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

318 P. van Nieuwenhuizen, Supergravity5.3. Flat superspace*Supertorsion and supercurvatures are defined by the (anti) commutator of two covariant derivativeswith flat indices. In global supersymmetry one can also define covariant derivatives, and from their(anti) commutators one finds that global supersymmetry has torsion but no curvature. This flat space limitof curved superspace was obtained in the last subsection by using gauge completion; in particular wefound that Va tm = ~ Here we will obtain the same result by deriving the covariant derivatives inglobal supersymmetry and by using the method of induced representations, following Salam andStrathdee.Consider a group element of the formgL(x, 0, i~)= exp(~Q+ xmP~+ ~maMmn) (1)Consider next a new group element, obtained by left multiplicationgL(x, 0’,~‘)= exp(~Q+ ~mPm ±~Amt2M~fl)g~(x, 0, ‘q). (2)Using the Baker—Hausdorfi formula, one finds an induced representation of the generators F, 0, M onthe coordinates x, 0, ~ of the group manifold, which is 14-dimensional and infinitely nonlinear.To find an induced representation on superspace with coordinates xm, 0” we consider the cosetspacesexp(~Q+ XtmPm) mod M. (3)By mod M we mean exp ~Atm”Mmn with any Am~Any group element of the form (1) belongs to a coset.To see this explicitly, note that, using= e_ RJ/2+ eAeB (4)where A = 00 + x P and B = . M, we can factor the M dependence out. This is possible, since inthe multiple commutators [A,B],B]... one always encounters only P and 0 while [M,M} -~M (Inother words, M ~ (P + 0) is a semidirect sum. The decomposition (M + QL)~ (P + OR) yields asymmetric algebra.) We now considergL(x,0)=exp( Q+~P+2A .M)exp(O. Q+x .P) modM (5)with antihermitian 0, P. M. Usingt2~ if A””’” =0 (6)eAeB =e~~~”eAeB =e~j~~eA jf~_~:_Ø (7)* I thank Dr. Ro~ekfor useful discussions concerning this subsection.t Eq. (6) is the usual form of the Baker—Hausdorif theorem. Eq. (7) follows if one expands eAeBe_A = exp(eAB e~4)in terms of B. Eq. (4)follows from (6) if one uses (6) twice.