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

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;?) in one dimension. We find that SU(2) deformations can be achieved using n pair of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2,1;?) 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;?) 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 Kähler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4. d, N=2 sigma models coupled to supergravity and minreps of their isometry groups.