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

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394 P. van Nieuwenhuizen, Supergravity[578]J. Wets, R. Grimm and B. Zumino, A Complete Solution Of The Bianchi Identities In Superspace With Supergravity Constraints, NucI. Phys.B152 (1979) 255.[579] 1. Wess, Supergravity, talk given at the IX Texas Symp. on Relativistic Astrophysics, Munich 1978; Karlsruhe preprint 1979, to be published.[580] J. Wets, Methods of Differential Geometry In Gauge Theories and Gravitational Theories, talk given at the Einstein Symp., Berlin 1979,Karlsruhe preprint, to be published.(581] J. West, Lectures given at the Boulder Summer School 1979; Karlsruhe preprint, to be published.[581a]i. West and B. Zumino, NucI. Phys. B70 (1974) 39.[582] P.C. West and KS. Stelle, Auxiliary Fields For Supergravity, Phys. Lett. B74 (1978) 330.[583] P.C. West and K.S. Stelle, Vector Multiplet Coupled to Supergravity, Phys. Lets. B77 (1978) 376.[584] P.C. West and KS. Stelle, Relation Between Vector And Scalar Multiples And Gauge Invariance In Supergravisy, NucI. Phys. B145 (1978)175.[5851P.C. West and KS. Stelle, Maxwell Multiplet Coupled To Supergravity and B.R.S. Invariance, NucI. Phys. B140 (1978) 285.[586] P.C. West and A.H. Chamseddine, Supergravity As A Gauge Theory Of Supergravity, Nuel. Phys. B129 (1977) 39.[587] P.C. West, A Geometric Gravity Lagrangian, Phys. Lest. 76B (1978) 569.(588] P.C. West and K.S. Stelle, Spontaneously Broken de Sitter Symmetry And The Gravitational Holonomy Group, prepnnt ICTP/78-79/19.[589] P.C. West and KS. Stelle, de Sitter Gauge Invariance And The Geometry Of The Einstein—Cartan Theory, i. Phys. A, No. 8, L205 (1979).[590] T. Yoneya, Background Metric In Supergravity Theories, Phys. Rev. D17 (1978) 2567.[591] C.K. Zachos, Extended S.G. With A Gauged Central Change, Cal. Tech. Ph.D. thesis, April 1979.[592] C.K. Zachos, N = 2 Supergravity With A Gauged Central Change, Phys. Lett. 76B (1978) 329.[593] B. Zumino, Symmetry, Proc. Conf. at Northern University, Boston, Sept. 26 and 27, 1975, eds. R. Arnowitt and P. Nath (the MIT PressCambridge, Mass. and London, England).[594] B. Zumino, Non-Linear Realization Of Supersymmetry In Anti De Sitter Space, Nuci. Phys. B127 (1977) 189.[595]B. Zumino, Supersymmetry And The Vacuum, NucI. Phys. B89 (1975) 535.[596] B. Zumino, Supergravity, Annals of the NY Academy of Sciences 302 (1977) 545.[597] B. Zumino, Supersymmetry and Supergravity, Physica 96A (1979)99.[598] B. Zumino, Some Recent Developments In Supergravity, Czech. J. Phys. B29 (1979) 245.[599] B. Zumino, Supergravity, Colloques Nationaux du Centre de Ia Recherche Scientifique, No. 937.[600] B. Zumino, Relativistic Strings And Supergauges, reprint from Renormalization and Invariance In Quantum Field Theory, ed. ER. Caianiello(Plenum Publishing).[601] B. Zumino, Euclidean Supersymmetry And The Many-Instanton Problem, Phys. Lets. 69B (1977) 369.[602] B. Zumino, Supersymmetry And Kahler Manifolds, to be published in Phys. Lett. B.[603] B. Zumino and S. Ferrara, The Mass Matrix of N = 8 Supergravity, to be published in Phys. Lett. B.[604] B. Zumino, Topics In Supergravity And Supersymmetry, Coral Gables Conf., Jan. 1977.[605] B. Zumino, in: Proc. 1979 Cargése school (Plenum Press).[606] MT. Grisaru, P. van Nieuwenhuizen and CC. Wu, Phys. Rev. D12 (1975) 3203.[60718. Lee and J. Zinn-Justin, Phys. Rev. D7 (1973) 1049.[608]0. Senjanovic, Phys. Rev. D16 (1977) 307.
