Reduction Redux of Adinkras
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November 6, 2018 UMDEPP-013-019arXiv:1312.2000 [hep-th]
Reduction Redux of Adinkras
S. James Gates, Jr. † , Stephen Randall † , and Kory Stiffler ∗† Center for String and Particle TheoryDepartment of Physics, University of MarylandCollege Park, MD 20742-4111 USA and ∗ Department of Chemistry, Physics, & AstronomyCollege of Arts and SciencesIndiana University NorthwestGary, Indiana 46408 USA
ABSTRACT
We show performing general “0-brane reduction” along an arbitraryfixed direction in spacetime and applied to the starting point of min-imal, off-shell 4D, N = 1 irreducible supermultiplets, yields adinkraswhose adjacency matrices are among some of the special cases proposedby Kuznetsova, Rojas, and Toppan. However, these more general reduc-tions also can lead to ‘Garden Algebra’ structures beyond those describedin their work. It is also shown that for light-like directions, reduction tothe 0-brane breaks the equality in the number of fermions and bosons fordynamical theories. This implies that light-like reductions should be doneinstead to the space of 1-branes or equivalently to the worldsheet. [email protected] [email protected] kmstiffl@iun.edu a r X i v : . [ h e p - t h ] D ec Introduction
A number of years ago [1], we began the study of a ubiquitous mathematical sub-structure that appears in all off-shell linear representations of supersymmetry in oneand two dimensions. The topic was developed in a series of papers which uncoveredincreasingly complex intricacies such as a series of mapping operations acting onthese structures. In the works of [2], a procedure,“0-brane projection”, was describedwhereby these same structures could be derived from supersymmetric theories in allhigher dimensions. Eventually, these structures were named “adinkras” [3].It is rather simple to prove that when any standard field-theoretic formulation of asupersymmetric field theory is examined under such a projection, forcing all dynami-cal fields to only depend on a single temporal coordinate, this results in the revealingof adinkras. So these networks appear to be universal to all supersymmetric fieldtheories. Shortly after the christening, there was assembled an interdisciplinary col-laboration of mathematicians and physicists (the ‘DFGHILM group’ ) which workeddiligently to define more rigorously the properties of adinkras.By now there has been established a substantial literature that treats aspects ofthis approach to understanding the fundamental structure of off-shell spacetime su-persymmetric representations. Along the way many unexpected developments haveoccurred. The most recent such surprise [4] is the discovery that adinkras are di-rectly related to super Riemann surfaces with no reference to Superstring/M-Theory!Adinkras are equivalent to very special super Riemann surfaces with divisors.We have long advocated [2] that adinkras may be regarded as the equivalent ofgenes for biological systems. That is, we have suggested it is possible to begin solely with adinkras, which are intrinsically one dimensional networks, and then reconstructhigher dimensional supersymmetric representations from this starting point. We havenamed this process “SUSY holography” as should it succeed, this would imply thatthe information to reconstruct the higher dimensional supersymmetric systems mustin some way be holographically encoded within the starting point of one dimensionaladinkra networks.Within the last year, two presentations have advanced the possibility of creatinga proof of the “SUSY holography conjecture” at least regarding the relation between1D, N = 4 adinkras and 4D, N = 1 classes of theories.First, it was established [5] that given some specific 1D, N = 4 adinkras, their A large number of these publications can be identified by using the webpage resource at http://math.bard.edu/DFGHIL/index.php?n=Main.Publications on-line. ⊗ SU(2)algebra . There is a group theoretical understanding of this result that leads tothe conclusion that for a general adinkra with N colors, the associated adjacencymatrices will carry some representation of SO( N ). Since locally the algebra of SO(4)is isomorphic to an SU(2) ⊗ SU(2) algebra, the demonstrated results consequentlyfollow.Though it appears not generally appreciated, the covering algebra of the SO(1, 3)Clifford algebra also carries a representation of an SU(2) ⊗ SU(2) algebra. It is pos-sible to identify these two algebras as one and the same structure and re-construct,solely from the 1D, N = 4 adinkras, matrices in the covering algebra of the SO(1, 3) γ -matrices. Thus, information about the SO(1, 3) spin bundle associated with a 4D, N = 1 supermultiplet can indeed be encoded within the network of adinkras!Second, it was established [7] that the sets of all possible quartets of 4 × N = 1 supermultiplets (the chiral, vector, and tensorsupermultiplets) can be identified with the 384 elements of the Coexter Group BC .Taking absolute values of the elements of these matrices then leads to a furtheridentification with S , the permutation group of four elements.Next, it was shown there exists a discrete transformation (acting on these adinkrasidentified with the elements of the Coexter Group BC ) with the property of defininga class structure of three distinct parts. It was then argued that these three sub-classesshould be identified with the respective three distinct minimal off-shell representationsof 4D, N = 1 supersymmetry and that the operation provides (in the space of 1D, N = 4 adinkras) a realization of a ‘one dimensional Hodge star operation’ acting onforms in the covering algebra of the SO(1, 3) Clifford algebra!These two results together establish a fairly well defined path for the reconstruc-tion of 4D, N = 1 supermultiplets from one dimensional four-color adinkras:(a.) using the putative and proposed ‘one dimensional Hodge staroperation’ any such adinkra can be associated with one ofthe minimal 4D, N = 1 supersymmetric representations, and(b.) the SU(2) ⊗ SU(2) content associated with the given adinkra de-termines the SO(1, 3) spin bundle representation carried by thenodes of the adinkra. The presence of this SU(2) ⊗ SU(2) algebra was first observed in the work of [6] where it wasnoted that while off-shell representations are representation of this algebra, on-shell represen-tations only carry representations of an SU(2) ⊗ SO(2) algebra.
