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The Kahler 2-Form in D=11 Supergravity D=11 Supergravity with Isometry Group (SU(3)/Z3)×U(2) B Dolan to Cite This Article: B P Dolan 1985 Class

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The Kahler 2-Form in D=11 Supergravity D=11 Supergravity with Isometry Group (SU(3)/Z3)×U(2) B Dolan to Cite This Article: B P Dolan 1985 Class Classical and Quantum Gravity Related content - An S2×S1 bundle over CP2-a solution of The Kahler 2-form in D=11 supergravity d=11 supergravity with isometry group (SU(3)/Z3)×U(2) B Dolan To cite this article: B P Dolan 1985 Class. Quantum Grav. 2 309 - Hopf fibration of eleven-dimensional supergravity B E W Nilsson and C N Pope - Kaluza-Klein theories View the article online for updates and enhancements. D Bailin and A Love Recent citations - On the stability and spectrum of non- supersymmetric AdS5 solutions of M- theory compactified on Kähler-Einstein spaces Jonathan E. Martin and Harvey S. Reall This content was downloaded from IP address 149.157.61.64 on 20/02/2019 at 15:25 Class. Quantum Grav. 2 (1985) 309-321. Printed in Great Britain The Kahler 2-form in D = 11 supergravity Brian P Dolan Department of Natural Philosophy, University of Glasgow, Glasgow GI2 8QQ, UK Received 28 September 1984 Abstract. Solutions of D = 11 supergravity are presented, in which the internal space consists of Kahler manifolds and S'. The 4-form field is constructed from bilinears in the Kahler 2-forms as well as the volume element of four-dimensional spacetime. The group of isometries of the internal metric is (SU(3)/Z3) x(SU(2)/Z2) xU(1). Four-dimensional spacetime is either anti-de-Sitter or Robertson-Walker. 1. Introduction There is much current interest in non-Abelian Kaluza-Klein theories in which the gauge group of internal symmetry acts on a compact, n-dimensional manifold, which is a submanifold of a larger (4+ n) dimensional spacetime [ 1-41. Supergravity supplies an attractive method of incorporating fermions into Kaluza-Klein theories. In par- ticular, eleven-dimensional supergravity seems to have the smallest number of dimensions in which one can hope to fit a gauge group with the algebra of SU(3) x SU(2) x U( 1). There are now many known solutions of the bosonic equations of motion of D = 11 supergravity [5]. All of these solutions rely on the Freund-Rubin ansatz [6], in which the 4-form field, 9,consists of a term which is a constant multiple of the volume element of four-dimensional anti-de-Sitter spacetime, possibly with additional terms on the internal space. This results in a manifold which is a direct product M4XB, where M4 is anti-de-Sitter space and E a compact, seven-dimensional manifold. In this paper some other solutions will be explored, in which the internal space is constructed from Kahler manifolds and S', and the internal part of 9 is quadratic in the Kahler 2-forms. Abbreviated versions of some of the results have already appeared in [7] and [8]. Here more details will be presented, along with some new results. Kahler manifolds and Kahler 2-forms have been considered in the context of ten- dimensional Einstein-Maxwell systems by Watamura [9], [ 101 and in the dimensionally reduced N = 2, non-chiral D = 10 supergravity in references [ 111 and [ 121. In 9 2 the bosonic Lagrangian and equations of motion are presented in differential form language. In 8 3 solutions are presented in which the internal space is CP2x S2 x S' and four-dimensional spacetime is of Robertson-Walker type. In 9 4 the problem of spinor structures on CP2 is discussed and modified metrics for fibre bundles of S' and S2 over CP2 are presented. In 8 5 and 9 6 these 'twisted' metrics are used to construct more solutions. Section 7 contains a summary and conclusions. Notation and conventions are summarised in appendix 1 and the properties of orthonormal 0264-9381/85/030309 + 13$02.25 0 1985 The Institute of Physics 309 3 10 B Dolan l-forms for the Fubini-study metrics and Kahler 2-forms on CP2 and CP' = S2 are presented in appendix 2. Appendix 3 contains some details of calculations nekded in 0 S and § 6. 2. Equations of motion The bosonic part of eleven-dimensional supergravity has the following action density ll-form [13] A = R~~ , E AB - 9, 9+ 3 A -I 9,9,d (1) where RAE are the curvature 2-forms obtained from the orthonormal l-forms EA via the torsion-free condition (see appendix 1). 