Harmonic Superspaces and Superconformal Fields

TL;DR

This paper develops a systematic method to construct and realize unitary irreducible representations of the four-dimensional superconformal group $SU(2,2|N)$ by embedding representations as holomorphic superfields on coset (superflag) spaces of the complexified group $SL(4|N)$. Using parabolic induction, the authors show that any representation can be realized on suitable harmonic or analytic superspaces, where short multiplets are naturally encoded on analytic spaces without extra constraints; massless multiplets and their Dynkin-label content are treated explicitly. The framework provides concrete realizations for both bosonic and supersymmetric cases, with detailed discussion of superindices in $N=2$ analytic superspace and explicit examples such as the hypermultiplet, Maxwell multiplet, and stress-energy multiplet. Key contributions include the characterization of short representations via Levi subalgebra content, the use of $(N,p,q)$ harmonic/analytic spaces to realize and tensor representations, and the explicit construction of on-shell multiplets consistent with superconformal symmetry. The results have potential implications for AdS/CFT, enabling component-free construction of correlation functions and providing a connection to oscillator-based approaches to superconformal representations.

Abstract

Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is argued that any representation can be given as a superfield on many superflag manifolds. Representations on analytic superspaces do not require constraints. We discuss short representations and how to obtain them as explicit products of fundamental fields. We also discuss superfields that transform under supergroups.