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

366 P. van 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)