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

358 P. van Nieuwenhuizen, Supergravityturn to the mass generation. One writes only the fields with curved indices in the form of (1). namely~ (x, x 5) = U” 0 (x 4)U~(x5)A,,”~’(x)U”~(x5)= exp(Mx5) (10)where M is a Sp(8) matrix, which can be taken as0 m1-m1 0M= ••..• . (11)0 m4-m4 0In principle any E6 matrix could have been chosen as mass matrix, but in order to exclude a cosmologicalconstant, one selects the maximal compact subgroup of E6. As it happens, this is also a Sp(8) group. (This isthusaglobal Sp(8) andhas nothing to do with the Sp(8) local which extendedthe 0(6) local.) Since Sp(8) hasrank 4, there are precisely four arbitrary mass parameters generated.Inserting (10) into the Maxwell action, one finds as mass term for the photons“2“2~(photons)= A~~A~(( — )“m10121 + (— )~m~12j). (12)This leads in d = 4 dimensions to 3 massless photons 2 where(A12, i, j =A34, 1, 4 A56 (eachand mass A78, is satisfying doubly degenerate). ~ =A0)and fourth 24 massless massive vector photonfields is obtained with masses by the(m1 reduction± rn,) of the vielbein from 5 to 4 dimensions. One can takern1 = m2 = m3 in which caseonefinds sixmore massless photons, yielding atotal of 10. These represent the 8gluons, the ordinary photon and the Z°or the antigraviton, see Scherk in ref. [391].For the fermions, masses are due to the Sp(8) connections Q~in the covariant derivatives, while thescalars obtain masses through P5. We recall that5) = Vsa~~(x)(U ‘(x5))~ 1(X5))8p (13)0(U—c/t~(x xso that one finds for ‘V’35’V the result9$’)°Va$ = e~M A e” = M . A + [M A, c.~]+... (14)— Vcd~(2MaO’in the special Sp(8) gauge where there are no A generators in the exponent of V. Clearly, 05 starts withM A and hence Q~yields masses to the fermions.The masses of the scalars come from P 1D”‘V = — tr P,,P”and because P 5 because the scalar kinetic matrix is tr D,.7/~5 is linear in masses and scalar fields: P5 = [M A, c.X1 +.... In fact, the E6 algebra hasa Z2 grading: [A, A] -~ A, [A, £] .~, [I, ~]-~ A. (This is thus a Cartan grading of an ordinary Liealgebra. Superalgebras have also a Z2 grading, but the bracket relation of two odd elements is forsuperalgebras symmetric whereas it is here still, as usual, antisymmetric.) Thus the Q,. are even in thenumber of scalar fields c, while the P,. are odd.This concludes our discussion of the way in which one can introduce masses into the N = 8 model in