Volume 148B, number 6 PHYSICS LETTERS 6 December 1984 CAN THE SQUASHED SEVEN-SPHERE PREDICT THE STANDARD MODEL? M.J. DUFF, I.G. KOH 1 and B.E.W. NILSSON The Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 16 July 1984 We argue that in Kaluza-Klein supergravity the breaking of the elementary SO(8) IN = 8] symmetry of the round S 7 down to the SO(5) X SO(3) symmetry of the squashed S 7 IN = 1 or N = 0] corresponds to a breaking of a composite SU(8) down to either SU(4) X SU(2) IN = 1] or SU(5) X SU(3) X U(1) IN = 0]. We suggest that forN = 0 the SO(5) X SO(3) acts as a confining force yielding bound states of SU(5) × SU(3) × U(1). By demanding that the effective bound state theo- ry be both anomaly free and asymptotically free (in the SU(8) sense), we find with the standard SU(5) embedding a fermion spectrum with 4 generations of (5* + 10) together with a realistic Higgs sector. In the search for a realistic Kaluza-Klein theory, it is customary to look for the SUc(3 ) × SUw(2 ) × Uy(1) of the standard model inside the isometry group G of the extra dimensional ground state metric. However this leads to severe problems with chirality [1 ]. Alternatively one could argue that if the isometry group is non-abelian and asymptotically free, it should rather be interpreted as a confining force (i.e. a meta- color force), and the particles of the standard model are bound states formed from the Kahiza-Klein preons. By applying this idea to the S 7 compactification of d = 11 supergravity and invoking the composite SU(8) invariance of the N = 8 theory, we obtain, under the assumptions described below, an SU(5) GUT with 4 generations of (5* + 10) together with a realistic Higgs sector. We begin by recalling that in four dimensions the elementary SO(8) symmetry ofN = 8 supergravity does not contain the SUc(3) × SUw(2) × Uy(1) of the standard model. It has been suggested that the composite symmetry of SU(8) under which the fermions are chiral might become dynamical in the quantum theory [2]. Ellis, Gaillard, Maiani and Zumino [3] focused their attention on the ungauged 1 On leave of absence from Physics Department, Sogang University, Seoul, Korea. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) N = 8 theory of Cremmer and Julia [2] for which the SO(8) is global, whereas de Wit and Nicolai [4] con- sidered the gauged N = 8 theory with local SO(8) × SU(8) suggesting that the SO(8) might act as a con- fining force. Though attractive in many ways, neither scheme met with complete success. An alternative Kaluza-Klein origin of the standard model was suggested by Witten [5] starting from N = 1 supergravity in d = 11 by noting that the isometry group G of the extra 7 dimensions could contain SUc(3 ) × SUw(2 ) × Uy(1) but that the fermions could never be chiral. However, this work prompted Duff and Pope [6] to observe that a gauged SO(8) N = 8 supergravity in d = 4 could be obtained by a Freund-Rubin [7] compactification of d = 11 super- gravity on the seven-sphere, describing a massless N = 8 supermultiplet coupled to an infinite tower of massive N = 8 supermultiplets. They conjectured [6,8] that the sector describing only the massless supermul- tiplet of spins (2, 3 1 :, 1, :, 0 ÷, 0-) lying in the (1, 8S, 28, 56S, 35V, 35C) representation of SO(8) coincides with the de Wit-Nicolai theory and hence contains a hidden SU(8) as well. This conjectured equivalence has still not been verified beyond the linearized level [9,10], but is supported by cosmological constant cal- culations [ 11 ] and other more general arguments. The SO(8) representations of the massive fields are ob- tained by taking the tensor product of the massless 431