P. van Nieuwenhuizen, Supergravily 395Addendum6.6. Supergravity and phenomenologyThe nongravitational forces around 102 GeV seem to be adequately described by SU 3 (color) x SU2 xU1. If one uses the renormalization group 2, they techniques seem to come determine together the dependence at iO’~GeV of(the “grand corresponding unifiedthree mass” running M). Although couplingthis constants is not on far Qfrom the Planck mass (h/G)1”2 c512 = 1019 GeV, it is commonlyassumed that the nongravitational forces first combine at M into the simple group SU5. Since SU5 (orother candidates like E6 and SO~~) have many free parameters, while also gravity is left out, theseresults cannot be the conclusive end point of unification.Supergravity opens the possibility to include also gravity; moreover, it has very few free parameters.The massless N = 8 model has only a global SU8 symmetry of the physical particles on-shell, while it isnot even clear whether the subgroup SO8 can be gauged off-shell. Even if it could, this local SO8 is toosmall to contain SU3 x SU2 x U1. However, there are now several recent proposals to circumvent this.One fascinating proposal by Ellis, Gaillard, Maiani and Zumino, though containing many loose endsand based on rather uncertain assumptions, nevertheless comes up with the result that we have seen allquarks (except, at this moment of writing the top quark). The argument goes as follows.The basic ansatz is that the local off-shell SU8 symmetry of the classical (preon) action in d = 4dimensions (comparable to the local Usp(8) symmetry in d = 5 dimensions, see section 6.3) can give riseat the nonperturbative quantum level to bound states which form an N = 8 supersymmetry multiplet atthe Planck mass, in which particles of a given spin form St]8 multiplets, such that the theory with boundstates plus the graviton (which is a preon) has a local SU8 symmetry. At the classical level there is only alocal SU8 symmetry if one adds to the 70 physical spin 0 fields 63 extra nonphysical fields (just as onecan add to the metric ~ extra antisymmetric fields such that one ends up with tetrads). This is thus afake symmetry. The gauge fields of this local SU8 are composites of scalar fields (compare with the spinconnection w (e)). However, it is known that 2’ = ~ + iA5.)~2 which has the same features (one cansolve A5. from its algebraic field equation, but with A~,the theory has a local U(1) symmetry), yields aspin 1 massless bound state (poles in the 2-point Green’s function in a 1/N expansion in thepath-integral) and that the fermions also form bound states. Actually, in this model the fermionic boundstates have the global SU(N) symmetry of the classical action, not only the local symmetry, which forthe N = 8 model would mean that the bound states form E7 multiplets. Since E7 is noncompact, thesemultiplets would be infinite dimensional; we will not discuss this possibility any further.Thus one assumes that the bound states form an N = 8 massless supersymmetry multiplet. Whichmultiplet? To answer this question, note that the N = 1 massive multiplet [2, 3/2, 3/2, 1] splits into themassless [2, 3/2] + [3/2, 1] multiplets, while for N = 2 the massive [2, (3/2)~,(1)6, (1/2)~,0] contains amassless [(3/2)2, (1)~,(1/2)2]. These cases suggest the following N = 8 supersymmetry multiplet in whichthe vectors are in the adjoint representation of SU(8) rather than U(8) (see subsection 6.3)helicity 3/2 1 1/2 0 —1/2 —1 —2 —5/2jiCtuç’A TJA r’A ~A r’Ai,j V ~ F BC ~) BCD 1’ BCDEcomponents 8 63 216 420 504(+1) (+8) (+28) (+56)Since one expects St]8 rather than U8 multiplets, one drops the traces; these are indicated in brackets.
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In memoriam Joel ScherkJoel Scherk
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(A V,.”)00 = ~ s*a = (D,. + ~ A,.
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