Adinkra ‘color’ confinement in exemplary off-shell constructions of 4D, N $$ \mathcal{N} $$ = 2 supersymmetry representations

Abstract

Evidence is presented in some examples that an adinkra quantum number, χ o (arXiv: 0902.3830 [hep-th]), seems to play a role with regard to off-shell 4D, N = 2 SUSY similar to the role of color in QCD. The vanishing of this adinkra quantum number appears to be a condition required for when two off-shell 4D, N = 1 supermultiplets form an off-shell 4D, N = 2 supermultiplet. We also explicitly comment on a deformation of the Lie bracket and anti-commutator operators that has been extensively and implicitly used in our work on "Garden Algebras" adinkras, and codes.

A proposal on culling & filtering a coxeter group for 4D, N $$ \mathcal{N} $$ = 1 spacetime SUSY representations: revised

“…But the work in [19,20] emphasizes that codes also play a role in understanding dimensional enhancement from 1D to 2D. In the light of the result in [17] on fermionic dimensional enhancement, it would be an interesting investigation to see how codes play a role in that result. It has been noted [27] the mathematical object shown in figure 2 is the case of p = 1 of the more general partitions given by S 4p /{Z (1) 2 ×· · ·×Z (p+1) 2 } of S 4p .…”

Examples of 4D, 𝒩 = 2 holoraumy

“…These matrices form two mutually commuting su (2) In this paper, we consider 4D, N =2 supermultiplets with 8 bosons and 8 fermions as these are the minimal representation presented among the results shown in Table 1. In the work of [46] these results were presented. The transformation laws and anti-commutator algebra are reproduced here for the convenience of the reader.…”

“…The actual derivation of all of these results follow from the forms of the 4D, N = 2 supercovariant derivative operator when evaluated on each field. The explicit expressions for this evaluation were presented in [46] and in order to streamline our presentation, we have given these in an appendix.…”

“…In the work of [46], an exhaustive study was made of the most general construction of a 4D, N = 2 supermultiplet based on pairings of the 4D, N = 1 chiral, vector, or tensor supermultiplets. As this involved the three distinct 4D, N = 1 supermultiplets, when taken in pairs, there might have existed as many as six combinations that form off-shell N = 2 supermultiplets.…”

“…By this means, we can reinterpret the condition for when two 1D, N = 4 adinkras can be combined to form a single 1D, N = 8 adinkra. In the work of [46], it was shown that this condition was determined by the calculation of the quantity χ o on the two 1D, N = 4 adinkras and demand that the some of this quantity for the two 1D, N = 4 adinkras must vanish. The discussion surrounding (8.9) and (8.10) show an equivalent condition that the chiral 1D, N = 4 adinkra can be combined with another 1D, N = 4 adinkra (R) if the quantity at the end of the equations shown in (8.10) is an element of the so(8)/so(4) coset.…”