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

348 P. van Nieuwenhuizen, Supergravityin d = 10, i/i,, has 16 components and becomes in d = 4 equal to four gravitinos. However, one finds ind = 4 matter multiplets, and splitting in d = 4 these matter couplings from the pure N = 4 gauge actionis rather hard and has not been attempted. Rather, the SU(4) model was constructed directly. Theappearance of an off-shell SU(4) symmetry is obvious since the Lorentz group 0(9, 1) splits into theusual Lorentz group 0(3, 1) of d = 4 and an 0(6) which is equivalent to SU(4). Since the pseudoscalarfield B is obtained by a duality transformation on an antisymmetric tensor field (which can only appearpolynomially in d = 10 and d = 4), B necessarily has opposite parity and appears polynomially;however, i/i does appear nonpolynomially.At the classical level the two theories are equivalent because one can find a transformation law of thefield which turns the one action into the other. For the axial and vector fields the correspondence is thefollowing. Denoting by G,.,.” the tensor which gives the equation of motion of V~,hence D’G,,,,” = 0(since only the curls F~,.of V~appear in the action, G,,,.” can be defined) one has for thetransformation laws of the curls of A,.” and B,,”~ = (F~,.+ ~F~’,, ”,,)(14)[3~B~,. =(G~,,.—where a and /3 are certain numerical matrices. The first equations can be carried over directly onto thegauge fields themselves, but the second equation together with the Bianchi identity D”B,W~= 0 yields(D”G~,.)—~ ‘k,(D”G,,~) = 0. (15)Hence, one-half of the V’~field equations must be satisfied and one is partly on-shell. It is now alsoclear why in the SU(4) model there are three axial vector fields, namely due to the e,.J’~symbol in (14).The SU(4) model is the simpler model. In the action one only finds nonpolynomiality in the scalarfield 4,, and only by means of exponentials of 4,. In the SO(4) model, on the other hand, the scalarkinetic term reads [1 — .c2(A2 + B2)]1 [(8,.A)2+ (3,B)2], which, incidentally, limits the range of KA and KBbetween —1 and +1 [221,222].In the SU(4) model there is a group of global SU(4) invariances off-shell, and a further global SU(1, 1)on-shell. The off-shell SU(4) transformations transform the fields A~’,. and B”,, into each other andcontains also chiral rotations of i/i,.’, and of x~while 4,, and B and em,, are inert.The SU(1, 1) global symmetry group leaves the scalar kinetic terms invariant. These are given by(8,. Y8’Y) (Y + Y)2 where Y = exp(—24,) — 2iB and the transformations Y -* (aY + if3) (iyY + 5)_lwith aS + fly = 1 leave it invariant. To see that these transformations are really SU(1, 1), note that thetransformations8(z) = = (a i/3’1(zi\Z21 \1y Sj\z2can be written as exp(iw1r1 + a2r2 + w3’r3) and leave z * r1z invariant. Since this noncompact group isrealized nonlinearly it does not lead to ghosts. Again, this SU(1, 1) contains duality transformations of theform SA,.j’ = G,,,”(A) and idem for B,.”.