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

P. van Nieuwenhuizen. Supergravity 269charge. Although the chiral charge of the ghost C~is fixed by requiring this on-shell invariance, itagrees with the chiral charge which keeps the effective quantum action invariant (note that there areterms (CôMy~çfj)CM in the Faddeev—Popov ghost action).One is left with the contribution from the gauge fixing term and the new Nielsen—Kallosh ghost. Thefirst yieldsj~= (i/4)4r. ~~ M75~ 4,. At this point we use the Ward identity derived in subsection 6 whichstates that constructing two gravitinos with gamma matrices is equivalent to minus twice the sameprocess with a spin 1/2 field. (Insert in (8) of subsection 6 a factor 8MYMYSE_l and integrate over x = y.To check sign and factors consider free propagators.) Thus, at the one-loop level the chiral charge isBRST invariant if the chiral charge of the Nielsen—Kallosh ghost is opposite to that of the Faddeev-Fopovghosts. (Note that both closed ghost loops do not require a minus sign.)The result is now easy. For the gravitino loop one straightforwardly finds —20A, for the Faddeev—Popov ghost —2A and for the Nielsen—Kallosh ghost +1A, where A is the axial anomaly for a realelectron. It would be interesting to analyze what happens at the two-loop level, but this requires first tosolve the problem how to define BRST invariance with y” in ~F,,y”F~being dependent on dynamicalfields. It would also be interesting to redo the calculation for different gauge choices, to check that theaxial anomaly is gauge-invariant.Trace anomaly [172]—This anomaly can, in principle, be calculated by using again the Adler—Rosenberg method. It has also been calculated by using ‘t Hooft’s lemma for one-loop divergences,usingTi,, = _2g”2gMP 6W/ôgTM’ = £2’ (7)where W is the total one-loop effective action (W but not Ti,, contains nonlocal terms). The £2’result in (7) is valid for massiess theories. If the classical action is not invariant under local scaletransformations, as for example supergravity, which is only globally scale invariant as far as the termsquadratic in quantum fields are concerned, then the Ti,, in (7) includes the non-anomalous part as welland is infinite (in fact its infinity is proportional to 1L1R). Using dimensional regularization, oneobtains the anomaly as the difference between first letting n tend to 4 and then taking the trace, and thereverse order of these two operations.Here, however, we will present a method which gives the trace anomaly for all spins at the sametime. The method yields Ta,, off-shell, but only on-shell (with respect to the background field which mayinclude the extended supergravity fields) is Ti,, proportional to a total derivative (since £Sff in (7) isquadratic in field equations up to total derivatives). To obtain the anomaly due to a physical massless ormassive spin J particle, one considers fields 4’(A, B) which transform irreducibly under Lorentztransformations as the (A, B) representation and takes linear combinations such as to end up with thecorrect number of helicities. For example, for spin 3/2 one considers 4,(1, ~)and subtracts twice thecontribution from 4)(~,0). Working with 4’(1, ~)means choosing the unweighted gauge y~4, = 0, while24’(~,0) is the usual Faddeev—Popov ghost. One obtains the desired anomaly by computing thecontributions from the fields 4’(A, B) as obtained from the asymptotic expansion of the heat kernel; fordetails see ref. [172,174].The results for N massless spin J physical fields areI .~4— J.., [.....~L... I ,.~4 * D * DM”P”~ x g A — ~ L3~iT2~ ~ x g 1’MVP” ~x[—~ (N2 — N312 + N, — N,,2 + ~N0)+ (58N2 — 17N,,,2 — 2N1 — N112 + No)] (8)