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

306 P. van Nieuwenhuizen, Supergravityl6em,. — 10 spacetime — 1 dilational gauge = 5 Bose fields16q/’,. — (4+4) ordinary and conformal supersymmetries = 8 Fermi fields4A,. — 1 chiral gauge = 3 Bose fields4b,. —4 conformal boosts = 0.Thus there are equalnumbersof Bose and Fermi fields off-shell, andno auxiliary fields are needed from thispoint of view. Indeed, as discussed in subsection 4.3, the gauge algebra of conformal simple supergravitycloses by itself. This has an important consequence. Since all fields of conformal simple supergravity aregauge field, they couple minimally to matter.Thus, one can at once write down the coupling of the globally superconformal multiplet discussed insubsection 1 to conformal supergravity. All one has to do is replace ordinary derivatives by superconformalderivatives, where the gauge connections are given in the usual minimal way.To see the relevance of this fact for ordinary supergravity, let us recall that the Hilbert action ofgeneral relativity (R) can be written in a locally scale invariant form by adding a scalar kinetic term withunphysical signs2’(Brans—Dicke) = — ~ R4o2 + ~ 3”4~34~ (1)Fixing the new degree of gauge freedom by fixing p = K1, one regains the Hilbert action. Orequivalently, one can choose a gauge in which the regauged metric g,.~p-2 is locally scale invariant.In supergravity the analogue of a scalar field ~ is a scalar multiplet I = [A, B, X, F, G] which wetake to be a superconformal scalar multiplet with weight A = 1 in order to interpret A as the analogueof ~. The action for I is in flat space= —~[(3,.A)2 + (3,.B)2 + ~/A — F2 — G2] (2)and it is invariant under 6(A + iB) = ë(1 + ys)X,= ~J(A — iy5B)e + ~(F + iy5G) and 8(F + iG) = ~E/(1+ y5)X.To couple we first obtain the supercovariant derivatives D,.C. From subsection 1 we find easilyD,.CA = 3,.A - ~ c,.x - Ab,.A + A,.B1- AD,.CB = 3,.B + ~~I’,.Y5X — Ab,.B — ~ A,.A= ~ øc(A_iy5B)cb,. —~(F+iy5G)iIi,.(3)—(A +~)b~~—(~—~)A,.(i~s~)—A(A +iy5B)ço,..