is a discussion offour-dimensional spinors useful for understanding details of our calculations.We use below some specialized terminology. Accordingly, we finish this introduc-tion section by providing the following three-term glossary, for reference purposes: Shadow : We refer to the one-dimensional multiplet which results from dimensional re-duction of a higher-dimensional multiplet as the “shadow” of that higher-dimensionalconstruction.
Adinkra : The term Adinkra refers to 1D supermultiplets represented graphically, asexplained in Appendix B. We sometimes use the terms Adinkra, supermultiplet, andmultiplet synonymously.
Valise : A Valise supermultiplet, or a Valise Adinkra, is one in which the componentfields span exactly two distinct engineering dimensions. These multiplets form rep-resentative elements of larger “families” of supermultiplets derived from these usingvertex-raising operations, as explained below. Thus, larger families of multiplets maybe unpacked, as from a suitcase (or a valise), starting from one of these multiplets
It is easy to derive a one-dimensional theory by dimensionally reducing any givenhigher-dimensional supersymmetric theory. Practically, this is done by switching offthe dependence of all fields on the spatial coordinates, by setting ∂ a →
0. One wayto envision this process is as a compactification, whereby the spatial dimensions arerendered compact and then shrunk to zero size. Alternatively, we may envision thisprocess as describing a restriction of a theory onto a zero-brane, which is a time-likeone-dimensional sub-manifold embedded in a larger, ambient, space-time. Using thislatter metaphor, we refer to the restricted theory as the “shadow” of the ambienttheory, motivated by the fact that physical shadows are constrained to move upon awall or a wire upon which the shadow is cast.
Supersymmetry transformation rules can be written in terms of off-shell degrees offreedom, by expressing all fields and parameters in terms of individual tensor orspinor components. Thus, without loss of generality, we can write the set of boson6omponents as φ i and the set of fermion components as ψ ˆ ı , without being explicitlycommittal as the the spin (1 , D − spin (1 , D − δ L φ i = θ µν ( T µν ) i j φ j δ L ψ ˆ ı = θ µν ( ˜ T µν ) ˆ ı ˆ ψ ˆ , (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 ) ˆ ı ˆ represents the spin algebra as realized on the fermion fields, while θ a parameterizesa boost in the a -th spatial direction and θ ab parameterizes a rotation in the ab -plane.According to the spin-statistics theorem, ( ˜ T µν ) ˆ ı ˆ should describe a spinor represen-tation and ( T µν ) i j should describe a direct product of tensor representations. Thespin representations may also involve constraints. For example, boson componentsmay configure as closed p -forms.In four-dimensions the N = 1 supersymmetry algebra is generated by a Majoranaspinor supercharge with components Q A subject to the anticommutator relationship { Q A , Q B } = 2 i G µAB ∂ µ where G µAB = − ( Γ µ C − ) AB . A parameter-dependent su-persymmetry transformation is generated by δ Q ( (cid:15) ) = − i ¯ (cid:15) A Q A , where (cid:15) A describesan infinitesimal Majorana spinor parameter, and ¯ (cid:15) A = ( (cid:15) † Γ ) A is the correspondingbarred 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. Furthermore, in this basis, we have the nice result G AB = δ AB . Withthis choice, we can re-write our supersymmetry transformation as δ Q ( (cid:15) ) = − i (cid:15) A Q A ,where Q A = ( Γ ) A B Q B . The four-dimensional N = 1 supersymmetry algebra,written in terms of the operators Q A , is then given by { Q A , Q B } = 2 i δ AB ∂ τ − i G aAB ∂ a , (2.2)where x := τ is the time-like coordinate parameterizing the the zero-brane to whichwe intend to restrict, and x a := ( x , x , x ) are the three space-like coordinatestransverse to the zero-brane. To dimensionally reduce a four-dimensional field theoryto a one-dimensional field theory, we set ∂ a = 0. In this way, the second term onthe right-hand side of (2.2) disappears, and we obtain the one-dimensional N = 4supersymmetry algebra. See Appendix E for specifics related to this basis. This merely technical re-organization facilitates dimensional reduction of 4D multiplets, asdone in Appendices C and D.
7t proves helpful to add a notational distinction, by writing δ Q φ i = − i (cid:15) A ( Q A ) i ˆ ı ψ ˆ ı and δ Q ψ ˆ ı = − i (cid:15) A ( ˜ Q A ) ˆ ı i φ i , appending a tilde to ˜ Q A when this describes a fermiontransformation rule. The supercharges may be represented as first-order linear differ-ential operators, as ( Q A ) i ˆ ı = ( u A ) i ˆ ı + ( ∆ µA ) i ˆ ı ∂ µ ( ˜ Q A ) ˆ ı i = i ( ˜ u A ) ˆ ı i + i ( ˜∆ µA ) ˆ ı i ∂ µ , (2.3)where u A , ˜ u A , ∆ µA , and ˜∆ µA are real valued “linkage matrices” which play a centralrole in our discussion below.The matrices ( u A ) i ˆ ı describe “links” corresponding to supersymmetry maps fromthe bosons φ i to fermions ψ ˆ ı having engineering dimension one-half unit greater thanthe bosons. Therefore, these codify “upward” maps connecting lower-weight fermionsto higher-weight bosons. Similarly, the matrices ( ˜ u A ) ˆ ı i codify “upward” maps con-necting lower-weight fermions to higher-weight bosons. The matrices ( ∆ µA ) i ˆ ı and( ˜∆ µA ) ˆ ı i codify “downward” maps accompanied by their respective derivatives ∂ µ .The component fields may be construed so that the linkage matrices conform toa special structure, known as the Adinkraic structure. This says that there is atmost one non-vanishing entry in each column and at most one non-vanishing entry ineach row. Moreover, the non-vanishing entries take the values ±
1. All known higher-dimensional off-shell representations in the standard literature satisfy this condition. The supersymmetry algebra (2.2) implies( u ( A ˜ u B ) ) i j = 0 ( ˜ u ( A u B ) ) ˆ ı ˆ = 0( ∆ ( µ ( A ˜∆ ν ) B ) ) i j = 0 ( ˜∆ ( µ ( A ∆ ν ) B ) ) ˆ ı ˆ = 0 , (2.4)which describes a higher-dimensional analog of the Adinkra loop parity rule describedin [ ] and below, and also implies( u ( A ˜∆ µB ) + ∆ µ ( A ˜ u B ) ) i j = Λ µAB δ i j ( ˜ u ( A ∆ µB ) + ˜∆ µ ( A u B ) ) ˆ ı ˆ = Λ µAB δ ˆ ı ˆ , (2.5) The term “weight” refers to the engineering dimension of the field. We sometimes use theterm weight in lieu of dimension, to avoid confusion with spacetime dimension. The weightof a field correlates with the vertex “height” on an Adinkra diagram. The only counterexamples that we know of were contrived by us in [ ] , as special defor-mations of one-dimensional Adinkraic representations. And we suspect that these do notenhance. Further scrutiny will be needed to ascertain any relevance of non-Adinkraic multi-plets to physics. We find it sensible for now to focus on Adinkraic representations, especiallysince all known field theoretic multiplets are in this class. µAB = ( Γ G µ Γ ) AB = ( Γ Γ µ Γ C − ) AB , whereby Λ AB = G AB and Λ aAB = − G aAB . The equations (2.5) play a central role in this paper.The classification of representations of supersymmetry in diverse dimensions isequivalent to the question of classifying and enumerating the possible sets of reallinkage matrices which can satisfy the algebraic requirements in (2.4) and (2.5), andidentifying the corresponding spin representation matrices ( T µν ) i j and ( ˜ T µν ) ˆ ı ˆ . The one-dimensional N = 4 superalgebra is specified by { Q A , Q B } = 2 i δ AB ∂ ,which corresponds to (2.2) in the limit ∂ a →
0. In this case, the supercharges arerepresented as ( Q A ) i ˆ ı = ( u A ) i ˆ ı + ( d A ) i ˆ ı ∂ τ ( ˜ Q A ) ˆ ı i = i ( ˜ u A ) ˆ ı i + i ( ˜ d A ) ˆ ı i ∂ 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 re-named by writing ∆ A as d A and ˜∆ A as ˜ d A . Asmentioned above, the fields may be configured so that each linkage matrix has notmore than one non-vanishing entry in each row and likewise in each column, andthe non-vanishing entries are ±
1. This specialized structuring enables the faithfultranslation of 1D supercharges in terms of helpful and interesting graphs known asAdinkras, as mentioned in the Introduction. The reader should consult Appendix Bfor 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 spacetime indices µ and ν . Thus, the linkagematrices are constrained by( u ( A ˜ u B ) ) i j = 0 ( ˜ u ( A u B ) ) ˆ ı ˆ = 0( d ( A ˜ d B ) ) i j = 0 ( ˜ d ( A d B ) ) ˆ ı ˆ = 0 . (2.7)These relationships imply a “loop parity” rule, described in our earlier papers, whichsays that any closed bi-color loop on an Adinkra diagram must involve an odd numberof edges with odd parity. The linkgage matrices are further constrained by( u ( A ˜ d B ) + d ( A ˜ u B ) ) i j = δ AB δ i j ( ˜ u ( A d B ) + ˜ d ( A u B ) ) ˆ ı ˆ = δ AB δ ˆ ı ˆ . (2.8)9n this context, the algebra defined by (2.8) was called a “Garden algebra” by Gateset al., in [ ] , and the the matrices u A and d A were called Garden matrices. Thelarger algebra given in (2.4) and (2.5) generalizes this concept to diverse spacetimedimensions, 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. Givena 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 andinvariant action functionals from which one can study one-dimensional physics. Thelinkage matrices associated with any Adinkra satisfy the algebra (2.7) and (2.8) bydefinition. The requirement that the linkage matrices appearing in the supercharges (2.3) are spin (1 , D − aA are completely determined by the “time-like”linkage matrices ∆ A . The proof of this assertion is given as Appendix A, with theresult ( ∆ aA ) i ˆ ı = − ( Γ Γ a ) A B ( ∆ B ) i ˆ ı ( ˜∆ aA ) ˆ ı i = − ( Γ Γ a ) A B ( ˜∆ B ) ˆ ı i . (3.1)It is interesting that the matrix ( Γ Γ a ) A B is precisely twice a boost operator inthe a -th spatial direction, in the spinor representation. It is also interesting thatthe result (3.1) holds irrespective of the spin (1 , D − T µν ) i j and ( ˜ T µν ) ˆ ı ˆ defined in (2.1) does not influence (3.1). These nontrivial consequences are derivedexplicitly in Appendix A. It is worth mentioning that the form of (3.1) agrees precisely with the linkagematrices derived from Salam-Strathdee superfields. Also, the appearance of Γ a onthe right hand side is tied closely to the appearance of the Γ a in the defining super-symmetry algebra. The Lorentz invariance of the linkage matrices does imply interesting and interlocking con-straints on the allowable choices of ( T µν ) i j and ( ˜ T µν ) ˆ ı ˆ . These are exhibited in AppendixA. Such correlations are certainly expected, and we suspect that