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Register a new userMinimal unitary representation of D(2,1;lambda) and its SU(2) deformations and d=1, N=4 superconformal models
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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 Kahler 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.
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We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as deformed D=4 $AdS$ supersymmetries. Subsequently we perform suitable contraction of quantum Jordanian $AdS$ superalgebra and obtain new $kappa$-deformation of D=4 Poincare superalgebra, with the bosonic sector describing the light cone $kappa$-deformation of Poincare symmetries.
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. 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 zeta, 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 arXiv:hep-th/9806042.
We present the class of deformations of simple Euclidean superalgebra, which describe the supersymmetrization of some Lie algebraic noncommutativity of D=4 Euclidean space-time. The presented deformations are generated by the supertwists. We provide new explicit formulae for a chosen twisted D=4 Euclidean Hopf superalgebra and describe the corresponding quantum covariant deformation of chiral Euclidean superspace.
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
In this paper, we study quantum group deformations of the infinite-dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite-dimensional subalgebras we classify all possible Lie bialgebra structures and for selected examples, we explicitly construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known $kappa$-Poincare Hopf algebras constructed on the finite-dimensional Poincare or (anti) de Sitter algebra. The dual $kappa$ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the $kappa$-Poincare can not be extended consistently to the infinite-dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations.
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