Superfield representations of superconformal groups

TL;DR

The paper develops a comprehensive framework to realize all unitary irreducible representations of four-dimensional superconformal groups as (unconstrained) fields on various superspaces, notably analytic superspace, by allowing fields to transform nontrivially under supergroups. It systematically combines parabolic induction, harmonic and analytic superspace formalisms, and the oscillator method to show how every representation can be realized on some superspace, with the N=4 case realized from copies of the Maxwell multiplet on $(4,2,2)$ analytic space. A key result is that analytic superspace provides unconstrained, irreducible realizations of UIRs, and that the oscillator construction naturally corresponds to these analytic-space representations. The work unifies multiple approaches and provides explicit constructions and mappings between representations, superspaces, and oscillator realizations, with significant implications for organizing operators in superconformal field theories and AdS/CFT contexts.

Abstract

Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N=4 complex superconformal group $PSL(4|4)$ with integer dilation weight from copies of the Maxwell multiplet on $(4,2,2)$ analytic superspace. This construction is compared with the oscillator construction and it is shown that there is a natural correspondence between the oscillator construction of superconformal representations and those carried by superfields on analytic superspace.