9 is a 4-form derived, at least locally, from the 3-form potential d,9 = dd,whose presence in the full Lagrangian is necessary to balance the bosonic and fermionic degrees of freedom of eleven-dimensional supergravity [ 131. A is a constant and the eleven-dimensional Hodge duality operator. The Einstein equations obtained from (1) by varying the metric are R~~,T~E~~~=.FAjc(~,9)-(ic9),,~,S (2) For the definition of i,, see appendix 1. Note that the third term of (1) does not contribute to (2) since it does not contain any information about the metric. The matter field equations, obtained by varying the 3-form d in (1) are d;, %= .!FA9/A. (3 1 The potential d does not appear in the equations of motion (2) and (3) and in the following 9 will be such that no global potential exists (though one will always exist locally, on each coordinate patch, and on the coordinate overlaps two potentials can always be matched by a gauge transformation). Hence a third condition is imposed on 9 d9=Q (4) (2), (3) and (4) are the bosonic equations of D= 11 supergravity. 3. B=CP2xSZxS' Let the seven-dimensional internal space be CP2x S2 x SI and take E" = e", E "' = e"' where ea and em are orthonormal 1-forms for the Fubini-study metrics on CP2 and S2 respectively (see appendix 2). The gauge symmetry is (su(3)/z3)x(SU(2)/Z2) x U(1). With K2 and K1 the Kahler 2-forms on CP2 and S2 respectively, take the following ansatz for 9 9=PE'*~'+ (412) K,, K2+ (r/&) K,, K2 (5) with p, q and r constants. Then ;,9=-(p/2)E: KIAKZn K2+ qE:2304K,+ ( r/J2)E'2304,KZ. (61 If the volume form of four-dimensional spacetime and E4are independent of position The Kahler 2IJorm in D = 11 supergravity 311 on CP2x S2 (dE 12304 = 0) then d9=0, (7) since the Kahler 2-forms are closed. By inspection d;, 9=A-'9, 9#0 is impossible, hence (3) and (4) are satisfied provided d;,9=9,9=0 e pr=0,pq=OandpdE4=0. (8) Therefore, either p # 0, r = q = 0 and dE4= 0 or p = 0. The case p # 0 is the usual Freund-Rubin ansatz, but CP2x S2 x SI will not give a solution in this case since it does not admit an Einstein metric, due to the SI factor. The second case, p = 0, is different from the usual Freund-Rubin ansatz since 9 then contains no term on four-dimensional spacetime. For the remainder of 0 3 it will be assumed that p = 0. Now consider the Einstein equations (2) with p = 0: (For index conventions and definitions see appendices 1 and 2). Using (Al.l) and (A1.2), equations (9) give (with R5 the curvature scalar for E") R, = 3 ( q2+ r2- RCp2 - RS2), RCp2=-2(q2+ RgS RS2), q2-r2= R5+RCP2 (10) + RCP2= 3 ( r2+ 4q2), Rs2 = 3 (2r2- q2), R5=-$(r2+q2). Thus (9) are solved if E" is an orthonormal basis of 1-forms for a five-dimensional Einstein metric with negative scalar curvature Rpy,;EPY"=-(r2+q2);Ea (11) and the curvature scalars on CP2 and S2 are 3(r2+4q2)and 2(2r2-q2) respectively (note that this requires q2< 2r2). All that remains to obtain a complete solution is to solve (1 1) with the fifth dimension topologically SI. This problem has been considered in [7] and [14]. With the ansatz 0 6 ( < 2~ and gt3) a metric for a maximally symmetric 3-space with k = (+ 1, - 1,0) for positive, negative or zero 3-space curvature respectively, equations (1 1) reduce to 312 B Dolan (. =d/dt) R/R+ (R/R)’+ k/R2= -( r2+ q2)/6 (R/R)’+RdJ/R@ + k/R2= -( r2+ q2)/6 2R/ R + (R/R)2+klR2+ 6/@+2RdJ/RQ, = -(r2+ q2)/2 which have general solutiont { w = [f ( r2+ q2)]’/’} R( t)= [ P sin(wt + 6)-2k/~~]’/~ QW cos(wt + 6) Q,(t)=- 2 [Psin(wt+ 6)-2k/wZ]”’ where P, Q and S are constants. The special case k = 0 and r2 = 29’ was presented in [ 111 (r2= 2q2 makes the metric on CP2x S2 Einstein). The limits of t must be chosen so that (Psin(wt+ S) -2k/w2)30 at all times. For k = -1 (3-space hyperbolic), if 1 PI < 2/w2then t can run over many cycles and R2will always be positive. However, if k = 0, +I or /PI > 210’ then t must be restricted. For example, the case of flat 3-space, k = 0, with S = 0, must have 0 == t < r/w and it is not possible to take multiple copies of this range, as the metric would not be differential everywhere. Since in Kaluza-Klein theories it is usually assumed that the size of the internal dimensions is of the order of a few tens or hundreds of the Planck length, CO-’ will be a few tens or hundreds of the Planck time. Hence solutions with k = 0, +1 or IPI > 2/w2 would correspond to universes lasting only s: Thus the flat 3-space solution must be rejected on physical grounds, unless some way can be found to patch together many cycles in such a way as to make the joins smooth.
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