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

P. van Nicuwenhuizen, Supergravity 243A direct calculation shows that (6) is indeed true, provided one uses as basic fields in addition to e~,I/ia S, P the contravariant vector field A’4 and not the world-scalar Am [446,556]. Perhaps a moresignificant basis is4,’ = (em,.,, Il/rn”, 5, F, Am), Il/rn” = e~’4i/í~”. (7)Also on this basis (6) holds, since6(&fr,,,a )/ôç(, 2= 6(6~~” )/6IlJ~”+ 6(ôe,n’4)/ôem’4 (8)m~with a similar relation with i/i~,~ Am and 4,~-~ A’4 so that the vierbein-terms cancel. In equations such as(6), tracesover fermionicsymmetries or fermionic fields acquire an extraminus sign. Right-differentiations6/64,’ in (6) are equivalent to left-differentiations, see appendix H.On the basis (etm14, l/i,~”, S, F, Am) there is thus a measure in the path-integral given by (det e) 1 since6(ln e’) = (—6e~)e~”while öem~= —em~(6e~~)e”~. If one integrates out 5, F, Am one finds theintegration measure[dem~ di/i~”(detg)2]. (9)The precise form of the measure has no physical significance, but it is important that one can choose ameasure at all. For example, if one would not have a closed gauge algebra (by dropping 5, P, Amaltogether), (6) does not hold and four-ghost terms must be added to the action.Later on we will derive Ward identities from the BRST invariance of I(qu). To that purpose onemust show that also the Jacobian equals unity for a BRST transformation [514].It is not alwaysappreciated that the criterion which allows Slavnov—Taylor identities which was given in (6) is also thecriterion for unit BRST Jacobian. Indeed, the first term is the contribution to the Jacobian from 4,’ ifone replaces r by C”A while the second term is precisely the contribution of the ghost fields. Theanti-ghosts do not contribute, since their BRST-law does not contain antighost fields. In Yang—Millstheory each term in (6) vanishes, but in supergravity only the sum vanishes, but it vanishes for all threelocal symmetries separately.Finally we come to an interesting aspect of path-integral quantization. If one considers a quantizedgravitino in a background gravitational field, one integrates only over [dlfr~”].The classical action is then~‘(cl)=and it is invariant under &/i,~ = D, alone, provided the gravitational field satisfies Ru,, = 0 and the spinconnection c~,.,”’(e) is free from torsions. Under these circumstances one may choose as gauge fixingterm~2~(fix) = ~ ~ yø~y.Il’so that at all stages one has kept the space-time symmetries manifestly preserved. In this case thenormalization factor (det ,Ø)~112leads to a new ghost, first discovered by Nielsen and Kallosh[343—345, 304]. Since a commuting Majorana spinor has vanishing action, one circumvents problemswith spin and statistics by replacing (det Ø)_1~l2 by (det 0)’ (det Ø)1~l2~In this case one finds a