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

P. van N,euwenhuizen, Supergravity 307To obtain the supersymmetric extension of (2), one replaces 0,. by D,.c and uses the same productrules for multiplets as in global supersymmetry (since covariant derivatives satisfy the Leibniz rule, justas ordinary derivatives). Thus one constructs the multiplet I® 11 and uses the following actionformula, valid for any superconformal scalar multiplet I (in particular I ® 11) with weight A = 3 (sothat the F component has weight 4) [527]I =fd~x[eF+ ~ey,y + ~ei/i,.o-’~”(A — iy 5B)i/i~], (A = 3). (4)In this way one obtains supersymmetric Brans—Dicke theory.An alternative way to derive this action is as follows. The action 22(1) of global (conformal)supersymmetry can be obtained by first writing I as a vector multiplet [4481V(1) = [A, ~75,1~’,F, G, 8mB, 0, 0] (5)5-component) which isthen givenmultiplying by V(I) times itself, and finally taking the last component (the .12C2ZA+H2+(~B2m(8mC)2~IZ (6)Replacing 8,. by D,.C, and using the following action formula for a vector multiplet with weight A = 2 (sothat D has weight 4) [447]I =fd~xe[I3 — ~K2C(—~R— ~ + more] (7)onefinds the action from V(1) x V(1) + V(par 1) x V(~parI), wherepar I is the parity reflected multiplet(B, —A, —i-y5~,G’, —F’). The result is2(supersymmetric Brans—Dicke) = (—4e)x2+ B2XR + ~ + ~D,.cA)2+ ~D,.cB)2 — ~F2— ~G2+ .. •]. (8)[—MATo reduce to the case of ordinary supergravity, on~fixes the local scale invariance by A = constant, thelocal chiral invariance by B = 0, the local boosts by b,. = 0, and a linear combination of Q andS supersymmetry such that x = 0 as well as 8~= 0. The only linear combination of 0 and S supersymmetrywhich satisfies 8,~= 0 is 6oc(2E) + o~((—F— iy5G — (iI2)A~y5A)e)and the action reducesto2= —~(R+ Rr)_~(F2+ G2)+~A~A2. (9)If one identifies the chiral gauge field A,. with —2/3 times the auxiliary field A~X, and further F =G = —~.P,A = 1 then one recovers exactly ordinary supergravity. For example, 80(2 )F= eøcX andsince ÔSCF = 0 for A = 1, one finds, using that y = —~y,.R~ + ~ the result for ÔS in