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

224 P. van Nieuwenhuizen, Supergravitvtransformations in flat space8B,.—~iy,.A, SA~cr~F , oFo”~F,.~ (2)into a total derivative, S2°=3,.[~F”~ëy,Jt+-Lky”cr . F ]. To show this, use the Bianchi identity~VPO8,F =0. The Noether current is thus given by —~o~F-y”A. Hence, we begin by adding theNoether term= ~ ~ Fy”A. (3)This ensures that the order ic°terms in 8(10 +1N) cancel,2)iy”i/f,.even in 1°and in curved fromspace.SB,. and SA in JN~Considerfirst Tothe order terms K, one of the has terms form ,cF2 coming . The fromMaxwell Se”,. = (K/action yields (K/2Xiy”l/i~)T,.~(B) while the Noethercoupling yields (K/2)l/i,.cr Fy”o- . F . The sum is equal to~uI \Iz~ _..i L I~a$4 kV~’#’Y5Y~ ~a rn4g,.~alSlSince, however, the second factor is identically zero, these variations cancel. (This identity can beproved by using that ~ antisymmetrized in (va/pcr) vanishes.)To ordcr K there are also K l/iAOA terms in 8(1°+ 1N)~From the Dirac action one finds5J1/2 = —e ~ ~ i/i(AØ°A)+~ ( y”/ia)Oty”1-T),.°A) ~ (Ay~cr”A)(2ëybl/i,.,,— 7,.l/iab). (5)The first term is clearly cancelled if one adds to Sit a new term SA = —(K/2)Ey . i/iA since this newvariation produces in 211(2 precisely the opposite. Clearly, quite generally terms in the v~zriedactionproportional to a field equation can always be cancelled by adding an extra term to the transformation lawof the field whose field equation appears. Using this observation,m is proportional we can to cancel the the gravitino last terms field in equation. eq. (5),since A Aymcr~I~A= second general ~ ~““Ay5y,~A mechanism and which *hI~z&yai/1bc= one uses 2y5R over and over again in the Noether method is theobservation that terms in the varied action with 3 can be cancelled by adding an extra term to the actionobtained by replacing 3,. by —i/i,., provided all gravitinos appear symmetrically in the end result. Forexample, the variation of B,. in the Noether coupling in eq. (3) yields~ ycr”~A8~ (ëy,3A) — ~(i~”y”A~t9a (ëy,.A) — 0,. (ëyait)]. (6)Partially integrating the first two terms, they are proportional to R” since Iy~ — 3 . i/i = ~ R and/i/i,. — 3,.y• i/i = ~y~3,.R”so that they are cancelled by an extra Si/i,.. But the last term is equal to~- (ii”y~AXa,.e)yaA + ~ (~“y”A)(ëy~3,.A) (7)and one clearly sees that the 3 terms can be cancelled by adding to the action ~(K 2I4Xl/i,.y,,AXi/i”y”A)since both gravitinos yield a 3,, variation and appear symmetrically.