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

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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
P. van Nieuwenhuizen. Supergravity 321The connection with the covariant derivatives D..~of curved superspace is now easily establishedDA = (~4~ + ~(~/~a, ~D 4 =(~,~ (22)Note that we consider (x, 0) as “cartesian” coordinates for which no connection is needed in the basemanifold. (Even for rigid translations in curvilinear coordinates one needs connections.) The flatsupervielbein reads thusVAA(X,0)=(l(~m:) 30). (23)Since supervielbeins can be considered as determining the orientations of local superirames withrespect to the base manifold, this result states that a particular orientation (which is supercoordinatedependent) leads to derivatives DA which are very convenient (they commute with supersymmetry).This is indeed the flat limit of the supervielbein as obtained by gauge completion. This flat supervielbeinand the vanishing spin connection are a solution of the fields equations of supergravity (but not of thoseof gauge supersymmetry at the classical level). If one computes the supertorsion and supercurvatureaccording to the formulae in subsection 1, then one finds that there is a nonvanishing component ofsupertorsion (see page 316)tm = _~(Cym)ab. (24)TabThus, flat supersymmetry has torsion but no curvature.5.4. Construction of multiplets using the closed gauge algebraWe show here how one can use superspace methods to extend a tensor in ordinary space withexternal Lorentz or internal indices to a full multiplet of which this tensor is the first component. Forexample, in the supersymmetric Maxwell—Einstein system there is a scalar multiplet I,, which starts withA,,, but the other components do not rotate into each other in exactly the same way as the componentsof the canonical multiplet I = (A, B, x~F’, G’) of subsection 4.6. This one might already expect fromthe fact that the commutator of two local supersymmetry variations contains a local Lorentz rotationwhich does not act on A but does act on A,,. Hence local supersymmetry transformations for I~must bemodified.The method is based on the known anticommutation relations of the covariant derivatives DA. Sinceto order 0 = 0 in global supersymmetry D” is equal to the supersymmetry generator G” (see theprevious subsection) while in local supersymmetry this is still true for superfield tensors with flatindices, one defines, to begin with5~~~( A(x, ) 0 = 0) = (EC~D)A(x, ~ (1)We use complex notations where A stands for A — iB, and repeat that A may have external indices.Using a Fierz rearrangement (see appendix) one finds for the commutator of two spinorial covariant
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