equations (A.4) and (A.5)have deep and useful implications, which we hope to explore in future work. µ assumes only the value 0. To probe whether that multiplet de-scribes a shadow, one creates “provisional” off-brane linkages using the powerful ex-pression (3.1). Since there is no algebraic guarantee that the transformation rulesso-extended will properly close the higher-dimensional superalgebra, nor that theboson and fermion vector spaces will properly assemble into representations of thehigher-dimensional spin group, the higher-dimensional superalgebra itself, applied tothis construction, provides the requisite analytic probe of that possibility: if the one-dimensional multiplet is a shadow then the provisional construction will close thehigher-dimensional superalgebra; if it is not possible, then it will not.The supersymmety algebra in D -dimensions closes only if ( Ω µAB ) i j ∂ µ φ j = 0 and( ˜Ω µAB ) ˆ ı ˆ ∂ µ ψ ˆ = 0, where we define the following useful matrices,( Ω µAB ) i j = ( u ( A ˜∆ µB ) + ∆ µ ( A ˜ u B ) ) i j − Λ µAB δ i j ( ˜Ω µAB ) ˆ ı ˆ = ( ˜ u ( A ∆ µB ) + ˜∆ µ ( A u B ) ) ˆ ı ˆ − Λ µAB δ ˆ ı ˆ . (3.2)This requirement is a minor re-structuring of (2.5). In this way, we have written thesupersymmetry algebra as a linear algebra problem, cast as matrix equations.Many important supemultiplets exhibit gauge invariances, manifest as physicalredundancies inherent in the vector spaces spanned by the component fields. In thesecases, the matrices ( Ω µAB ) i j and ( ˜Ω µAB ) ˜ ı ˜ , are not unique. Instead these describeclasses of matrices interrelated by operations faithful to the gauge structure. Wedescribe this interesting situation below, in Section 5. It is useful, however, to beginour discussion with what we call non-gauge matter multiplets, which do not exhibitredundancies of this sort. For this smaller but nevertheless interesting and relevantclass of supermultiplets, the higher-dimensional supersymmetry algebra is satisfiedonly if ( Ω µAB ) i j = 0( ˜Ω µAB ) ˆ ı ˆ = 0 . (3.3)We refer to these equations as our non-gauge enhancement criteria. These enablea practical algorithm for testing whether a given 1D supermultiplet represents theshadow of a 4D non-gauge matter multiplet.We use the linkage matrices for a given 1D supermultiplet in conjuction with the4D Gamma matrices to compute all of the d × d matrices Ω µAB and ˜Ω AB , defined11n (3.2). If (3.3) is satisfied, i.e. , if all of these matrices are identically null, thenthe 1D multiplet passes an important, non-trivial, and necessary requirement forenhancement to a 4D non-gauge matter multiplet. If these matrices do not vanish,then the 1D multiplet cannot enhance to a 4D non-gauge matter multiplet. In thelatter case, further analysis must be done to probe whether this multiplet can enhanceto a gauge multiplets. Equation (3.3) represents a useful “sieve” in the separation of1D multiplets into groups as shadows versus non-shadows.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 inpairs related in-part by a Klein flip, which is an involution under which the statisticsof the fields are reversed — boson fields are replaced with fermion fields and viceversa. Thus, we can organize those multiplets which pass the test (3.3) into suchpairs. We then ascertain which elements of each pair satisfy the requirement thatfermions assemble as spinors and the bosons as tensors. Those multiplets that donot pass this test describe another class of multiplets which do not describe ordinaryshadows. Typically, one multiplet out of each pairing satisfies the spin-statistics testwhile the other multiplet fails this test. In the explicit examples analyzed below in this paper, it is obvious when certainmultiplets which pass the first enhancement test (3.3) fail the spin-statistics test. Thisoccurs when the multiplicity of fermions with a common engineering dimension is nota multiple of four, thereby obviating assemblage into 4D spinors. In fact our analysisbelow is remarkably clean. In more general cases, we suspect that more carefulattention to the implications of the Lorentz invariance of the provisional supercharge,codified by equations such as (A.4), will provide the requisite sophistication neededto address enhancement at higher N and higher D . We think this will be a mostinteresting undertaking. In this section we impose our enhancement equation (3.3) on the linkage matricesassociated 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 non-gauge matter multi-plets. Since the represention theory for minimal irreducible multiplets in 4D N = 1supersymmetry is well known, this setting provides a natural laboratory for testingour technology. The principal result of this section is that our enhancement equation The important role of the Klein flip in the representation theory of superalgebras was ad-dressed by one of the authors (G.L.) in previous work [ ] . N = 1 super-symmetry, thereby passing an important consistency test. Another principal result ofthis section identifies our enhancement equation as the natural algebraic sieve whichdistinguishes the Chiral multiplet shadow from its “twisted” analog.By the term “non-gauge matter multiplets” we refer to 4D supermultiplets whichinvolve component fields neither subject to gauge transformations nor subject todifferential constraints, such as Bianchi identities. This excludes the Vector and theTensor multiplets, as well as the the corresponding field strength multiplets. Wepostpone a discussion of these interesting cases until the next section. In fact, theonly non-gauge matter multiplet in 4D N = 1 supersymmetry is the Chiral multiplet. As we will see, among the 60 different minimal N = 4 Adinkras there are exactly fourwhich satisfy our primary enhancement condition (3.3). For two of these, the fermionsconfigure as a spinor and the bosons configure as Lorentz scalars. For the other twothe fermions configure as Lorentz scalars while the bosons configure as a spinor. Thelatter case fails the spin-statistics test, which says that fermions must assemble asspinors, and bosons must assemble as tensors. Thus, our method identifies the twoAdinkras which can provide shadows of 4D minimal non-gauge matter multiplets. 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 edgecolors. Thus, the shadow of the Chiral multiplet has a twisted analog which cannotenhance 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. As it turnsout, the linkage matrices for the Chiral multiplet shadow satisfy the enhancementequations (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 subjectthe corresponding linkage matrices to the space-like subset of the equations in (3.3). An Antichiral multiplet, which can be formed as the Hermitian conjugate of a Chiral multi-plet, is not distinct from the latter as representation of the 4D N = 1 supersymmetry algebraseparate from inherent complex structures; the assignment of possible U (1) charge assign-ments represents “extra” data not considered overtly in this paper. Ignoring the complexstructure, the Chiral and the Antichiral multiplets have indistinguishable shadows. Conceivably, the fact that there are two such enhanceable N = 4 minimal Adinkras mayrelate to the fact that there are two complementary choices of complex structure, related tothe Chiral and Antichiral multiplets, as mentioned in the previous footnote. We learned about this interesting curiosity from Jim Gates, in the context of a formercollaboration. Figure 1 : The two N = 4 Valise Adinkras. The Adinkra on the right is obtainedfrom the Adinkra on the left by implementing a “twist”, toggling the parity of thegreen edges.(The time-like equations are satisfied automatically, since an Adinkra is a represen-tation of 1D supersymmetry by construction.) In the case of testing enhancement to4D N = 1 supersymmetry, each of the two conditions in (3.3) describes 30 matrixequations for each of the 60 Adinkras to be tested, since for each of the three choicesfor a , the corresponding symmetric matrices G aAB = G a ( AB ) have ten independentcomponents. Thus, according to the crudest counting argument, in order to test boththe bosonic and fermionic conditions in (3.3) for all the minimal N = 4 Adinkras,we need to check 60 × × × The smallest Adinkras which can possibly enhance to describe 4D supersymmetryare N = 4 Adinkras describing 4+4 off-shell degrees of freedom. We therefore startby considering N = 4 bosonic 4-4 Valise Adinkras. There are exactly two of these notinterrelated by cosmetic field redefinitions. These are exhibited in Figure 1. In thispaper 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 , , , . For purposesof setting a convention for ordering the rows and columns of our linkage matrices,we sequence the boson fields φ i and the fermion fields ψ ˆ ı using the obvious faithful More generally, testing enhancement to D dimensions will involve at least ( D − d ( d +1)independent matrix equations per Adinkra, where d is the number of fermions or bosons inthe 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 supersymmetrywould involve (9) ( 128 ) ( 129 ) = 74 ,
304 equations, each involving products of 128 × φ i withcorresponding index choices, while the black vertices labeled 1, 2, 3, 4 representthe fermion fields ψ ˆ ı with corresponding index choices. This allows us to readilytranslate each Adinkra into precise linkage matrices, using the technology explainedin Appendix B.The linkage matrices ( u A ) i ˆ ı corresponding to the first Adinkra in Figure 1 areexhibited in Table 1. These codify the “upward” links connecting the bosons φ i to the fermions ψ ˆ ı having greater engineering dimension. Since there are no edgeslinking downward from any of the boson vertices, it follows that ( d A ) i ˆ ı = 0 in thiscase. Similarly, we have ( ˜ u A ) ˆ ı i = 0, reflecting the fact that none of the fermions haveupward directed edges. Finally, we have ( ˜ d A ) ˆ ı j = δ AB δ jk ( u B ) k ˆ k δ ˆ k ˆ ı and ( d A ) i ˆ = δ AB δ ˆ ˆ k ( ˜ u B ) ˆ k k δ ki , schematically ˜ d A = u TA and d A = ˜ u TA , reflecting the fact that everyedge describes a pairing of an upward directed term and a corresponding downwarddirected term. The relationships ˜ d A = u TA and d A = ˜ u TA are characteristic of “standardAdinkras”. Non-standard Adinkras, which can include “one-way” upward Adinkraedges, appear in gauge multiplet shadows, as explained below, and in also in othercontexts of interest. N = 4 bosonic 4-4 Adinkras Using the features ˜ u A = d B = 0 and ˜ d A = u TA , and using the matrices G aAB in (E.10),we can begin to analyze the enhancement equations associated with the left Adinkra inFigure 1. Consider the first equation in (3.3) for the index choices ( a | A, B ) = ( a | , = − G = 0, that 4 × u ˜∆ = 0. We then use (3.1),along with the Gamma matrices in (E.3) to determine ˜∆ = − ˜∆ , which is equivalentto ˜∆ = − ˜ d using the nomenclature ˜∆ 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) tothe simple matrix equation u u T = 0, where we have also used ˜ d = u T . Using Table1, it is easy to check that this simple requirement is not satisfied. This tells us that theleft Adinkra in Figure 1 cannot enhance to a 4D non-gauge matter multiplet. Sincethe linkage matrices associated with the right Adinkra in Figure 1 are obtained fromTable 1 by toggling the overall sign on u only, and since the enhancement equation u u T = 0 is unchanged by such an operation, it follows that neither Adinkra in Figure1 can enhance to a 4D non-gauge matter multiplet. The diligent reader should verify the correspondence between Figure 1 and Table 1 usingthe simple technology explained in Appendix B. Some considerations involving “one-way” Adinkra edges were described in both [ ] and [ ] . = u = − − u = − −
