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

P. van Nieuwenhuizen, Supergravity 211problems to understand it. If the spin connection (or any field for that matter!) satisfies O1/0w,.~~0,then in the variation of the action one has according to the chain ruleOI(e, t/i, w(e, ç!i))= 8e~I + 51~!L1 +~, (Sw(e~~) ö~+t5~~(e, t/’~Se). (4)Se ~ e.w(e,~P) e.çi~ Si/i SeSince, however, w,.a~(e,t/i) satisfies its own field equation, one can drop the last term due to 01/Sw 0identically after inserting w = w(e, i//). Thus, one can replace in the variation of the action thecomplicated expression which results when one applies the chain rule to (o(e, i/i) by zero! In otherwords, one only needs to vary the explicit tetrad fields and gravitino fields, but, although w(e, t/’) iscertainly not invariant, one may put Ow = 0 in the action. This is the same result as found in subsection2 by gauging the super Poincaré algebra, where we found that Sw =0. Hence, it is the 1.5 orderformalism that makes contact with group theory.It should be noted that in the extended supergravities, this observation has been crucial in obtainingthe actions. When one deals not with actions, but, for example, with gauge algebras, then one still needsthe non-zero law 8w(e, i/i) in eq. (1).One small technical detail 1.l/i,..These in our notation. two differ When sincewe D,.e~is write ~I’abwe non-zero. mean ea e~(D,.ç(i~— Dpi/i,.) and notDaili~— The result D~çfr~with in (2) il,~= will e~ be needed when we discuss constrained Hamiltonian systems, hence we brieflydiscuss its derivations. Varying the spin connections in the Einstein action, one finds (see subsection 4,eq. (3))~ mn~pp~,o- \ira r\I!sUW~ I~ mnrs jy....,.e ,,,~2e,,Varying St/i,. = (1/K)D,J in the Rarita—Schwinger action and partially integrating, one picks up a term~ .P1~u(D,.empXiysymDpI/i,,). (6)The sum of both terms must vanish, and one solves Ow,.,,,, from501pm,, — ~ = E757m1/ipg,S00pab = abCdS(0p (7)in the same way as one solves for the Christoffel symbol..1.7. Explicit proofof gauge invarianceWe now present an explicit proof of the gauge invariance of the gauge action. Using the 1.5 orderformalism, we must vary the fields indicated by arrows. From now on we will put K = 1,2’ = ~ — ~ ~“ii,.y5y,,Dp(w)ili,,. (1)1~1~1’ t1~ 1’As we shall see, the variation of the Hilbert action yields as always the Einstein tensor times 8e”,.. Thevariation of the gravitino fields yields the commutator [Dr,D0.] ,hence a curvature times . Since there