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Register a new userA Note On Exemplary Off-Shell Constructions Of 4D, ${mathbf {cal N}}$ = 2 Supersymmetry Representations
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We continue the search for rules that govern when off-shell 4D, $cal N$ = 1 supermultiplets can be combined to form off-shell 4D, $cal N$ = 2 supermultiplets. We study the ${mathbb S}_8$ permutations and Height Yielding Matrix Numbers (HYMN) embedded within the adinkras that correspond to these putative 4D, $cal N$ = 2 supermultiplets off-shell supermultiplets. Even though the HYMN definition was designed to distinguish between the raising and lowering of nodes in one dimensional valises supermultiplets, they are shown to accurately select out which combinations of off-shell 4D, $cal N$ = 1 supermultiplets correspond to off-shell 4D, $cal N$ = 2 supermultiplets. Only the combinations of the chiral + vector and chiral + tensor are found to have valises in the same class. This is consistent with the well known structure of 4D, $cal N$ = 2 supermultiplets.
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We investigate the representation theory of some recently constructed N=2 super W-algebras with two generators. Except for the central charges in the unitary minimal series of the N=2 super Virasoro algebra we find no new rational models. However, from our results it is possible to arrange all known N=2 super W-algebras with two generators and vanishing self-coupling constant into four classes. For the algebras existing for c >= 3 which can be understood by the spectral flow of the N=2 super Virasoro algebra we find that the representations have quantized U(1) charge.
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