We study the minimal unitary representation (minrep) of SO(5,2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5,2) describes a massless conformal scalar field in fiv
e dimensions and admits a unique deformation which describes a massless conformal spinor. Scalar and spinor minreps of SO(5,2) are the 5d analogs of Diracs singletons of SO(3,2). We then construct the minimal unitary representation of the unique 5d superconformal algebra F(4) with the even subalgebra SO(5,2) X SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS_6/CFT_5 (super)-algebras. The Joseph ideal of the minrep of SO(5,2) vanishes identically as operators and hence its enveloping algebra yields the AdS_6/CFT_5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a deformed higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS_6/CFT_5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.
Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analo
g of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.
Massless conformal scalar field in d=4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2,2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and i
ts deformations were obtained by quantization of the nonlinear realization of SU(2,2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS_5. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_5 is simply the enveloping algebra of SU(2,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS_5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2,2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS_5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in AdS_4 where the 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.
Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under ce
rtain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelbergs generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality.
Candidate counterterms break Noether-Gaillard-Zumino E_{7(7)} current conservation in N=8 supergravity in four dimensions. Bossard and Nicolai proposed a scheme for deforming the subsector involving vector fields in a Lorentz covariant manner, so as
to restore duality. They argued that there must exist an extension of this deformation to the full theory that preserves supersymmetry. We show that it is not possible to deform the maximal supergravity to restore E_{7(7)} duality, while maintaining both general covariance and N=8 supersymmetry, as was proposed. Deformation of N=8 supergravity requires higher spins and multiple gravitons, which presents a concrete obstacle to this proposal.
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 S
U(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.
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 minre
p 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.
