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

270 P. van Nieuwenhuizen. Supergravitvf d~x\/~DMj~=~~ J d~xV~*R RMV~~~][21N 112 — N,,2]. (9)The result in (8) holds for RMV = 0 but (9) is valid even off-shell. Readers who are not sympathetic toheat kernels could have derived these results by direct though tedious computations. For supersymmetrictheories, the first term in (8) cancels, and one finds that 0(4) supermatter (N~= n, N,,2 = 4n,N,) = 6n) and 0(3) supergravityhave vanishing trace anomaly (note that although for theories which arenot locally scale invariant Ti,, (x) is infinite, its infinity, proportional to ER disappears upon integration).We remark that for self-dual spaces the left-hand side of (8) should be an integer, so that theEuler number x between square brackets in (8) is an integer times 24; also the numbers ~ are thenumber of zero spin 2 modes in a self-dual compact gravitational background. Vanishing of the axialanomaly is harder to achieve (although one might 4= define 0 evenappropriate in supersymmetric chiral weights) theories. and one notes inparticular that f Ti,, = 0 does not imply f DMJM~2.16. Chirality, self-duality and counter termsThe gauge action of N = 1 supergravity has a global chiral invariance under &/IM = i’Y54’M. This can beused to find restrictions on possible counter terms [175]. Since the curls DM4’,, —D4,,, = 4’M’ satisfyon-shell c~MV+ b’s’l’MV = 0, the combinations ~ = 4,,,,,. ± y54’,,,~transform as follows4’~,.—*(exp ~ iq)4i~,, if ~I’M—* exp(ifl’yS)4’~. (1)If the leading fermionic terms are only products of 4’MV (which is the case in all known models), then thischirality symmetry only allows as counter terms (4’±)k(4’_)k. In superspace, the possible counter termsare constructed from the Weyl superfield WABC (symmetric in the three dotted or undotted indices),and introducing again W~and W as above, one finds for the generic counterterms1~(DWr + w~w~[(Dw÷)~ +~. •]. (2)£2’= W+Wk+(DW+)This result is due to the fact that DW±is invariant under (1). This follows from the fact that W+ scalesby a factor, as in (1), while the derivative D scales in the opposite direction (see [498]).Since thesuperfields W±contain only self-dual Weyl tensors (and W_ only anti-self-dual Weyl tensors), it isclear that if W+ = 0 then the only possible counter terms are of the form (DW+)” with any number ofderivatives D. These, it is claimed, are total derivatives, and hence the only possible counter termshave as leading bosonic parts the following product of (anti) self-dual Weyl tensorsC~C~+C~C~+,k2 (3)possibly with arbitrarily many derivatives (parity requires the sums in (3)).If the theory would also have a chirality invariance of the Weyl tensorC±—*C+, C—*-C (4)then in (3) one only has k and n even. However, there is no reason to suppose that such an invariance ispresent, the only duality invariance being the one for 4,,,,,..