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

222 P. van Nieuwenhuizen, SupergravitvSPAmO (2)= eG”” — lIAyyai/A~ — ~ e”’(S2 + P2 — A~,,) (3)= ~“~(y5y~D,.,i/i,, + h’syaili,,(Dpe”,,)). (4)The latter term is obtained by partially integrating i/i,.y5’y~D~Oifr,, = (D~8~,,)y5y4,. and vanishes (seeappendix C). Thus the field equations reduce toSPAmO, R~as ~”’~ysyvDpçbcr0 (5)eG”” — ~4’AySya~/iAv = ~, ~ = ~ — ~ (6)We now discuss the question of consistency [120].Since R~= 0 has open indices, one can once moredifferentiate. To show what might go wrong, consider first the case of a complex spin 3/2 field coupledminimally to electromagnetism. The field equation ~““’y~y~(3~— ieA~)i/i,,= 0 yields upon differentiationwith 3,. — ieA,. that F~”y,.i/i~=0 which puts as extra constraint for consistency that either i/i~=0or that the photon be a gauge excitation. In supergravity, no such problems arise. FromM = ‘~“ ys[ yvD,,Dpi/1,, + (yaDpi/i,,XD,.e°~)] (7)D,.Rand the torsion equation D,.e”,. — Due”,. ~l/I,.yal/,~ one findsDLLR~= ~ ~““’y5[y,.o~dlII,,R,.P’4 + (y0D~~,,Xi~,.y”ip~)]. (8)If one expands the product y,~o~cdas explained in appendix A and uses the cyclic identity with torsion2Ge,,y’i/i~+ ~‘/Ydt/1,, ~~”R,.P~(757 UcdlIJ,,) ~”R,.p’~=r~,.pvd~.W) — i~(f’A7d aVJ~9,lE~ i \_ IT ~ \ a$A,, (9)all terms in DIR” cancel on-shell. (Use the Einstein field equations and Fierz both undifferentiatedgravitinos together.) Hence supergravity is consistent.It is usually believed that consistency is a consequence of gauge invariance. This is not always so; forexample [125],adding to the gauge action of simple supergravity a mass term ei/I,.o””i/I~leads again to aconsistent theory although it is not locally supersymmetric. (For this one also needs a cosmologicalterm, see subsection 6.1.)Finally we give a list of different forms of the gravitino field equationR” = ““°ysy~Dpçlçr, YAIPA,. = R,. — . R, yA~A,.= —275R,.‘Il . R = 2o”~t/i,.~, t/i,.~,= D,.i/i~ — D~t/i,.7a*~,y + 7~i/Aya + )~ylf1~= ,.,~,y5R” (10)I 7 —— I — P”I~ 275~P,w — ~ , ‘~‘~‘— c” ~R,. — ~y,,y• R = ~ysy”çb,.,.— ~7”I/i,.,,.