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

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280 P. van Nieuwenhuizen, SupergraviyW. Nahm, V. Rittenberg and M. Scheunert, Phys. Lett. 61 B (1976) 383 and papers in J. Math. Phys.A. Pais and V. Rittenberg, J. Math. Phys. 16 (1975) 2062.3.2. The gauging of (super)algebras(Conformal) supergravity is the gauge theory of the super(conformal) Poincaré algebra. In order tobe complete we will therefore first discuss how one gauges in general a superalgebra.Let the algebra be given by generators XA which satisfy the following commutators or anticommutators[XA, XB] =!ABCXC. (1)The structure constantsf,~ are ordinary numbers.Corresponding to any generator, one introduces a gauge field, as shown by the following superalgebra-valuedvector field for the super Poincaré algebra= hMAXA = eMF 0, + ~cL Mmn + ~I’M0Q° (m
P. van Nieuwenhuizen. Supergravily 281Under arbitrary variations, curvatures transform as8RM,.A = D,.8hMA — DM6h,.A. (8)Consider now a Yang—Mills type of action,1— IA4 0 A— ~ x “~L~’0 Rf~M~po- 91~’po~s..IABwhere 0 may depend on fields. Under local gauge transformations81 fd~x{RM,.’~?R (—fCB”Q~~”+ fBA”Q~’)+ RM,.AR,,,~8gaugeQ~~}. (10)Under arbitrary variations81 = Jd4x [4(D,.6hM~R~Q~T + RM,,ARP~B6QMAT]. (11)Partially integrating D,,6hMA and using that always fCD~~~h,.l)8hMC’ = _6h,,Ch,.DfDj~~one finds [308]forarbitrary variations= +R:Ro;A PO~T~48hMARPU8 3~Q~T]. (12)In the frequent case that Q~A~B~” is a constant times ~ the last three terms vanish, due to the Bianchiidentity D,.R,,~ ~°°’ = [D,.,[D0,D~]] ~°~ = 0.Finally we derive a theorem, which is useful when constraints on curvatures are not gauge invariantunder all local symmetries [523].Suppose one can solve a constraint RMA,.I~MA~= 0 for a given field hM 4°.Then hMA0 is a functional of other fields, and transforms no longer as (DM )A0, but according to the chainrule. However, after solving the constraint, its variation vanishes identically. Denoting by ô’ thedifference between the actual transformation law (using the chain rule) of hMA0 and the group lawo(gauge), one has the followingTheorem:(8’h,/~°) 8h8A, (RM,.”F~~) = [_.f~AR,,,B~C— R,,,,.” 8(gauge)]F~,”. (13)In particular, if the constraint is invariant under 8(,gauge), then hMA0, although no longer an independentgauge field, still transforms according to o(gauge) just as when it was an independent field.Using the formalism of this subsection, McDowell and Mansouri [308] showed that N = 1 supergravitywith cosmological constant follows from gauging OSp(1/4) and Townsend and the author [517]showed the same for N = 2 and OSp(2/4). In the latter case, however, a term 84’~— FM,. ’ did not follow
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