11 1 u = − − Table 1 : The boson “up” linkage matrices for the 4-4 Valise Adinkra shown in Figure1. The methodology explained in the last paragraph can be applied systematically foreach possible index choice ( a | A, B ) for any selected Adinkra. In each case the time-likelinkage matrices ∆ aAB are determined using (3.1), so that the enhancement equationcan be translated to a matrix statement involving the linkage matrices specific to the1D multiplet directly corresponding to the Adinkra. In the following discussion wedo not repeat most of these steps. But the reader should be aware that equation (3.1)is used in each example which we discuss, and the use of this equation is what allowsus to cast the enhancement equation in terms of the matrices u A , d A = ∆ A , and theirtransposes. N = 4 bosonic 3-4-1 Adinkras Consider next those Adinkras obtained by raising one vertex starting with eachAdinkra 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 exam-ple, if we raise the boson vertex labeled “4” starting from each Valise, what resultsare the two Adinkras in Figure 2. In these cases, we end up with three bosons atthe 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 3-4-1Adinkras, where the sequence of numerals faithfully enumerates the sequence of ver-tex multiplicities at successively higher levels. (These alternate between boson andfermion multiplicities, naturally.) It is easy to see that there are exactly eight bosonic3-4-1 N = 4 Adinkras, and that these split evenly into two groups interrelated by atwist operation.We should point out that two Adinkras are equivalent if they are mapped intoeach other by cosmetic renaming of vertices, equivalent to linear automorphisms on16 Figure 2 : The two 3-4-1 Adinkras obtained from the Valise Adinkras in Figure 1 byraising one vertex. Here we have raised the boson vertex labeled 4.the vector spaces spanned by the bosons φ i or fermions ψ ˆ ı , in cases where these mapspreserve all vertex height assignments. Such transformations have been called “innerautomorphisms”. The simplest examples correspond to re-scaling any component fieldby a factor of −
1. This manifests on an Adinkra by simultaneously toggling the parityof every edge connected to the vertex representing that field, i.e. , by changing dashededges into solid edges and vice-versa. (This is referred to as “flipping the vertex”, andwas described already in [ ] .) Our observation that there are two distinct familiesof 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. (Thecurious reader might find it amusing to draw Adinkra diagrams, and investigate thisstatement for his or her self.) It is also true that there are only two twist classes ofminimal N = 4 Adinkras, despite the fact that there are four different colors whichcan be used to implement a twist. This is so because a given twist applied using anychosen edge color can be equivalently implemented as a twist applied using any otheredge color augmented by a suitable inner automorphism.When we raise an Adinkra vertex, the up and down linkage matrices accordinglymodify. For example, consider the the 3-4-1 Adinkra on the left in Figure 2, obtainedfrom the Adinkra on the left in Figure 1 by raising the φ vertex. The correspondingboson 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 longer17 = d = u = − − d =
00 01 u = − −
11 0 d = u = − − d = Table 2 : Linkage matrices for the left 3-4-1 Adinkra shown in Figure 2. The linkagematrices for the right Adinkra in that Figure are obtained from these by changingthe sign of u and d . 18anish as they did in the case of the Valise. This is because the φ vertex obtainsdownward links after being raised. The fermion up matrices, which are determinedfor this standard Adinkra using ˜ u A = d TA , are also non-vanishing after this vertexraise, since each of the fermions obtains an upward link to the boson φ .Given a standard Adinkra, it is possible to raise the n -th boson vertex if and onlyif the n -th row of each boson down matrix is null; i.e. ,, provided ( d A ) n ˆ ı = 0 for allvalues of ˆ ı . This criterion ensures that the n -th boson vertex does not have any down-ward links which would preclude the vertex from being raised. (Since for standardAdinkras we have ˜ u A = d TA , this criterion also implies that there are no lower fermionswhich link upward to the boson in question.) Absent such a tethering, the boson isfree to be raised. This operation is implemented algebraically by interchanging the n -th row of each boson up matrix u A with the n -th row of the respective boson downmatrix d A . Thus, we implement the matrix reorganizations ( u A ) n ˆ ı ↔ ( d A ) n ˆ ı . At thesame time, we must interchange the n -th column of each fermion up matrix ˜ u A withthe n -th column of the respective fermionic down matrix ˜ d A , via ( ˜ u A ) ˆ ı n ↔ ( ˜ d A ) ˆ ı n .The latter operation preserves the standard relationships ˜ u A = d TA and ˜ d A = u TA . Itis easy to check that the linkage matrices in Table 2 are obtained from the linkagematrices in Table 1 by appropriately interchanging the fourth rows of the boson upand down matrices according to the above discussion.We now use the enhancement equation to analyze the eight standard N = 4bosonic 3-4-1 Adinkras to ascertain if any of these can enhance to a 4D N = 1 non-gauge matter multiplet. To begin, we start with the left Adinkra in Figure 2, by usingthe boson linkage matrices in Table 2 and the fermion linkage matrices determinedby ˜ u A = d TA and ˜ d A = u TA . Using the G aAB given in (E.10), the first condition in (3.3)reduces for the choice ( a | AB ) = (1 |
11) to the matrix equation u u T + d d T = 0.Using the explicit matrices in Table 2, it is easy to see that this requirement is notsatisfied. This tells us that the left Adinkra in Figure 2 cannot enhance to a 4Dnon-gauge matter multiplet.Since the right Adinkra in Figure 2 is obtained from the left Adinkra in that figureby twisting the green edges, corresponding to replacing Q → − Q , the linkage ma-trices for that second 3-4-1 Adinkra are obtained from those in Table 2 by scaling thematrices u , d , ˜ u and ˜ d each by a multiplicative minus sign. The ( a | AB ) = (1 | , u u T + d d T = 0, is unchanged by this operation. So weconclude that neither Adinkra in Figure 2 can enhance to a non-gauge 4D mattermultiplet. It is straightforward to repeat this analysis for all cases associated withraising any possible single boson vertex starting with either of the Valise Adinkras inFigure 1. It follows, after careful analysis of each case, that the non-gauge enhance-19 Figure 3 : The N = 4 2-4-2 Adinkras may be obtained from 3-4-1 Adinkras byraising one vertex. Here we have raised the third boson vertex starting with the twoAdinkras shown in Figure 2.ment equation (3.3) is not satisfied for any of the eight bosonic 3-4-1 Adinkras. N = 4 bosonic 2-4-2 Adinkras Things become more interesting when we raise one of the lower bosons in 3-4-1Adinkras to obtain 2-4-2 Adinkras. In the end there are twelve minimal N = 4bosonic 2-4-2 Adinkras — six obtained by two vertex raises starting from the leftAdinkra in Figure 1 and six obtainable by two vertex raises starting from the rightAdinkra in Figure 1. The six possibilities in each class correspond to the six differentways to select pairs from four choices. For example, if we raise φ and φ in eithercase then what results are the two 2-4-2 Adinkras shown in Figure 3. For the leftAdinkra in Figure 3, the boson linkage matrices are shown in Table 3. (It is straight-forward to read these matrices off of the Adinkra. It is also straightforward to obtainthese matrices algebraically, as explained above, by interchanging the third rows ofthe 3-4-1 up and down matrices shown in Table 2.)We now use the enhancement equation (3.3) to analyze the twelve standard N = 42-4-2 Adinkras to ascertain if any of these can enhance to a 4D N = 1 non-gaugematter multiplet. To begin, we start with the left Adinkra in Figure 3, equivalentlydescribed by the boson linkage matrices specified in Table 3 and by the fermion linkage20 = d = u = − d = − u = − −
10 0 d = u = − d = − Table 3 : Linkage matrices for the left 2-4-2 Adinkra shown in Figure 3. The linkagematrices for the right Adinkra in that Figure are obtained from these by changingthe sign of u and d . 21atrices determined from these by ˜ u A = d TA and ˜ d A = u TA .We found above that for each of the two N = 4 Valise Adinkras and for each of theeight 3-4-1 Adinkras the enhancement equation corresponding to ( a | A, B ) = (1 | , × u u T + d d T does not vanish in these cases. The reader should compute thiscombination in those cases, and then also compute this combination using the linkagematrices in Table 3. It is noteworthy that in this latter case, i.e. , using the matrices inTable 3, the computation of u u T + d d T does indeed produce the 4 × a | A, B ) = (1 | ,
