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

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 ζ, one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0, ζ/2)((−ζ/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 ζ, 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 hep-th/9806042.

Minimal unitary representation of 5d superconformal algebra F(4) and AdS6Fernando1, Günaydın2 2015Nuclear Physics BSelf Cite

“…In previous work we gave a construction of the minimal unitary realization of the Lie algebra of four-dimensional conformal group SU (2,2) and showed that it admits a one-parameter family of deformations [1]. The minimal unitary representation (minrep) and its deformations describe all the massless conformal fields in four dimensions, and the deformation parameter is simply the helicity which can take on continuous values.…”

“…These results were also extended to the superalgebras SU (2, 2|N), and it was shown that minimal unitary supermultiplet and its deformations describe massless N -extended superconformal multiplets in four dimensions. For PSU (2, 2|4), the minimal unitary supermultiplet is simply the N = 4 Yang-Mills supermultiplet [1].…”

Notes on higher-spin algebras: minimal representations and structure constants

“…Notice also that for the sl N series, the quantization or equivalently the minimal representation is not unique but is of one-parameter family labeled by λ. For the sl 4 case, the minimal representation has been studied also in [52].…”

“…Much progress has been made on the subject of minimal representations, see for example [40][41][42][43][44][45][46][47][48][49][50] and references therein. In physics literature, the minimal representations of isometry algebras are explored to a large extent by Gunaydin and collaborators [51][52][53][54][55].…”

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

“…We choose a particular type of (massless discrete series) positive-energy unitary irreps 10 H n known as "minireps" or doubletons [34][35][36], which have the remarkable property that they remain irreducible 11 under SO(4, 1) ⊂ SO(4, 2). They have positive discrete spectrum…”