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

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; λ) 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; λ) 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; λ) 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 4d, N = 2 sigma models coupled to supergravity and minreps of their isometry groups.

New realizations of the supergroup D(2, 1; α) in N = 4 $$ \mathcal{N}=4 $$ superconformal mechanics

“…At α = 0, −1 it reduces to the semi-direct product P SU(1, 1|2) ⋊ SU (2) and at α = − 1 2 to the supergroup OSp(4|2) 1 . The realizations of D(2, 1; α) in the models of supersymmetric mechanics were a subjects of many works (see, e.g., [7]- [12], [17]- [22], [24], [25], [28], [23] and references therein) 2 . As a rule, the realizations on one or another fixed type of the irreducible N = 4, d = 1 supermultiplet were considered.…”

Deformations of large N = (4, 4) 2D SCFT from 3D gauged supergravity

“…The holographic study of the conformal symmetry D 1 (2, 1; α) is not only useful in the context of AdS 3 /CFT 2 correspondence but also in AdS 2 /CFT 1 correspondence. This is because the symmetry D 1 (2, 1; α) also arises in superconformal quantum mechanics [18,19,20]. The isometry of AdS 2 is SO(2, 1) which is a subgroup of the AdS 3 isometry SO(2, 2) ∼ SO(2, 1) × SO(2, 1).…”