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

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366 P. <strong>van</strong> Nieuwenhuizen, Supergravitvusual SO 3 generators (cr1, cr2, cr3). This representation is unique up to the signs of y~and y5. We choose/0 1cr,,\ /0 1\ 7~~j /1 —1 0Ykl~j~,, 0 )‘ ~ o)’The Lorentz generators o,.,, are diagonal, and under Lorentz transformations (A P)0 = (exp ~w cr)”,,A b(x’Y = [exp(w+ iv)(io./2)]~~BxB, (~‘)A= [exp(4~,— iv)(iu/2)]A8~B. (1)We define (fl’)A and(A’~by ~x’Y(n’)A = X”fl~and (A’Y(C’)A = AA~A.Note that w’ = k11~wJkand v’ = iw4are real. Thus (xF)A = LABXB with L E SL(2, C) and ~A transforms as ~‘A = 1AB~Awith lAB the Hermitianinverse of L.it is straightforward to prove that ~, ,.~, ~, ~4j~are invariant tensors under SL(2, C). (Afortiori, they are invariant tensors of the subgroup SU(2), and ~A = ~B BA transforms as ~ ForSU(2) one therefore denotes tensors which transform as the complex conjugate of tA by tA, and forSU(N) a similar convention is adopted although for N >2 no EBA exists. Instead one has the sameresult that (~A)*transforms as ~A =11BA~For SL(2, C) the transformation~Bçj~~ properties if oneof considers the groups USp(N) with symplectic metric(~~~)* ~.A differ from those of ~A, but here anotherrelation exists, namely (A,j* transforms as ‘~~B.)We make the at first sight unusual normalization— A.B...... •. AE.....1 ~ — —~—f —--~ ~—- —i vOI’ — , —6AB and C is a Hermitian matrix. The tensors are what the metric ~ is in fourThus ( AB)tdimensions.=They raise=and lower indices as follows XA = XBEBA XA = ~~XB (idem with A, B). In orderthat CAB = cA DB, one finds that 12 = 12, and we always contract as indicated: from left-above toright-below (defining XA = XB BA would give the opposite sign).It is straightforward to prove that XA transforms as (~A)*, and ~A as (x~4~*.This is really thejustification of denoting complex conjugated spinors by dotting their indices. Thus we define generaltensors XA and ~A by(x’Y = (L~~~B)*xB _LAi~xBIr’\ _iT.B\*r _i Bs-~)A2,’A )~BtA~B.(3)Another invariant tensor is OAB and SAB. In fact ABEBC = —O’~’c,so that the 0 and tensors arereally the same abstract tensors. The charge conjugation matrix in this special representation is diagonal/ jU3 0\C=y4~y2=k0 ~AA). (4)Thus a Majorana spinor satisfies ~A = (C4*.The gamma matrices define 2 x 2 matrices which map an upper dotted index into lower undottedindex, and vice-versa,— ./ 0 ~(5)
P. <strong>van</strong> Nieuwenhuizen, Supergravity 367One thus finds= (o’, il)’ 48 and (U,,)AB = (u, —iI)~ 8. (6)Since (cr,. )AB are Hermitian (which means antiHermitian for s = 4 in our conventions), lowering itsindices to (O,,)c~one finds the transpose (or complex conjugate) of (0,L)AB, namelyNote that(u~)co (cr” )‘4~ ~ = U~Dc. (7)48(o~”)nc= 0”~c (no sum over it). (8)(cr,.)’Raising of the index ~ is done by 0”~= (+, +, +, +). with ,a, ii = 1,4 in our conventions.The matrices (o~J’~and (o~,.)’48are invariant tensors of SL(2, C). That is to say that, for example,(cr~’~48)’ = L’4cL~L~,.u~’~ = U~~AE (9)One can thus express tensors with Lorentz indices into tensors with spinor indices as followsT~..:-* T~: (U,. rM(O-”)RRT~..:. (10)Often one writes TRR ~ as TRSRS, but one should not confuse this with TSRRS.The connections (cr,.~8 satisfy the following completeness relations~CD D C(o,,)An(o ) —20 A0 g(cr,. )AB(U~)cD = 2 A~BD= (ci,,)BA(O~)Dc (see (7))(U,.)AB(U~~)BA= 20~= (u,.)AB(ci~)~~ = (0,.)~8(0.o)~4B(cr,,Y~ 8(cr~)’~ = 2 ~ B1~~ = (cr,,)~(o.1~f The reason we defined ~~B= —1 for A = 1, E = 2, is that with this definition no minus signs occur inthese relations. One can use them to go back and forth:r’ I AR CI)i,r,,,. — CT,. FABCD(12)FARCD = F,.,XJ~AE1Y = ~(11)where FBADC= F,.,.(o~”)8’4(o-”)DC.In fact, quite generally VAJ3 = VEA, as one easily proves usingUMAB = ciBA and o~~4B=0RA (13)An antisymmetric tensor GAB = — GRA is clearly proportional to EAR. Hence, for an antisymmetric
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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