Volume 121B, number 2,3 PHYSICS LETTERS 27 January 1983 SPONTANEOUS GENERATION OF Osp(4/8) SYMMETRY IN THE SPONTANEOUS COMPACTIFICATION OF D = 11 SUPERGRAVITY R. D'AURIA l Ddpartement de Physique Thdorique, Universitd de Gendve, 121 l Geneva 4, Switzerland and P. FRI~ lstituto di Fisica Teotqca, Universitd di Torino, Turin, Italy and INFN, Sezione di Torino, Turin, Italy Received 15 October 1982 In this letter we show that the compactified vacuum solution ofD = 11 supergravity currently used to derive the Kaluza-Klein interpretation of the extra dimensions has an Osp(4/8) symmetry. The solution describes the rigid superspace Osp(4/8)/SO(1,3) ® SO(7). Recently there is renewed interest in Kaluza-Klein theories motivated by the special features ofD = 1 1 super- gravity [ 1,2 ]. This model, which is the largest possible supersymmetric extension of Einstein theory, exhibits the spontaneous compactification mechanism. This means the following: there is an exact solution of its field equa- tions (Freund-Rubin solution [3]) where the fermion field is zero, the electromagnetic-like field F~vpo is a con- stant and the metric of the 11-dimensional space is the metric of a cartesian product of two Einstein spaces of 4 and 7 dimensions respectively. When the first is chosen to be anti de Sitter space and second a 7-sphere the solu- tion exhibitsN = 8 supersymmetry [5,1 ]. The interest of this solution is that it provides a natural way of inter- preting D = 11 supergravity in 4 dimensions as a possible realistic theory. Since the 7-space is compact one can expand systematically all fields in spherical harmonics on this space, which happens to be a coset SO(8)/SO(7) [4]. This expansion yields fields belonging to SO(8) representations. The zero modes of this expansion yield a massless supergravity theory in 4 dimensions which has been recently identified in ref. [5] with de Wit and Nicolai'sN = 8 theory [6]. Further work is now in progress concerning more general compactifying solutions where the SO(8) in- variant metric of the internal 7-dimensional space is replaced by other less symmetric affine structures (squashed spheres with torsion) which might generate the partial or total supersymmetry breaking [7]. In this letter we want to show that the symmetry of the compactifying solution described above is much larger than the SO(8) group of the 7-sphere; actually it is the Osp(4/8) supergroup (plus the gauge symmetry of the 3- form field). This is important because the zero and higher modes in the harmonic expansion around the solution should be classified as Osp(4/8) multiplets: so many SO(8) representations will be patched together in a smaller number of supermultiplets. Moreover we remark that Osp(4/8) is the natural group one would start from when constructingN = 8 supergravity in 4 dimensions. To prove our result we rely on the geometrical formulation of the D = 11 theory obtained in ref. [8] whose conventions and notations we shall use throughout the paper. In the geometric approach, the fundamental fields appearing in the lagrangian are Yang-Mills potentials for a group, or a more general algebraic structure of the type 1 On leave from lstituto di Fisica Teorica dell'Universit~ di Torino, Turin, Italy. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 141