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

320 P. van Nieuwenhuizen. Supergravin~which again constitutes a representation of the supersymmetry algebra on (x””, 0”). Finally, with (9) andintroducing O’~,p~ M’~as in (10) one findsSR~(X0) = [~OR+ ~ + ~A””’”M~,P,.s~f(x,0)]= [~QR ~] = +~(-~._~m9~)~ = —( ~G)4~ ~(15)= ~ ~ — (~A. o,O~+A m~x’”As one may verify, the operators0R, pR MR satisfy the super-Poincaré algebra. In particular{G”, G~}= ~(,a’C’)”~= ~ (16)Also the operators in (12) form a representation of the supersymmetry algebra on fields. In particular{D”, D~}= ._J(ymc~-l)a~i3 = +~(ymC_I)a$p~,. (17)In fact, D” and G~anticommute, as one may easily verify directly. This is no accident but due to thefact that the induced representations on the coset space are associative[(exp ~O)g] exp ëRO = exp eLoEg(exp jRQ)], (18)Hence on fields(exp ERG) (exp — ELD) 4(exp ELD) (exp — ERG) = (same with D G) (19)from which it indeed follows that {D”, G’~}= 0.The interpretation of these results is the following. G generates supersymmetry transformations.(Note that on ordinary fields S~A= [A, P,.] = 8,,A.) For example, on asuperfield V(x, 0) = C + OZ + ~00Hetc. one has from 8( )V= EGV, upon equating the coefficients of definite powers of 0SC = EZ, SZ = (20)[S(e,), 5( 2)]C= ~E2y”” 13,,,C (21)and one has indeed a representation of the algebra of global supersymmetry on the components ofV(x, 0). Since D” commutes with G~,we identify D” with the spinor part of the covariant derivative.Indeed, covariant derivatives are per definition derivatives such that, for any tensor T, D”T transformsas T, and this means that D” TM. must (There commute are better with but G~.For morethe complicated vector part arguments.) of the covariant derivativeone easily finds that D,. = 8/ôx