1) equation, it remains to analyze all of theother possible choices for ( a | A, B ) and check the enhancement equations (3.3) in eachcase. It is interesting to comment on the case ( a | A, B ) = (1 | , u u T + d d T + u u T + d d T = 0 . (4.1)Note that this is satisfied using the matrices in Table 3. So the left 2-4-2 Adinkrain Figure 3 passes this second enhancement test. (Thus, this Adinkra passes two outof 60 different tests, counting the both the bosonic and the fermionic enhancementconditions for each of the 30 index choices ( a | A, B ).)It is interesting that, unlike the left Adinkra in Figure 3, the right Adinkra inFigure 3 fails the test (4.1). This can be seen by noting that (4.1) is sensitive to theparity on any one of the four edge colors since the overall signs on the first two termsflip upon toggling the sign on Q or Q while the sign on the third and fourth termsflip upon toggling the sign on Q or Q . More specifically, the linkage matrices forthe second Adinkra in Figure 3 are obtained from Table 3 by toggling the sign on Q ,which toggles the overall sign on the matrices u and d . If we substitute the linkagematrices for the right Adinkra in Figure 3, obtained in this way, into (4.1) we findthat this equation is no longer satisfied. Thus, we conclude that the right Adinkrain Figure 3 cannot enhance to a 4D non-gauge matter multiplet. Again, the readershould check these assertions by doing a few simple matrix calculations.Further analysis of all of the remaining 58 enhancement conditions shows that thematrices in Table 3 pass every one of these tests. This is a non-trivial accomplishment,which indicates that the left Adinkra in Figure 3 does represent the shadow of a 4D N = 1 non-gauge matter multiplet. As a representative example, consider the bosonic22nhancement condition for the choice ( a | A, B ) = (2 | , u u T + u u T + d d T + d d T = 2 , (4.2)where the factor of 2 on the right hand side means twice the 4 × N = 1Chiral multiplet is easy to check by performing a direct dimensional reduction of theChiral multiplet. This is done explicitly in Appendix C. In that Appendix we derivethe shadow Adinkra, shown in Figure 4, by direct translation of the 4D Chiral multi-plet transformation rules. The left Adinkra in Figure 3 is obtained from the Adinkrain Figure (4) by reorganizing fields according to the following four permutation op-erations: φ ↔ φ , φ ↔ φ , ψ ↔ ψ , and ψ ↔ ψ . This describes a cosmetic innerautomorphism, indicating that the two Adinkras are equivalent.The fact that the left Adinkra in Figure 3 passes the enhancement criteria whilethe right Adinkra does not identifies the left Adinkra as the Chiral multiplet shadowand identifies the right Adinkra as the so-called Twisted Chiral multiplet. We alsofind that the 2-4-2 Adinkra obtained by raising the vertices φ and φ starting fromthe left 4-4 Valise in Figure 1 also passes all of the enhancement criteria whereas thetwisted analog of this does not. Scrutiny of all twelve bosonic 2-4-2 Valises confirmsthat only those two cases in the non-twisted family, associated with the left Valisein Figure 1, obtained by raising either the pair ( φ , φ ) or the pair ( φ , φ ) canenhance to non-gauge matter multiplets in 4D. N = 4 Adinkra families
It is straighforward to systematically check the enhancement conditions for all 30Adinkras in each of the two families — one family associated with each of the twoValises in Figure 1. In each case, the 30-member family consists of the bosonic 4-4Adinkra (the Valise), four bosonic 3-4-1 Adinkras, six bosonic 2-4-2 Adinkras, fourbosonic 1-4-3 Adinkras, and the Klein flip of each one of these 15 representatives. (TheKlein flipped Adinkras are the fermionic 4-4 Valise and the 14 fermionic Adinkrasobtainable from this by various vertex raises.)In total there are exactly four out of the 60 minimal Adinkras which pass our non-gauge enhancement criteria (3.3). The first two are the bosonic 2-4-2 Chiral multipletshadows obtained by raising either φ and φ or by raising φ and φ starting from23he left Adinkra in Figure 1. The other two Adinkras reside in the other (relativelytwisted) family, and are obtained from the right Adinkra in Figure 1 by first raisingall four bosonic vertices, then raising either the fermionic vertices ψ and ψ or byraising the fermionic vertices ψ and ψ . These operations produce fermionic 2-4-2Adinkras corresponding to twisted Klein flips of the two enhanceable bosonic 2-4-2Adinkras. Since in these cases there are at most two fermions at any given heightassignment, it is clear that these cannot assemble as 4D spinors. As we explainedabove, the Adinkras which pass the enhancement criteria come in pairs, one element ofwhich passes the spin-statistics test and one which does not. In this way we concludethat of the 60 minimal Adinkras specified above only the two bosonic 2-4-2 cases candescribe shadows of non-gauge 4D matter multiplets.It might appear curious that the four bosonic 2-4-2 Adinkras obtained from theright Adinkra in Figure 1 by raising ( φ , φ ) or ( φ , φ ) or ( φ , φ ) or ( φ , φ )do not pass the enhancement criteria whereas the two Adinkras obtained by raising( φ , φ ) or ( φ , φ ) do pass this test. The reason why certain combinations ofcomponent fields appear favored relates to the fact that we have made a choice of spinstructure when we selected the particular Gamma matrices in (E.9). It is interestingthat we lose no generality in making such a choice, however, since the freedom tochoose a 4D spin basis is replaced by a corresponding freedom to perform innerautomorphisms 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 re-sult that exactly two out of the 60 minimal N = 4 Adinkras enhance to non-gauge 4Dsupersymmetric matter, along with the observation that those 1D multiplets which doenhance have 2-4-2 component multiplicities says something salient about 4D super-symmetry representation theory. Specifically it says that any 4D N = 1 non-gaugematter multiplet must have two physical bosons, four fermions, and two auxiliarybosons. This corroborates what has long been known about the minimal represen-tations of 4D N = 1 supersymmetry. What is remarkable is that we have herebyshown that this information is fully extractable using merely 1D supersymmetry anda choice of 4D spin structure — that this information lies fully encoded in the 1D su-persymmetry representation theory codified by the families of Adinkras, and that the24ey to unlocking this information is contained in our enhancement equation (3.3). The non-gauge enhancement condition (3.3) relies on the result (3.1), which is derivedin Appendix A. An important part of that derivation uses the assumptions ∆ A = ˜ u T and ˜∆ A = u TA . These translate into the statement that every Adinkra edge codifiesboth an upward-directed term and a downward-directed term in the multiplet trans-formation rules. (In other words, this result applies to “standard” Adinkras.) But thepresence of gauge degrees of freedom or Bianchi identities obviates this assumption.This is demonstrated explicitly by dimensionally reducing the 4D N = 1 Maxwellfield-strength multiplet, as described in detail in Appendix D. In field strength multiplets, the vector space spanned by the boson components φ i islarger than the vector space spanned by the fermion components ψ ˆ ı . The physicaldegrees of freedom balance, however, owing to redundancies in the space of bosons,related to the constraints. This feature manifests in non-square 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 intoan “electric” sector, including components with a time-like index, and a “magnetic”sector involving components which have only space-like indices. The electric sectorand the magnetic sector are correlated by the differential constraints implied by theBianchi identity. Upon reduction to one-dimension, the magnetic sector decouples.The reason for this is that locally the magnetic fields are pure space derivatives, whichvanish upon restriction to a zero-brane.Thus, in order to enhance a one-dimensional gauge multiplet to a higher-dimensionalanalog, not only do we have to resurrect the spatial derivatives, ∂ a , but we also haveto resurrect the gauge sector. In the case of field strength multiplets, this meansre-instating the magnetic fields. Since these are physically decoupled on the shadow, We believe that the extra structures, namely that the bosons complexify and that thefermions assemble into chiral spinors, is also encoded in our formalism, using the equa-tions (A.5). We also believe that deeper scrutiny of those equations should provide analgebraic context for broadly resolving natural organizations of supermultiplets, includingcomplex structures, quaternionic structures, and so forth, in diverse dimensions. But thislies beyond the scope of this introductory paper on this topic. p -form multiplet components. Since closed 1-formsdo not involve gauge invariance, it follows that the simplest case involves closed 2-forms, such as F µν subject to ∂ [ λ F µν ] = 0. This allows access to the important casesinvolving Vector multiplets. The higher- p cases may be treated similarly, but theseinvolve additional subtlety. In order to keep our presentation relatively concise, wewill not address cases p ≥ P A ) i ˆ ı := ( ˜ u TA − ∆ A ) i ˆ ı , (5.1)where ˜ u TA is the transpose of the A -th fermion “up” matrix. A nonvanishing phantommatrix indicates the presence of one-way upward-directed Adinkra edges. If P A isnon-vanishing then this modifies the analysis in Appendix A precisely at the pointwhere (A.9) is introduced as the transpose of (A.8). If the Phantom matrices areincluded and the analysis is repeated, it is easy to show that (3.1) generalizes to∆ aA = − ( Γ Γ a ∆ ) A − ( Γ Γ a P ) A + T a P A − P A ˜ T a . (5.2)Note that the final three terms will contribute nontrivially to this equation only inthe gauge sector.It is helpful to briefly review the particular phantom sector associated with theshadow of the Maxwell field strength multiplet. This provides the archetype forgeneralizations, and motivates what follows. The 4D N = 1 super Maxwell multiplet involves four boson degrees of freedomoff-shell. These organize as the auxiliary scalar D plus the three off-shell “electro-26agnetic” degrees of freedom described by F µν . It is natural to write E a = F a and B a = ε abc F bc . The Bianchi identity ∂ [ λ F µν ] = 0 correlates E a and B a . Lo-cally, we can solve the Bianchi identity in terms of a vector potential A µ , so that E a = ∂ A a − ∂ a A and B a = ε abc ∂ b A c . Upon restriction to the zero-brane we take ∂ a →
0, so that E a → ∂ A a and B a →
0. Since the magnetic fields vanish onthe zero-brane, it is natural to think of the E a as more fundamental for our pur-poses. The shadow is described by a fermionic 4-4 Adinkra where the bosons are( E , E , E , D ) and the fermions are ( λ , λ , λ , λ ). To enhance this multipletwe must re-introduce ∂ a and also re-introduce 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 vectorspace, φ i = ( E , E , E , D | B , B , B ) in conjunction with the fermionic vec-tor space ψ ˆ ı = ( λ , λ , λ , λ ). 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 , , = E , , and φ , , = B ¯1 , ¯2 , ¯3 ,respectively. In this way, phantom fields are designated by an over-bar on the relevantindex. Boson fields not in the phantom sector are indicated by underlined indices, sothat φ , , , = ( E , E , E , D ). Matrices with two boson indices then divide intofour sectors, X i j , X i ¯ a , X ¯ a j , and X ¯ a ¯ b .The shadow transformation rules associated with the Maxwell multiplet can bewritten as (2.3), but the linkage matrices are not square! Instead, ( ˜ u A ) ˆ ı j is 7 × µA ) i ˆ ı is 4 ×
7. We exhibit the precise linkage matrices associated with the Maxwellmultiplet in Appendix D. For the Super Maxwell case, the first enhancement equationin (3.3) is a 7 × × Owing to the derivatives in the enhancement condition (3.3), we may use the Bianchiidentity, ∂ [ λ F µν ] = 0, usefully re-written as ∂ B ¯ a = ε ¯ abc ∂ b E c ∂ ¯ a B ¯ a = 0 , (5.3)27o define “canonical reorganizations” of the matrices in (3.2) under which (3.3) re-mains unchanged. Specifically, the first equation in (5.3) allows us to redefine( Ω AB ) i ¯ a →
0( Ω aAB ) i b → ( Ω aAB ) i b + ε ab ¯ c ( Ω AB ) i ¯ c , (5.4)whereby we exchange each appearance of ∂ B ¯ a in a supersymmetry commutator withan equivalent expression involving spatial derivatives on the electric field components.Similarly, the second equation in (5.3) allows us to redefine( Ω AB ) i ¯1 →
0( Ω AB ) i ¯2 → ( Ω AB ) i ¯2 − ( Ω AB ) i ¯1 ( Ω AB ) i ¯3 → ( Ω AB ) i ¯3 − ( Ω AB ) i ¯1 . (5.5)In this way, we define a canonical structure of the matrices ( Ω µAB ) i j , ensured bythe transformations (5.4) and (5.5), enabled by the Bianchi identity (5.3), whereby( Ω AB ) i ¯ a = 0 and ( Ω AB ) i = 0. The first equation in (3.3) may now be interpretedas saying that ( Ω µAB ) i j → × µAB ) i j defined by (3.2),using the linkage matrices exhibited in Appendix D, do satisfy ( Ω µAB ) i j → p = 1 gauge enhancement conditions Based on the above, a means becomes apparent under which we can ascertain whichone-dimensional multiplets may enhance to 4D gauge field strength multiplets, basedonly on a knowledge of the one-dimensional transformation rules, or equivalentlygiven an Adinkra.For physical gauge fields, the bosonic field strength tensor has greater engineeringdimension than the corresponding gaugino fermions. Therefore, the ambient fermionstransform into the magnetic fields via terms in the fermion transformation rule δ Q λ given by ε abc B a Γ bc (cid:15) or by ε abc B a Γ bc Γ (cid:15) . These are the only Lorentz covariantpossibilities. The former case involves a vector potential and the latter case involvesan axial vector potential. We focus first on the former case, and comment on axialvectors afterwards. It is straightforward to determine the phantom “up” links andthe time-like fermion “down” links using δ Q λ = · · · + ε abc B a Γ bc (cid:15) . By rearranging28his term into the form δ Q λ i = · · · + (cid:15) A ( ˜ u A ) i ¯ a B ¯ a , we derive ( ˜ u A ) ˆ ı ¯ a = ε ¯ abc ( Γ bc ) ˆ iA ( ∆ A ) ¯ a ˆ ı = 0 , (5.6)whereby using (5.1) we determine the non-vanishing part of the phantom matrix as( P A ) ¯ a ˆ ı = ε ¯ abc ( Γ bc ) A ˆ ı . (5.7)The entire phantom matrix has non-vanishing entries only in its final three rows.We can resolve the ∆ aA matrices in two parts, using different methods. First weresolve the phantom part ( ∆ aA ) ¯ a ˆ ı . Then we resolve the non-phantom part ( ∆ aA ) a ˆ ı .We determine the phantom part of the space-like boson down matrices using thefact that ( ∆ A ) ¯ a ˆ ı = 0, which says that there are no connections linking downwardfrom the phantoms. Thus, equation (A.4) tells us ( ∆ aA ) ¯ a ˆ ı = − ( T a ) ¯ a i ( ∆ A ) i ˆ ı .Next, we use the fact that a boost shuffles magnetic fields into electric fields, via( T a ) b ¯ c B ¯ c = ε ab c E c , to determine( ∆ aA ) ¯ b ˆ ı = ε ¯ b ac ( ∆ A ) c ˆ ı . (5.8)This determines the ¯ b -th row in the phantom sector of the A -th space-like downmatrices in terms of “electric” rows in the time-like down matrices.We determine the non-phantom part of the space-like down matrices by consid-ering the non-phantom sector in (5.2). Thus, we allow only non-phantom values forthe suppressed boson index. In this case, the second and the fourth terms on theright-hand side vanish because ( P A ) a ˆ ı = 0. The third term on the right-hand sidecan be resolved by noting that a boost shuffles electric fields into magnetic fields, via( T a ) b c E c = ε b a ¯ c B ¯ c , whereby ( T a ) b c = ε b a ¯ c . Substituting this result, along with(5.7), we derive ( ∆ aA ) b ˆ ı = − ( Γ Γ a ) A B ( ∆ A ) b ˆ ı + ( Γ a b ) A ˆ ı . (5.9)This is the same as our non-gauge result (3.1), modified by the second term. Takentogether, (5.8) and (5.9) generate for us the entire space-like phantom-modified downmatrices, generalizing our earlier non-gauge result (3.1) to the case in which fieldscan assemble into closed 1-forms. Consistency of (5.6) with (5.1) implies usefully extractable information about the spin rep-resentation assignments. We will not explore this arena in this paper. Note that the second term in (5.9) can also be written as − ε b ac ( R c ) A ˆ ı , where R c gener-ates a rotation in the c -th spatial dimension. N = 4 Adinkras using the methods described inthis section. Presently, we explain a means to produce a representative in one ofthe two minimal N = 4 Adinkra families which demonstrably enhances to a 4DMaxwell multiplet. Importantly, we can verify that this Adinkra enhances to such a4D multiplet using only one-dimensional reasoning.This is done by starting with right Adinkra in Figure 1. (Note that this Adinkrais in the family twisted relative to the family which includes the Chiral multipletshadow.) From this starting Adinkra, we raise all four boson vertices, to obtain afermionic 4-4 Adinkra with the four bosons at the higher level. We then performa permutation of the first and the fourth boson vertices, and a permutation of thethe second and the fourth boson vertices. We then flip both the first and the fourthboson vertices. (To flip a vertex means to scale this by an overall minus sign.) Finallywe flip the third fermion vertex. We compute the time-like up matrices ˜ u A and thetime-like down matrices ∆ A using the resultant fermionic 4-4 Adinkra. We designatethe first three boson vertices in this final orientation as our designated “electric”components. We then apply (5.6), (5.8), and (5.9) to append phantom sectors tothese time-like linkage matrices, and to generate provisional space-like down links∆ aA , including phantom sectors. What results from these operations are precisely thematrices shown in Appendix D. We next compute the Ω matrices using (3.2). Finally,we apply a canonical reshuffling of these Ω-matrices, using (5.4) and (5.5). Happily,we find that under this operation all of the matrices ( Ω µAB ) i j and ( ˜Ω µAB ) ˆ ı ˆ vanish.This illustrates that this representative set of operations produces an Adinkra whichpasses our non-trivial gauge enhancement test.If we repeat the above search allowing for axial vector potential, we would modifyall equations in this subsection with an additional factor of Γ . What we find isthat every multiplet which enhances to provide a vector potential also enhances toprovide an axial vector potential. This is loosely similar to the situation involvingChiral versus Antichiral multiplets, which have identical shadows. It follows that theboth the Vector multiplet and the Axial Vector multiplet shadow lie in the family ofAdinkras relatively twisted relative to the Chiral multiplet. We will not exhibitseparate equations for the Axial Vector case, leaving that as an exercise for theinterested reader. We have already explained that it is possible to systematically generate the linkagematrices for each member of the family associated with a given Valise. For each30epresentative, we can sequentially select triplets of vertices as potential electric fieldcomponents, and use equations (5.6), (5.8), and (5.9) to generate postulate phantomsectors. For each of these, we compute the relevant Ω-matrices using (3.2), thenshuffle these into canonical form using (5.4) and (5.5). By selecting those Adinkrasfor which all the canonical Ω-matrices obtained in this way vanish, we thereby obtainan algorithmic search for all multiplets in which vertices can assemble into closed1-forms. This search is guaranteed to locate those multiplets which do properlyenhance. (N.B. we have explained in the previous paragraph an example which weknow works.)In the case of closed 1-form multiplets, an enhanceable Adinkra exhibits a syn-ergy between the postulated electric field components and the designated magneticphantom sector, vis-a-vis the assignment of the component basis φ i . This is be-cause the enhancement criteria are sensitive to the basis choice on the componentboson vector via the structure of our imposed phantom sector. Practically, thisrequires, for a comprehensive algorithmic search for enhanceable Adinkras, that inaddition to sifting through all possible vertex raises and all possible selections of ver-tex 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 the-ory involving closed 1-form multiplets is accessible and understandable using only 1Dsupersymmetry. We find this interesting.In summary, following is a method to test an Adinkra to see if it enhances to givea 4D multiplet with a closed 1-form gauge field strength:1) Select three boson vertices with a common engineering dimension as the presumedelectric components, and arrange the boson vector space so that φ , , correspond tothese.2) Compute time-like linkage matrices u A , ˜ u A , ∆ A and ˜∆ A from the Adinkra.3) Augment the boson vector space by adding on a phantom sector consisting of threenew fields φ ¯1 , ¯2 , ¯3 with the same engineering dimension as φ , , .4) Add phantom sectors to the up matrices ˜ u A , by adding three extra columns, andadd phantom sectors to the time-like down matrices ∆ A by adding three extra rows.5) Populate the phantom sector of the up matrices using (5.6). Note that designating the phantom sector using (5.8) and (5.9) does not remove generalityfrom the search, much as choosing 4D Gamma matrices does not remove generality, forreasons described above. Instead, this removes redundancies from the answer set.
