Short representations of SU(2,2/N) and harmonic superspace analyticity
S. Ferrara, E. Sokatchev
TL;DR
This work provides a uniform harmonic superspace framework to classify massless and short unitary irreducible representations of the superconformal algebra $SU(2,2|N)$. By employing Grassmann (G) analytic and harmonic (H) analytic conditions, the authors identify $[ rac{N}{2}]$ elementary ultrashort analytic superfields whose first components transform in distinct antisymmetric $SU(N)$ representations and possess protected top spins $J_{ m top}=ig( rac{N}{2}- rac{k}{2},0ig)$. For $N=2n$, tensoring the self‑conjugate ultrashort multiplet with top spin $ig( rac{n}{2},0ig)$ yields all possible UIRs with residual shortening, organized by analytic subspaces that omit certain fermionic coordinates. The approach clarifies the construction of short and composite operators (supersingletons and their products) and has direct relevance to AdS$_5$/CFT$_4$ and the spectrum of protected operators in extended supersymmetry theories.
Abstract
We consider the harmonic superspaces associated to SU(2,2/N) superconformal algebras. For arbitrary N, we show that massless representations, other than the chiral ones, correspond to [N/2] ``elementary'' ultrashort analytic superfields whose first component is a scalar in the k antisymmetric irrep of SU(N) (k=1... [N/2]) with top spin $J_{\rm\scriptsize top}= (N/2-k/2,0)$. For N=2n we analyze UIR's obtained by tensoring the self-conjugate ultrashort multiplet $J_{\rm\scriptsize top}$= (n/2,0) and show that N-1 different basic products give rise to all possible UIR's with residual shortening.