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

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294 P. <strong>van</strong> Nieuwenhuizen, SupergravityMm,, = (-Tm,,, Pm = —h~m(1 7~),Km ~A=diag-~(1,1,1,1,4), D=—~y 5(Qa)/~= —~[(1 + 75)C’]”’, (Qa)5 = +~(1—(S~5= +~[(1— 75)C’]”’, (S”) 5 1 = — ~(1+ ~ (6)We will now write down a representation of this superalgebra on fields. Consider the scalar multiplet= [A, B, x~F, G]. (We hope the reader will not confuse the scalar field A(x) with the chiral generatorA.) We already have a representation for the global super Poincaré algebra. Defining 80A(x) =[A, EQ] = ~Jx, one finds with i,, == ~x, 60B = —~iiy5,y, 80F = ~iJx, 80G = ~i~y!JX8oX = ~J(A — iy5B) + ~(F+ iy5G) .(7)Thus [8(e~),8(e2)]A = [A, [ë2Q, e~0]] so that if one definesmPm] = ~m’9mA (8)6~A(x)= [A, ~then the {Q, Q} — F relation is satisfied.In order to find the representation of Mmn on fields we first define how Mmn acts on fields at theorigin. The only nontrivial result is8~~~f(0) = [x”(O),Am”Mmn] = Amn(o~m,,)a~X$ (m
P. <strong>van</strong> Nieuwenhuizen, Supergravity 295i5j the boost generator can be nonzero. Specifically, with SDC = ACA 13, we shall see that[AKm]””2’~t3mC, [A,Km]~AymZ. (12)In order to avoid confusion with the dilation generator D, the field i5 carries a hat.Consider next S-supersymmetry. From S = /~[Km, (ymQ)~] one finds that A, B are S-inert. But for xone finds a nonzero result using the Jacobi identityOne finds[K,,,,8bA] = [Km, [‘4, Pb]] = [Km, A], Pb] — [Km, Pb], A]. (13)8~x”A(A+iy5B) s, o~(~)=(1_A)(~~~~). (14)8~, we determine 8A from (0, S} anticommutator. One findsSince we know now 5~andIA\ A 1—B\ 1—3 A\. /F\ 1—3 A\f—G\ÔA(~B)= 2 ~..A )AA~ 6~Y= ~T’)’~~~’ ÔAI~G)= ~ F )AA. (15)These results for D, Ka, A are again only for the fields at x 0. Just as for Mm,,, one can find thetransformation rules of the fields away from zero by using the commutation rules of Pm with thesegenerators. However, these results we will not need, because in order to construct covariant derivatives,one only needs the transformations of fields at the origin (“the spin parts”; the orbital parts are neededto prove, for example, the invariance of the Dirac action under global Lorentz transformations). Thuswe have derived a representation of the superconformal algebra on the scalar multiplet.A representation of the superconformal algebra on the coordinates of superspace (x”’, 0”) is given byMm,, = X0,D,, — Xnt9m + O(Tmnt9I8O, P,,, = 3,,,0” = ~(a/8O—J0)”, D = x ö + ~ëa/aëA = —~i~y53/8~ (16)Km = 2x0,x2t9m — OX’7m3/30 ~(O0)28m — O0(07m3130)a — X5” = ~(O03/3O+0750753/30)” + ~(0y5ymO)(75yin8/80)~ + (—~a/aë+~ejo + ~0J0)”.The derivatives are always left derivatives and .~= ytmx0,. One can always choose the scale of 0 and thesign of the charge conjugation matrix such that 0” is as above. (Taylor expanding as ç(x, 0) = q’(x, 0) +O”(3/30”)p(x,O)+... and using 3/30” = C~$3/80$.)For completeness we also determine the representation of the superconformal algebra on the vectormultiplet V = [C, Z, H, K, B0,, A, D]. The transformation rules follow from ö(supsym)V(x, 0) =(Eiy5G)V(x, 0) with G” = a/a0~— (JO)” (see subsection 5.3; we have rescaled 0 here by a factor 2).
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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