The 1980s the quest for high spin interactions

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Unformatted text preview: cetime as background. Using the light-cone gauge approach, higher-spin s-s0 -s00 cubic vertices in four spacetime dimensions were found by Bengtsson, Bengtsson, and Brink (1983a, 1983b), Bengtsson, Bengtsson, and Linden (1987), and Fradkin and Metsaev (1991). These results, in the light-cone gauge approach, were considerably generalized later by Metsaev, 1993a, 1993c, 2006, 2007) and Fradkin and Metsaev (1995) with a complete classification of cubic (self- and cross)couplings for arbitrary massive and massless higher-spin fields, bosonic and fermionic, in dimensions four, five, and six. Mixedsymmetry fields were also considered therein. Moreover, Metsaev (1993b) obtained a wide class of cubic interactions for arbitrary fields in arbitrary dimension. ´ As far as manifestly Poincare-invariant vertices in the Lagrangian approach are concerned, Berends, Burgers, and van Dam obtained a class of manifestly covariant, nonAbelian cubic couplings (Berends, Burgers, and van Dam, 1984, 1985). They used a systematization of the Noether procedure for introducing interactions, where the couplings are not necessarily of the form ‘‘gauge field times conserved current.’’ Berends, Burgers, and van Dam (1985) obtained consistent and covariant cubic couplings of the kind s1 -s2 -s2 , for the values of s1 and s2 indicated in Table I. Of course, some of the vertices were already known before, such as, for example, in the cases 1-1-1, 2-2-2, and 2- 3 - 3 corresponding 22 to Yang-Mills, Einstein-Hilbert, and ordinary supergravity theories, respectively. There is a class of cross interactions s1 -s2 -s2 for which the cubic vertices could easily been written. This class corresponds to the ‘‘Bell-Robinson’’ line s1 ¼ 2s2 and below this line s1 > 2s2 (Berends, Burgers, and van Dam, 1986); see Deser and Yang (1990) in the s1 ¼ 4 ¼ 2s2 case and some more recent considerations by Manvelyan, Mkrtchyan, and Ruehl (2010c). In the aforementioned region s1 ! 2s2 , the gauge algebra remains Abelian at first order in a coupling constant although the gauge transformations for the spin-s2 field are deformed. The reason is that the first-order 20 As a matter of fact, a nonstandard action principle for Vasiliev’s equations, which leads to a nontrivial quantization, was proposed by Boulanger and Sundell (2011). Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 995 deformation of the free spin-s2 gauge transformations involves the spin-s2 field only through its gauge-invariant Weinberg–de Wit–Freedman field strength (Weinberg, 1965; de Wit and Freedman, 1980).21 Although they do not lead to non-Abelian gauge algebras, it is interesting that the cubic interactions on and below the Bell-Robinson line (i.e., for s1 ! 2s2 ) have the form ‘‘spin-s1 field times current’’ where the current is quadratic in (the derivatives of) the spin-s2 field strength (Berends, Burgers, and van Dam, 1986; Deser and Yang, 1990) and is conserved on the spin-s2 shel...
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