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

4cr”‘’. To derive these results, use for example that 7,, ”””7s)’~= —2o,.~y~ ””~ = Also use— ~gvp~.j. ~Pav .j.. .~°~“P— (P. van Nieuwenhuizen, Supergravity 223,,,,E U~,, Ua$y Ua$y ~where S~= S~O”~S.It follows that on-shell GAA =0, but one should not conclude from this result that the gravitinoenergy momentum tensor is traceless. There are extra gravitino terms in G,.~which should beconsidered as part of the gravitino stress tensor, and their trace does not vanish. Thus, in second-orderformalism the action for the gravitino (including the torsion terms from the Hubert and Rarita—Schwinger actions) is not locally scale invariant on-shell. (We recall that a matter action 2(e, 4’) for amatter field 4’ varies into TAA, under Se”,. = Ae”,. and 84’ = A”çb with some power a, provided one ison-shell where the 4’ field equation is satisfied, so that 82/54’ = 0. See Weinberg’s book.)Sometimes one considers the gravitino as a fermion in an external gravitational field. The curl— D,~i/i~contains then only the connection w,.mn(e), not w,.m”(e, i/i). In this case the action isinvariant under Si/c,. = D,.(w(e)) provided the external gravitational field satisfies the Einstein equations(we leave the proof as an exercise). Also this action is not locally scale invariant, now becausealthough the gravitino field equation again reads R” = 0 (with w(e)), we can no longer use 1.5 orderformalism and there is an extra contribution to the stress tensor coming from varying 00(e) with respectto the tetrad field. However, in first-order formalism, the action —~ ““’°‘t/i,.yOw,.tm” = 0. This is obvious 5y~(30 since if the is2i/i,.,locally derivative scale 3 hits invariant A, theunder rest (4~’,.y t5e”,. = Ae”~and Si/i,. = A”+~w,,. cr)t/i,,5’y4,,) ””””vanishes due to the Majorana character of the gravitino.Again, one cannot say that the matter action is invariant, since the terms containing w coming from theHilbert action are not invariant.1.11. Matter coupling: the supersymmetric Maxwell—Einstein systemAfter having discussed the gauge action of simple (N = 1) supergravity, we now turn to the secondstage: matter coupling. As in any gauge theory, we start with a globally supersymmetric action, andcouple it to the gauge action by introducing couplings to the gauge fields. For ordinary gauge theoriesthe prescription is rather simple: replace ordinary derivatives by minimally covariant derivatives(3,. — ieA,. in electromagnetism). In supersymmetry this is not so simple and the analogous solution isbest obtained by the tensor calculus which we will discuss later. Here we will start with the Noethermethod since it is conceptually simpler (though algebraically more tedious).We will begin with the coupling of the spin (1, 1/2) photon—neutrino system to the gauge action ofsupergravity. Historically this was the first matter coupling [204].Another coupling exists betweenphotons and gravitons, namely the coupling of the spin (3/2, 1) systems to the gauge action (so-calledN = 2 extended supergravity) [203].This latter coupling comes nearer to a unification of gravity andelectromagnetism, since one can connect gravitons and photons by a series ofsymmetry operations. In themodel we are going to discuss, this is not possible. Forconvenience weconsider an abelian vectorfield, butall results easily generalize to the Yang—Mills case [202,237].The photon—neutrino Lagrangian density in curved space readscpO__1~’r ,.pvcr_~Av1oA E’ —~D_.~D— 4ez ,.~ ,.,,,g g 2 ‘‘ — ~ U,J.J,.where D~A= o,.A + ~w,.ab(e)cr~~ contains not yet i/i-torsion. 2°varies under the global supersymmetry