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

342 P. van Nieuwenhuizen, SupergravityWe come now to the well known k —*0 limit. In this limit the group G(N) is expanded into the group0(N), and one hopes to find the N-extended supergravity models. To see how this limit is obtained, westate without proof that one may place F” by ‘y” in (36) and (39) provided one replaces in g~, 91thematrix S,~by K,1. In that case the condition becomes [M, y”] = 0 and [M, kK] = 0. Thus M”~1=&~(M0)’.,where (M0)~is antisymmetric, so that the internal symmetry group expands into 0(N).We conclude our discussion of gauge supersymmetry with a short discussion of four topics.Vanishing cosmological term. While the supercosmological constant A in (30) is nonzero, the ordinarycosmological constant (in the Einstein sector) vanishes as a consequence of the global supersymmetry ofthe vacuum metric below (35). (It may be stressed that other vacuum solutions might not be globallysupersymmetric, but this might induce an enormous cosmological constant if such breakdown occurredat the tree or the first loop level.) The reason is that if one substitutes in (30) g = g~+ h, due to (36)only terms at least linear in the quantum fields h survive. In more detailV I \ — DEinstein( \ L r”(°)r$(°) —~ ~x1 ,.~ a —and the second term cancels Ag~.Ultraviolet finiteness. One quantizes with the globally supersymmetric gauge fixing termgAll (0)CB(z) CA(z) (46)where CA(z) is the linearized harmonic gauge. The Faddeev—Popov ghost action is then obtained asusual. The propagator of the metric is obtained by solving the Ward identity(ShgA (z1) h1~(z2) + h11A (z1) 8hi~(z2)) = 0. (47)The result is/iIIA.I1’I(P, z1, z2) = exp(oip,.F”~)~ F~x1u(p,wiP~(~”) (48)where F,,, is a polynomial or order m in ~“ = ~(0’~’ + 0~)and a” = 0~’— 0~.Furthermorem(p, w”)—~~ ~ . . Wa’. (49)Fii 0Let us now consider the n-polygon in fig. 1. There are n vertices, and each vertex goes like k2.Q Fig. I.Its degree of divergence is given byd,, =4+2n—~(2+4N+n,). (50)