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

P. van Nieuwenhuizen, Supergravity 225In this way one adds systematically new terms to action and transformation laws until one arrives at acompletely invariant action. There are general arguments that this iterative procedure must always stopexcept when there are spin 0 fields A present [202]; in that case the action can be infinite series in(KA) tm.The final result obtained in this way is given by2 —~F,.,F””—~,øA+~ çfr,.o .Fy”A+~—(çIi,.u”~y”AXi/i~y,,A)+~ K2(AAXAA)Sit = ~ Fco~~ , SB,. = — ~ëy,.A (8)Setm,. = ~- iymi/i,. Si/i,. = -~-D,. + ~- (~y5y”A)(g~,.+ o-,,,. )y~ .The spin connection in ØÀ contains only i/i-torsion, but no A-torsion. The symbol F~’ is thesupercovariant curlF~=8,.B~+~i/i,.y,A-,a4-*v. (9)The first suggestion that there should be an axial vector auxiliary field was due to this result [202].Clearly, the result for Si/i,. depends on the matter fields A and suggests replacing them by a new field A~.We know already from the gauge algebra that= i(D,. + ~A,.y5) + ~y,.(S — iy5P — iA~y5) (10)and hence one adds to the action the gauge action 2(gauge, e, i/i, 5, P, A,.) as well as a coupling+~~~y5yh1IA)A~. (11)Solving for A,,, and substituting the solution A,, =4 and A2 terms.—(31/8),c(Aysy,,A) back into the action and Si/i,., onereproduces Thus, in all theAsupersymmetric Maxwell—Einstein system, only A,. plays a role, but S and P do not.This is due to the fact that this system is superconformal invariant. As we shall see, A,. is the gauge fieldof chiral rotations. (In the example of the spin (0, 1/2) coupling, S and P do play a role.) The action canbe written in a suggestive manner as2=~2)+2’3/2_~1(S2+P2_A~j+21_~A(ø+~-1.4ys)A+~(~,.cr”~y”AXF~s +F~’)(12)SB,. = —~-iy11,A, Sit “‘so .Due to the auxiliary fields S, F, A,. one has only added matter terms to the action without changing the