What is meant by "singleton" representations of $SO(2, d)$ and 'small representations' in Witten's paper, Anti de Sitter Space and Holography?


The "singleton" and "doubleton" language comes from the oscillator method of finding group representations.

Given a (non-compact) group $G$ admitting lowest weight representation with maximal compact subgroup of the form $H\times\mathrm{U}(1)$ for some other compact group $H$, the oscillator method is to describe (unitary irreducible) representations 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 $H$.

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 - this obviously extends to the idea of "$n$-ton" representations.

A more thorough discussion is found for example in "Singletons, Doubletons and M-theory" by Murat Gunaydin and Djordje Minic (arXiv:hep-th/9802047v2).

  • $\begingroup$ Thank you for your response. I have a few questions. When does a non compact group not admit lowest weight representations? Why does the maximal compact subgroup have to be $H \times U(1)$ for the oscillator method to work? Also what do you mean by the anti-fundamental representation of H? Why does there have to be a U(1) for this to work? Also I know that SO groups have a fundamental representation which is the spinor representation so your comment in the definition of singleton makes sense to me. But what do you mean by two sets of oscillators. Can you give a simple example? $\endgroup$ – leastaction Apr 5 '15 at 14:56
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    $\begingroup$ @leastaction: I think that the existence of lowest weight representations actually forces the maximal torus to be of that form, but I can't find a reference or argument for it (you should probably try to obtain the paper which they refer to in the paper I linked as having developed the oscillator method if you really want to know). The anti-fundamental representation is the conjugate representation of the fundamental one. (If $a$ transforms in the fundamental, $a^\dagger$ transforms in the anti-fundamental) $\endgroup$ – ACuriousMind Apr 5 '15 at 15:02

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