2f course, more work understanding the details of these steps is required in the gen-eral case as well as a complete understanding of the filters that allow completelyconsistent dimensional enhancement. But the results in hand provide the essence ofan existence proof that we believe can be successfully constructed.All these results are within the context of adinkras associated with our traditional“0-brane reduction” of 4D, N = 1 supermultiplets. However, in this work, we turnto a different question. Up to this point previously, whenever we have constructedadinkras from higher dimensional supersymmetric field theories, the single bosoniccoordinate in the adinkra was associated with a purely time-like direction. It is a verynatural question to ask, “What might occur if instead the reduction was done withrespect to an arbitrary constant 4-vector direction among the spacetime coordinates?”This is the question probed in this current work.We shall show in the following work, using the familiar off-shell minimal 4D, N = 1 SUSY representations, that “0-brane reduction” utilizing a non-purely time-like direction yields adinkras whose associate L-matrices and associated R-matricesform representations that are generalizations of the “Garden Algebras” introduced inwork of [1]. The authors Kuznetsova, Rojas, and Toppan have previously introducedgeneralizations [8] of the “Garden Algebras” some time ago. We shall indeed see that“0-brane reduction” utilizing a space-like direction can yield structures first suggestedby these authors. However, we shall also see that the “0-brane reduction” utilizing ageneral space-like direction produces generalizations even beyond those considered intheir work. The starting point to revealing the adinkra sub-structure of higher dimensionalsupersymmetric theories in Minkowski space begins with the usual spacetime four-gradient operator ∂ µ ∂ µ = (cid:18) ∂∂t , ∂∂x , ∂∂y , ∂∂z (cid:19) = T µ ∂∂t + X µ ∂∂x + Y µ ∂∂y + Z µ ∂∂z , (2.1)where the constant four-vectors T µ , X µ , Y µ , and Z µ respectively denote T µ = ( 1 , , , , X µ = ( 0 , , , , Y µ = ( 0 , , , , Z µ = ( 0 , , , . (2.2)3ext a single real parameter τ may be introduced via the equations below ∂∂t = cos α ∂∂τ , ∂∂x = sin α sin β cos γ ∂∂τ ,∂∂y = sin α sin β sin γ ∂∂τ , ∂∂z = sin α cos β ∂∂τ , (2.3)whereupon the four-gradient operator ∂ µ takes the restricted form ∂ µ = (cid:96) µ ∂ τ or moreexplicitly ∂ µ = [cos α T µ + sin α sin β cos γ X µ + sin α sin β sin γ Y µ + sin α cos β Z µ ] ∂∂τ . (2.4)This restricted form of the spacetime four-gradient operator clearly generates motionin the Minkowski space along a straight line described by the four-vector (cid:96) µ . Wehave named this process ‘0-brane reduction.’ It has long been our assertion that thegroup theoretic structures obtained from 0-brane reduction in the context of spacetimesupersymmetric representation theory plays the same role as the Wigner ‘little group’for non-supersymmetric representation theory.In a four-dimensional N = 1 theory, the anti-commutator algebra for the super-symmetry generator can be written as { Q a , Q b } = i γ µ ) a b ∂ µ , (2.5)which under 0-brane reduction becomes { Q a , Q b } = i γ µ ) a b (cid:96) µ ∂ τ = i γ · (cid:96) ) a b ∂ τ . (2.6)At this stage, it does not appear that any particular 0-brane projection (for arbi-trary values of the parameters α , β , and γ ) possesses any distinguished behavior fromany other projection. However, in the following we will