31) Generate space-like down matrices, including phantom sectors using (5.8) and(5.9).7) Use the complete set of phantom-augmented linkage matrices to determine thematrices ( Ω aAB ) i j and ( ˜Ω aAB ) ˆ ı ˆ using (3.2).8) Reshuffle the boson matrix ( Ω aAB ) i j using the prescription (5.4) and (5.5), toobtain a new matrix, in canonical form,( Ω aAB ) i j → ( (cid:98) Ω aAB ) i j . (5.10)The presence of the hat indicates canonical form.9) The p = 1 gauged enhancement conditions now correspond to the original enhance-ment conditions (3.3) augmented by the addition of phantom sectors and a canonicalreshuffle. We conclude that a necessary requirement for an Adinkra to enhance to a p = 1 field-strength multiplet is ( (cid:98) Ω µAB ) i j = 0( ˜Ω µAB ) ˆ ı ˆ = 0 . (5.11)This is our p = 1 gauge enhancement condition. The key difference as compared tothe non-gauge case is that the linkage matrices are not square in the gauge case, owingto the presence of the phantom boson sectors. Furthermore, we must implement thecanonical reshuffling maneuver to generate the hatted (cid:98) Ω matrices which describe thenon-gauge enhancement condition.The way we have designed our formalism is tailored toward implementation viacomputer-searches. This may require supercomputers for cases with higher N , whichwill involve large matrix computations. We hope to enlarge our algorithms so thatsifting through one-dimensional multiplets is controlled by the relevant lists of doubly-even error-correcting codes which correspond to these, as explained in the introduc-tion. But this lies beyond the scope of the presentation in this paper. We find itsufficiently noteworthy that such algorithms exist, even in principle. Our hope isthat this will shed light on unknown aspects of supersymmetry which have defiedattack using previous conventions. We have presented algebraic conditions which allow one to systematically locate thoserepresentations of one-dimensional supersymmetry which may enhance to higher di-32ensions. Equivalently, we have explained how to discern whether a given one-dimensional supermultiplet is a shadow of a higher-dimensional analog. This allowsthe representation theory of supersymmetry in diverse dimensions to be divided intothe simpler representation theory of one-dimensional supersymmetry augmented withseparate questions pertaining to the possibility of enhancement into higher dimen-sions.We have shown through explicit examples how information pertaining to four-dimensional N = 1 supersymmetry may be extracted using only information fromone-dimensional supersymmetry. We did this comprehensively for the case of 4D N = 1 non-gauge matter multiplets. We have also explained how this systematicsgeneralizes to cases involving higher-dimensional gauge invariances, specializing ourdiscussion to the case of 4D N = 1 Super-Maxwell theory.We intend to use the formalism and the algorithms developed above to seek inroadstowards off-shell aspects of interesting supersymmetric contexts where the off-shellphysics remains mysterious but potentially relevant. A A Proof
In this Appendix we prove that demanding Lorentz invariance of the linkage matricesdefined in (2.3) completely determines all of the space-like linkage matrices ∆ aA interms of the time-like linkage matrices ∆ A , and does so in precisely the mannerspecified above as equation (3.1). We also show how this same requirements providesconstraints on the spin representation content of supermultiplet component fields.The linkage matrices ( ∆ µA ) i ˆ ı transform under spin (1 , D − δ L ( ∆ µA ) i ˆ ı = θ µ ν ( ∆ νA ) i ˆ ı + θ λσ ( Γ λσ ) A B ( ∆ µB ) i ˆ ı + θ λσ ( T λσ ) i j ( ∆ µA ) j ˆ ı − θ λσ ( ∆ µA ) i ˆ ( ˜ T λσ ) ˆ ˆ ı . (A.1)In (A.1), the first term indicates that on ( ∆ µA ) i ˆ the µ index is a vector index, the sec-ond term indicates that the A index is a spinor index, and the last line accommodatesthe representation content of the supermultiplet component fields.Using (A.1), we obtain the following boost and rotation transformations for the“time-like” linkage matrices ( ∆ A ) i ˆ ı , δ boost ( ∆ A ) i ˆ ı = θ a ( ∆ aA ) i ˆ ı + θ a ( Γ a ) A B ( ∆ B ) i ˆ ı + θ a ( T a ∆ A ) i ˆ ı − θ a ( ∆ A ˜ T a ) i ˆ ı rotation ( ∆ A ) i ˆ ı = + θ ab ( Γ ab ) A B ( ∆ B ) i ˆ ı + θ ab ( T ab ∆ A ) i ˆ ı − θ ab ( ∆ A ˜ T ab ) i ˆ ı , (A.2)and the following boost and rotation transformations for the “space-like” linkagematrices, δ boost ( ∆ aA ) i ˆ ı = θ a ( ∆ A ) i ˆ ı + θ b ( Γ Γ b ) A B ( ∆ aB ) i ˆ ı + θ b ( T b ∆ aA ) i ˆ ı − θ b ( ∆ aA ˜ T b ) i ˆ ı δ rotation ( ∆ aA ) i ˆ ı = θ a b ( ∆ bA ) i ˆ ı + θ bc ( Γ bc ) A B ( ∆ aB ) i ˆ ı + θ bc ( T bc ∆ aA ) i ˆ ı − θ bc ( ∆ aA ˜ T bc ) i ˆ ı . (A.3)We demand that the linkage matrices are Lorentz invariant. This imposes that eachof the transformations in (A.2) and (A.3) must vanish. Requiring δ boost ∆ = 0imposes ( ∆ aA ) i ˆ ı = − ( Γ Γ a ) A B ( ∆ B ) i ˆ ı − ( T a ∆ A ) i ˆ ı + ( ∆ A ˜ T a ) i ˆ ı . (A.4)This determines ( ∆ aA ) i ˆ ı in terms of ( ∆ A ) i ˆ ı and in terms of the representation as-signments of the supermultiplet component fields.The remaining consequences of imposing Lorentz invariance on the linkage matri-ces ( ∆ µA ) i ˆ ı are the following, ( Γ ab ) A B ∆ B = ∆ A ˜ T ab − T ab ∆ A δ b a ∆ A = ( Γ Γ b ) A B ∆ aB + T b ∆ aA − ∆ aA ˜ T b η a [ b ∆ c ] A + ( Γ bc ) A B ∆ aB = ∆ aA ˜ T bc − T bc ∆ aA , (A.5)where the ( · ) i ˆ ı index structure has been suppressed on each term. These correspond,respectively, to δ rotation ∆ = 0, δ boost ∆ a = 0, and δ rotation ∆ a = 0, for arbitrary trans-formation parameters θ a and θ ab . The equations (A.5) place significant restrictionson the spin representation content of the component fields. As explained above, wesuspect that these equations encode useful and extractable information regardingallowable complements of spin structures in supermultiplets in diverse dimensions.The linkage matrices ( u A ) i ˆ ı transform under spin (1 , D − δ ( u A ) i ˆ ı = θ µν ( Γ µν ) A B ( u B ) i ˆ ı + θ µν ( T µν u A ) i ˆ ı − θ µν ( u A ˜ T µν ) i ˆ ı (A.6) This is similar to “demanding” that the Gamma matrices appearing in a Salam-Strathdeesuperfield be Lorentz invariant — in that case they are, automatically, as a consequence ofthe Clifford algebra. ( Γ µν ) A B ( u B ) i ˆ ı = ( u A ˜ T µν ) i ˆ ı − ( T µν u A ) i ˆ ı . (A.7)This indicates correlations between the up linkage matrices and the representationcontent of the component fields.Similar conditions result from demanding invariance of ( ˜ u A ) ˆ ı i and ( ˜∆ µA ) ˆ ı i . Theseare obtained from the above constraints by placing tildes on all matrices which donot have tildes and removing tildes from those that do. For example, invariance of( ˜ u A ) ˆ ı i imposes ( Γ µν ) A B ( ˜ u B ) ˆ ı i = ( ˜ u A T µν ) ˆ ı i − ( ˜ T µν ˜ u A ) ˆ ı i . (A.8)Note that for standard Adinkras we have ∆ A = ˜ u TA . Thus, using (A.8) we determine ( Γ µν ) A B ( ∆ B ) i ˆ ı = ( T Tµν ∆ A ) i ˆ ı − ( ∆ A ˜ T Tµν ) i ˆ ı . (A.9)This equation can be used in conjunction with (A.4) to replace that equation withan analog in which the representation matrices are not included.The boost matrices ( T a ) i j and ( ˜ T a ) ˆ ı ˆ are generically symmetric Therefore,(A.9) can be re-written as( T a ∆ A ) i ˆ ı − ( ∆ A ˜ T a ) i ˆ ı = ( Γ Γ a ) A B ( ∆ B ) i ˆ ı . (A.10)Substituting this result for the last two terms in (A.4), we determine( ∆ aA ) i ˆ ı = − ( Γ Γ a ) A B ( ∆ B ) i ˆ ı . (A.11)Remarkably, this relationship is completely independent of the representation contentof the component fields. This is an interesting result, which says that the space-likelinkage matrices ∆ a are determined from the time-like linkage matrices ∆ . For non-standard Adinkras, such as those which accommodate gauge invariances, this rela-tionship does not hold. Instead, we define ˜ u A = ∆ A + P A , where P A is a so-called Phantommatrix, which encodes the nexus of one-way upward-directed Adinkra edges. This general-ization is addressed in section 5. For example, if the fermions assemble as spinors then ˜ T a = Γ Γ a . In the Majoranabasis Γ is antisymmetric and real while Γ a are symmetric and real, and since Γ andΓ a anticommute, it follows that ˜ T a is symmetric in that case. For vectors V a we have( T a ) b = δ a b and ( T a ) b = − η ab ; for our metric choice η ab = −
1, so that these T a aresymmetric. This reasoning generalizes to higher-rank tensors and to all products of tensorand 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 ∆ a = 0.) Adinkra Conventions
In this Appendix we give a very concise overview of the graphical technology ofAdinkra diagrams. These were introduced in [ ] , and have formed the basis of a mul-tidisciplinary research endeavor, aimed at resolving a mathematically rigorous basisfor supersymmetry [ ] . Some of the conventions, notably as regards signchoices, have varied in these references, in part because some of these papers aim ata physics audience and some at a mathematics audience. Thus, one reason for thisAppendix is to clarify our conventions, as used above, so that this paper can be ap-preciated without undue confusion. Another is to allow this paper to be functionallyself-contained.A representation of N -extended supersymmetry in one time-like dimension con-sists of d boson fields φ i and d fermion fields ψ ˆ ı endowed with a set of transforma-tion rules, generated by δ Q ( (cid:15) ), where (cid:15) A are a set of N anticommuting parameters,which respect the N -extended supersymmetry algebra specified by the commutator[ δ Q ( (cid:15) ) , δ Q ( (cid:15) ) ] = 2 i δ AB (cid:15) A (cid:15) B ∂ τ . The transformation rules can be written for bosonfields as δ Q φ i = − i (cid:15) A ( Q A ) i ˆ ı ψ ˆ ı and for fermion fields as δ Q ψ ˆ ı = − i (cid:15) A ( ˜ Q A ) ˆ ı i φ i ,where ( Q A ) i ˆ ı and ( ˜ Q A ) ˆ ı i are two sets of N abstract d × d matrix generators ofsupersymmetry. By definition these represent( Q ( A ˜ Q B ) ) i j = i δ i j ∂ τ ( ˜ Q ( A Q B ) ) ˆ ı ˆ = i δ ˆ ı ˆ ∂ τ , (B.1)where the symmetrization brackets are defined with “weight-one”. It is possible to use cosmetic field redefinitions to re-define the component fields φ i and ψ ˆ ı into a “frame” where the generators ( Q A ) i ˆ and ( ˜ Q A ) ˆ ı ˆ are first orderdifferential operators with a specialized matrix structure. Specifically, it is possibleto write ( Q A ) i ˆ ı = ( u A ) i ˆ ı + ( d A ) i ˆ ı ∂ τ ( ˜ Q A ) ˆ ı i = i ( ˜ u A ) ˆ ı i + i ( ˜ d A ) ˆ ı i ∂ τ , (B.2)where ( u A ) i ˆ ı , ( d A ) i ˆ ı , ( ˜ u A ) ˆ ı j , and ( ˜ d A ) ˆ ı j are four sets of N real d × d “linkagematrices” with the features that every entry of each of these matrices takes onlyone of three values, 0, 1, or −
1, and such that there is at most one non-vanishingentry in every row and at-most one non-vanishing entry in every column of each whereby X ( A Y B ) = ( X A Y B + X B Y B ).
36f these matrices. Remarkably, we lose no generality by specializing to generatorsof the sort (B.2) with these particular properties. A mathematical proof that anyone-dimensional supermultiplet can be written in this manner is provided in [ ] .A simple example in the context of N = 2 supersymmetry is given by the followingtransformation rules, δ Q φ = − i (cid:15) ψ − i (cid:15) ψ δ Q φ = − i (cid:15) ∂ τ ψ + i (cid:15) ∂ τ ψ δ Q ψ = (cid:15) ∂ τ φ − (cid:15) ∂ τ φ δ Q ψ = (cid:15) φ + (cid:15) ∂ τ φ . (B.3)It is straightforward to verify that these satisfy the commutator relationship specifiedabove.The operator ∂ τ carries unit engineering dimension, while supersymmetry param-eters (cid:15) A carry engineering dimension one-half. Thus, in order to balance units in thetransformation rules (B.3) it follows that the two fermions ψ , have a common engi-neering dimension one-half greater than φ , and that φ has an engineering dimensionone-half greater than the fermions, and one unit greater than φ .The transformation rules (B.3) can be expressed equivalently, in terms of linkagematrices, as ( u ) i ˆ = (cid:32) (cid:33) ( u ) i ˆ ı = (cid:32) (cid:33) ( d ) i ˆ ı = (cid:32) (cid:33) ( d ) i ˆ ı = (cid:32) − (cid:33) ( ˜ u ) ˆ ı j = (cid:32) (cid:33) ( ˜ u ) ˆ ı j = (cid:32) − (cid:33) ( ˜ d ) ˆ ı j = (cid:32) (cid:33) ( ˜ d ) ˆ ı j = (cid:32) (cid:33) , (B.4)where blank matrix entries represent zeros. This set of eight matrices is completelyequivalent 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. In a system where (cid:126) = c = 1, a field with engineering dimension q carries units of ( Mass ) q .
