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

P. van Nieuwenhuizen, Supergravitv 209Substituting (9), one arrives at~ ~/i)= ~ — ~,.ynl/im + ~rny,.ll’n). (ii)Using the second tetrad postulate, 8,.e~+ ,.a~ — F~,.e”~ = 0 one can obtain an expression for F”,.~from eq. (ii). In general one defines torsion as the antisymmetric part of F”,.~= ~(F~— F~,.). (12)Clearly, F”,.~— F~,.= —K,.”~+ K~a,.,so that the torsion is given byS”,.~= —~-(t/i,.y”çfi,,). (13)In a similar manner as in eq. (10), one can solve from D,.(w(e))e’ —~s~-* ii = 0 the dependence ofw~,b(e)on the tetrad field. (Another way is to solve directly the tetrad postulate 8,.etm,, + w,.m”(e)e~~ —= 0.) Putting this result together with eq. (ii), one finds~ i/i) = ~ — R,.mn + Rmn,.)R,.p,m = —8,.e~~+ 8~e,,,,.+ ~- çl’,.y~i/i~. (14)The symbol R,.,b.m is defined to be ~ similarly for the other two R-functions. Clearly, the spinconnection w~”’(e,t/i) is supercovariant, by which is meant that if one transforms it under localsupersymmetry transformations, then it contains no 8 terms. Indeed, the terms (—~8,.ë)y”i/i,.comingfrom 8(—8,.em~)cancel against the 8 terms from Oil’,.. The symbols R,.~.rnare supercovariant bythemselves. That the solution of SI/Ow = 0 is supercovariant is an accident. For example, in theextended supergravities this equality does not hold, see subsection 6.2.1.5. Flat supergravity with torsionIn Einstein—Cartan theory, the Hubert action has a well-known symmetry f d~xeR(e, w(e)+ r) =f d~xe[R(e, 00(e)) — T + 1. (rAA,. )2] under w,.”~’1.V~T’~~0)ab + r,.~thwith T,.”~’= — ~ba but further arbitrary.The proof is trivial if one writes the terms linear in r in R,.,~flb(w(e)+r) as D,.(w(e))rpab — p.~*-~ 71 sinceone may add a Christoffel connection to D,.r~~5(because it cancels anyhow in the curl) and having thusobtained the full covariant deriv,ative, one can partially integrate to obtain a total derivative.In supergravity a similar identity holds. It is a matter of a small computation to show that [207]2’~ 2~(e, w(e, ~/~)+r)+2’312(e, t/’, w(e, t/i)+ r)= 2’~2~(e, w(e, I/i)) + 2’312(e, i/i, w(e, t/J)) + ~-~--~ (r,,~~r”’~ — (TAA,.)2) + total derivative. (1)Since w(e, I/i) is an extremum of the action, terms linear in r cancel.