RevModPhys.84.987

Vertices and string theory generically the idea is

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Unformatted text preview: vertices obtained by Bekaert, Boulanger, and Cnockaert (2006) and Boulanger, Leclercq, and Sundell (2008) (which were shown to be related to the, appropriately taken, flat-space limit of the corresponding Fradkin-Vasiliev vertices) are also the seed for the construction of consistent massive higher-spin vertices in flat and AdS spacetimes. Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 From these non-Abelian flat-space vertices, one can systematically construct massive and massless vertices in ` AdS and flat spaces by switching on mass terms a la ¨ Stuckelberg and cosmological constant terms. This approach was used with success by Zinoviev (2009a, 2009b). See also the recent work by Zinoviev (2010) where the framelike formalism for higher-spin gauge fields is used. B. Cubic vertices in AdS spacetime As mentioned in the previous section, at cubic level (i.e., at first order in perturbative deformation) Fradkin and Vasiliev found a solution to the higher-spin (gravitational, self- and cross) interaction problem by considering metric perturbations around ðAÞdS4 background (Fradkin and Vasiliev, 1987b, 1987c). This was later extended to five dimensions (Vasiliev, 2001a), N ¼ 1 supersymmetry (Alkalaev and Vasiliev, 2003) and arbitrary dimensions (Vasiliev, 2011). For a recent analysis of the Fradkin-Vasiliev mechanism in arbitrary dimension D and in the cases 2-s-s and 1-s-s, see Boulanger, Leclercq, and Sundell (2008). The Fradkin-Vasiliev construction was the starting point of dramatic progress leading recently to fully nonlinear field equations for higher-spin gauge fields in arbitrary dimension (Vasiliev, 2003). We will not detail their construction here but we simply comment that the use of a twistor variable and a Moyal-Weyl star product is central, although historically the usefulness of the star product was not immediately recognized. In a few words, the main problem with the higher-spin gravitational interaction was that, introducing the Lorentz minimal coupling terms in the action and gauge transformations, higher-spin gauge invariance could not be satisfied anymore. The solution provided by Fradkin and Vasiliev was to introduce a nonvanishing cosmological constant à and expand the metric around an (A)dS background. The gauge variation of the cubic terms coming from the Lorentz minimal coupling around (A)dS are now canceled on the free shell, by the variation of a finite tail of additional nonminimal cubic vertices, each of them proportional to the linearized Riemann tensor around (A)dS and involving more and more (A)dS-covariant derivatives compensated by appropriate negative powers of the cosmological constant. In that gauge variation, the terms proportional to the free equations of motion are absorbed through appropriate corrections to the gauge transformations. This solution is the Fradkin-Vasiliev mechanism, and we call the gravitational cubic coupling they obtained the quasiminimal coupling, in the sense that th...
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