Superconformal interpretation of BPS states in AdS geometries

TL;DR

Ferrara and Sokatchev develop a unified, cross-dimensional framework for classifying and realizing unitary short representations of the superconformal algebras $SU(2,2/N)$, $OSp(8^*/2N)$, and $OSp(8/4,\mathbb{R})$ via Grassmann analyticity and harmonic superspace. By constructing UIRs as products of massless supersingletons and imposing G-analyticity and harmonic-analyticity constraints, they enumerate BPS states (e.g., $(p,q)$, $(0,q)$, and chiral) across $d=4,6,3$, including detailed shortening patterns and explicit harmonic cosets. The approach clarifies how protected dimensions arise and how multitrace boundary operators and AdS black holes preserving fractions of supersymmetry fit into the spectrum, providing explicit realizations in AdS/CFT contexts. The results establish a systematic map of short superconformal multiplets across dimensions, with direct implications for holography, brane worldvolume theories, and the classification of BPS configurations in AdS geometries.

Abstract

We carry out a general analysis of the representations of the superconformal algebras SU(2,2/N), OSp(8/4,R) and OSp(8^*/4) and give their realization in superspace. We present a construction of their UIR's by multiplication of the different types of massless superfields ("supersingletons"). Particular attention is paid to the so-called "short multiplets". Representations undergoing shortening have "protected dimension" and correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary conformally invariant theories as well as for the classification of AdS black holes preserving different fractions of supersymmetry.