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

P. van Nieuwenhuizen. Supergravitv 261For N = 1 supergravity, one finds that the R2-type invariants as far as kinetic terms are concerned,are given by (either by the Noether method or by the tensor calculus)~2’2=R2+R .yJy.R—4(~S)2—4(a,,P)2+4(a•A)2= (R,~,,2—~R2)—~I~(L16,W — 9,,~9,.XR”...~“y R)—~(8,~A,. —(2)For the gauge action we have, of course,~ =R+ifr R+~(S2+F2—A~,).To find the physical states in this system, it is advantageous to use spin projection operators in order todescribe the terms bilinear in fields. In this way one finds with (see subsection 1.13)RM,. = ~(EhM,. — h,,,. — h,.,, + h.,,,.) = ~(F2+ \/~p0.ts + PO.S)h,,,,,eR = L1O,~,.h,,,,.— ~h,,,.(L~F2— 2LIJFO.s),0.h,,,,. (3)R 2 = 3h(LI2P°”)h, R,~,.2—~R2 = ~h(LIFF2)hthat the graviton propagator becomes P2(~yE+ ~$L12Y’+ FOs(_~yE+ 3aE2)~’,so that the gravitonstates are:(i) a massive spin 2 with M2 = —~yIJ3and ghost if /3 0(iii) a massive spin 0 with M2 = ~y/a and physical if a > 0.For the gravitino part one finds (see subsection 1.13)4’ R = ~(P312— 2P1”2~)J4,~ yJY• R = ~121i(LIP1”2”)J4’ (4)4i~,LI&”~(R,, —~y,;y. R)~(EP3”2),~’4’that the propagator becomes —P3”2[J(f3L1 + ~y)]’ + P~’2[J(aEJ+ ~y)]~. Thus the gravitino states are:(i) two massive spin 3/2 with M2 = —~y//3and both ghosts if /3 0(iii) two massive spin 1/2 with M2 = i1syla and both physical if a > 0.Note that at E = 0 both P2 and ~O.S(and F3”2 and Pl~2~s) define the usual tetrad and gravitino states.To check these results, note that ~2’3 contains only highest spin (as expected from a conformalinvariant) while ~2’,yields the usual spin 2 and 3/2 propagators if sandwiched between conservedcurrents. The relative factors between the graviton and gravitino projection operators in ~2’2and ~2’3arethe same.If one considers the spin 1 and spin 0 states of the auxiliary fields (which propagate in 2’2 and 2’3!)one finds for the A,,, propagator P’(~y+ ~/3[])~’+F°(~y — 4aLJ). For the fields S and P one finds twomore massive spin 0 states. All these fields clearly constitute(i) the usual (2,~)massless physical multiplet;