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

P. van Nieuwenhuizen, Supergravity 221where the hatted symbols are functions of the components of A”. One recognizes the expected generalcoordinate and Lorentz transformation on the tetrad, and the local linear supersymmetry rotation onthe gravitino. It follows that one can use the gauge freedom to eliminate in H the components C,., Z,.,,and the transversal parts of F,., S,.. Hence, one expects as auxiliary fields, 0,.S~,0,LP~and A,..To find how these fields rotate into each other under linear global supersymmetry transformations,one first performs the transformation OH,. = ~GF~L. (see subsection 5.3). Since this will produce nonzeroC,., Z,., etc., one subsequently performs a gauge transformation, which restores the gauge C,. = 0 etc.The sum of both transformations is then the global supersymmetry transformation, which leads to thelinearized version of (2). —To explain why local (linear) gauge transformations are of the form OH,. = Diy,.y4x is invariant under Oh,.~= O,,4,, 5A + we 0,4,. recall since that the inlinearizedsource satisfiesgeneralo,.T”’relativity the coupling f h,.,, T”’ d= 0. In global supersymmetry T”’ is part of a superfield of sources JAA (intwo-component notation) containing also the axial current and the Noether current of global supersymmetry.These sources satisfyDAJ~ = DAS (25)where ~5 is a chiral superfield (DAS = 0). (For the definition of DA, see again subsection 5.3.) IfDAJ~ = 0 would hold, f (DAAA — DAAA)JAA = 0 and one would expect that HAA (the two-componentform of H,.) is invariant underSHAA = (DAAA — DAAA). (26)However, DAJ’~ = DAS, hence we must restrict AA such that AADAS vanishes. All these termsappear in the action and using well-known results of global supersymmetryJ d4x d4O AA DAS = —f d4x d4O (DAAA)S= Jd’~xd2O~D”DA[(D~”AA)S]one arrives at the quoted result for OH,..1.10. Field equations and consistency= fd4x d2~(1YDAD’~AA)S (27)The field equations for the gauge action are obtained very easily by using 1.5 order formalism.Indeed, in the action2’ = —~eR(e,w)—~ ~““l/i,.y 2 — A~,) (1)5y~D~t/f,, — e(52 + Pwe may again consider w,.m”(e, i/i) as an inert field. Thus one easily derives for the field equations