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

300 P. van Nieuwenhuizen, Supergravityaction (namely OII8w,.mn = 0). Both methods lead to the same result, but the former has the advantagethat it is independent of any particular action. In conformal supergravity, the spin connection ispropagating, and the torsion is here fixed by the constraint R,,,,m(F) = 0, and not by a field equation.However, for the conformal boost field ftm,. this duality is again present: solving the field equation81/of”,. = 0 for f”,. one finds a result which can also be obtained as a constraint on the curvatures. Thisconstraint isR,.i,mn(M)emAen,. —~RA,.(Q)y,,~I/ +1~ ,,,,~,,,R”°~(A)= 0 (1)and it is again K, M, D, S, A invariant. Thus under these symmetries, O’f = 0 but as we shall deriveshortly, 8’~fr0. (It should be noted that as far as invariance of the action is concerned, all O’f areallowed to be non-vanishing since f satisfies its own field equation. See subsection 1.6.) Eq. (1) is thecovariantization of the “Einstein equations” R,,,,(M) = 0. The term with R(Q) is the connection neededto covariantize the derivative 3w inside the M-curvature. If one introduces a connection~~mn = ~,.mn ~e,,kCk0,?,,~A~~ (AC = chiral gauge field) (2)then the constraint even simplifies toR~(M,th)=0. (3)Thus there are three m,,,,(P) constraints = 0, not in a field conformal equation. simple supergravity:-(i) (ii) Torsion: Duality-chirality R and tracelessness: R,.,,(0)-y” = 0 which is equivalent to R,.,,(Q) + ~R,.,,y5= 0plus R,,,,(Q)o”~= 0.(iii) Einstein equation: R ~M, ~) = 0.It is probably not a coincidence that the generators for which the curvatures are constrained (F, 0, M)form the super Poincaré algebra.All constraints are gauge invariant under all 24 local symmetries, except under P-gauge and 0-gaugetransformations. The extra terms 8~~w,.mnand ~ were mentioned before. In a similar way one findsc,’ zm — 1 covi \ mp lflm,CovI \OoJ ,. — ~ ,,,. ~ jO~ C~— 8r.,. ~ j 7~C~where the Q-covariantization is due to the 34 terms in R(S).One can now understand geometrically why these constraints are needed. The global commutator{0, 0) — P must turn into {Q, Q} — general coordinate transformation plus more if one evaluates thecommutator of the two ordinary supersymmetry transformations on fields. This is to be expected sinceour action was by construction invariant under general coordinate transformation, and invariance underalso P-gauges would be double-counting. How does the theory manage to do this? Note that a generalcoordinate transformation on agauge field hA,. is related to local gauge transformation by the followingidentity8gen.coord(~~)h’~,.(D,.(~. h)~+ ~‘R”,.A. (5)