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

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218 P. van Nieuwenhuizen. SupergravityIn this action S, P, Am are clearly nonpropagating, and one sees that, as in the Wess—Zumino model,1.~OVtheir dimension is 2. Hence they must rotate into the gravitino field equation. The symbolis theRgravitino field equation R1. = 1.”~y5y~D~t/i0 (which we will discuss in the next subsection) but with thesupercovariant derivatives. Hence1.,co~~= “ysy~(D~~~,,,R ~ A,,y — 5t/i,, + ~‘Yu7l1/i~). (4)One obtains the supercovariantization D,.COV of the derivative O,.A of some field A transforming asOA = EB by adding the connection —4iB. Indeed, in that case S(D,.COVA) = 8(3,,A — i/i,.B) is free from3,. terms. For the gravitino this rule yields all m” terms inside D~for eq. (4), two except reasons. that First, we did thenot curltry [Dr,Do.] to addconnections is already supercovariant for the variations by itself 8t/i~= (and Da covariantizing and Swa Si/i,, = D,, by adding a connection, the whole curlwould vanish). Secondly, the spin connection w~mn(ei/i) is supercovariant by itself because it wasalready supercovariant without S, P, A,. while in ~,.mn (e, i/i) no 8i/i terms appear, so that the extra termsin Si/i,. cannot produce 0 terms in 5~mn One finds upon explicit computation [224]= ~ë(y~i/’,.~’~°” — 7ai/’,.b’~°”— y~/i~~CoV)+ ~ë(u~~fl+ 7P~7ab)i/’,. (5)where i/i~’ is the curl between parentheses in eq. (4).Let us now consider the gauge algebra. Without auxiliary fields, all commutators on em,., i/i”,. closeexcept the commutator of two local sypersymmetry transformations on the gravitino [235]. For thiscommutator on the tetrad one finds[S( ~),0(2)]e~,.= ~i27”D,. 1— 1 ~ 2. (6)Guided by the global anticommutator [e~0, i20] = ~i2y~ 1P,.,we expect on the right hand side generalcoordinate transformations with parameters ~ = 2 2Y~ I. Thus we rewrite (6) as2eAO,.( 27e1) + ~(3,.e”AXe2YA i)+ 4 2[7,,,.cd]~1~cd= ~{3(i2~yAEl)}emA+ ~(E27 E1XOAe ,.) + ~(D,.e~A— DAem,.)i2yA l+ ~ 2yEIwAen,.. (7)The first two terms are clearly a general coordinate transformation on the tetrad, with parameter~ The last term we recognize as a local Lorentz rotation with parameter mA —DAem,. ~i2yA =~l/i,.yml/’A iwAmn. The asremaining ~~~~(e terms can be rewritten, using the torsion equation D,.e m,. with parameter~_~(e 2yA l)t/,Aymq,,. and clearly constitute a local supersymmetry variation of e2yA l)t/iA. Hence, we can summarize that, as far as the tetrad is concerned, the following commutatoris valid[8~(e~),So( 2)]= S~(~~)+ SL(~~w,.~”)+ ~(8)r 2 27~ 1.This is the local version of the super Poincaré algebra. As one sees, the P in {Q, 01 = P is replaced by
P. van Nieuwenhuizen, Supergravily 219general coordinate transformations, and this we anticipated in the beginning of this report when wederived supergravity. But there is more. One also finds on the right hand side the other two gaugesymmetries. We also see that the structure constants defined by this result are field-dependent. Onemight speak of structure functions. This is a property of supergravity not present in Yang—Mills theoryor gravity. The moral of this result is that one cannot simply deduce the local algebra from the globalalgebra.We now repeat this calculation for the gravitino, still without auxiliary fields[5( ~),S( 2)]çl’,. = ~(o~flE2XO( i)w,.) — 1 ~-* 2. (9)With the result of subsection 6, eq. (1) for 5~,.mn without auxiliary fields, one finds after Fierzing so thatthe parameters 2 and ~are in the same spinor traceth(ii 0i 2X2~ya~0jy~i/i,.a + OabOjY,.l/Jba] — 1 2. (10)Using that Ybl/i,,a — 7a1/1,.b is equal to7,~JI’ab+ R’ -terms (see the next subsection) one arrives atwhere[S( ~),S( 2)]lfr,. =~(e2yA lXDA4~,. —D,.4’A)+~(ëly” 2)~ +~(ëlo”’ 2)T,.P,,AR” (11)~ = ~ + 2eç~PAy5y (12)eT,.,,,,A = g,.pg,,A +~g,..Ao,,,,—~eE,,,,,.Ay5. (13)The functions V and T will be called the nonclosure functions. The first terms in eq. (11) produce againthe result in eq. (8). To see this, we rewrite these terms as~( 2y lXOAt/s,.)+~{O,.(e2yEl)}t/iA ~~~t9,.(e2~yA lt/,A)One again finds the general result of eq. (8) back, since5 lXwA. cri/i,. —00,. 01/iA). (14)+~(ë2y= ~A0Al/~,,. +(D,.~)l/iA, S0i/i,. =~i’.t~i —.1 si 1 5— tmflI~ ~0 ~ ~mnThus, without the auxiliary fields S, P, A,., the gauge algebra does not close, since there are extra termsproportional to the fermionic field equation in the commutator of two local supersymmetric variationsof the fermion. These results are identical to what happened in the Wess—Zumino model.The other commutators in the gauge algebra close and are given by[öo(fl”), oG(r)J = SG(~”3a77~— ~“3~~) (16)[OL(Wmn), OG(fli] = SL(flaO~Wmn) (17)(15)[SL(wm~), 5L([lmn)1 = oL(—W ~flPII + flm,.~~) (18)
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