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

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202 P. tan Nieuwenhuizen, Supergravitvparameter will in general become space-time dependent as a result of this Lorentz transformation.Hence local supersymmetry and gravity imply each other and this is the reason for the namesupergravity. However, we stress that this is a heuristic argument only, and that, for example, constanttranslations when written in terms of curvilinear coordinates look also like general coordinate transformations.Nevertheless, the results confirm this heuristic reasoning.A third argument is more solid but requires a little bit of work. If one considers the variation of IN in(15) then one finds new terms due to SA, SB, OA in 5jN~For the terms quadratic in A and B one finds1.~(B)) (17)3(I+It.~) f d~x~(~1.y~ )(T1.~(A)+ Twhere T,,~(A)= 8 is1.A 8~A— ~S1.~(8AA) 2 the energy momentum tensor of the field A. (The AB termswill be discussed in subsection 3, eq. (10).) Such variations can only be canceled by adding a secondNoether coupling, of a new field g1.,. to the Noether current of translations, ~T1.P,and requiring thatunder local supersymmetryK K—Sg1.~= — ~- t/i~y,, — ~- t/i~y1. . (18)Since 1915 we know how to do this kind of Noether method to all orders in iteration: by covariantizingwith respect to gravity (81. -4 D1., °1.v -~ g1.~and a V’~in front of the Lagrangian). Varying in—~V’~ 81.A 8~.Ag 1.~’the metric, one finds ~Og 1.~. T 1.~(A)and the field g 1.,. in eq. (18) can be identified withthe metric tensor.We have now learned two things: First of all, gravity must be present and since in supersymmetryone necessarily has fermions present, the gravitational field is described by tetrads ~ rather than themetric field g1.~.The relation between them is g1.1. = e m 1.e”~8mn~ For details of tetrad formalism, seeWeinberg’s book. Secondly, the transformation law of the tetrad must satisfy (18). This does notdetermine 8e~uniquely, but consideration of the AA terms in 01N yields as unique result8e~= ~K~yml/i,1. (19)(see under (32)). For the transformation law of the gravitino one expects1D= K 1.E, D1. = ~ + ~1.mnUmn (20)since we are in curved space. Here v1. m7’ is a spin connection (see appendix I).The gauge fields of supersymmetry should in particular form a representation of global supersymmetry.(More precisely: the physical states with helicities (+2, +3/2) and (—2, —3/2) described by e ‘ andt/’~.should form two doublets.) This puts another constraint on 8e~and Ot/I~P,which is in agreement with(19) and (20) as we now discuss.The helicity (2, 3/2) representation of global supersymmetry is given by= ~ (iy1.çb~+ ~yvt/ip,), ~*1. = (W1.mht)ijTmn (21)
P. <strong>van</strong> Nieuwenhuizen, Supergravity 203where L stands for linearized and are constant. Since boson fields have dimension 1, one defines thespin 2 field by h 1.~= (g1.~— 81.~)/K in which case the K in (21) disappear. (Thus h1.~is the quantumgravitational field.) Thus (21) agrees with (19). This result was derived in ref. [281]by using only thealgebra of global supersymmetry, the so-called super-Poincaré algebra, which we now derive anddiscuss.If one defines for the fields A, B, A in eq. (1) 80(e)A = [A, ~aQ”]mPm] and 8M (A mn = ~(Amn~Fmn)a~A$etc.,= [A,~Am”Mmn] and defines the onePoincarécan findgenerators the commutators by S~(~)A of Q,=P 81.A and=M[A, using ~ the Jacobi identities. For example[S~( ~),8 0( 2)]A = ~(j2ym~1) 8mA = ~~2ym 1[A, Pm]=[A, 20],ët0] — [A, iTO], 20] = [A, [ë20, eQ]] (22)where we used the Jacobi-identity the last equation. Removing ë2 and ë1 (using that ~ = t 0C$~andthat and 0 anticommute) one finds that the following equation holds when acting on A{Q~Q$} = ~(ym)a 8$pm (23)8(C_l)where Cap~ = Oa5’ and CF = —C (see appendix B). In this way one derives the followingalgebraic commutators and anticommutators when acting on A or B. (For A one only finds the sameresults if one adds auxiliary fields, see subsection 8.)[Mmn, Mrs] = SnrMms +3 more terms[Pm, Mrsl = SmrPs — SmsPr, [Pm, P~]= 0 (24b)F/la 7) 1_A F/la Al 1—( \a çL’I ~T~JU, [~‘ ,lVJmnJk(Jmfl) gt~./(Qa Q$}(ymC_l)a$pm. (26)This is the super-Poincaré algebra. It is a closed algebra since all Jacobi identifies are satisfied (or since anexplicit matrix representation will be given in section 3).The results in (24) are of course the ordinary Poincaré algebra, but (25) states that the charges ~aare constant in space and time, and carry spin 1/2. An explicit representation of 0” for the model in (1)is 0” = fd3xj°~~” with the Noether current given in (11). The really interesting relation is (26); it is thealgebraic analogue of eq. (9). One may check (24—26) by expressing all charges in terms of theircorresponding Noether currents, and using canonical (anti)commutation rules. (For these rules forMajorana spinors, see subsection 2.14.) Since 0” have spin 1/2, a physicist is not too astonished to see in(26) anticommutators, but the appearance of anticommutators leads one outside the domain of ordinaryLie algebras and into superalgebras.1 =The matrix C in (26) must satisfy CYC 7m.T in order that the following Jacobi identities aresatisfied{[Mmn, 0”], Q$} + {[Mmn, Q$], Q’~}= [Mmn, {Qa, Q~}]. (27)Physicists recognize this matrix as the charge conjugation matrix; for further details see appendix B.(24a)
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