see that though the changeabove is mild, when the questions of dynamics are engaged, depending on whether (cid:96) is time-like or space-like versus light-like, subtle differences do emerge.As discussed in [6], our gamma matrices are chosen to be real and explicitly givenby ( γ ) = i ( σ ⊗ σ ) , ( γ ) = ( I ⊗ σ ) , ( γ ) = ( σ ⊗ σ ) , ( γ ) = ( I ⊗ σ ) , (2.7)thus describing a mostly plus Minkowski spacetime metric η µ ν that appears in γ µ γ ν + γ ν γ µ = 2 η µ ν I (2.8)so that the purely imaginary ‘gamma-five’ matrix takes the form( γ ) = i γ γ γ γ = − ( σ ⊗ σ ) . (2.9)4rom the definitions above, it follows that the generators of spatial rotations Σ i j = i/ γ i , γ j ] take the explicit formsΣ = ( σ ⊗ σ ) , Σ = ( σ ⊗ σ ) , Σ = ( I ⊗ σ ) , (2.10)and it is easily seen that these form an SU(2) algebra. In our discussions, we oftenrefer to this as the SU α (2) algebra.However, using the gamma matrices described above, it is possible to constructa second SU(2) algebra that we denote as the SU β (2) algebra. The generators ofSU β (2) algebra take the explicit forms ( σ ⊗ σ ) = − i γ , ( σ ⊗ σ ) = − γ , ( σ ⊗ I ) = γ γ . (2.11)Due to the defining properties of the gamma matrices, these two SU(2) algebrascommute. This implies that the complete set of sixteen elements in the coveringalgebra of the gamma matrices carry a representation of SU α (2) ⊗ SU β (2).On the other hand, as noted in [6] among other places, a four-color adinkra insome representation R has a set of L-matrices and R-matrices that satisfy the “GardenAlgebra” conditions( R ( R ) I ) ˆ ıj ( L ( R ) J ) j ˆ k + ( R ( R ) J ) ˆ ıj ( L ( R ) I ) j ˆ k = 2 δ I J ( I d ) ˆ ı ˆ k , L ( R ) = (cid:2) R ( R ) (cid:3) t = (cid:2) R ( R ) (cid:3) − . (2.12)From these, one can construct a set of objects called “adjacency matrices” using astandard definition from graph theory that we can denote by (cid:98) γ I where (cid:98) γ I = (cid:34) I R I (cid:35) . (2.13)Due to the defining property of the L-matrices and R-matrices it follows that thequantities (cid:98) γ I defined this way form the generators of a four-dimensional Euclidean-signatured Clifford algebra. Thus, the quantities defined by (cid:98) Σ I J = i/ (cid:98) γ I , (cid:98) γ J ]correspond to a set of hermitian generators of SO(4).Owing to the fact that SO(4) is locally isomorphic to SU α (2) ⊗ SU β (2), the resultsof (2.7) - (2.13) imply for every adinkra in any representation R with four colors, the N = 4 version of the “Adinkra/ γ -matrix Holography Equation” [5](R ( R ) I ) ˆ ıj (L ( R ) J ) j ˆ k − (R ( R ) J ) ˆ ıj (L ( R ) I ) j ˆ k = 2 (cid:104) (cid:96) ( R )1 IJ ( γ γ ) ˆ ı ˆ k + (cid:96) ( R )2 IJ ( γ γ ) ˆ ı ˆ k + (cid:96) ( R )3 IJ ( γ γ ) ˆ ı ˆ k + i (cid:98) (cid:96) ( R ) IJ ( γ ) ˆ ı ˆ k + i (cid:98) (cid:96) ( R ) IJ ( γ ) ˆ ı ˆ k + (cid:98) (cid:96) ( R ) IJ ( γ γ ) ˆ ı ˆ k (cid:105) (2.14)5ust be valid for some set of constant coefficients (cid:96) ( R ) I IJ and (cid:98) (cid:96) ( R ) I IJ . Given that theAdinkra/ γ -matrix Holography Equation, for every four-color adinkra, generates aset of matrices ( γ γ , Σ i j ) solely from the data in the adinkra, we simply note theequation γ i = (cid:15) i j k γ γ Σ j k , (2.15)yields the remaining spatial gamma matrices. After these are constructed we multiplyby γ , and then γ respectively. Finally I must be included the to construct all sixteenelements of the covering algebra. We can generalize the original notion of a Garden algebra seen in [1] to