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

P. van Nieuwenhuizen, Supergravity 325(1 — ys)a,,J~b4,(x 0) = DA~(X, 0) = 0 (6)then also {DA, DB}4 = 0. From the definition of torsion and curvature this implies‘7’ C ‘1’1AB ——The first constraint” is part of T~= 0, the second follows from Ti,,,’ + ~(Cy’)~b= 0. Actually, ifDA~CD= 0 one also finds R.i,~m’”= 0 and also this constraint follows from the torsion constraints as aconsequence of the Bianchi identities. Thus the set of constraints imposed on the supertorsion seemsreasonable, but they are postulated, not derived from some general principle.The Bianchi identities follow from[DA, DB}, D~}(— )AC + 2 cyclic terms = 0 (8)by taking separately the terms proportional to DE or DE. Thus one findsin,, ,‘,‘r D ,nn —A BC TLi4~~ DC —DA TBCD + 2TABETEc~)+ ~RABCD = 0 (9)where one must add terms cyclic in ABC with the proper signs. The symbol RABc’~DD is RABm’”Dn ifDD is bosonic and RABm’”DD( kmn)°cif DD is fermionic. The minus signs are due to the fact that theindex D of DD is a lower index.As an example, consider the following identity[{DA, DB}, D~]+ [{DB, D~},DA] + [{D~, DA}, Dfi] = 0 (10)where A,B, C are 2-component spinor indices (see appendix). The terms proportional to DD yield with(C-2’ ),~— 0RABmn~[X~fl,D!~.]+ [(Cym)BCT~AI) + A 4*B]DD. (11)Since [Xmn, Da] =OmnY’a1)t, one has in 2-component notationMoreover[Xmn, DE] = ~(0m, ~ — m ‘-~ n)DF. (12)D mn — I m,EF n.FF~ .P~AB — 4O~RAB,EE.FE = EERABEF + EFRAB,EF. (13)We arrive at (symmetric parts are underlined)~ -= TBC,A.O + TAeBD. (14)Decomposing the torsion as