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

The Bel—Robinson tensor is defined byP. van Nicuwenhuizen, Supergravity 255T”’3”4 = R~~PAUR$P’4O. + *RaPAU *R$’4 + R*aPk~~R*$~’4~ + *R*aPAu *R*$’4 (2)where the stars denote left and right duals. Since *R*~ + Ra 0p,~,is proportional to R~,.— ~ and** = —1, it follows that on-shell (i.e., Ru,. + Ag,,,. = 0), left and right dual of the Riemann tensorcoincide. Also *R*‘~$ is on-shell proportional to the Einstein tensor. The B.R. tensor is symmetric inaf3, in ~ under pair exchange (a$) ~-* A~and on-shell it is conserved, Da Ta$A~~=0. (See the thesis ofL. Bel, for example.) If there is no cosmological constant A, one has on-shell that T~~0AMis completelysymmetric, traceless and conserved.It is now possible, either by the Noether procedure or (to all orders5to in Ka and full locally orders supersymmetricin ~fi,.~)by thetensor invariant. calculus, Thus, to at show the three that loop one can levelextend there the is a invariant £2’ for N T~03,8T”’3”’ = 1 supergravity on-shell which is on-shellinvariant under local supersymmetry and which poses a danger for the finiteness of supergravity.The crucial question whether N =8 supergravity is finite beyond two loops has been analyzed bymeans of superspace methods by Howe and Lindström. Just as for N = 1, 2, 3, 4 supergravity, alltorsions and curvatures are on-shell functions of only one superfield. For N = 1 this is a 3-spinor WABC(see section 5), for N = 2 a 2-spinor WAB, for N = 3 a one-spinor WA, for N = 4 a scalar W, while forN = 8 one has a scalar W,1k, where i, ..., 1 are SU(8) indices. Moreover, W,jkl is totally antisymmetricand self-dual. The 0 =0 components of W,Jk, represent the scalars in the N = 8 model after fixing thelocal SU(8) gauge (see section 6). The constraint D~,Wjk~m= öbAk~m1aputs W on-shell. These results arelinearizations of results obtained by Brink and Howe.The Weyl tensor C appears in W asD~D~D~D~Wmnpq~ (3)(In two-component notations, C is totally symmetric, see the appendix.) It is now easy to write down asupersymmetric invariantI = K~~Jd’~x ~ (4)2088h8where Note that g~,.= from y,.,’, (3) + Kh~,..Performing it follows that Wthe is dimensionless, 9-integration one so that endsone up finds with ainnonvanishing I a term of the product form of K Weyltensors (together with its supersymmetric extensions). Thus it would seem that N = 8 has nonvanishinginvanants which could serve as infinities in the S-matrix from 7 loops onwards. Increased understandingof how to build invariant actions with measures d”8 where n