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

(A V,.”)00 = ~ s*a = (D,. + ~ A,.ym) —314 P. van Nieuwenhuizen. Supergravitywith the ordinary space transformation= 8,.re~”+ ~r8,.m + ~fiytmf// + A ~ (3)In order to do so, one identifies the parameters as followsL””’(O=O)=Am”. (4)Compatibility for the terms with ~“ and A tmfl is manifest, but for the terms in “ one finds(8,. ~1)Vam(0= 0) + ~a (8aV,im ~ = ytm~f; (5)Thus, V~m(0 = 0) = 0 and V,.tm = e,.m + ~ëy”i~j,. + . One must thus choosea gauge such that these resultshold. Whether this can be done in all order 0 cases considered below, has not been proven but seemsplausible.Repeating the same procedure for the gravitino, we have (see page 217)(AV,.i)= (8,.Er)V,~~+(8~E”)Va°+Er8rV,.a +Ea8aV,.ui +~(L. ~7)abIIIb(6)In this case, 8(sup)*,. must be equal to the 0 = 0 terms in (8,.Ea)V,.~~+ E~8V,.~ and one findsva”(0 = 0) = ~a ~ = ~a + ~ — ~,. )o” +... (7)where m~= —~(S+ iy5F — 2i,.4y5) and ~uses ,~and ~i instead of ~ and w)is the improved spin connection (the algebra simplifies if onema — ma( ,\_ mnp~~ ‘8(O~ —Ui,. l~C~I~I~J3 ~L p~m~to order 0, and finds hatm “ (0 = 0)= 0. The gauges VaA(0 = 0) = h~~(oSimilarlyone can find h,. = 0)= 0can always be implemented by choosing the order 0 parts of EA and A ma approximately.Since Va”(O = 0) = 6,,” and V,,m(0 = 0) = h,,m~(0= 0) = 0, one can require also compatibility for thesecomponents. One finds then to order 0 =0= 0 = (0a~”)~II,.” + (8,,E~)6fl°+ r8,.&r,a + ~8~V,,”+~Atmh2(OmflY~b6,,b. (9)Clearly one needs to know first the parameters to order 0, and then one can determine Vaa to order 0from (9).The superparameters to order 0’~’follow from the knowledge of the superparameters to order0k byrequiring compatibility of the parameter composition rules in ordinary space and in superspace. Insuperspace a symmetry operation with (E2A L2 m~)followed by an operation with (,E 1A, L1 tm~)minus14*2 yields again a symmetry operation with composite parameters