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

P. van Nieuwenhuizen, Supergravity 279two possibilities for e -+0 and J 3 =(( 0), (e0)} ‘—‘ (~2~)+ e2(M) + 2(G), [( 0), G”] ... (EQ)(( 0),(~O)} (~2J3)+ 2(M)+ (~2Q) [( 0),( 2G”)] C2( O)(29)In the first case the internal charges G become outside charges: they are not reproduced on theright-hand sides, but they rotate the 0 under internal symmetry transformations. In the second case onehas central charges: now the G commute with all other generators, but they are produced on theright-hand sides. For example, OSp(2/4) has an 0(2) charge which rotates the two 0”’ into each other.Contracting to the N = 2 super Poincaré algebra with central charge, this central charge still produces the-~i’4,~, ,jtransformations of the photon under supersymmetry, but all physical states become inertunder the central charge (only ÔA,. = 8MA remains after contraction).Other super algebras. There exist other super algebras. Since they are little used in supergravity wewill be brief. The matrices with vanishing supertrace form of course a closed system since strM 1M2 = str M2M, (considering entries with anticommuting variables one has only commutators in thealgebra). This is the super algebra SL(M/N) which can be viewed as acting on a carrier space with Mfermionic and N bosonic coordinates. If M = N, the unit element has zero super trace, and one canomit it to obtain SSL(N/N) = SL(N/N)/AI. These algebras are simple.An example of a non semisimple algebra is for example given by the fermion operators E, a, at, ata.This is GL(1/1) and the ta, unit andelement SSL(1/1) Ebyisomitting the ideal the (anunit invariant E. subalgebra). Again one can constructSL(1/1) Somebyexceptional omitting asimple super algebras are F4 with 3 + 21 + 2 x 8 generators (with bosonic partSU2 x spin (7) where spin (7) are the twentyone 8 X 8 matrices a~ constructed from the gammamatrices in 7 dimensions), G3 with 3+ 14+2 x 8 generators (its bosonic part is SU2 X G2 the ordinaryexceptionalLie algebra G2 is obtainedby fixing one componentof spin (7) and hasthus 14 generators). Alsothere is an algebra D(2/1; a) (=OSp(2/4) if a = —1), as well as W, 5, S, H.In addition there are hyperexceptional algebras P(n), Q(n). For example 0(3) contains as evengenerators the SU3 generators along the diagonal, and as odd generators again the SU3 matricesEven = (‘~ ,~,), Odd = (,~‘b). (30)The bracket relations are usual commutators, except that the bracket for two odd generators is given by2}—~tr(O’O2) (31)(Q1 02)= {0’, O(which yields the d1,. coefficients of SU3). We have given this example to show that the bracket relationsneed not be only commutators or anticommutators, when one has a representation in terms of matrices.For further literature, seeL. Corwin, Y. Ne’eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573.P. Freund and I. Kaplansky, J. Math. Phys. 17 (1976) 228.V.G. Kac, Funkc. analiz 9 (1975) 91 and Comm. math. Phys. 53 (1977) 31.Lecture notes in Math. 676, Advances in Math. 26 (1977) 8.