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

214 P. van Nieuwenhuizen, Supergravityml/Iwith the Einstein tensorand a term with the dual of the Riemann tensor‘qk,m A 11~’~A ëyA Dt/I A 77~b~1-Using 00~= ~ the terms with the_Einstein tensor cancel. The w field equationexpresses the torsion DOtm (=—D,.e”, — Due”,.) in terms of t/’yt/iDOtm mi/i. (17)—~-~,~k!m1/i A ysycrkllIi t/yThus one is left with02 7,)’ A Dt/i A flab11 — ~iYSYm A Di/1DOtm — ~I/i A ysy~DçfrA ~ym~fr (18)From here on, the proof proceeds in the same way as before.The Lagrangian density varies into the following total derivativeS2”d,.K~, ,c1.—ëy1.a~”D~tfr,,. (19)This shows clearly that supersymmetry is not an internal symmetry, but a spacetime symmetry, justlike general coordinate transformations, where 02 = 8,.(2~~). In both cases 82 = 0 on-shell.Finally, we note that the linearized action is, of course, invariant under global transformations whichare linearized in the fields. However, not every action quadratic in fields which is invariant under globaltransformations which are linear in fields, can always be extended to an action which is invariant underlocal transformations (which usually then becomes nonlinear in fields). A counter example [478] isN = 1 supergravity in d = 11 dimensions with as fields a tetrad, a gravitino and a photon ~1.8. Auxiliary fields in global supersymmetlyIn globally supersymmetric models such as the Wess—Zumino model, one needs auxiliary fields inorder that the algebra of global symmetries closes. It is only with auxiliary fields that one can obtain atensor calculus, see section 4, while also the quantum theory becomes much simpler with auxiliaryfields, since in this case the transformation laws are linear and one can easily derive Ward identities forone-particle irreducible Green’s functions. For supergravity the auxiliary fields are not only needed forthe same reasons, but in addition one needs them in order to apply the usual covariant quantizationmethods of Feyman, De Witt, Faddeev—Popov, ‘t Hooft and Veltman and others.Let us begin with global sypersymmetry. That one needs two scalar auxiliary fields for theWess—Zumino model might be expected by counting field components4 fermionic components A”2 bosonic components A, B.Thus one needs at least two extra bosonic fields in order to have equal numbers of bosonic andfermionic field components off mass shell. The action reads2= —~[(o,.A)2+ (a,.B)2 + £8’A — F2 — G2] (1)and it is rather easy to show that it is invariant under