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

P. van Nieuwenhuizen. Supergravitv 227SA=~iA, SB=—~y 5A, SF=~Ø°A,OG = ~ iy510°A, Sit = ~[I(A — iy5B)] + ~(F+ iysG)e, (2)02 = — ~ (D,,ë)(JA +4’iy5B)y”A + 3,. [—~ ~y”(IA —Ii75B)A + ~ (F + iy5B)y”AFrom now on we will omit the auxiliary fields F and G, but reinstate them at the end of thissubsection.Since the action 1°varies for local (x)into the Noether current, (see subsection 2) we add theNoether coupling= ~ ~,.[I(A÷ iy5B)]y”A. (3)The variations of A, B, A yield(i) AA and BB terms. These cancel with the vierbein variation in 2°(both are proportional to theenergy-momentum tensor, see subsection 2). —(ii) AB terms. These are of the form K(i/,.y~ ) ”’~3,,A 3,,B and upon partial integration the 8terms are cancelled by adding a term ~while the remainder is proportional tothe gravitino field equation and can tm,. be absorbed in 2°andby from an extra SA andSi/i,. SB-~y5 in 2NA3,,B cancel (see (after subsection tedious3).algebra),if one (iii) adds the AA Sit terms-~K1/IAcoming and Si/i fromAA Se terms to the action. One needs Fierz rearrangements, the identityIv. i/i — 3. i/i = b~,.R”and the identityDp(Ay”yay,.A)Irr AIyay,iA + AYa71JA + 2A(Oay,. — )‘aO,.)lt. (4)The new Sit and Si/i variations as well as the old variations must now be used in the new, order ,c 2 termsin the action. One finds -(iv) that the new Si/i -.-.‘cA OB ~fl 2N are cancelled by adding a term K2AAA t9B to the action.(v) that the new Si/i AA in 2’N yields K2A3(OA) variations while also the old SA ëA in the newterm in(iv) yields such terms. The sum is cancelled by adding a K2(AA)2 term to the action, and aSit = K2AAA term.(vi) Finally, many sources contribute to K21/12A OA variations, but only one source yields a term with3 . Partially integrating this term, all variations cancel if one adds a Si/i —~K2AA1/1 term.The complete answer is given by2= 2(gauge, e, i/c)+2°+2°’+ 2(contact)2(contact) = ~ ~y5y,,A — ~ A~~B]+ eIc2(Ays,flA)[_j~i/f~ysy,4i” ~A3TB ~4Ays7~AJ (5)where 2’(gauge) contains i/i-torsion, but the Dirac action for A contains no torsion. The transformation