thefollowing form. Denoted by GR ( d, N, (cid:96) µ ) this generalized algebra is defined byL (I R J) = R (I L J) = 2[ E ( (cid:96) µ )] IJ I d for I , J = 1 , , . . . , N , (2.16)where (cid:96) µ is the “direction” of the reduction, a four-vector parameter of the algebraon equal footing with d and N , and the E ( (cid:96) µ ) determine the ‘metric’ on the space ofSUSY parameters.In the following, we shall show that reduction along any spacetime axis satisfiesthe generalized Garden algebra . Finally, some explicit examples of the values of thisSUSY parameter space matric for GR (4 , , (cid:96) µ ) are[ E ( T µ )] IJ = [ I ] IJ , [ E ( X µ )] IJ = [ σ ⊗ σ ] IJ , [ E ( Y µ )] IJ = [ σ ⊗ I ] IJ , [ E ( Z µ )] IJ = − [ σ ⊗ σ ] IJ , (2.17)with the eigenvalues of E ( T µ ) all being +1 and the eigenvalues of E ( X µ ), E ( Y µ ), and E ( Z µ ) being ± E ( T µ ) is equal tofour, and the traces of E ( X µ ), E ( Y µ ), and E ( Z µ ) are all equal to zero.For the case of E ( X µ ), this is precisely an example of the structure presented byKuznetsova, Rojas, and Toppan [8] who advocated considering this type of metric inthe parameter space of the 1D, N -extended SUSY QM systems. However, in the caseof E ( Y µ ), and E ( Z µ ), we see the metric in the SUSY parameter space can be evenmore general than they proposed. For general values of the angles α , β , and γ , onefinds: [ E ( (cid:96) µ )] IJ = cos α [ E ( T µ )] IJ + sin α sin β cos γ [ E ( X µ )] IJ +sin α sin β sin γ [ E ( Y µ )] IJ + sin α cos β [ E ( Z µ )] IJ . (2.18)6owever, we also find a surprising result. If we impose the condition(L I ) t = [ E ( (cid:96) µ )] IJ (R J ) , (2.19)as the natural generalization from the temporal 0-brane reduction, this forces theangle α to either take the value of α = 0, π/
2, or α = π !Let us introduce one other notation in order to illustrate another result. We canwrite E ( n ) = E ( X µ ) , E ( n ) = E ( Y µ ) , E ( n ) = E ( Z µ ) . (2.20)It now follows from (2.10) that we have the nice algebraic relations (cid:2) E ( n i ) , E ( n j ) (cid:3) IJ = i i j ) IJ . (2.21)Let us note that we can define V IJ = [ L I R J − L I R J ] , (cid:101) V IJ = [ R I L J − R I L J ] . (2.22)and from these it follows that[ V IJ , V KL ] = [ E ( (cid:96) µ )] IK V JL − [ E ( (cid:96) µ )] JK V IL − [ E ( (cid:96) µ )] IL V JK + [ E ( (cid:96) µ )] JL V IK , (cid:104) (cid:101) V IJ , (cid:101) V KL (cid:105) = [ E ( (cid:96) µ )] IK (cid:101) V JL − [ E ( (cid:96) µ )] JK (cid:101) V IL − [ E ( (cid:96) µ )] IL (cid:101) V JK + [ E ( (cid:96) µ )] JL (cid:101) V IK . (2.23)where (cid:96) is picked along each of the coordinate axes. These commutator algebrasclearly do not describe SO(4) for α (cid:54) = 0. This is turn implies that the adjacencymatrices also do not form a representation of SO(4) for α (cid:54) = 0.In closing, we note that each supermultiplet when subjected to 0-brane reductionis associated with its own adjacency matrices. For distinct multiplets, these adjacencymatrices are different. In the following we will prove that the results in (2.19) and(2.21) hold independent of which multiplet on which the evaluation is made. In order to investigate in concrete detail the procedure of general 0-brane reductionand any subtleties that arise along the way, the standard 4D, N = 1 chiral multipletis considered. Using the conventions of [6], the Lagrangian that is invariant withrespect to supersymmetry variations takes the form L = − ∂ µ A∂ µ A − ∂ µ B∂ µ B + i ( γ µ ) bc ψ b ∂ µ ψ c + F + G (3.1)7ith the SUSY transformation laws in component form realized by the supercovariantderivative D a operators acting on the fields asD a A = ψ a , D a B = i ( γ ) ab ψ b , D a ψ b = i ( γ µ ) ab ∂ µ A − ( γ γ µ ) ab ∂ µ B − iC ab F + ( γ ) ab G , D a F = ( γ µ ) ab ∂ µ ψ b , D a G = i ( γ γ µ ) ab ∂ µ ψ b , (3.2)in four dimensions. Under the 0-brane reduction where ∂ µ = (cid:96) µ ∂ τ these becomeD a A = ψ a , D a B = i ( γ ) ab ψ b , D a ψ b = i ( γ · (cid:96) ) ab ∂ τ A − ( γ γ · (cid:96) ) ab ∂ τ B − iC ab F + ( γ ) ab G , D a F = ( γ · (cid:96) ) ab ∂ τ ψ b , D a G = i ( γ γ · (cid:96) ) ab ∂ τ ψ b , (3.3)and the Lagrangian becomes L = − (cid:96) µ (cid:96) µ (cid:2) ( ∂ τ A ) + ( ∂ τ B ) (cid:3) + i ( γ · (cid:96) ) bc ψ b ∂ τ ψ c + F + G . (3.4)Furthermore, utilizing the explicit form of the γ matrix to find η = − (cid:96) µ , this becomes L = cos(2 α ) (cid:2) ( ∂ τ A ) + ( ∂ τ B ) (cid:3) + i ( γ · (cid:96) ) bc ψ b ∂ τ ψ c + F + G . (3.5)A next step involves defining the valise related to these equations. This is doneby making the re-definitions F → ∂ τ F and G → ∂ τ G so that we findD a A = ψ a , D a B = i ( γ ) ab ψ b , D a ψ b = i ( γ · (cid:96) ) ab ∂ τ A − ( γ γ · (cid:96) ) ab ∂ τ B − iC ab ∂ τ F + ( γ ) ab ∂ τ G , D a F = ( γ · (cid:96) ) ab ψ b , D a G = i ( γ γ · (cid:96) ) ab ψ b , (3.6)and the Lagrangian becomes L = − (cid:96) µ (cid:96) µ (cid:2) ( ∂ τ A ) + ( ∂ τ B ) (cid:3) + i ( γ · (cid:96) ) bc ψ b ∂ τ ψ c + (cid:2) ( ∂ τ F ) + ( ∂ τ G ) (cid:3) . (3.7)Implementing the further definitionsΦ i = ABFG , Ψ ˆ k = − i ψ ψ ψ ψ , (3.8)8akes it clear that (3.6) can be written in the concise formsD I Φ i = i (L I ) i ˆ k Ψ ˆ k and D I Ψ ˆ k = (R I ) ˆ ki ∂ τ Φ i , (3.9)for some matrices (L I ) and (R I ). The Lagrangian then can be written as: L = δ i j ( ∂ τ Φ i ) ( ∂ τ Φ j ) − i δ ˆ k ˆ l Ψ ˆ k ∂ τ Ψ ˆ l , (3.10)where numerically we have δ ˆ k ˆ l = − ( γ · (cid:96) ) ˆ k ˆ l (3.11)for purely time-like reduction (cid:96) µ (cid:96) µ = −
1. The condition in (3.11) is characteristic ofMajorana representations and we have assumed its use in all our previous discussionsof adinkras.In this section, we have given a detailed discussion of the steps required to ob-tain the valise formulation of the chiral supermultiplet. Similar discussions could beundertaken for the vector multiplet starting from its equationsD a A µ = ( γ µ ) ab λ b , D a λ b = − i [ γ µ , γ ν ] ab F µν + ( γ ) ab d ,D a d = i ( γ γ µ ) ab ∂ µ λ b , (3.12)or the tensor multiplet starting from its equationsD a ϕ = χ a , D a B µν = − [ γ µ , γ ν ] ab χ b , D a χ b = i ( γ µ ) ab ∂ µ ϕ − ( γ γ µ ) ab (cid:15) µρστ ∂ ρ B στ . (3.13)However, the end point of such discussions is precisely the same as in (3.9) and (3.10)but with different L-matrices and R-matrices describing the distinct supermultiplets. Explicit reduction of the chiral, vector and tensor supermultiplet in a purelytime-like direction has been presented before [6]. Once the valise form of each super-multiplet is obtained, the supersymmetry variations are completely described by theresults in (3.9). Thus, all that remains is to give the explicit form of the L-matricesand the R-matrices. Below these are written in our compact matrix notation asexplained in the appendix. A straightforward calculation with the matrices below9eveals that our assertion in section 2 is correct: that Eqs. (2.19) and (2.21) holdindependent of multiplet or reduction coordinate.
