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

P. van Nieuwenhuizen~Supergravity 345[2821). Some chiral—dual symmetries were first found in another model which was not finite butdisplayed miraculous cancellations [510]. For a complete treatment of dual—chiral symmetries, see ref.[3821.An explanation of this finiteness was given by using these symmetries in this model which wenow discuss.According to the analysis of Haag, Lopuszanski and Sohnius (subsection 3.3), the maximal symmetrygroup of the S-matrix of supersymmetry algebras containing the Poincaré algebra is U(2). In fact, thefield theory has this global symmetry only on-shell (i.e., only the field equations have an U(2) globalsymmetry). Off-shell, only an SU(2) remains. These symmetries read [382]547,.L = iw $,.‘- — ~ = ~(1+ y~)i/i,.= ~ ~ ~I’~— kg)’ ~,.R = ~(1— y 5)çb,.U(1): Si/i,. = —iy5~’,, and SE,.,, = ie ,.,.~F”. (4)Thus the SU(2) part rotates ~,L as (2) and i/,’~as (~),while the U(1) part is a combined chirality-dualitytransformation. To prove the U(2) invariance, note that in the gravitino action i/’L couples to çIITRC sothat these terms are SU(2) and of course chirally (U(1)) invariant. The torsion is separately U(2)invariant, and, finally, the remaining terms can be analyzed by eliminating the F~,.kinetic terms throughthe equation of motion for F,.,.. This cancels half of the Noether coupling with bareF,.,, and the result isthat the action is the sum of the tetrad and gravitino actions plus the following term4V2 çi’,.[eP”~ + ~y5F”~]çb,,’E”. (5)Clearly, also this last term is U(2) invariant. The SU(2) invariance holds off-shell, but for N >2 only the0(N) invariance holds off-shell.An interesting connection between the coupling of photons and the cosmological constant was foundby Das and Freedman [108]and Fradkin and Vasiliev [225]. When one couples the photons minimallyto the fermions one needs at the same time a cosmological constant and a mass like term in the action.Thus it would seem that electromagnetism is due to the curvature of a de-Sitter universe. Actually, thiscosmological constant is much too large, and can be eliminated altogether by spontaneous symmetrybreaking, as we shall discuss in subsection 5.For the N = 2 model one can find this extension of the action by adding a cosmological constant tothe action and finding the further modifications by the Noether method. The complete set of extra termsis given by.~‘(cosm.)= 6eg 2 + 2eifr,.’o-”~47,.’— ~ 1F””°(D~i/i,,.’+ ge”A,,i/i,~,”) (6)8./i,.’ = D,.(a(e,i/J))E’ + gy,. ’+ gE”A,.E”where we added the terms with D,. for comparison. Clearly, g is a dimensionless gauge coupling, but itis remarkable that also a mass-like term is needed (as well as a cosmological term) when one couples thephoton to the gravitinos in an electromagnetic manner. Note that the theory still has the same numberof local invariances. The mass term is actually needed in order that i/i,.’ still be massless in de-Sitter