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

330 P. van Nieuwenhuizen Supergravitysuper Jacobians of each chiral space be constraint. This adds global chiral or scale transformations tothe symmetry group. If one has no constraint on AF’ and A” at all, one finds that this leads to conformalsupergravity.Thus the transformations in the chiral superspaces can be written as (see appendix F)= x” + ~A~(x”+ iH~(x,0), ~(1+ y~)0)+~AF’*(x~— iH”(x, 0), OR) (5)and similarly for 0””. For the axial super field one hasrJ’~’I ‘ a’\—_I ‘,. — ‘F’ \—_( F’ — F’ii ~X,v 1—2.~X L X L X R= H”(x, 0) + A~(x+ iH(x, 0), ~(1+ y~)o) + c.c.}. (6)The crucial point is now that by expressing x’ and O’in terms of x, 0 one gets an equation for SH”(x, 0)which determines how the components of H” (x, 0) transform. After choosing a gauge which puts thefirst two components of H” (x, 0) equal to zero, there remains still a class of transformations whichmaintains this gauge. As we shall see, these transformations are just general coordinate, local Lorentzand local supersymmetry transformations, while the remaining components of H” are the fields e m,.,~,a s~P, A,,, of simple pure supergravity with auxiliary fields. Thus the minimal set of auxiliary fieldsarises naturally from this chiral superspace approach. However, although this approach is perhaps theclosest possible to the ordinary approach in normal space-time, one still has to fix the gauge such thatH” takes the form mentioned above. On the other hand, no constraints are needed here, whereas inother superspace approaches they are needed (for one reason, to exclude spin 5/2, 3 etc.).The geometric picture is thus that one has in the combined 8+ 4 dimensional space (XL, XR, OL, OR) a4+4 dimensional hypersurface defined by,. — ,. ,.,. ,.1 ~ j~. ~ LIX L X R — “ ~L X R, ~L, (FRWhen one shifts points by a general coordinate transformation, the hypersurface itself is deformedaccording to (6) such that the new points lie on the new hypersurface.The idea to consider the imaginary part of the bosonic coordinates of the chiral superspaces as adynamical field, i.e. eq. (7), comes from global supersymmetry. There, a chiral superfield is defined byDa(1 — 7S)ab4,(x 0) = 0, D~= (8/80” + ~(ö%) 0) (8)whose genefal solution is easily seen to be47 = 47(x1., OL), X”L = X~— ~O7”7~0, OL = ~(1+ y~)0. (9)Usually one considers the X”L and X”R = x” + ~0y”y~Oas some convenient basis inside the 4+4dimensional space (x~,0”). However, in order to generalizeX”LX”R” 40770 (7a)