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

226 P. van Nieuwenhuizen, Supergravitvgauge field transformation laws. Since the A field equation in the absence of A,. can only rotate into theB,. field equation and this latter field equation requires two derivatives, the variation of the A-fieldequation cannot contain 8 terms. Thus the A-field equation must be supercovariant by itself, and sincethe terms t/JAF in the action contain only one field A, they require an extra factor of 2 in order that theA field equation becomes ~ØCOVA.Finally, we discuss the role of the matter auxiliary fields. The globally supersymmetric spin (1, ~)system has a closed algebra if one adds a pseudoscalar field D (see subsection 8)SB,. = —~ëy,.A, 5A = ~cr F + ~ y 5D , SD = ~ eyJA. (13)One finds uniformly for A and D the usual result [S( ~),5( 2)]D= ~(i2y” 1)3,.D, but for B,. there is aMaxwell gauge transformation added: —3,. (~i2B~1). The flat space action becomes2. (14)2= —F~—~/A +~DIn curved spacetime, the Noether current and hence the Noether coupling are unchanged. Thematter transformation laws are now given bySB,. = —~iy,.A, SA = . F~°”+ iysD),. coy KSD = 7s7 ~ A + ~-A,.y5A0~+ iysD)ili,.. (15)D~VA= D,.A — ~- (o . PSince D is a matter field, one expects SD to vary without 8 , and indeed one finds a supercovariantderivative in SD.Summarizing, the total action reads2= ~2)+23/2+21+2h/2(D,. +~A,.y5)_~(S2+P2_A~)+D2+~ i~,.(cr F+u Fco~o)yMAand is invariant under eq. (15). The derivatives D,. contain only i/i-torsion.(16)1.12. The scalar multiplet couplingThe next example of the coupling of a globally supersymmetric matter system to the gauge action ofsimple supergravity is the coupling of the spin (~0~0) Wess—Zumino model [202,509]. The action,already covariantized with respect to curved spacetime, is given by2°= —~eg”(3,.Ac9,,A + 8,,B 8MB) — ~eAy”D,.°A+~e(F2+ G2) (1)where D,,°is the gravitationally covariant derivative without torsion. In curved space it varies into