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

P. van Nieuwenhuizen, Supergravity 263One can also add masses and consider non-renormalizable models such as the Lang—Wess model. Inall these cases one finds the same phenomenon. Thus there are three possibilities for counterterms:(i) £2’ is off-shell invariant;(ii) £2? =0 on-shell but not off-shell invariant;(iii) 8(&2’) = 0 but &.2’~0 on-shell, nor off-shell invariant.It is easy to construct many examples of case (iii). Take any invariant action, thus 61 = ôçb’I,61,g + (&/i”,g)I,j 1 =0.Differentiating so that this identity with respect to some parameter g in the action, one finds81’.g vanishes on-shell but I~gitself is not invariant in general.Thus to prove the finiteness of the S-matrix it is incorrect to demonstrate that all invariant countertermsvanish on-shell. It is correct to show that on-shell no £2? exists which is on-shell invariant. In that case,one must use the on-shell transformation rules and, for example, to show that no £2’ exists withoutpurely gravitational sector, requires very detailed analysis. Details of such analysis have not yet beenpublished.From a more general point of view [165],the generating functional in the background field methodZ(~)=fdx exp i[I(cb +x) — I(cb) — I(~).~x] (4)explains these examples. If the original action is invariant under 4’ -+ g(4’) then in general thebackground functional is invariant under background symmetry transformations 4’-~g(çb) and x -~g(4’ + x) — g(çb), provided g(4’) is linear in 4’. (In that case I.~xis again an invariant.) In our examples,however, g(4) was after elimination of auxiliary fields no longer linear in fields.In order to connect these considerations to supergravity, one must find arguments which show thatcase (iii) counterterms do not occur there. In most of the analysis of on-shell invariants, auxiliary fieldswere not considered since theywere unknown at the time. Now, however, one can discuss the influenceof nonclosure of the algebra on invariance of counterterms. Since the commutator of two symmetrytransformations is again a symmetry of the original action, in supergravity with open algebra (withoutauxiliary fields) there are infinitely many field-dependent transformations under which any trueinvariant of the theory should be invariant. In most cases this leaves the original action as the only trueinvariant, and dangerous counterterms should then be of type (iii) with respect to these symmetries.It should be noted that in supergravity the transformation laws with auxiliary fields are still non-linear.However, the one- and two-loop finiteness remains true, since it was proved on-shell using helicityarguments [512,283] or Noether methods [123].2.13. Canonical decomposition of the actionThe Hamiltonian approach to supergravity, using path-integrals, first demonstrated four-ghostcouplings in supergravity (Fradkin and Vasiliev [226]).As an introduction we discuss the “p, q”decomposition of the action. This action analysis is the counterpart of the analysis of the field equations,and, of course, we will again find that there are two physical modes. However, unlike in the fieldequations, one does not need to choose a gauge for the actions.We consider the free linear spin 3/2 action first, and make a spacetime split [126]11= ~E”fIIiys( 7j494 — 749j)lIJk + E”~4y5%84rk. (1)