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

P. van Nieuwenhuizen, Supergravity 311Consider a base manifold consisting of four bosonic coordinates x~and 4N fermionic coordinates O~where a = 1, 4 and i = 1, N. The index N refers to N-extended supergravity. For simplicity (and sincefor N> 1 only partial results are known) we restrict ourselves to N = 1. As in general relativity, oneerects at each point (xv, O~~) a local tangent frame. In the base manifold one considers generalcoordinate transformations with parameters E’ t where A = (JL, a). As we shall see, E’~generate theordinary general coordinate transformations but ~ generate local supersymmetry transformations! Thedifference between the various superspace approaches is often that a different symmetry group is chosenin tangent space. For example, in the theory of Arnowitt and Nath the tangent group is Osp(3, 1/4N),while in most other approaches the tangent group is the purely bosonic Lorentz group times unitary (ororthogonal) transformations acting on the index i, hence 0(3, 1) x SU(N).There are two geometrical superfields in superspace, namely the supervielbein (viel = many inGerman) and the superconnection. (In the chiral approach, both are functions of an axial vectorsuperfield. In the Wess—Zumino approach, the superconnections can be expressed in terms of thesupervielbein after imposing constraints, see subsection 5. All these approaches use thus second orderformalism.) In addition, there may or may not be matter superfields. The supervielbein VA~’(x,0)depends on all 8 coordinates x~,0’~,and the curved (world) index A runs over A = (u, a), while the flatindex A runs over A = (m, a). As the indices indicate, V,.tm, and vaa are bosonic objects, and V,.” andV~mare fermionic. That is to say, they are even and odd elements of a Grassmann algebra, respectively.The superconnection hAm~(x,0) is bosonic for A = ~uand fermionic for A = a. The two flat indices mnrefer to the Lorentz group. For N> 1 one has also an 0(N) connection. It follows that hAm~isantisymmetric in (mn).The base manifold and tangent manifold symmetries act on these two geometrical fields in astraightforward generalization of general relativity. Under general coordinate transformations G withparameters .E” (x, 0) one has= (8nE~r)V,~A+ .E~7r8ITVAA= (8AE~)h,,.m~+ S~8,rh,~”(1)where 8.,~ = 8/0x’~ is a left derivative. Contractions are defined without extra signs if the upper index ison the left. Each contraction is thus a sum over four bosonic and four (or 4N) fermionic indices, and 8Astands for (8/OxTM, 8/OOa). A scalar density D transforms as SD = 8A (EJID)(_)A as one easily checks forsdet V.The local Lorentz rotations L in the tangent frames with parameters L’3(x, 0) act on the two basicsuperfields asSL(L) VAtm =Lm~V;= ~(Lm~mfly1bVfl?~ (2)~ Ir~i. mn__~ tmnJL mryn~L aiim~.‘LI.. Jftfl — ~UflJ_~ nA 1 “A ~ rTensors with indices A upstairs and indices A downstairs transform per definition such that TA TA andT’~T4 are scalars. Thus if STA = LABTB and I5TA = LABTB, then LAB = _LBA. Using our contractionrule we rewrite this as STA = — TBLBA. The matrices LAB are diagonal (have no Bose—Fermi parts) and