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

P. van Nieuwenhuizen. Supergravity 2934. Conformal simple supergravity4.1. Conformal supersymmetryConformal supergravity is the supersymmetric extension of Weyl’s action R2,,,. — ~R2 which isinvariant under local scale transformations 3g,.,~(x)= A(x) g,,,,(x). The theory is thus a higher derivativetheory, and for the time being it does not seem to be of physical interest. However, that may be due toour present underdeveloped knowledge how to treat higher derivative field theories. At the moreformal level, conformal supergravity is of the utmost interest, since it explains the structure of ordinarysupergravity: where the auxiliary fields come from and how to obtain a tensor calculus. Since(super)conformal methods are little known, we will give here a rather complete treatment. We begin bydiscussing conformal rigid symmetry.At the basis lies the superconformal algebra [581, 519]. It contains as bosonic part the conformalalgebra with (P0,, M0,,,, K,,,, D). Since F,,, and the conformal boosts Km play a rather symmetrical role, itcomes perhaps as no surprise that there are two fermionic spinorial charges, namely the usual squareroot of P0, called 0”, and a square root of K,,,, called henceforth S”9Pm (1){Q”, Q~}”’ +~(y”C~)”’{s”, S~}= _~(ymC_I)asK0,. (2)(In the literature on conformal supergravity one often finds the special representation where (C’)”” =~ see appendix.) The rather unexpected feature is that if one adds 0” and S”, one needs at thesame time one more bosonic generator A for chiral rotations{Q”, S~}= +~C~1”’~D— ~ — (iy5C’)”’~A (m > n). (3)The conformal algebra is given by the ordinary Poincaré algebra, discussed in subsection 2.1, eqs. (24)together with[Pm,D] Fm, [Km,D]” ~ [Km,Pn] 2(6mnD+Mmn). (4)The nonvanishing commutators linear in fermionic charges are[~:~Mmn] = (omn)”s(~:)~ ~ A] =rQ”1 I “ .. IC” D 1 — I \a $I DI_HJI I.”’ mj~7mj$ 3. (5)1s”’ — ~ [0”, Km] = (Ym)”$S’Rather than check explicitly the Jacobi identities, we prove that this is a closed algebraic system bygiving an explicit matrix representation. We construct 5 x 5 matrices, of which the 4 x 4 part containsthe conformal group, in the fifth row and fifth column one finds Q and S, and along the diagonal thereisA,