SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

278 P. t’an Nieuwenhuizen, SupergravityThe superconformal algebra SU(2, 2/1) will be discussed in subsection 4.1. In order to obtain theextension to SU(2, 2/N), one endows ~a and 5°with an extra index i, as in (24). If one evaluates thecommutators of 0”’ (and S°’)with G’, one finds structure constants which are a product of y~times thesymmetric matrices r’ (the antisymmetric matrices are multiplied by the unit matrix). Hence it isadvantageous to introduce chiral charges. One finds when with Q~’= ~(1+ y 5)°,3Q13, etc.[Q’L G”— ~ 3R/ [QRI LJLI G’—— J jkfQRkIc”’ ~—( ~ ~‘k J’~ i— (r) ,,, . (L~R J \Thus, QL and SR transform in the vector representation of SU(N), while OR and S1. transform in thecomplex conjugate of this representation. As explained in appendix E, we therefore have written theindices i of the (N) representation upstairs but those of the N* representation as subscripts. Moreover,0 ~ anticommutes with S ~ (and QR,i with SL.I). Since there are no generators which transform astensors under G’, this is to be expected. However, the product of (N) and (N*) contains scalars andvectors, as shown by{QL, SLJ} = ~(1+ y6~C~D— umnC~M,,,,,— iC’A) + 2(rkYJC’G~]{QRI, Sk} = ~(1—y5)[8~(~C~’D _O.mnC_ItvI,,,,, + iC’A) — 2(Tk)’JC’G~].(26)Also this is clear, since the invariant tensors 6~and (Tk)’j are the Clebsch—Gordon coefficients for thetwo irreducible representations contained in (N) x (N*). The anticommutator with right-handed charges1’ are antiHermitean.)followsOf course from complex the superconjugation. Poincaré algebra (Usingis Q* contained = —S, y~= in the—y~and superconformal that the Talgebra; therefore we havechosen the representations for Pm, Mmn and 0”’ the same. However, also the de-Sitter algebra iscontained in SU(2, 2/N) [197].To see this, take a general linear combination of all charges0’ = aQ~+/3QRI + YSLI + SSk (27)and require (0’, ‘} = ~(ymC_l)Pm6i1 + more. One finds Pm = af3Fm — ySKm. Next require that [P, 0] isagain proportional to 0. One finds only one condition, namely ay = /38. However, if one now evaluatescompletely {Q, Q} one finds that only the antisymmetric generators T~ appear on the right-hand side{Oai, 0~’}= {~ymC’’P6” — (ay + /36)O.m0C_lMmnö~i+ 4ay(rk,A)”C~’G~J°~. (28)This suggests that the maximal internal symmetry group of the super de-Sitter algebra is an SO(N)and not an SU(N) group. It also explains why the de-Sitter models are of the 0(3, 2) type and not0(4, 1): because the commutator [Pm,F,,] = kMmn has k = 4af376 > 0. The spin 1 fields of N-extendedde-Sitter supergravities gauge these 0(N) subgroups, and one has in these models always a cosmologicalconstant and a masslike term for the gravitinos. - -If one now goes from de-Sitter space to Minkovski space (i.e., [Q”,Pm] = 0) by means of a groupcontraction, the 0(N) charges can become central charges which commutewith all other generators. Thus,asexplained in the sectionon how to gauge algebras, the gauge coupling ofthe vector fieldsdisappears andone finds ~N(N — 1) Abelian vector fields. The process of contraction leads one to the super Poincaréalgebra. To this purposeone multiplies the {0, 0} relation by ~2, defines new charges0 = eO andhas then