Volume 138B, number 1,2,3 PHYSICS LETTERS 12 April 1984 STABILITY OF THE CLASSICAL SOLUTION WITH SU(3) X SU(2) X U(1) SYMMETRY IN ELEVEN-DIMENSIONAL SUPERGRAVITY Nobuyoshi OHTA Institute of Physics, College of General Education, Osaka University, Toyonaka 560, Japan Received 3 January 1984 Eleven-dimensional supergravity admits a spontaneous compactification on the coset spaces SU(3) X SU(2) X U(1)/ SU(2)' × U(1)' × U(1)" characterized by a rational number p/q. It is shown by extending our previous method that this class of solutions is stable against perturbation in the sizes of each space for any p/q. Eleven-dimensional supergravity [1 ] has recently attracted much interest as a possible candidate for the theory unifying all interactions including gravity. It is considered to be of more fundamental significance than the ungauged N = 8 supergravity obtained by simple dimensional reduction [2] since relevant infor- mation may well have been lost in this process. In fact it has been shown that, in addition to the ungauged version, this theory admits many effective four-dimen- sional theories through a spontaneous compactifica- tion of eleven-dimensional space-time into the prod- uct of four-dimensional space-time and several kinds of seven-dimensional manifold MT, because of the nonvanishing field strength of a third-rank antisym- metric tensor field [3-5]. The manifold for M7 studied most widely so far is the seven-sphere S7 [4]. It is believed that this com- pactification leads to the N = 8 extended supergravity with gauged SO(8) symmetry in four dimensions [6]. Stability of this classical solution has also been estab- lished [7,8]. Unfortunately the gauge group SO(8) does not contain the phenomenological SU(3) X SU(2) X U(1) and hence this solution does not seem to lead to a realistic Kaluza-Klein theory. If one still wishes to stick to $7, one would have to rely on the hidden symmetry SU(8) which does not possess ele- mentary gauge fields [2,9]. In this approach the auxil- iary composite gauge fields for SU(8) must become dynamical but this does not seem to be an easy task [10]. It thus appears that we should search for a more 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) realistic compactification scheme. Recently it has been shown that there are also SU(3) X SU(2) X U(1) invariant metrics in seven di- mensions as solutions ofD = 11 supergravity [5,11], thus providing us with the possibility to construct real- istic models. In this article we wish to examine the classical stability of this class of solutions by extending our previous method [7] to this case. Establishing sta- bility of this solution is clearly very important in the application of supergravity to realistic models along the above line of thought. The bosonic action of the theory is given by (, S = f 4g ill) dllx - -~R - AFMNpQFMNPQ X/~ 1 eMl ... M11 + 6(4!)~ x/g ~ FM 1 ...M4FMs...MsAM9MIoMll ) " X We use the metric (+, -, - .... ), e 0123 .... +1 and FMNPQ = 4!O[MANpQ]. The field equations are 1 RMN -- ~gMN R _ 1 48 (8FMPQRFNPQR _ gMNFSPQRFSPQR) , FMNPQ M = Vr2 1 ; 2(4!)2 x/gOD X eMI'"MsNPQFM1 ...MaFMs...Ms • (1) (2) (3) 63