Minimal unitary representation of 5d superconformal algebra F(4) and AdS6

Sudarshan Fernando

¹

,

Murat Günaydın

²

Abstract

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 five 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 Dirac's 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) × 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.

Exceptional F (4) higher-spin theory in AdS6 at one-loop and other tests of duality

“…More recent work has shown that Rac's, which are singletons of SO(d, 2) for odd d and scalar doubletons of SO(d, 2) for even d, are simply the minimal unitary representations of SO(d, 2). For odd d they admit a single deformation (spinor singleton), Di, and for even d they admit an infinite family of deformations (doubletons) [8,[24][25][26]. Furthermore there exists a one-to-one correspondence between the minimal unitary representations of SO(d, 2) and their deformations and massless conformal fields in d dimensional Minkowskian spacetimes [8].…”

“…The Hilbert space of the minimal unitary representation of SO(d, 2) obtained via the quasiconformal method is spanned by states in the tensor product of the Fock space of (d − 2) bosonic oscillators with the state space of the Calogero Hamiltonian or of conformal quantum mechanics [8,39,86]. The explicit expressions for minimal unitary realization (minrep) of SO(5, 2) were given in [26]. To show that the minrep of SO(5, 2) is a positive energy unitary representation that describes a massless conformal scalar field in 5d one uses the compact 3-grading of SO(5, 2) with respect to the Lie algebra of its maximal compact subgroup SO(5) × SO(2) whose covering group is Spin(5) × U(1) ≡ USp(4) × U(1):…”

“…In AdS 3,4,5,/ 6,7 one does find infinite families of anti-de Sitter superalgebras with any number of supersymmetries. We find it remarkable that there exists [26] an AdS 6 higher-spin algebra whose maximal finite-dimensional subalgebra is the exceptional Lie superalgebra F (4), which is a unique supersymmetric extension of the 5d conformal algebra SO (5,2). Even subalgebra of F (4) is SO(5, 2) ⊕ SU (2) and the corresponding HS algebra can be realized as the universal enveloping algebra of the minimal unitary supermultiplet 1 We should note that the simple exceptional Lie algebra of rank 4 is denoted as F 4 and does not contain the even subalgebra SO(7) ⊕ SU (2) of the exceptional Lie superalgebra F (4).…”

“…The simplest free conformal fields provide the basic examples of HS AdS/CFT dualities: free scalar field is dual to Type-A HS theory with spectrum made of totally-symmetric HS fields and free fermion is dual to Type-B whose spectrum contains specific mixed-symmetry fields that include totally-symmetric HS fields too. The results of [8,[24][25][26] establishing a one-to-one correspondence between higher-spin theories and supersymmetric extensions thereof and massless conformal fields and conformal supermultiplets imply that this duality extends to all unbroken higher-spin theories and their supersymmetric extensions.…”

Twisted Flato-Fronsdal theorem for higher-spin algebras

“…7 In dimensions 3, 4 and 6, the higher symmetries of super-Laplacians were studied in [56], thereby extending Eastwood's approach to the supersymmetric case. 8 See also [26,39,[58][59][60] for the quasiconformal approach to higher-spin (super)algebras.…”