nLab singleton representation

Idea

The anti de Sitter group $SO(2,D-1)$ has “ultra-short” unitary representations that admit no Minkowski spacetime-limit (along the Lie algebra contraction). For $D = 4$ these were first discussed by Dirac 63 and named singletons.

More generally, given a (non-compact) Lie group $G$ admitting lowest weight representation? with maximal compact subgroup of the form $H \times U(1)$ for some other compact Lie group $H$, the oscillator method is to describe unitary representation by realizing the generators of $G$ as creation and annihilation operators on some Fock space, and have them transform in the (anti-)fundamental representation of $H$. Then a singleton representation is one where the generators are realized as the creators and annihilators of a single oscillator, and there are typically two of these (usually the scalar and spinor representations, in a physical language).

A doubleton representation is one where the generators are realized as two sets of oscillators, and so forth.

In AdS/CFT

In (Fronsdal 81, Flato-Fronsdal 81, Angelopoulos-Flato-Fronsdal-Sternheimer 81) it was observed that these representations may naturally be understood as arising in free field theory on the asymptotic boundary of anti de Sitter spacetime.

Analogous statements hold true for the super anti de Sitter group (e.g. Gunyadin 89)

Based on this observation, it was conjectured in Duff 88, p. 29-30 that the singleton representation of $SO(3,2)$ is realized by the field content of the worldvolume-theory of a fundamental M2-brane stretched along the asymptotic boundary of anti de Sitter spacetime factor $AdS_{4}$ in a Freund-Rubin compactification of 11-dimensional supergravity.

This conjecture was shown to be true in … and is a pre-cursor of what is now known as the AdS-CFT correspondence (see Duff 98 for review). See also at super p-brane – As part of the AdS-CFT correspondence.

From Gunyadin 98, p. 2

The ultra-short singleton supermultiplet sits at the bottom of this infinite tower of Kaluza-Klein modes and decouple from the spectrum as local gauge degrees of freedom [25]. However , even though the singleton supermultiplet decouples from the spectrum as local gauge modes, one can generate the entire spectrum of 11-dimensional supergravity over $S^7$ by tensoring the singleton supermultiplets repeatedly and restricting oneself to “CPT self-conjugate” vacuum supermultiplets.

References

• Paul Dirac, A remarkable representation of the 3+2 de Sitter group, Journal of Mathematical Physics, 4(7), 901-909, 1963 (doi:10.1063/1.1704016)

• Christian Fronsdal, Dirac supermultiplet, Phys. Rev. D 26 (1982) 1988 [doi:10.1103/PhysRevD.26.1988]

• Moshé Flato, Christian Fronsdal, Quantum field theory of singletons. The rac, J. Math. Phys. 22 (1981) 1100 [doi:10.1063/1.524993]

• E. Angelopoulos, Moshe Flato, Christian Fronsdal, Daniel Sternheimer, Massless Particles, Conformal Group, and De Sitter Universe, Phys. Rev. D23 (1981) 1278

• Sergio Ferrara, Christian Fronsdal, Conformal Maxwell theory as a singleton field theory on $AdS_5$, IIB three-branes and duality, Class.Quant.Grav.15:2153-2164, 1998 (arXiv:hep-th/9712239)

• Murat Günaydin, Singleton And Doubleton Supermultiplets Of Space-time Supergroups And Infinite Spin Superalgebras, 1989 (spire:282501)

• Mike Duff, Supermembranes: The First Fifteen Weeks, Class.Quant.Grav. 5 (1988) 189 (spire:248034)

• Eric Bergshoeff, Mike Duff, Christopher Pope and Ergin Sezgin, Supersymmetric supermembrane vacua and singletons, Phys. Lett. B199, 69 (1988)

• Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, The $Osp(8|4)$ singleton action from the supermembrane, Nucl.Phys.B542:157-194, 1999 (arXiv:hep-th/9807115)

• Paolo Pasti, Dmitri Sorokin, Mario Tonin, Branes in Super-AdS Backgrounds and Superconformal Theories (arXiv:hep-th/9912076)

• Mike Duff, Anti-de Sitter space, branes, singletons, superconformal field theories and all that, Int. J. Mod. Phys. A 14 (1999) 815-844, [arXiv:hep-th/9808100]

• Murat Günaydin, Unitary Supermultiplets of $OSp(1 \vert32,\mathbb{R}$) and M-theory, Nucl. Phys. B 528 (1998) 432-450 [arXiv:hep-th/9803138]

• Henning Samtleben, Ergin Sezgin: Singletons in supersymmetric field theories and in supergravity [arXiv:2409.03000]