37s an example, to illustrate what these matrices mean, consider the matrix ˜ u defined in (B.4). This is the “second fermion up matrix”, where the qualifier “second”refers to the subscript on ˜ u and indicates that this matrix encodes a mapping underthe second supersymmetry, while the qualifier “fermionic” refers to the tilde, andindicates that this matrix encodes transformations of the fermions. The single non-vanishing term in this matrix is in the first row, second column, which indicatesthat, of the two fermions, only the first fermion ψ transforms under the secondsupersymmetry, into the second boson φ . The fact that this matrix entry is − δ Q ψ on the term proportional to φ , i.e. , δ Q ψ = · · · − φ (cid:15) , 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 engineeringdimension — ψ in this case— to a field with higher engineering dimension — φ inthis case.The matrices in (B.4) exhibit the properties ˜ u A = d TA and ˜ d A = u TA . It is easy tosee that this indicates a symmetric feature in the transformation rules (B.3), wherebya fermion appearing in a boson transformation rule is correlated with that bosonappearing in the transformation rule for that fermion. In other words, terms in thesetransformation rules come paired. This feature is satisfied by a wide and importantclass of supermultiplets, which we call “standard”. (These are also called “Adinkraic”in the literture.)There is a third equivalent way to represent the supersymmetry transformationsgiven by (B.3) and by (B.4). This method uses the observation that the genericproperties of linkage matrices facilitate a concise system under which the entire col-lection of linkage matrices for a given multiplet can be faithfully represented by agraph. Such a graph, called an Adinkra, consists of d white vertices (one for eachboson) and d black vertices (one for each fermion). Two vertices are connected by an A -th colored edge if the two fields corresponding to those vertices are inter-related bythe A -th supersymmetry. The edge is rendered solid if the corresponding Q A matrixentries 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 correlatefaithfully with the respective engineering dimension.Thus, if we designate Q using purple edges and Q using blue edges, then theexample multiplet described by (B.4), equivalent to (B.3), would have the following38dinkra,
121 2 (B.5)where the numerals on the vertices specify the fields, e.g. , the black vertex labeled2 represents the fermion field ψ . As an easy exercise, the reader should confirmthat (B.4) can be recovered from (B.5) using the rules described above. There is astriking economy exhibited by this graphical method, empowered by the fact thatthese graphs completely encode every aspect of the transformation rules, in a waywhich allows for ready translation from any Adinkra into linkage matrices or intoparameter-dependent transformation rules.As another example, consider the following Adinkra, (B.6)This describes a supermultiplet distinct from the previous example, as evidenced bythe 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 d and d obviously vanish because the two bosons do notconnect “downward” to any lower fermion vertices. Similarly, the two fermion upmatrices ˜ u and ˜ u also obviously vanish, since there are no links “upward” from theblack vertices. We can determine the non-vanishing linkage matrices by “reading”the diagram. For example, the boson up matrix u encodes blue edges connectingupward from boson vertices. Thus, since the boson φ links upward via blue edge39nly to the fermion ψ , and does so with a solid edge, this tells us that the matrixentry ( u ) = +1. In this way, we can translate the Adinkra (B.6) into the linkagematrices described by d A = 0, ˜ u A = 0,( u ) i ˆ = (cid:32) (cid:33) ( u ) i ˆ ı = (cid:32) − (cid:33) , (B.7)and ˜ d A = u TA .Note that the Adinkra (B.5) can be obtained from (B.6) by an interesting opera-tion: by moving the vertex φ to a new position located one level above the fermions,while continuously maintaining all inter-vertex edge connections, so that the edgesswivel upward during this process. This macrame-like move encodes a transformationwhich maps one supermultiplet into another, and is called a vertex raising operation.One of our results concerning Adinkras is a mathematical proof that any standardsupermultiplet can be obtained by a sequence of vertex raising operations startingfrom an Adinkras with vertices which span only two different heights, e.g. , (B.6).Accordingly, the representation theory of 1D standard multiplets breaks naturallyinto two parts; first to classify all of the possible two-height Adinkras for a givenvalue of N , and then to systematize the possible sequences of vertex raises using eachof these as a starting point.Owing to the special role played by the two-height Adinkras, we have given thesea special name. Standard Adinkras which span only two height assignments are calledValise Adinkras, or Valises for short. The reason for this nomenclature is based on theobservation that a large number of multiplets can be “unpacked”, as from a suitcase(or a valise), by judicious choices of vertex raises. Using the information above, the reader ought to be able to verify the relationshipsbetween the Adinkras shown in Figures 1, 2, and 3, with the corresponding linkagematrices exhibited in the respective Tables 1, 2, and 3, and should appreciate our useof 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 toextract its shadow.The 4D N = 1 Chiral multiplet has the following transformation rules, δ Q φ = 2 i ¯ (cid:15) L χ R We credit Tristan H¨ubsch for inventing this catchy and useful term. Q χ R = ∂/ φ (cid:15) L + F (cid:15) R δ Q F = 2 i ¯ (cid:15) R ∂/χ R , (C.1)where φ is a complex scalar, F is a complex auxiliary scalar, and χ R is a right-chiralWeyl spinor field. The parameter (cid:15) R is also a right-chiral spinor, while (cid:15) L describesthe corresponding Majorana conjugate, i.e. , (cid:15) L = C − ¯ (cid:15) TR . The transformation rules(C.1) satisfy [ δ Q ( (cid:15) ) , δ Q ( (cid:15) ) ] = 4 i ¯ (cid:15) [2 L ∂/ (cid:15) L on all component fields φ , F , and χ R .Note that we can define a Majorana spinor parameter via (cid:15) = (cid:15) R + (cid:15) L , so that (cid:15) R,L = (1 ± Γ ) (cid:15) are the corresponding right- and left-chiral projections. In termsof the Majorana spinor, the algebra is [ δ Q ( (cid:15) ) , δ Q ( (cid:15) ) ] = 2 i ¯ (cid:15) ∂/ (cid:15) .We express spinors in the Majorana basis described in Appendix E. Accordingly,we write the spinor field and the spinor supersymmetry parameter as χ R = χ + i χ χ − i χ χ + i χ χ − i χ (cid:15) R = (cid:15) + i (cid:15) (cid:15) − i (cid:15) (cid:15) + i (cid:15) (cid:15) − i (cid:15) , (C.2)where χ , , , are each real anti-commuting fields and (cid:15) , , , are each real anti-commutingconstant parameters. We also write the complex boson fields as φ = φ + i φ , where φ , are real bosons, and F = F + i F , where F , are real auxiliary bosons.Using these definitions, setting ∂ a = 0, and using the spinor identities in AppendixE, the transformation rules (C.1) become δ Q φ = − i (cid:15) χ + i (cid:15) χ + i (cid:15) χ − i (cid:15) χ δ Q φ = − i (cid:15) χ − i (cid:15) χ + i (cid:15) χ + i (cid:15) χ δ Q χ = F (cid:15) − F (cid:15) − ˙ φ (cid:15) − ˙ φ (cid:15) δ Q χ = F (cid:15) + F (cid:15) − ˙ φ (cid:15) + ˙ φ (cid:15) δ Q χ = φ (cid:15) + φ (cid:15) + F (cid:15) − F (cid:15) δ Q χ = φ (cid:15) − φ (cid:15) + F (cid:15) + F (cid:15) δ Q F = − i (cid:15) ˙ χ − i (cid:15) ˙ χ − i (cid:15) ˙ χ − i (cid:15) ˙ χ δ Q F = − i (cid:15) ˙ χ + i (cid:15) ˙ χ − i (cid:15) ˙ χ + i (cid:15) ˙ χ . (C.3) The choice of basis is immaterial for the computing the dimensional reduction; we obtainidentical results using any other basis. We use the Majorana basis here in order to maintainconsistency with other derivations in this paper. Figure 4 : The Shadow of the Chiral multiplet, expressed as an Adinkra equivalentto the transformation rules (C.3.)These rules describe the shadow of (C.1). We organize the boson fields so that φ i = ( φ , φ , F , F ) and the fermion fields so that ψ ˆ ı = ( χ , χ , χ , χ ). Usingthe Adinkra conventions described in Appendix B, along with our edge colorationscheme whereby Q , , , are respectively described by purple, blue, green, and redcolored edges, we can unambiguously represent (C.3) as the Adinkra shown in Figure4. It is easy to translate the Adinkra in Figure 4 into equivalent up and down ma-trices. For example, we determine the boson up matrix u by looking at the purplecolored edges extending upward from boson vertices. There are two such edges: asolid edge connecting φ with ψ and solid edge connecting φ with ψ . Thus, thereare two non-vanishing entries in u : one in the first row, third column, and the otherin the second row, fourth column. These both take the value +1 because both edgesare solid.We can also determine the time-like down linkage matrices directly from (C.1).For example, to determine the femion 1-sector down matrices, ˜∆ A we isolate thoseterms in the fermion transformation rule δ Q ψ R involving the derivative ∂ . These aregiven by δ (1) Q χ R = Γ (cid:15) L ∂ φ . We then use the explicit matrix Γ specified in (E.9),the spinor components specified in (C.2), and we write φ = φ + i φ . This allows us,after a small amount of algebra, to re-write the 1-sector fermion transformation rule42 = − u = − u = − − u = − Table 4 : The boson up linkage matrices for the Chiral multiplet shadow. The fermiontime-like down matrices are determined from these via ˜ d A = u TA .∆ = ∆ = − ∆ = ∆ =
