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

312 P. van Nieuwenhuizen. SupergravilyLAB = ~Lmn(X~flY’Bacton tensors as in (2). One can noweasily show that SAB is an invariant tensor. (Sincethe parameters Lrs(x, 0) are bosonic it does not matter whether one writes them in front or in the back. Inthe theory of gauge supersymmetry with tangent group Osp(3, 1/4N)thereare fermionic parameters, whichleads to complicated extra signs.)There are 8 x 8 x 16 ordinary fields in VAA (since expansion in 0 yields sixteen components) and8 x 6 x 16 ordinary fields in hAm~.The number of local symmetries is 8 X 16 (for ~) plus 6 x 16 (forA ~fl)~ These numbers are much larger than for the ordinary space approach to supergravity, where onehas 22+ 16 fields (or 22 + 16 + 4 x 6 fields if W,.m~is an independent field) and 4+4 + 6 local symmetries.The question now arises what the relation between the superspace approach and the ordinary spaceapproach is. As we shall see, if one chooses a certain gauge in superspace in which to order 00V,.m = e,.m, v,.’~= ~ h,.ma = W,.mi~(e,~‘) (3)and if one imposes in addition constraints on the supertorsions by hand (i.e., from the outside; they arenot field equations), then one recovers the ordinary space approach.Let us conclude this introduction by defining torsion and curvature. The inverse of the supervielbeinis defined by~z A11 B — ~ BVA VA —11 = SAH. However,As a consequence, also VAB VB~r H~j B —‘ \(B+I1XA±B)T, B11 TIVB VA I~) VA VBwhere in the exponent H = 0 for bosonic values and 11 = 1 for fermionic values.Covariant derivatives are defined byDA =8A + ~hAmu1Xmfl,DA = VAD.4 (6)where Xmn are the Lorentz generators. These derivatives are covanant with respect to local Lorentzrotations, but only with respect to general coordinate transformations if one considers curls. They arethe kind of derivatives first introduced by Cartan, and one can introduce p-forms in superspace whichavoid any extra minus signs. We prefer to exhibit indices explicitly for pedagogical reasons.It should be stressed that these Cartan covariant derivatives are not the same as the covariantderivatives which one sometimes introduces in gauging groups. For example, in gauging the superPoincaré algebra, we defined in section 3= (D,.E)A O,. A + h,.BeCfcB~~ (7)and one sums over all connections h,.B, not only over the Lorentz connections. Thus, in this way ofgauging groups there is double counting: translations appear both in the tangent group and in the basemanifold (as general coordinate transformations).Torsion and curvature are defined by[DA, DB} = RABm~X~fl) — ~ (8)