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

356 P. van Nieuwenhuizen, Supergravityhigher dimension, say D + E + F, and to use local symmetries there to obtain global symmetries inD + E dimensions. For our applications we start with the N = 1 model in D + E + F = 11 dimensions,and want to obtain a global symmetry of as high a rank as possible in D + E = 5 dimensions. The localsymmetry we use is general coordinate invariance in d = 11. (In order that the method works also forsystems without fermions, we do not study whether local Lorentz invariance can be used.) As wediscussed in the last section, the expected global symmetry group is SL(6, R). Now no extra scaletransformations are present which extend this SL(6,R) to GL(6, R). (Note that E 6 does not containGL(6, R) but that E(7) contains GL(7, R.) Basically the reason is that there are two types2(F~)2 of termsandintheE~ abcdefF,,vFpgArt.action which cannot Thus be the expected made scale-invariant global symmetry at isthe SL(6, same R). We time, refer namely to the previous 3” section.The Lorentz group S0(D + E + F — 1, 1) splits up into SO(D + E — 1, 1) x S0(F) if we fix part of thislocal symmetry by casting the vielbein in triangular form. This we discussed in the previous section.Then, naively we expect for the N = 8 model in d = 5 dimensions as internal symmetry group SO(6)local® SL(6, R) global.However, things are more interesting. The particle spectrum in d = 5 consists of irreduciblerepresentations of Sp(8) rather than of SO(6). This particle spectrum is: one tetrad e~(m,ji = 1, 5), eightgravitinos ~i~(a = 1, 8), 27 vector fields, 48 spin ~fields and 42 scalar fields. Indeed, all fields can berepresented by completely antisymmetric tensors which are traceless with respect to the symplecticmetric. For example, the photons are described by the 27-dimensional Sp(8) representation A,,~”9=—At” with ~ = 0 (a, j3 = 1, 8) which is not an irreducible S0(6) representation.This particle spectrum is obtained from the fields enM, t/.~,AAfII(M, A, [1, ~ = 1, 11 and B = 1, 32)by ordinary dimensional reduction as follows:-+ e~,6 vectors e~,~(6x 7) = 21 scalars e~—*8t/i~and 48 spin ~fields tfr~.(i = 1, 8; a = 5, 11) 3—~ sca ar ~ vec ors ,~,, vec ors ~20 scalars A~,3,,.The field A,,,~,,has as field equation DMF,LP,,., = 0 and since F,,~ = 8[,,A~,,,,1= ~ it follows that= 3,.~’and A,,,,,,, describe vectors (since O[p~AvpIa = F,,,~pa= ~ one has ~‘p~= O~”pa~. Insertingthis into F yields a Maxwell action). As a check one may verify that, as in d = 11, there are 128 bosonicand 128 fermionic states (in d = 5 the little group of a massless particle is SO(3) and hence a particle ofspin J has 2J+ 1 polarizations).Thus one expects that one can write the fields as completely antisymmetric and symplectic tracelesstensors e~,~, Afl”, x”, 4)abcd where 1lai,A~= 0 and I? = 14 X ir2 is the antisymmetric 8 x 8 metric ofSp(8).The reality conditions of the bosonic fields are (A,,~~!~)* = A,,ab and idem for 4)abcd where indices areraised and lowered by 11. Note that under Sp(8) a vector va transforms as ov” = A bV” and that= 0. Hence, OVa = — Vi,A”a where V,, = Vaflab. Only if (V~)*transforms as V0 can weimpose the reality conditions mentioned above. Now O( V°)* = (A ab)*( V~~)*. Hence one must requirethat (A ab)* = ~j b In other words, in addition to being symplectic, the matrices must be antihermitian. Thelocal symmetry group is Usp(8), not T[2 Sp(8). + tiM However,= 0. Multiplying they have bythe Ii same one has generators. also tiMT + Mfl = 0 sinceSymplectic = —1 and taking matrices the satisfy real andMimaginary parts of the sum and differences of these equations, one seesthat Usp(N, C) has as many generators as Sp(N, R). In fact, for N = 8, Usp(N, C) = Sp(P’~,R).