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

P. van Nieuwenhuizen, Supergravity 361In both, inequivalent, (40 + 40) sets, Poincaré supersymmetry follows a 0 + S rule (except that for (1)it becomes a Q + S + K rule and for the set which yields the de-Sitter action, it becomes even aQ+5+K+U(1)rule).Aconformal tensor calculus has been developed forN = 2, from which onededuces the N = 2 Poincarétensor calculus. Similar results for the Poincaré tensor calculus based on the set (1) have been obtainedby Breitenhohner and Sohnius from superspace methods [72].It is gratifying to see how starting from a larger symmetry (N = 2 superconformal theory) with fewerfields, one can descend to smaller symmetries with more fields in a systematic fashion. Although rathertechnical, all these results reflect the N = 1 results, and the reader should have no conceptual problemsafter studying section 4.Subsection 6.6 is added as an addendum7. AppendicesA. Gamma matricesThe Dirac matrices we use satisfy {y,,, y~}= 2g,,~with p, i’ = 1, 4 and ys = Y1Y2Y3Y4 with y~= 1. Allfive matrices are Hermitean. A basis for 4 x 4 matrices is given by the 16 elementsO,=(1,y,.,2io,.,,iy 5y,.,y5) with p’(v (1)which satisfies tr(O,O,) = 4S~.We define r,.,, = ~[y,.,y~]so that they satisfy the Lorentz algebra[o,.~, cr,,~,]= g,,~ a~,.,,+ 3 other terms. (2)tm0Umn. Gamma matrices with Latin indices areFurthermore, constant, but with we Greek define indices 0 = D,.y”, one has and~,,. A .~ = A The matrix y~is always constant.Some useful formulae areC abcdU = 2Y50~cd (3)YaYbYc + YcYbYa = 2(c50acyb) (4)0bYC + ObcYa —YaYbYc — YcYbYa = 2 abcdY5Y’~, (~1234= +1). (5)The result for YaYbYC follows by adding the last two equations. ThusY,’Uab = ~(e0,.y6 — eb,.ya + ec,,eO&dy5y~i). (6)Other identities often used are= U”~YaUcd = 0 (7)= ~3, O~”tT~dO~b = (8)