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

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336 P. van Nieuwenhuizen, SupergravityContravariant vectors with fiat and curved indices transform per definition as follows= 47B~A + 47A~A (4)5471% = ~~ 1%E47’ + 47A~I (5)where ERA are completely arbitrary functions of z. The law for 5471% follows from requiring that41%gn 11be a scalar. The internal symmetry group ~1I is at this point arbitrary.11~iCovanant vectors are defined by requiring that VJA47A, 47Acu,A cu/Açb1% and çb’~fr.jbe scalars. A scalar Stransforms as SS = ~81%S= (8AS)e. One finds, for example,— Bj ~j.(NPA~EA (PB~(PA.Ac— A Bf_\(A±B)A~ ~A(‘(PA ‘PBEA )Note that (— )A47A transforms as 47A. This allows one to ,switch to and from hatted indices, bymultiplication by (~)A.1?AE is introduced by requiring thatNext a tangent space metricA B47 ~i/i with ~ = (~ flBA (7)be a tangent and world scalar. This determines 5~j~and shows that ~moreover that the tangent metric be an invariant tensor, one findsis a tensor, but requiringe g~_ C ,‘ \B(B+C) Cj— U — EA 11CB — ?JAEEB vClearly, flAt is a constant tensor, and clearly, the tangent space parameters ~= EA~i~( — )B, must satisfy the following symmetrywhich are defined byEAB+ BAO. (9)This thus restricts the parameters for given flat metric.One can lower flat and curved indices of contravariant tensors in two ways, by putting gIJA and ~on the left or on the right of a given tensor. We define= 471g1,4, cbA = 47BnBA47A11A1147— 11 — Bçb1%—gnnçb ,Clearly, 47A = (— )‘~47~. These objects are good covariant tensors as defined above (6). For completenesswe give the transformation rules of curved covariant tensorsS4~A= 47~S~1,A+ 47A~SS~1— I Ii \jt±1 ( )8471% —çb1%g +ç6~~ A~)The last term can be rewritten as 47~I~1%in which case all indices match on both sides of the equation.
P. van Nieuwenhuizen, Supergravity 337In general one contracts indices as in the four contractions above (6). One first writes down anequation so that it is correct in the bosonic sector, and then adds signs so that it is correct for all sectors.As an exercise the reader may derive47An = (— ~ (12)We now come to the important point of specifying the group in tangent space, i.e., the determinationof ~. At this point a (weak) assumption is made: the tangent group contains ordinary constant Lorentzrotations. Under ordinary Lorentz rotations Sv = A m,,v” and Sv” = ~(A. ~y)~2~~b• In superspace 55A =SBEBA per definition. Hence for the bosonic part Am~= ,,m while for the fermionic part Eb” = ~(A .Notice that we always use gamma matrices with indices as in (ym )cd.O-)Taking in 8i~~in (8), A and B equal to m and n, one easily derives that urn,, is a multiple of theMinkowski metric. On the other hand, taking A = m and B = b one finds that the Bose—Fermi part ofi~ vanishes. For the Fermi—Fermi part of the tangent metric one finds with A = a, B = b from (8) withthese special Eb0b.— i,0~( ~ )“b = 0. (13)Hence, i~must commute with the Lorentz generator. The general solution is therefore1lcd = a[C(1 + ibys)]~d (14)where Cat, is the charge conjugation matrix.Hence, there is a 3-parameter class of solutions of ~ in (8) for the special case that is equal toconstant Lorentz rotations, namely [d~mn (Mink.), aC0~+ ib(Cy5)~~].We now pick one element of thisclass as the metric in tangent space. It is at this point that the difference with supergravity occurs,because in supergravity one keeps all 3 invariant metrics, as a result of which the tangent group ofsupergravity is the Lorentz group. Also the coset approach of page 319 leads to this result.Substituting this form for the tangent metric, one finds that the tangent group is defined by leavinginvariant d47 m n~,,(Mink.)~!i~ + 0(1 + iy 5b)~i.This shows that the local tangent group is Osp(3, 1/4N).(For b = 0 this is the standard definition for b~ 0, see below.)In gauge supersymmetry, the only field is the metric g,~. There are no independent matter fields; thecomponent matter fields appear in the 0 expansion of the metric. Therefore one can describe the theoryequally well in terms of the metric or of supervielbeins. However, in order to make contact with otherformalisms, we introduce a supervielbein V1%~’’by the definition— A \I7(I+B) Bgnn— Vn ua~() V,i . (15)The transformation properties of V1%” are as indicated by the indicesSVAA = ~(A+IXA+l)vA~I + ~ + VABER” (16)Knowing how V4”, g~and ,~ transform, one finds that the relation between g and V is a correcttensor relation. One can now scale the y~in the tangent metric away by rescaling the super-vielbeins.To see this, note that (a + iby5) = (a 2 + b2)1”2 exp(iay 2 + b2)112, and redefine 47 =5) with cos a = a(a
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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