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

P. van Nieuwenhuizen, Supergravity 267result note that~ ~ = _~EkIm(Cy 5ym)a0. (18)First class constraints generate gauge transformations. For each first class constraint one chooses agauge condition. For our case we choose(4,°’~, ök4’I’~) = gauge conditions. (19)Defining A~= (wa, A~O~)as the first class constraints, gauge choices and second class constraints, thematrix C,,, = {A1, A} must be nonsingular. In that case one defines the Dirac bracket by{A, B}D = {A, B} — {A, A1}C~.0{A1,B}. (20)Clearly {A1, B}D = 0 for arbitrary B. Thus, asfar as the Dirac bracket is concerned, it does not matter thatfor Majorana fields the conjugate momenta are proportional to the fields (see (10)).Since the pair (4i°”, ff~) commutes with all other constraints, we may drop them. To evaluate theDirac bracket, one first evaluates it with respect to the 4”~,and then with respect to the other variablesA in A. Denoting the first bracket by { },, one has{A, B}D = {A, B}1 — {A, A,},({A, A},)”~’{A1,B},. (21)In this way one finds for the Dirac brackets{4,7(x, t), ‘ff~(y,t)}D = —~(F”’ 2)~~6~63(x —y) (22)where F312 is the spin 3/2 projection operator. The anticommutator for two 4, fields or momenta iTfollows from (21) and {A1, B}D = 0; in particular, the result for {4’, ‘fr}D is the same as when evaluatedusing (1). (In the gauge (19), the constraints imply y 4’ = 0 [107].For comparison one must use in (1) thepolarization tensorssatisfying y~u = lu,,, = 0.) Thus the total result is that replacingthe Poisson bracket bythe Dirac bracket replaces &b by ~(F”” 2)”b. (The factor ~is needed in order that 4,7 = {4,7, HT}D.)For path-integral quantization, one starts withZ =f d4i,, dñ~’d 3x (14M.frM4!i0 d~°ô(coa) ô(~a)ô(Oa) SdCt{~a,~b} (sdet{Oa, Ob}) exp if— ~‘T) (23)dif the gauge conditions are in convolution, {~,,,~}= 0. The product of both superdeterminants is just(sdet{A,, A1})”2 as one easily verifies, using that the Poisson bracket of p with 0 and itself vanishes.2.15. AnomaliesIn global supersymmetry, the axial and trace and supersymmetry anomalies (a . ~.4, T,,~and y Js)are part of a multiplet [299J.We discuss below the first two anomalies for a gravitino in an externalgravitational field.