Chiral Multiplet
Reduction for (cid:96) = T gives the following L and R matricesL = (1 4 2 3) , L = (2 3 1 4) , L = (3 2 4 1) , L = (4 1 3 2) , R = (1 3 4 2) , R = (3 1 2 4) , R = (4 2 1 3) , R = (2 4 3 1) . (4.1)Reduction for (cid:96) = X gives the following L and R matricesL = (1 4 2 3) , L = (2 3 1 4) , L = (3 2 4 1) , L = (4 1 3 2) , R = (1 3 4 2) , R = (3 1 2 4) , R = (4 2 1 3) , R = (2 4 3 1) . (4.2)Reduction for (cid:96) = Y gives the following L and R matricesL = (1 4 4 1) , L = (2 3 3 2) , L = (3 2 2 3) , L = (4 1 1 4) , R = (3 2 2 3) t , R = (4 1 1 4) t , R = (1 4 4 1) t , R = (2 3 3 2) t . (4.3)Reduction for (cid:96) = Z gives the following L and R matricesL = (1 4 1 4) , L = (2 3 2 3) , L = (3 2 3 2) , L = (4 1 4 1) , R = (2 3 2 3) t , R = (1 4 1 4) t , R = (4 1 4 1) t , R = (3 2 3 2) t . (4.4) Vector Multiplet
Reduction for (cid:96) = T gives the following L and R matricesL = (2 4 1 3) , L = (1 3 2 4) , L = (4 2 3 1) , L = (3 1 4 2) , R = (3 1 4 2) , R = (1 3 2 4) , R = (4 2 3 1) , R = (2 4 1 3) . (4.5)Reduction for (cid:96) = X gives the following L and R matricesL = (2 4 13) , L = (1 3 2 4) , L = (4 2 3 1) , L = (3 1 4 2) , R = (3 1 4 2) , R = (1 3 2 4) , R = (4 2 3 1) , R = (2 4 1 3) . (4.6)Reduction for (cid:96) = Y gives the following L and R matricesL = (2 2 1 1) , L = (1 1 2 2) , L = (4 4 3 3) , L = (3 3 4 4) , R = (4 4 3 3) t , R = (3 3 4 4) t , R = (2 2 1 1) t , R = (1 1 2 2) t . (4.7)Reduction for (cid:96) = Z gives the following L and R matricesL = (2 2 4 4) , L = (1 1 3 3) , L = (4 4 2 2) , L = (3 3 1 1) , R = (1 1 3 3) t , R = (2 2 4 4) t , R = (3 3 1 1) t , R = (4 4 2 2) t . (4.8)10 ensor Multiplet Reduction for (cid:96) = T gives the following L and R matricesL = (1 3 4 2) , L = (2 4 3 1) , L = (3 1 2 4) , L = (4 2 1 3) , R = (1 4 2 3) , R = (4 1 3 2) , R = (2 3 1 4) , R = (3 2 4 1) . (4.9)Reduction for (cid:96) = X gives the following L and R matricesL = (1 3 4 2) , L = (2 4 3 1) , L = (3 1 2 4) , L = (4 2 1 3) , R = (1 4 2 3) , R = (4 1 3 2) , R = (2 3 1 4) , R = (3 2 4 1) . (4.10)Reduction for (cid:96) = Y gives the following L and R matricesL = (1 1 2 2) , L = (2 2 1 1) , L = (3 3 4 4) , L = (4 4 3 3) , R = (3 3 4 4) t , R = (4 4 3 3) t , R = (1 1 2 2) t , R = (2 2 1 1) t . (4.11)Reduction for (cid:96) = Z gives the following L and R matricesL = (1 1 3 3) , L = (2 2 4 4) , L = (3 3 1 1) , L = (4 4 2 2) , R = (2 2 4 4) t , R = (1 1 3 3) t , R = (4 4 2 2) t , R = (3 3 1 1) t . (4.12) There is one other subtlety that we need to address shortly. We start by notingthat a simple calculation using the parametrization in terms of α , β , and γ shows (cid:96) · (cid:96) = (cid:96) µ η µ ν (cid:96) ν = − cos(2 α ) . (5.1)Clearly the constant four-vector (cid:96) µ defines three distinct regimes for 0-brane reduc-tion. − cos(2 α ) ∝ < , for time-like (cid:96) µ = 0 , for light-like (cid:96) µ > , for space-like (cid:96) µ (5.2)It turns out that light-like reduction has subtle differences from the other two cases.We will briefly discuss this here.Given any fixed four-vector of the form that appears in (2.4), it is always possibleto construct a second such four-vector from this one. The second four-vector islinearly independent of the first and obtained from it by reversing the signs of all thespatial components of the first such four-vector. For light-like 0-brane reduction thisis important for reasons having to do with dynamics.11hus, even for a four-vector, denoted by (cid:96) ( ) µ , that satisfies (cid:96) ( ) · (cid:96) ( ) = 0, thereexists a second four-vector, denoted by (cid:96) ( ) µ , constructed as described immediatelyabove. These two satisfy the conditions (cid:96) ( ) µ η µ ν (cid:96) ( ) ν = 0 , (cid:96) ( ) µ η µ ν (cid:96) ( ) ν = 0 , (cid:96) ( ) µ η µ ν (cid:96) ( ) ν = − , (5.3)using the parametrization provided by (2.4).Under light-like 0-brane reduction we use ∂ µ = (cid:96) ( ) µ ∂ + (cid:96) ( ) µ ∂ where ∂ and ∂ are independent derivative operators. Since they are independent, this is no longer a0-brane reduction because one is actually reducing to a 1-brane where the two vectorfields ∂ and ∂ (in the mathematical sense) generate motion on a world-sheet. Inthis circumstance the algebra of SUSY generators take the form { Q a , Q b } = i γ · (cid:96) ) a b ∂ + i γ · (cid:96) ) a b ∂ . (5.4)The form of the SUSY variations in (3.6) do not exhibit any pathological behaviorfor any values of the parameters α , β , and γ . The Lagrangian is a very different storyin this regard and picks out a special value in the parameter space that requires amore careful analysis.As long as (cid:96) µ (cid:96) µ (cid:54) = 0 (or alternately α (cid:54) = π /4 or α (cid:54) = 3 π /4), the Lagrangian in (3.5)shows that by re-scaling the bosons and possibly reversing the sign of the τ -derivativeall such reduced action can be brought to the same form. However, if (cid:96) µ (cid:96) µ = 0, thenthe form of (3.5) shows there is a complication — the A and B terms are absent fromthe Lagrangian and the off-shell equality in the number of bosonic versus fermionicdegrees of freedom is lost.In the case of a light-like (cid:96) -parameter, the reduction must be done to a 1-brane(not a 0-brane) or equivalently to a world-sheet. But it should be clear that it isthe requirement of being able to write an appropriate Lagrangian, i.e. the dynamics,that forces these changes in analysis.For these we note ∂ µ = (cid:96) ( ) µ ∂ + (cid:96) ( ) µ ∂ so the Lagrangian we then find is L = ( ∂ A ) ( ∂ A ) + ( ∂ B ) ( ∂ B )+ i ( γ · (cid:96) ( ) ) bc ψ b ∂ ψ c + i ( γ · (cid:96) ( ) ) bc ψ b ∂ ψ c + F + G (5.5)and the SUSY transformation laws in component form realized via the supercovariant12erivative D a operator acting on the fields now becomeD a A = ψ a , D a B = i ( γ ) ab ψ b , D a ψ b = i ( γ · (cid:96) ( ) ) ab ∂ A + i ( γ · (cid:96) ( ) ) ab ∂ A − ( γ γ · (cid:96) ( ) ) ab ∂ B − ( γ γ · (cid:96) ( ) ) ab ∂ B − iC ab F + ( γ ) ab G , D a F = ( γ · (cid:96) ( ) ) ab ∂ ψ b + ( γ · (cid:96) ( ) ) ab ∂ ψ b , D a G = i ( γ γ · (cid:96) ( ) ) ab ∂ ψ b + i ( γ γ · (cid:96) ( ) ) ab ∂ ψ b . (5.6)So one can see the distinctive features that light-like 0-brane reduction possesses incomparison to the other cases. In fact, light-like 0-brane reduction is inconsistentwith maintaining the off-shell equality of bosons and fermions. Though we have investigated the 0-brane reduction only of the minimal off-shell4D, N = 1 supermultiplets, we expect our results to hold more generally. One of themost fascinating revelations of this work, is how a purely time-like 0-brane reductionis distinguished from all others. In the following, we give a detailed discussion of thispoint.The results for the various on-axis reductions of the chiral supermultiplet (4.1 -4.4), vector supermultiplet (4.5 - 4.8), and tensor supermultiplet (4.9 - 4.12), canall be substituted into equations of section 2 to show that these latter equations arevalid for each of the supermultiplets, independent of which set of L-matrices andR-matrices are used for the various supermultiplets.It is useful to reflect on analogous results for the group SU(3). The standardGell-Mann matrices λ A are known to satisfy a commutator algebra[ λ A , λ B ] = i f A BC λ C (6.1)for a very well-known set of structure constants f A BC . Upon taking the complexconjugate of this equation, one learns that − ( λ A ) ∗ satisfies the same equation. Thequantities λ A are the SU(3) generators acting on the quark states while − ( λ A ) ∗ are the SU(3) generators acting on the anti-quark states. This is similar to fact thethe distinct sets of L-matrices and R-matrices satisfy the same algebra.13e know from the results in [5] that for temporal reduction, the Adinkra/ γ -matrixHolographic Equation (2.14) holds. The discussion surrounding (2.22) and (2.23)shows this not the case if α (cid:54) = 0. As well, from [7], using temporal reductions, we havefound a possible realization of the Hodge star operator of 4D, N = 1 supermultipletswhich appears to have a ‘shadow’ realization acting on 1D, N = 4 adinkras. Thislast point is critical for the definition of a class structure to exist that allows a purelyone dimensional distinction between adinkras associated with the chiral, vector, andtensor supermultiplets, independent of any reference to higher dimensions. It is notat all clear whether these results can be extended to reductions that possess valuesof the angle α (cid:54) = 0.The 0-brane reduction along the temporal direction for supermultiplets and adinkrasappears to be distinguished as being different from other directions. “There is no difference between time and of the three di-mensions of space except that our consciousness movesalong with it.” - The Time Travellerfrom H. G. Wells’ The Time Machine
Acknowledgements
This work was partially supported by the National Science Foundation grantsPHY-0652983 and PHY-0354401. This research was also supported in part by theendowment of the John S. Toll Professorship and the University of Maryland Centerfor String & Particle Theory. We also thank T. H¨ubsch for the alternative descriptionsuggested below and F. Toppan for pointing out an incorrect reference in a previousversion of the paper.
