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

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324 P. van Nieuwenhuizen. Supergravitytransformation. Only if A has no external indices, one can see that F* + RA again satisfies the samechiral constraint as A does, namely eq. (6). Hence, it can be viewed as the lowest component of a newmultiplet, namely the kinetic multiplet.Similar ideas can be used for vector multiplets (non-chiral superfields A) or for linear multipletscorresponding to superfields and satisfyingDC 1DA = (Dy 1D)A 0. (17)5C~D)A= (D~~mnCFor applications of these ideas to supergravity see P. Breitenlohner [68,71].5.5. The Wess—Zumino approach to superspaceIn this approach, the tangent group is the ordinary Lorentz group. The geometry is thus: generalsupercoordinate transformation in the base manifold, local Lorentz rotations in the tangent manifold,and one has supervielbeins VAA and superconnection hA’”. These superfields are not arbitrary, butsatisfy the following constraints (as we deduced earlier from the ordinary spacetime approach but nowpostulate)pt_~r:_pc_pm~Yc’m\ —1rs — 1as — 1ab — 1ab ~4~’-~Y)ab —We now discuss how one might arrive at postulating them. First of all, notice [605]that for a tensor8A(sdet VwAVAA)(_)A = sdet V(DAw~ô+2wBTBAA)(_Y. (2)To prove this relation, note that 5(sdet V) = (sdet V) (5 VAA) VA’~(—)Atransformationso that for a general coordinate6(sdet V) = 8.,~(~A sdet V)(~)A (3)This expression can be written as11 VIJA)VA.4( )A(SV/~)VAA(_ )A =(Sfo11V1A + 8A5= 8A(SVfJ)VA’(~) +EH(8mV4A_(_)~~8nV,~(*)VA1t(_)1t. (4)LOne can replace 8A by DA in both terms, while the extra terms needed in replacing Oj~by D~cancel~nce (Xmn)AB is traceless in (A.B). Identifying wA with 511 Vil4 the result in (2) follows.Hence in order that one has just as in general relativityJ d4xd4Osdet VDAwA(_)~=0 (5)oj~ehas the requirement that TBAA( — )A = 0. For B = b this follows from the constraints, for B = m it istIue jf Tma” = 0. This is indeed the case and follows from the Bianchi identities [578]if one assumes theconstraints on the torsion. We will discuss this below.Another interpretation is that chiral superfields can live in superspace [224,421,452, 453]. Indeed, if
P. van Nieuwenhuizen, Supergravity 325(1 — ys)a,,J~b4,(x 0) = DA~(X, 0) = 0 (6)then also {DA, DB}4 = 0. From the definition of torsion and curvature this implies‘7’ C ‘1’1AB ——The first constraint” is part of T~= 0, the second follows from Ti,,,’ + ~(Cy’)~b= 0. Actually, ifDA~CD= 0 one also finds R.i,~m’”= 0 and also this constraint follows from the torsion constraints as aconsequence of the Bianchi identities. Thus the set of constraints imposed on the supertorsion seemsreasonable, but they are postulated, not derived from some general principle.The Bianchi identities follow from[DA, DB}, D~}(— )AC + 2 cyclic terms = 0 (8)by taking separately the terms proportional to DE or DE. Thus one findsin,, ,‘,‘r D ,nn —A BC TLi4~~ DC —DA TBCD + 2TABETEc~)+ ~RABCD = 0 (9)where one must add terms cyclic in ABC with the proper signs. The symbol RABc’~DD is RABm’”Dn ifDD is bosonic and RABm’”DD( kmn)°cif DD is fermionic. The minus signs are due to the fact that theindex D of DD is a lower index.As an example, consider the following identity[{DA, DB}, D~]+ [{DB, D~},DA] + [{D~, DA}, Dfi] = 0 (10)where A,B, C are 2-component spinor indices (see appendix). The terms proportional to DD yield with(C-2’ ),~— 0RABmn~[X~fl,D!~.]+ [(Cym)BCT~AI) + A 4*B]DD. (11)Since [Xmn, Da] =OmnY’a1)t, one has in 2-component notationMoreover[Xmn, DE] = ~(0m, ~ — m ‘-~ n)DF. (12)D mn — I m,EF n.FF~ .P~AB — 4O~RAB,EE.FE = EERABEF + EFRAB,EF. (13)We arrive at (symmetric parts are underlined)~ -= TBC,A.O + TAeBD. (14)Decomposing the torsion as
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