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

Acting with the complex conjugate of (4) on A yieldsP. van Nieuwenhuizen, Supergravity 323Dt~mn~(1— Ys)x = — ~R*X(L)A (10)From (9) and (10) one knows how Da acts on y m~,YmYsX and ~define a new superfield F as followsThus, since x is chiral, we canF(x, O)= +D~(1—ys)~ (11)where the complex F(x, 0 = 0) corresponds to F — iG of the multiplet I of Poincaré supergravity. Forthe product rule we now find2). (12)F(12) = A(1) F(2) + A(2) F(1) — ~(1) (1 — y~) x(We can now determine how X” transforms by using (1). Combining (9), (10) and the definition in (11),we get— ‘ys)X = ~(DmA — ~ GnX~A)ym~(1+ y~)+ ~(F + R *x(L)ci.mn) ~(1— ~ (13)One sees here clearly that if A has extra Lorentz indices, then Ssup( )X, as given by the 0 =0 part of(13), is modified by terms proportional to LQrentz rotations on A.Next we determine how F transforms. As before we apply (3) and the complex conjugate of (4) to— Vs)X. We do this by acting on (13) with Dytm and D~mn~(1— 75) respectively and find(DytmC~D)~(1— ys)X = [Dytm(—y’”D~A+ . . . F)~(1— y5)(~C’)]T (14)(Do-”””~(1± y 1)]~. (15)5)C’D)~(1— Ys)X = [Do.m~(1± y~)(yD~A+ . . .) (—~c-Using on the left-hand side (3) and (4), one finds that DaF is fully determined (already by (14) alone,(15) yields a consistency test). The only unknown at this point is the double derivative on A, which,however, can also be obtained using thatDaDmA = DmDaA + [Da, D~]A. (16)From FDa, D~]= 2 Tam”Db + Ram~Xpq one sees that in the transformation law of x a term enters,proportional to the gravitino curl when A has external Lorentz indices.In this way we have a multiplet of superfields which transform into each other under the action ofD~.The correspondence with the multiplets in ordinary spacetime is simply that the order 0 = 0components of the superfields are the components while the action of D,, is a supersymmetry