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

298 P. van Nieuwenhuizen, SupergravityI = J d~x[aR,,~, m”(M)R,,j”(M) 0,,,~+ $R,,,,(Q)y5(—C’)R,,,,~(S)+ yR,,,~(A)R~(D) + ~R,,.,(P)R,,.,.(K)]&”’” (4)(note that R,~(S)starts with ~ + ~ For internal symmetries such as electromagnetism, theaction is unique and non-affine, namely the Maxwell action. Hence, since the internal symmetriesconsist here of only the U(1) chiral invariance, we add its actionI = J d~x[8R,,~(A)R,,~(A)g”~g”~’Vg] (5)Under local M, D, A gauge transformations (given by oh”,. (D,,,e)”’ with e equal to Amn, AD, AA) theaction is invariant if one uses the rule of transformation of curvatures. Under local K-gauges, the actionis invariant if the P curvature vanishes and if the 0-curvature is chiral-dualR,.,, m(P) = 0, R,.~(Q)+ ~ ,.~“=R,,,~(Q)y = 50. (6)Local S-invariance follows then iffl=2iy=6=—8a, ~=0. (7)As we will discuss shortly, local 0-invariance follows if one imposes one more constraintR,,,,(0)o” = 0. (8)Thus all constraints are derived from a dynamical model by requiring various gauge invariances. Inprinciple, the transformation rules of fields are modified when one solves these constraints. However, asone easily verifies, all constraints are M, K, D, A, S invariant, and thus for these symmetries all fieldskeep their transformation rules 3h/~= (DE)A and the action remains M, K, D, A, S invariant.One can solve the constraints above. The P-curvature yields the torsion equation,my4’~)+(e”btm — em,,b”). (9)(0,, = _w mn(e)_.i(4’ — * yn4’m + ~pThus there is the same torsion as in ordinary supergravity, induced by gravitinos, plus torsion inducedby the dilation field.The solution of the 0-constraints eliminates the conformal supersymmetry gauge field= (y”(S~v+(7~~~v), 5,.,, = ~ /J,