nLab
singleton representation

Context

Representation theory

Ingredients

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Definitions

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Geometric representation theory

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Theorems

Contents

Idea

The anti de Sitter group SO(2,D1)SO(2,D-1) has “ultra-short” unitary representations that admit no Minkowski spacetime-limit (along the Lie algebra contraction). For D=4D = 4 these were first discussed by Dirac 63 and named singletons.

More generally, given a (non-compact) Lie group GG admitting lowest weight representation? with maximal compact subgroup of the form H×U(1)H \times U(1) for some other compact Lie group HH, the oscillator method is to describe unitary representation by realizing the generators of GG as creation and annihilation operators on some Fock space, and have them transform in the (anti-)fundamental representation of HH. 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)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 4AdS_{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 7S^7 by tensoring the singleton supermultiplets repeatedly and restricting oneself to “CPT self-conjugate” vacuum supermultiplets.

References

Last revised on August 1, 2018 at 18:10:28. See the history of this page for a list of all contributions to it.