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

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326 P. van Nieuwenhuizen, SupergravityTDGOA = T~j~+ TDO GA+ T~ DO+ oO GA(2iR) (15)and substituting this back and multiplying by ~AO~AG one finds TDO = 0. Similar manipulations lead toTGA = T~~j= 0. Finally, with D~in (10) replaced by Dm, a similar analysis yields~ =~(EEAEDO + DA EO)RREDGA=0. (16)This example shows how to use Bianchi identities; much work in superspace is based on deducingconsequences of Bianchi identities.After working out all Bianchi identities, one arrives at the following remarkable result [578]: allsupertorsions and supercurvatures can be expressed in terms of three superfieldsR, G~, WABC.Moreover G~.is Hermitean, WABC totally symmetric while1-’A1~— o — J—’A rs . ui ‘TBCD — g~ U, ~-‘ y-~Aç~ ~—‘AA . — o *DAW~C=DBEG~+B4*C(17)The conditions with DA state that R and WABC are chiral superfields; the last condition shows thatdifferentiating ~ (so that for example the curl of A,. enters without 0 factor) yields the same resultas differentiating G~.(Indeed, G,. (x, 0 = 0) = A,. and again its curl is produced.) The second relationhas a similar meaning; it corresponds in ordinary space-time to the fact that the trace of the Einsteintensor is proportional to the scalar curvature.The formulas expressing TA~and RABtm” in terms of R, G~and WABC are complicated and willnot be reproduced here.In Einstein theory the geometry is specified by stating that it is Riemannian. In superspacesupergravity we defined the geometry by the restriction that the tangent space parameters are theLorentz parameters (to appreciate how drastic this choice is, see subsection 7)n_rm b_1rmn~ ~a a— m_— ~ fl, ~a — 21~~ (9mn) b~ Em — Ca —and by the torsion constraints.Since the tangent group is degenerate the constraints in (1) on the supertorsion lead to two results(i) all spin connections can be expressed in terms of supervielbeins. Thus enough components ofare given to solve the torsion equation. Specifically, from Tr,’ = 0 one obtains hrs’, just as ingeneral relativity, while from T~.,j= 0 one finds h~,.,j(and from ~ = 0 one finds h,~,j).From thesix constraints ~ + TACB = 0 one finds the six ~ using that hBAC = h~ = 0.(ii) The other constraints are therefore constraints on the supervielbein. (In general relativity such aresult is not present since there the tangent group is non-degenerate.) In particular one can express V,,,’5in terms of Va’s. The V,,A can in turn be expressed into prepotentials [273].We now turn to dynamics. The action of N = 1 superspace supergravity is simply [574]1Id4~405~tV.(19)
P. van Nieuwenhuizen, Supergravity 327Let us see what its field equations are. From sdet V = expstr In V one has5I=Jd4xd4O(VA~~5V 4A)(_)*2sdetV. (20)but the variations SVA~must respect the torsion constraints. From the definition of TBCA given earlier itis straightforward to derive (using the index convention of page 313)STBCA = !DHA — 1( — )BCD HA + TBCDHDA H°TDCA + ( )BCHD TDBA + QBC — ( )13C1)A1BCwhere HAB = VAASVAB while ‘= VBA&,bACAi ‘~A(B±11)8AVfJ +U) VII ~PAB1jB~ A (22LJ4Vfl —11 Ti II .~VA — ~14 VA — (PAAThe signs follow by first making the purely bosonic sector agree, and then moving indices such that theyappear on the left and right hand side in the same order. As a consequence, the Leibniz rule8’5(vAuA) = (8’5vA)uA + (_)AAvA8u (23)(21)holds also for covariant derivatives. Of course the ~spin connections h~m~(compare with (18)).One can solve this equation for HAB and finds thencan be expressed in terms of the earlier definedSJd4xd4osdet V=fd4xd40sdet V[VtmTma” _RU_R*U*1 (24)where vtm and U are arbitrary superfields. Since Tm,,” is proportional to Gm, one has as field equationsGmRO. (25)These are the N = 1 superspace supergravity field equations. Since, as we shall show presently,R = S + iP +... and G~= A~X~ they agree with the ordinary space approach.To show how the Wess—Zumino approach can reproduce the ordinary space approach we establishthe bridge by requiring compatibility as before [577].For any superfield V (with any number of tangent space indices) a transformation S is defined bysv = ~DA V = ~A VAA 8A V+ ~~!hAmn(~X~,,V). (26)Hence, S is the sum of a general coordinate transformation with field-dependent parameter e” = ~A VAAand a local Lorentz transformation with field-dependent parameter ~AhAmn The advantage of thisdefinition is that the commutator is proportional to only supertorsion and super curvaturer.g.~1~z—~B~r1~B~A(0 mnly IT CL 1, 2J 525T1[ A, BfS2S1~ AB 2 mn AB C (27)
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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