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

360 P. van Nieuwenhuizen, Supergravity(em,., ~ x” A1, S, P11, V,,,, A~,,1,A~)(‘I’~,x2~A2, B,1,2, t,1,~,,,P52. M12, N’2, ~ (3)The first multiplet is, in fact, the nonminimal set of auxiliary fields of refs. [68, 69, 160] for N = 1supergravity. The second multiplet is a spin (~,1) multiplet, which, however, does not coincide with thespin (~,1) multiplet of ref. [347],one characteristic difference being that in the latter there are auxiliarygauge fields (see subsection 1.8, eq. (11)). A check on the set in (1) is that by covariantly quantizing thetheory with auxiliary fields according to the procedure of subsection 2.2, and then eliminating theauxiliary fields by means of their field equation, one obtains the same result as when one quantizes thetheory without auxiliary fields [473] (see subsection 2.8). This equivalence was the basis of theconstruction of refs. [228,229].As expected from the N = 2 model without auxiliary fields, the transformation rules as well as thePoincaré action (but not the de-Sitter action) are manifestly SU(2) invariant. From now on we use SU(2)notation.Based on the auxiliary fields one can define the following multiplets(i) scalar multiplets: (A”,, x”. F”,). These multiplets have nonvanishing central charges (a = 1, 2);(ii) chiral multiplets (in superspace: (1 + y5)D”14) = 0) with complex and chiral (A,, B,, C, Fmn =~,A,);(iii) vector multiplets (the sum of (0, ~)and (~,1) multiplets) containing (A”, B”, F”, G”, tfr’, W,.);(iv) linear multiplets: which are a sum of an ordinary (0, ~)multiplet and a (0, ~)muitiplet in which anantisymmetric tensor represents the spin 0 field (the simplest case is (T~,A, B”, di’, F, G));(v) nonlinear multiplets: which is the first known case where matter fields transform into squares ofmatter fields.Due to space limitations, we cannot discuss these multiplets in further detail, see refs. [170, 171].We now discuss how the notion of N = 2 superconformal gravity sheds light on the structure of theseresults. In N = 2 conformal supergravity one expects as fields which gauge the algebra SU(2, 2/2)(em,., wm0 b,,, f,., ip~,~, V,,’,, A,,) with V,.’, = _(V,,)*~,. (4)One imposes the same constraints on the (modified) R(P), R(M), R(0) curvatures as discussed insection 4. Elementary counting then reveals that one lacks 1 auxiliary spinor field (at least). The auxiliaryfields which one must add, turn out to be: T,d,” (6 fields) + D~(1 field) +x~(—8 fields). Thus one has atthis point a closed algebra for N = 2 conformal supergravity with (24 + 24) fields. (The Weyl multiplet.)All previously mentioned multiplets can be coupled to this gauge action in a locally N = 2 supercovariantway. If one couples the vector multiplet, one finds (32 + 32) fields and a field-dependentcentral charge in the {Q, Q} anticommui,ator which only acts on the photon (as a Maxwell transformation).One can then also couple a scalar multiplet (which needs a central charge). Fixing thenon-Poincaré symmetries by fixing most of the components of the scalar multiplet and some of the Weylmultiplet, one finds a different set of auxiliary fields from the one in (1), although it also contains(40 + 40) fields. As action one finds the sum of the vector multiplet action and the kinetic multipletaction for the scalar multiplet (see section 4). The de-Sitter N = 2 action results if one adds the massterm of the scalar multiplet.The original set in (1) results if one couples a vector and a nonlinear multiplet to superconformalN = 2 gravity.