Volume 148B, number 4,s PHYSICS LEITERS 29 November 1984 ELECTROVAC COMPACTIFICATION OF GAUGED d = 7 SUPERGRAVITY Fernando QUEVEDO zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ’ Theory Group, Department of Physics, The Universityof Texas at Austin, Austin, TX 78712, VSA Received 19 April 1984 Solutions of the classical field equations of gauged N = 2, d = 7 supergravity are found, even if the scalar field potential has no critical points. This leads to the splitting of the seven-dimensional spacetime to a product (Ads)* X Mz X Ma, where Mz and Ma may be compact or noncompact. For a wide range of parameter values, compactification on S3 from d = 7 to d = 4 occurs. For none of these solutions supersymmetry is preserved. Higher dimensional supergravity theories have recently become popular in the context of Kaluza- Klein theory. In particular, various solutions of the classical field equations of d = 11 supergravity have been found for which seven dimensions are compacti- tied [ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1] as a consequence of a non-vanishing four-index field strength [2]. No analogous compactification to d = 4 has been found for d < 11 supergravity, despite some attempts [3-S]. The most promising candidate is N = 2 (simple), d = 7 supergravity, as this also has a four-index field strength [5], but the usual ansatz, in this case (Ads), X S, , does not yield a solution. How- ever, the N = 2, d = 7 supergravity is not unique; there is a class of “gauged” N = 2, d = 7 supergravity theories with a two-parameter potential [S]. For some choices of these parameters there is a stable (Ads)7 ground state at a maximum of the scalar potential [6] but for other choices there are no critical points of the poten- tial. It has been found in a model of N = 4 gauged supergravity in d = 4 [7] that a non-maximally sym- metric “electrovac” background (non-vanishing con- stant electromagnetic field) may provide a stable ground state in the case of a potential without critical points [8], and it has been suggested that similar solu- tions might exist in the d = 7 case [6]. In this note we show that gauged N = 2, d = 7 super- gravity indeed has “electrovac” type solutions for ’ Research supported by the Robert A. Welch Foundation and NSF PHY8304629. 0370-2693/84/$03.00 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) which the d = 7 spacetime has the generic form (Ads)2 X M2 X Ms. According to the choice of parameters, M2 and M, may be compact or non-compact but there is a choice for which the solution is (Ads), X R2 X S3 ; i.e. for which compactitication on S, from d = 7 to d = 4 occurs. The price that is paid to achieve this com- pactification is that the effective four-dimensional spacetime is not maximally symmetric. For none of these electrovac solutions is supersymmetry preserved, so that their stability is not guaranteed. The field content of the N = 2, d = 7 supergravity theory is: the graviton e$, the gravitino ~Mi, an anti- symmetric tensor AM,,, a triplet of gauge vector fields A,i’, a spinor Xi and a scalar 4. The gauge group is SU(2). The spinors are eight-component complex SU(2) Majorana spinors. Since we are interested in configura- tions for which the Fermi fields vanish, we consider only the bosonic lagrangian, which is *r [S] x, = -ieR - fea -4(FMNPQ) 2 - aeo2(FMNii>2 - ie(a&)2 - eV(@ ) + (ie/48&) X ,+WRSTF M NmcFRSi ‘ATi’ - $iqARijA&‘ATik) - & iht?eMN~RSTFMNNpeARsT, (1) where u(G) = exp(-5-1/2#) _ (2) *l We are using the (+ + .. . +) convention for the metric, i.e., the “ict” convention. 301