Spin recently general results on the structure of

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Unformatted text preview: 00 ) non-Abelian already at this order were given, showing, in particular, that the maximum number of derivatives involved in a non-Abelian coupling is 2s0 À 1 or 2s0 À 2, depending on the parity of the sum s þ s0 þ s00 (Bekaert, Boulanger, and Leclercq, 2010). It was also shown that the cubic vertices saturating the upper derivative bound have a good chance of being extended to second order in the deformation parameter, as far as the Jacobi identity for the gauge algebra is concerned. Later on, the generic nonAbelian vertices were studied and explicitly built by Manvelyan, Mkrtchyan, and Ruehl (2010a, 2010b). Some classification results were also obtained about the structure of the Abelian cubic vertices. A posteriori, the approach (Manvelyan, Mkrtchyan, and Ruehl, 2010a, 2010b) to the writing of covariant non-Abelian vertices can be seen as the covariantization of the vertices already obtained in the lightcone approach by Bengtsson, Bengtsson, and Brink (1983a, 1983b) and Metsaev (2006, 2007) where, on top of the cubic coupling given by the light-cone gauge approach, terms are added which vanish in the spin-s De Donder gauge. With the advent of string field theory in the second half of the 1980s, the construction of higher-spin cubic vertices in flat space was carried out by Koh and Ouvry (1986), Bengtsson (1988), and Cappiello et al. (1989) in the socalled Becchi-Rouet-Stora-Tyutin (BRST) approach. This approach was indeed motivated by the BRST first quantization of the string and by the tensionless limit of open string field theory. More recently, this analysis was pursued by Bonelli (2003) and Buchbinder et al. (2006), Fotopoulos and Tsulaia (2007), Fotopoulos et al. (2007) [a review of the last three works plus other works can be found in Fotopoulos and Tsulaia (2009)]. The results obtained in this framework are encouraging, for instance, in the case of non-Abelian s-0-0 interactions (Fotopoulos et al., 2007), although the higher-spin gauge field (self- and cross) interactions found in Fotopoulos and Tsulaia (2007) are Abelian, and therefore can hardly be related to the non-Abelian higher-spin theory of Vasiliev. Before turning to the cubic interactions in AdS background, we continue with our review of positive results for higher-spin cubic vertices in flat space. Important results have recently been obtained by analyzing the tree-level amplitudes of the tensile (super)string. In what could be called a string and S-matrix approach, Polyakov (2009, 2010), Taronna (2010), and Sagnotti and Taronna (2011) obtained a plethora of vertices and recovered the vertices obtained in the previously cited approaches, thereby creating a direct link between open string theory and higher-spin gauge theory at the dynamical level. Moreover, in light of the uniqueness results of Boulanger, Leclercq, and Sundell (2008), one has a precise relation between the Fradkin-Vasiliev vertices and string theory. Generically, the idea is that the non-Abelian flat-space cubic...
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