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

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286 P. van Nieuwenhuizen. SupergravilyIntroducing one-forms hA = dy~~hAAand flat superparameters ~ = ~one has8gc(~)hA= (D~~+ h~~BR 8c~~. (2)These transformations will become in the end the local supersymmetry and general coordinatetransformations in ordinary four-dimensional spacetime.As we shall see, it sometimes happens as a result of the field equations that curvatures with one orboth lower indices in the subalgebra H vanish (RH = 0, horizontality). Moreover, sometimes R0 can beexpressed in terms of R11~(rheonomic symmetry) where I are the generators in the final base manifoldand 0 the generators which are neither in H nor in I. In such cases the transformations o~acting onh1” become transformations which involve only these fields themselves and their I-derivatives and arethus ordinary symmetries defined in the final base manifold I. For example, in N = 1 supergravity, G isthe super Poincaré algebra and the coordinates of the group manifold are y’ 4 = (XM, 0”, 77ab) Thesubgroup H is the Lorentz group and I = P0,, 0 = 0”.6gc becomes indeed the sum of general In this coordinate case oneand has horizontality local supersymmetry and rheonomic transformations.andsymmetryThose final symmetries can be systematically traced, once G, H and the action are given. Onechooses as action an expression linear in curvatures (this is thus an essential difference with the methodof subsection 2),I = J RA A h” A hK~CA.K, K, (3)where A ranges over the whole (super) algebra G and K, lie in G/H. The C are field-independentconstants such as ~ijk~and YSYm. The fields hA’4 are, for example, equal to emM(x, 0, re). In order that theaction be H-invariant, the C must be invariant tensors under H.Varying hKi the h Ki A CA... ~, do not transform in the coadjoint representation of the full group G.(Thus, fora tensor TA one has ÔTA = I~CT~Bsothat TATA is invariant.) Note that in general D,ICA K,,is not zero since the C’s are not invariant tensors under G.The manifold M over which one integrates is (n + 2)-dimensional since the integrand is clearly an(n +2) form. One has n +2< dimK = dim G/H. One uses a generalized variationalprinciple to obtainthe field equations: one varies both fields and surface. (Thus one considers all possible surfaces M in thegroup manifold, and not one particular surface.)The heart of the matter are the constants C. These are determined by requiring that the vacuum be asolution of all field equations, since this fixes the C to be the coefficients of a cohomology class of theLie algebra G. The equation for C to be solved is thusD[(g~dg)” A (g’ dgf”CA,K, K,~]= 0. (4a)Indeed, for the vacuum h = g’ dg and RA vanishes, so that one need only to vary RA yieldingÔRA = D5hA, and if (4a) holds, then 81 = 0. (One may partially integrate to obtain (4a) if the object in(4a) is in the coadjoint representation.) The solution of (4a) is obtained if the C in the integrand in (3)form a cohomology class of the G superalgebra. In fact, cohomology class means thatD(h” A A h”CAK K,,)”O (4b)
P. van Nieuwenhuizen, Supergravily 287is proportional to curvatures and is not itself the exterior covariant derivative of some other forms. Thefirst condition guarantees that while varying the action, one finds field equations linear and homogeneousin curvatures, while the second condition guarantees that the action does not vanish as aconsequence of the Bianchi identities. The useful mathematical result is that cohomology theory givesall solutions to (4b) and that there are only few such solutions.The h = hAXA are super Lie algebra valued one-forms, but h are not yet the usual connections(“principal connections on a fiber bundle”) because in that case the H-part of h should have the formh’~’= (g~ dg)’~’+ dyKhK~~ (5)with the curved K in G/H, meaning simply that the H-part of h’~is a gauge transformation under Hitself (gauge transformations with flat parameters ‘~ are defined as usual, 8h”~= (D y~).One calls theh” therefore pseudoconnections. If (5) holds, h” defines a principal connection on the fiber bundleG(G/H, H) and the rest of h, namely ~h”(with flat K) is what is called the soldering form (“thesuper-vielbein”).One now proceeds as follows. Varying the action in (3) with respect to all h 4’4, one finds the completeset of field equations. From here on we discuss the particular case of supergravity; this should enable thereader to deduce also the general case. Assume that one has found the coefficients C such that theaction readsI = J (R” A E” A E’ qk!+ 4’ A ys7,~4’A E,). (6)The R’ are the Lorentz curvatures dw” + ~w”A wj~’,E” is the vielbein one-form, 4”’. the gravitinoone-form and ~ is covariant with respect to H, i.e., it contains only the Lorentz connection. Theintegration is over a four-dimensional manifold since for the super Poincaré algebra there are only threecohomology classes of the algebra (not to be confused with the cohomology classes of manifolds), one ofwhich does not contain gravity, a second gives a theory with only the vacuum solution while the third isthe one used in (6). This proves the uniqueness of supergravity from the point of view of differentialgeometry.The field equations become8118w” = CilkiR n E’ = 0 (7)8118E’ = ~k,R” A E’ + ~çfrA y5y,R = 0 (8)6II8~fIa= (y5y,R)” A E — 1(yy4’)a A R’ = 0. (9)The symbols R” and R” denote the curvaturesof ifr” and E’~(thelattercontains a ihifr term). In deriving (9)we used the identity which we also used in the proof of the gauge invariance of the action insection 17~4’A 4’y’4’ = 0. (10)t. Aftersome Thelengthy interesting algebra point onenow finds is that the following one can project results: (7,8,9) onto the various components of dy’
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