On Harmonic Superspace

TL;DR

This survey presents harmonic superspace as a geometric framework for extended supersymmetry, detailing flat and curved constructions and the role of CR-analytic (and G-analytic) fields. It shows how key on-shell multiplets in low dimensions—namely $D=4$, $N=4$ Yang–Mills, $D=6$, $(2,0)$ tensor multiplets, and $D=3$, $N=8$ scalars—can be reformulated as single-component CR-analytic fields on suitably chosen harmonic superspaces, with their conformal operators arising as holomorphic sections and linked to Kaluza–Klein spectra in AdS/CFT contexts. The work connects harmonic superspace with twistor theory via a group-theoretic double fibration and extends the formalism to curved, superconformal geometries, suggesting a pathway to non-perturbative studies of correlation functions in highly supersymmetric theories. Overall, the paper emphasizes the utility of analytic superspace techniques for encoding and manipulating highly constrained multiplets in string/M-theory settings, including implications for the Maldacena conjecture.

Abstract

A short survey of some aspects of harmonic superspace is given. In particular, the $d=3, N=8$ scalar supermultiplet and the $d=6, N=(2,0)$ tensor multiplet are described as analytic superfields in appropriately defined harmonic superspaces.