(N,p,q) Harmonic Superspace
G. G. Hartwell, P. S. Howe
TL;DR
This work unifies and extends harmonic superspace by introducing the $(N,p,q)$ family in four dimensions, embedding previous constructions as special cases and situating them within complex flag supermanifolds and CR structures. It develops the machinery of G-analytic and CR-analytic fields, double fibrations, and harmonic integration to formulate massless multiplets, super Yang–Mills, and conformal supergravity in a single geometric framework. A key result is the equivalence between CR-analytic harmonic fields and on-shell massless superfields, and the demonstration that harmonic integrals reproduce superactions, including a manifestly supersymmetric form of the linearised $N=8$ three-loop counterterm via $\int d\mu\, W^4$ in the $(8,4,4)$ case. The framework highlights how conformal constraints in higher-$N$ theories can be captured as curvature/torsion conditions on CR-harmonic superspaces, with implications for potential integrability and off-shell formulations in extended supergravity.
Abstract
A family of harmonic superspaces associated with four-dimensional spacetime is described. Some applications to supersymmetric field theories, including supergravity, are given.