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

P. van Nieuwenhuizen, Supergravity 285n=(~ ~). (12)It now follows from (10) and (5) that the internal symmetry representation b isUSp(2n, 2n)Ø U(m). (13)Thus, for example, the N = 2 supergravity with U(2) invariance cannot (and indeed does not) have acentral charge on-shell.For super de-Sitter algebras we are not aware of similar complete results.As far as the super conformal algebras are concerned, if one considers only internal charges whichare Poincaré scalars again, then in the {Q, Q} and {S, S} relations one finds no B, for dimensionalreasons alone, but in the {Q, S} anticommutators a group U(N) (but SU(4) for N = 4) must be producedand central charges are absent. For N = 4, an outside charge U(1) (i.e., a charge which appears on theleft-hand side but not on the right-hand side) can be present, so that Q and S still rotate under a fullU(4). This special role of the N = 4 case may explain certain cancellations of infinities in the N = 4globally supersymmetric Yang—Mills theory.For literature, see:S. Coleman and J. Mandula, Phys. Rev. 159 (1967) 1251.R. Haag, J. Lopuszanski and M. Sohnius, Nucl. Phys. B 88 (1975) 257.3.4. Supergravity as a gauge theory on the group manifoldAn approach to the gauging of superalgebras which differs from the approach in subsection 2 is basedon the notion of the group manifold [334,1]. In this approach one considers as “base manifold” a spacewith as many (bosonic and fermionic) coordinates as there are generators in the full superalgebra G.The superalgebra G is arbitrary, and can be non-semi-simple. Fields depend initially on all coordinates,but as a consequence of the field equations they actually turn out to depend on fewer coordinates thanare present in the group manifold. One chooses a particular sub(super) algebra H of the full algebra Gand requires that the action be invariant under general coordinate transformations in the full basemanifold and under local gauge transformations generated by H. It then turns out that these symmetriesbecome symmetries of the final theory in a smaller base manifold [608].It is at present not known when the program works and when not. Also, one knows at present onlyhow to obtain the final theory on-shell, but no guiding principle is known how to obtain the auxiliaryfields. Nevertheless, the approach seems promising.General coordinate transformations can be written as gauge transformations plus curvature~terms*,, jp’,, A....j-l1,, ~ A -, p11, AOgcI~c)flit — ç unnn ~ c’nc flu= Dn(~Hh 1j4)+~~~RrjnA (1)(D11 )”= ~3~A + hnB ~fCB~~.‘This result seems due to F.W. Hehl. P. von der Heyde and GD. Kerlick, Rev. Mod. Phys. 48 (1976) 393.