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

234 P. van Nieuwenhuizen, Supergravily1.14. Higher spin theoryOnly for N> 8 do the extended supergravity theories have a SU 3 x SU2 x th internal symmetrygroup. This is obvious since 0(9) contains 0(6) x 0(3) and 0(6) is isomorphic to SU(4) which containsSU3 x U,. However, for N> 8 one has also spin 5/2 fields and more than one graviton. The questionarises whether one can repeat history and construct consistent field theories for spin 5/2 and higher.Free field actions for any higher spin do indeed exist (for massive fields, see Hagen and Singh; formassless fields see Fang and Fronsdal [258,259]).We consider as spin 5/2 field a symmetric tensor spinor i/,~,,,= i/ic,.. Coupling it to an external sourceJ”,.,,, one can ask whether an action exists which leads to a propagator without ghosts or tachyons. (Thisis equivalent to requiring positive energy of the classical theory, and easier to prove.) The result is thatthere is an action, given for massless Majorana spinors by2’= —~i).j,,,1i/i,.,,— i/i,.,,yJy~ifr~,. + 2i/J,.,,y,,3Al/JA,.+~i/I~.Il/J,.,. i/IAA3,.7,,i/i,L,, + LJl~ (1)It is invariant under the following local gauge transformationsSi/i,.,, = 8,. ,,+ 8,,e,. with y” ,.= 0, (2)as first found by J. Schwinger.In fact, this action is unique as an analysis based on spin projection operators shows. The fieldequation can be written in a Christoffel-like way as~I_~i I —~ I —~ I —7 ~V,.I/)a$ OaI,V~/3 U$I~,.a —Since the free field action is invariant under local spin 3/2 gauge transformations, one might repeatthe analysis of supergravity and see whether minimal coupling to gravity by means of 3,. —* D,. leads to avariation of the action proportional only to the Einstein tensor. In that case one can add the Einsteinaction and define the tetrad variation such that the total system is invariant to lowest order (afterwardsone should then follow the Noether procedure). However, a simple calculation [153]shows that under= D,. ,,+ D,, ,. with y =0 one finds4i,.YA1PAb +SI = G,.b[~i,,y”l/i”~— ?‘y”i/i~+4j,,YSYcrl/i,.p t~P~~~ëA7vlfrnAeIJ~]+ Ep7alIJAbC — ~ p757al/JAb ”~’°’Cpgab. (4)The last two terms are proportional to the Weyl tensor and seem to exclude a consistent couplingbetween spin 2 and spin 5/2.Approaching the problem from the matter end, no matter systems have been found which areinvariant under transformations with constant parameters ,. satisfying y = 0 [153] (for such systemsone could couple i/i,.,, to the Noether current and iterate), while also taking for spin 5/2 a vierbein spinor(nonsymmetric i/i~,,.)did not improve matters [144].The massive spin 5/2 system contains a mass term for i/i,.,, plus an extra spin 1/2 field x and is, again,unique. Quite generally, for any higher spins, these extra auxiliary fields x couple only in the mass term