00 1 − Table 5 : The time-like dowm linkage matrices for the Chiral multiplet. The fermionup matrices are determined from these via ˜ u A = d TA .43s δ Q ψ ˆ ı = (cid:15) A ( ˜∆ A ) ˆ ı j ∂ φ j , from which we can read off the four matrices ( ˜∆ A ) ˆ ı j . Itis straightforward to perform this calculation, and then to verify that the matricesthereby obtained satisfy ˜∆ A = − ( Γ Γ ) A B ˜∆ B , where ˜∆ A = u TA . Similar calculationscan be done for all of the time-like down matrices, providing a nice consistency checkon 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 fieldstrength multiplet. This provides a means to exhibit precisely how 1D phantomsectors arise upon restriction of a p = 1 gauge multiplet to a zero-brane. ThisAppendix is complementary to section 5 in the main text, above.The 4D N = 1 super-Maxwell field-strength multiplet has the following transfor-mation rules, δ Q λ = F µν Γ µν (cid:15) − i D Γ (cid:15)δ Q F µν = − i ¯ (cid:15) Γ [ µ ∂ ν ] λδ Q D = ¯ (cid:15) Γ ∂/ λ , (D.1)where λ is a Majorana spinor gaugino field, D is a real auxiliary (pseudo)scalar, andthe field strength tensor F µν is subject to the Bianchi identity ∂ [ λ F µν ] = 0. We obtainthe linkage matrices equivalent to (D.1) by de-constructing these rules using a specificspinor basis, and re-writing them in terms of individual degrees of freedom as specifiedin (2.3). It follows simply that that ˜∆ aA = 0 and u A = 0, since the fermions λ A sharea common engineering dimension of 3/2 while the bosons F µν and D share a commonengineering dimension of 2.We use the specific Majorana basis defined in Appendix E by the Gamma matricesgiven in (E.9). To determine the “up” linkage matrices, it is helpful to re-write thefermion transformation rule in (D.1) as δ Q λ = 2 E a B a (cid:15) + 2 B a R a (cid:15) − i D Γ (cid:15) , (D.2)where we have used the definitions E a = F a and B a = ε abc F bc , for the electricand magnetic fields, respectively. We have also used the definitions B a = Γ Γ a ,and R a = ε abc Γ bc for the boost and rotation generators also given in AppendixE. We now use the explicit matrices B a , R a , and Γ specified in (E.11) and (E.9),to re-cast (D.2) in matrix form: the left side as a four-component column matrix44 A = ( λ , λ | , λ , λ ) T and the right-hand side as a 4 × (cid:15) A = ( (cid:15) , (cid:15) | (cid:15) , (cid:15) ) T . A small amount ofalgebra then allows us to re-write the result in the form δ Q λ ˆ ı = (cid:15) A ( ˜ u A ) ˆ ı j φ j , where φ i := ( E , E | E , D || B , B , B ) T , whereupon we can read off each of the fourmatrices ( ˜ u A ) ˆ ı j . The result is shown in Table 6. The fact that λ transforms non-trivially into B a manifests in the non-triviality of the rightmost three columns inthese results.We then do a similar thing to the boson fields to determine the down matrices∆ µA . We do this separately for each of the four choices for µ , referring to these asthe µ -sector down matrices. For example, to extract the 0-sector down matrices, weisolate those terms in the boson transformation rules in (D.1) proportional to thederivative ∂ λ . These are given by δ (0) Q E a = i ¯ (cid:15) Γ a ∂ λδ (0) Q D = ¯ (cid:15) Γ Γ ∂ λδ (0) Q B a = 0 . (D.3)Note that the magnetic fields B a = ε abc F bc do not transform into time derivativesof the gaugino field. This is not surprising since the magnetic field is expressiblelocally as B a = ε abc ∂ b A 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 downwardAdinkra 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. Inthis way the magnetic fields de-couple from the multiplet upon reduction to one-dimension. By utilizing the specific Gamma matrices given in Appendix E we canuse the same techniques described above to re-write the 0-sector transformation rules(D.3) as δ Q φ i = − i (cid:15) A ( ∆ A ) i ˆ ı ∂ λ ˆ ı , and then read-off the the matrices ( ∆ A ) i ˆ ı . Theresult of this straightforward process is exhibited in Table 7. By isolating the terms in (D.1) respectively proportional to ∂ , , λ , writing theseexplicitly using the Majorana basis Gamma matrices shown in (E.9), and then re-configuring the rules as δ Q φ i = − i (cid:15) A ( ∆ aA ) i ˆ ı ∂ a λ ˆ ı , allows us to read off the remainingspace-like linkage matrices. The results of this straightforward process are exhibitedin Tables 8, 9, and 10. It is easy to see that ˜ u TA (cid:54) = ∆ A , so that in this case the Phantom matrix defined in (5.1) isnon-vanishing. 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 itsfull ambient analog. u = − − ˜ u = −
11 0 0 0 − − − ˜ u = − − − − ˜ u = − − Table 6 : The four “up” linkage matrices associated with the Maxwell field strengthmultiplet. ∆ = −
10 0 0 00 0 0 00 0 0 0 ∆ = − ∆ = − − −
10 0 0 00 0 0 00 0 0 0 ∆ = −
110 0 0 00 0 0 00 0 0 0
Table 7 : The time-like down matrices associated with the Maxwell field strengthmultiplet. 46 = − − ∆ = −
10 0 −
10 0 0 00 − − ∆ = − − ∆ = − −
10 0 0 00 0 0 11 0 0 0
Table 8 : The 1-sector space-like down matrices for the Maxwell field strength mul-tiplet. ∆ = − −
11 0 0 00 0 0 00 0 − ∆ = − −
10 1 0 00 0 0 00 0 0 − ∆ = − − − ∆ = −
10 10 0 0 −
10 0 0 00 − Table 9 : The 2-sector space-like down matrices for the Maxwell field strength mul-tiplet. 47 = − −
10 0 1 00 0 0 0 ∆ = − −
10 0 1 00 0 0 10 0 0 0 ∆ = − −
10 1 0 01 0 0 00 0 0 0 ∆ = − − Table 10 : The 3-sector space-like down matrices for the Maxwell field strengthmultiplet.
E 4D Spinor bases
Gamma matrices satisfy the Clifford relationship { Γ µ , Γ ν } A B = − η µν δ A B , where η µν = diag(+ − −− ). These act from the left on spinors ψ A and from the righton barred spinors ¯ ψ A := ( ψ † Γ ) A . In four-dimensions, the minimal solution involves4 × A takes on four values. The 4D charge conjugationmatrix C is defined by C Γ a C − = − ( Γ a ) T . In addition, the matrix C is real,antisymmetric, and has unit determinant. A chirality operator is defined by Γ := i Γ Γ Γ Γ .We can change bases by replacingΨ A → M A B ψ B ( Γ µ ) A B → ( M Γ µ M − ) A B C − AB → (cid:112) det( M ) ( M C − M T ) AB . (E.1)where M is any nonsingular 4 × G µ = − Γ µ C − G µ → (cid:112) det( M ) M G µ M T (E.2)Given any basis, this allows us to find a similarity transformation to render all spinorcomponents real (a Majorana basis), and moreover one for which G AB = δ AB . Theresultant basis is then specially tailored for dimensional reduction to 1D, for the simplereason that the four real components of the Majorana supercharge operator Q A supplynatural real 1D shadow supercharges, which satisfy the algebra { Q A , Q B } = i δ AB ∂ τ . E.1 The Weyl basis
In the Weyl basis we choose 4 × = (cid:32) − (cid:33) Γ a = (cid:32) σ a σ a (cid:33) C = (cid:32) ε ε (cid:33) Γ = (cid:32) − (cid:33) , (E.3)where 1l is the 2 × a = 1 , ,
3, and σ a are the Pauli matrices and ε = i σ .Right- and left-handed Weyl spinors satisfy the respective constraints Γ ψ R,L = ± ψ R,L . In terms of the Weyl basis (E.3), this means that right- and left-handedspinors are respectively configured as χ R = χ χ ϕ L = ϕ ϕ , (E.4)where χ , χ , ϕ and ϕ are complex anticommuting fields. Note that Weyl spinorstake an especially tidy form in the Weyl basis, since half of the four complex spinorcomponents vanish.A Majorana spinor satisfies ψ = C − ¯ ψ T . In terms of the Weyl basis (E.3), thismeans ψ = ψ ψ ψ ∗ − ψ ∗ , (E.5)where ψ and ψ are complex anticommuting fields. Note that Majorana spinors arerelatively awkward in the Weyl basis. 49 .2 The Majorana basis Change bases from the Weyl basis to the Majorana basis, using (E.1), by choosing M = − − i − i − i i . (E.6)Using the transformation (E.1), right- and left-handed Weyl spinors in the Weyl basistransform into right- and left-handed Weyl spinors in the Majorana basis, as specifiedrespectively by χ R,L = χ ∓ i χ χ ∓ iχ , (E.7)where χ and χ are complex fields. Note that the difference between left- and right-handedness in this basis manifests in the relative phases appearing in (E.7). Notethat Weyl spinors are relatively awkward in the Majorana basis.Using the transformation (E.1), a Majorana spinor in the Weyl basis, (E.5), trans-forms into a Majorana spinor in the Majorana basis, as given by ψ A = Re ψ Im ψ Re ψ Im ψ . (E.8)Note that Majorana spinors take an especially tidy form in the Majorana basis, sinceall four components are independent and real. In this basis, the Gamma matricesand the charge conjugation matrices areΓ = − − Γ = − − Γ =
11 11 Γ = − − = − − Γ = i − i i − i , (E.9)as obtained by transforming (E.3) using (E.1). The corresponding G -Matrices G aAB = − η ab ( Γ b C − ) AB are G = G = G = − − G = − − . (E.10)Note that G -matrices are symmetric, real, and traceless.Also useful are the “boost” operators B a = Γ Γ a and the “rotation” operators R a = ε abc Γ bc . In the Majorana basis (E.9) these are B = R = − − B = − − R = − − B = − − R = −
11 1 − . (E.11)Note that the boost operators are symmetric while the rotation operators are anti-symmetric. Note too that G a = 2 B a . The operators in (E.11) satisfy the Lorentzalgebra [ R a , R b ] = − ε ab c R c B a , B b ] = ε abc R c [ B a , R b ] = − ε a bc B c . (E.12)The Lorentz algebra (E.12) can be written concisely, and in a manner which is man-ifestly covariant, as[ M µν , M λσ ] = δ µ λ M ν σ − δ µ σ M ν λ + δ ν σ M µ λ − δ ν λ M µ σ , (E.13)where M a = B a and M ab = ε abc R c . Acknowledgements
The authors are thankful to their long-time collaborators Tristan H¨ubsch, CharlesDoran, and S. J. Gates, Jr., for many relevant discussions which spawned this analysis.M.F. is thankful also to his muse, Adriana, and to the the Slovak Institute for BasicResearch (SIBR), in Podvaˇzie Slovakia, where much of this work was completed,for gracious hospitality. The authors also thank the President’s office at the SUNYCollege at Oneonta, for providing hospitable use of a campus guest house, where theideas for this paper congealed.
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