Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions

TL;DR

This paper constructs the minimal unitary representation of SU(2,2) and its deformations, linking them to massless conformal fields and their supersymmetric extensions, including supermultiplets relevant to N=4 Yang-Mills theory.

Contribution

It introduces a one-parameter family of deformations of the minimal unitary representation of SU(2,2) and develops their supersymmetric extensions to SU(2,2|N), connecting to massless conformal fields and supermultiplets.

Findings

Deformation parameter e yields massless conformal fields with specific spins.

Constructed supersymmetric extensions for SU(2,2|N), including the N=4 case.

Identified supermultiplets corresponding to higher spin massless fields.

Abstract

We study the minimal unitary representation (minrep) of SO(4,2) over an Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of SO(4,2), which coincides with the minrep of SU(2,2) similarly constructed, corresponds to a massless conformal scalar in four spacetime dimensions. There exists a one-parameter family of deformations of the minrep of SU(2,2). For positive (negative) integer values of the deformation parameter \zeta one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0,\zeta/2) ((-\zeta/2,0)) representation of the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the massless N=4 Yang-Mills supermultiplet in four dimensions. For each given non-zero integer value of \zeta, one obtains a unique supermultiplet of massless conformal fields of higher spin. For SU(2,2|4) these supermultiplets are simply the doubleton supermultiplets studied in arXiv:hep-th/9806042.