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

P. van Nieuwenhuizen, Supergravity 349We now turn to the SO(4) version of the N = 4 model. Here the global SU(4) consists of a manifestS0(4) symmetry plus transformations generated by symmetric antihermitian matrices iAj”. More indetail~,L,j_~4j1L.kLap, — 111 k’(’,.*SXLSI = —i(ASA=BtrA,1k — S’k tr A )x” (16)SB=-AtrASF~,.= A G ik —The tensor D’G,,,.”‘ is again the field equation of V,,”‘.All SU(4) symmetries except the 0(4) hold only on-shell. The same is true for the N 3 model, asone easily sees by reduction (i.e., A = B = 0, etc.). There does not seem to be a consistent truncation toN = 2 or N = 3 supergravity theories of the SU(4) version of the N = 4 theory.One can gauge both the SO(4) and the SU(4) models. The SO(4) model was gauged by Das, Fischlerand Roéek [111]and has a potential with an indifferent equilibrium. Of the SU(4) model, having only 6spin 1 fields, one can at best gauge an SU(2) X SU(2) subgroup and this was done by Freedman andSchwarz [244].However, in this case the potential has no stationary point.Let us now jump to the N = 8 theory. One might expect that this theory is prohibitivelycomplicated — but one can formulate it in a very simple way in d = 11 dimensions as N = 1 (simple)supergravity. If one then uses dimensional reduction to d = 4 dimensions, the full theory emergesautomatically.6.2. The N = 8 model in 11 dimensions [93]Dimensional reduction means that all fields are assumed to depend only on x1,. . . , x~instead ofx1,.. . , x~.A tetrad in d = 11 splits up then in d = 4 into one tetrad e,.m (m = 1, 4), 7 vectorsea,. (a = 5, 11) and 7 X 7 scalar fields e~(a = 5, 11) which describe ~8 x 7 scalar particles (because the local0(7) symmetry, the residue of the 0(10, 1) Lorentz symmetry, eliminates the antisymmetric parts of e~).Dimensional reduction has been extremely fruitful in supergravity (and in global supersymmetry).We now explain why the simplest form of N = 8 model is in d = 11 dimensions, and then constructthis theory in d = 11 in the remainder of this subsection. Dimensional reduction and spontaneoussymmetry breaking will be discussed in following subsections.The example of the tetrad just given shows that by going to higher dimensions, supergravity theoriescan be reformulated in terms of fewer fields. The N = 8 theory in 4 dimensions has 8 gravitinos, henceone expects that its simplest form appears in d = 10 or d = 11 dimensions. Indeed, in d = 10 or d = 11,spinors have 32 components. The simplest version is the d = 11 theory, with only an 11 x 11 “elfbein”e,.m, a32 x 11 gravitino i/i,.” anda “photon” A,.,.~(antisymmetric in ~tvp).That one needs A,,,,0 follows fromcountingm transversalof statesand traceless: ~9x 10— 1 = 44e,.,/,,.~transversal in gauge y i/i = 0: (9 x 32—32) x = 128A,.,,,, transversal: (~)= 84.