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

Understand

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.