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

328 P. van Nicuwenhuizen, SupergravitvFor the vielbein itself one has an extra term due to the index A,SV/’ = DA~ + 2VA~TBCA. (28)As in subsection 2, we choose at 0 = 0 a gauge withV,.” = em,., V,.,° 14f a V,, 0 = 5a V,,m = 0. (29)m”(O = 0) agrees withUsing the connection the constraints ordinary on the supergravity. supertorsion, one finds that h~”(0= 0) = 0 while h,.Supersymmetry transformations are now defined by ~tm(0= 0) = 0 and ~0(0 = 0) ~a with °(x)theusual four-component spinorial parameter of ordinary supergravity. These transformations maintain thegauge V,tm = 0 if~ ~m~jC,~b~-’ Pfl_t1aS VaSlbChence with Tbam = — ~(Cym )b and Tb,, = 0 one finds~m(x,0)= ~m(x)+~iym + C(02) (31)precisely as before. On the other hand, the condition ~,,a= 5~is maintained provided the order 0 termin r(x, 0) vanishes. This also coincides with the Caltech approach, since ~a = ~A T~/~a=and e” = —~Ey~Owhile e’”(x, 0) = “ + ~,çb,.”(Ey~0).Let us now see how the tetrad and gravitino at order 0 transform. One finds for the tetrad:5t!~,a+ s,,”~”= ~ + 2e,.~~ l~Tb,,m. (32)m= _~(Cym)~and Tb,,tm = 0 this precisely reproduces the law Setm,. = ~Eym~/J,,.With Forthe theconstraints gravitino one T~ finds an interesting result= (D,. ~’+ 2em,. L~Tbm0. (33)The torsion components Tbm”fields R and Gmcan be expressed by means of the Bianchi identities in terms of the superTB,MM,AE13J~’4MAR‘7’ . . ~ . . ( )I B,MM.A ~‘ EMA BM — .)EBA~..JMM— .)EBM AM~Identifying at 0 = 0 the auxiliary fields by R -~ S + iP and Gm A~IXone finds the result of ordinarysupergravity back= (D~+ ~ A~~ys) + ky,. (S — iy5P — i,475) . (35)Thus the minimal set of auxiliary fields 5, P, Am follows from the superspace approach if one imposes