On Harmonic Superspaces and Superconformal Fields in Four Dimensions

TL;DR

This work develops a coherent framework to realize 4D superconformal representations through harmonic superspace and parabolic induction, revealing that short multiplets admit multiple, analytically efficient descriptions across different superspaces. It shows how massless on-shell multiplets serve as fundamental building blocks to generate all short representations via tensor products, while carefully tracking their dilation and R-symmetry weights. The paper also provides explicit transformation rules that preserve analyticity and demonstrates how N=4 composites reproduce a broad class of protected operators with AdS/CFT relevance, including multi-trace and KK-like states. Overall, it unifies geometric, algebraic, and field-theoretic perspectives on short superconformal representations in four dimensions and clarifies their realization in both complex and real harmonic superspaces.

Abstract

Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is shown how various short representations can be obtained by parabolic induction. It is also shown that such short multiplets may admit several descriptions as superfields on different superspaces. In particular, this is the case for on-shell massless superfields. This allows a description of short representations as explicit products of fundamental fields. Superconformal transformations of analytic fields in real harmonic superspaces are given explicitly.