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

240 P. van Nieuwenhuizen. Supergravitva -*p~) = (c~~c~) = ~ (6~6~+ ~ — ö~)k2. (3)For comparison we note that the spin 0 propagator reads P°= —ik2. The gravitino propagator isobtained from the terms bilinear in çl’,.~~3/2 = — ~ ç~[y5~yPy~ — yC~yPyIL]~ + ~= (4)where e is the determinant of e~.The field equation is thus ~Y~IY~~/’c. and its inverse yields thepropagator [510]P~L~ = (I4~L~C) = ~ (5)For comparison we note that the spin 1/2 propagator reads P”2 = —Kk2.Instead of (1), one sometimes uses the part linear in h,~,.of V~g’~”. Also one often chooses instead of‘y~”i/i,.. the linearization y’26”ac(’,,. with constant y~.They yield the same propagators. For a —~, theantisymmetric part of the vierbein is frozen out. However, as we shall see, the Lorentz ghosts derivedfrom the antisymmetric vierbein should not be neglected, even in the limit that a tends to infinity.In n dimensions with the indices of h,.~running from 1 to n, the graviton propagator acquires a factor2/(n —2) in front of ~ The gravitino propagator in eq. (4) in n-dimensions yields the n-dependentresult [556]= [y,.k’y,..+ (4— n){6,.~,,Jc’— 2kMk~k2X}]k~2. (6)However we keep all indices of l/J,.~and h,~always four-dimensional and only regularize by letting x~’andmomenta p’4 become n-dimensional (see subsection 7).We now turn to the ghost action. From ~‘(fix)we find the gauge choicesF~= {_a~(\/~g~u~), eav. — ~ — y ~‘}. (7)The ghost action is obtained by varying F~with respect to all gauge symmetries and sandwiching theresulting matrix with an antighost row and a ghost column. Thus one finds for infinitesimal variations6(Vgg’4”) = ~+ (~9,~~)g’4’~] + a(~Vgg’4~’)— s/~(ey’4~fr”+ ëy”~Y’)+~ey- fr\/~g’4’~. (8)6(y ~)= ~t9a(y ~)+ (0 +~,4y~- 2~)e+ ~A cry ~. (9)t5(ea~—ebC3,.~öa)={(9~~ )ea~+r~9,,ea~+Aa1’eb~~ (10)