Minimal unitary representation of D(2,1;\lambda) and its SU(2) deformations and d=1, N=4 superconformal models

TL;DR

This paper extends the minimal unitary representation framework to the N=4 superconformal algebra D(2,1;) in one dimension, exploring SU(2) deformations and their relation to superconformal models.

Contribution

It introduces SU(2) deformations of the minimal unitary representation of D(2,1;) and establishes a mapping to N=4 superconformal models in harmonic superspace.

Findings

SU(2) deformations achieved with boson and fermion pairs.

Generators commute with dual superalgebra OSp(2n*|2m).

Mapping between superconformal models and minimal supermultiplets established.

Abstract

Quantization of the geometric quasiconformal realizations of noncompact groups and supergroups leads directly to their minimal unitary representations (minreps). Using quasiconformal methods massless unitary supermultiplets of superconformal groups SU(2,2|N) and OSp(8*|2n) in four and six dimensions were constructed as minreps and their U(1) and SU(2) deformations, respectively. In this paper we extend these results to SU(2) deformations of the minrep of N=4 superconformal algebra D(2,1;\lambda) in one dimension. We find that SU(2) deformations can be achieved using n pairs of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2,1;\lambda) commute with the generators of a dual superalgebra OSp(2n*|2m) realized in terms of these bosons and fermions. We show that there exists a precise mapping between symmetry generators of N=4 superconformal models in harmonic superspace studied recently and minimal unitary supermultiplets of D(2,1;\lambda) deformed by a pair of bosons. This can be understood as a particular case of a general mapping between the spectra of quantum mechanical quaternionic K\"ahler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4d, N=2 sigma models coupled to supergravity and minreps of their isometry groups.