Added Note In Proof
T. H¨ubsch has suggested an alternative prescription for 0-brane projection thatwhen utilized modifies our results. In the discussion presented in section 3, the 0-brane projection for the superspace covariant derivative takes the formD a → D I . (6.2)The alternate suggestion is to use a projection of the formD a → [ F ( (cid:96) )] I J D J , (6.3)14here the factor [ F ( (cid:96) )] I J has the form of a particular Lorentz transformation. If (cid:96) is in the forward lightcone, then this Lorentz transformation may be chosen so as totransform (cid:96) µ to the purely time-like axis. As there exists no Lorentz transformationthat can transform a light-like (cid:96) µ to a time-like (cid:96) µ , it is clear that light-like 0-branereduction would remain distinct. Similarly, there is also no Lorentz transformationthat can transform a space-like (cid:96) µ to a time-like (cid:96) µ , so space-like 0-brane reductionwould remain distinct as well.Under this alternate prescription for 0-brane reduction, the only effectively distinctcases correspond to the angular parameter α in (2.4) being restricted to the valuesof 0, π/ π/
2, 3 π/
4, or π . The first and last of these are time-like, the second andfourth are light-like, and the third is space-like.The result of (2.14) can only occur for time-like 0-brane reduction. That ‘bridge’between four-color adinkras and the γ -matrices of SO(1, 3) only exists for time-like0-brane reductions. So all time-like 0-brane reductions are equivalent to a purelytemporal reduction and this case remains distinct from the other cases. A Defining A Compact Matrix Notation
In order to present our result most compactly, we have used a notation thatwe initiated in the work of [7]. In this work, we showed that since the L-matricesand R-matrices associated with purely time-light 0-brane reduction may be thoughtof as being constructed by multiplying a set of unimodal matrices of order two byelements of S , the permutation group of four elements, one of the standard notationsfor permutations may be adapted to our discussions.This notation is simply a nice way to represent our matrices, most easily definedthrough example: − −
11 0 0 00 0 − ≡ (cid:0) (cid:1) . (A.1)In the expression ( ijkl ), i represents the column in which the non-zero entry of thefirst row sits; j represents the same but for the second row, and so on. A bar overthe index signifies that the element in that spot should be − (cid:0) (cid:1) t to refer to the transpose of the matrix in (A.1).15 eferences [1] S. J. Gates Jr., and L. Rana, “Ultra-Multiplets: A New Representation of Rigid2D, N = 8 Supersymmetry,” Phys. Lett. B342 (1995) 132-137; S. J. Gates andL. Rana, “A Theory of Spinning Particles for Large N-extended Supersymmetry(I),” Phys. Lett.
B352 (1995) 50, arXiv [hep-th:9504025]; S. J. Gates, Jr., andL. Rana, “A Theory of Spinning Particles for Large N-extended Supersymmetry(II),” ibid. Phys. Lett.
B369 (1996) 262, arXiv [hep-th:9510151]; S. J. Gates,Jr., and L. Rana, “Tuning the RADIO to the off-shell 2-D Fayet hypermultipletproblem,” arXiv [hep-th:9602072].[2] S. J. Gates, Jr., W. D. Linch, III, J. Phillips and L. Rana, Grav. Cosmol. (2002) 96, arXiv [hep-th/0109109]; S. J. Gates, Jr., W. D. Linch, III, J. Phillips ,“When Superspace Is Not Enough,” Univ. of Md Preprint D71 (2005) 065002, [hep-th/0408004v1].[4] C. Doran, K. Iga, G. Landweber, and S. Mendez-Diez, “Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras,” private communication.[5] S. J. Gates, Jr., “The Search for Elementarity Among Off-Shell SUSY Repre-sentations,” Korean Institute for Advanced Study (KIAS) Newsletter, Vol. 51012, p. 19; S. J. Gates, Jr., T. H¨ubsch, and K. Stiffler, “Adinkras and SUSYHolography,” Aug 2012, 12 pp. PP-012-019, e-Print: arXiv:1208.5999 [hep-th].[6] S.J. Gates, Jr. ˙J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V.G.J.Rodgers and L. Wassink, “4D, N = 1 Supersymmetry Genomics (I),” JHEP , 008 (2009), e-Print: arXiv:0902.3830 [hep-th].[7] I. Chappell, II, S. J. Gates, Jr, and T. H¨ubsch, “Adinkra (In)Equivalence FromCoxeter Group Representations: A Case Study,” Oct 2012. 23 pp. UMD-PP-012-014, e-Print: arXiv:1210.0478 [hep-th].[8] Z. Kuznetsova, M. Rojas, and F. Toppan, “Classification of irreps and invari-ants of the N-extended Supersymmetric Quantum Mechanics,” JHEP0603