PLB34614S0370-2693(19)30294-110.1016/j.physletb.2019.04.060TheoryA uniqueness theorem for warped N > 16 Minkowski backgrounds with fluxesS.Lautzsebastian.lautz@kcl.ac.ukG.Papadopoulos⁎george.papadopoulos@kcl.ac.ukDepartment of Mathematics, King's College London, Strand, London WC2R 2LS, UKDepartment of MathematicsKing's College LondonStrandLondonWC2R 2LSUK⁎Corresponding author.Editor: M. CvetičAbstractWe demonstrate that warped Minkowski space backgrounds, Rn−1,1×wMd−n, n≥3, that preserve strictly more than 16 supersymmetries in d=11 and type II d=10 supergravities and with fields which may not be smooth everywhere are locally isometric to the Rd−1,1 Minkowski vacuum. In particular, all such flux compactification vacua of these theories have the same local geometry as the maximally supersymmetric vacuum Rn−1,1×Td−n.KeywordsSupergravity backgroundsFlux compactificationsRecently, all warped anti-de-Sitter (AdS) backgrounds with fluxes that preserve N>16 supersymmetries in d=11 and d=10 supergravities have been classified up to a local isometry in [1–3]. In this note, we extend this result to include all warped Rn−1,1×wMd−n backgrounds of these theories. In particular, we demonstrate that all warped Rn−1,1×wMd−n, n≥3, solutions with fluxes of d=11, IIA d=10 and IIB d=10 supergravities that preserve N>16 supersymmetries are locally isometric to the Rd−1,1 maximally supersymmetric vacuum of these theories. Massive IIA supergravity does not admit such solutions. A consequence of this is that all N>16 flux compactification vacua, Rn−1,1×wMd−n, of these theories are locally isometric to the maximally supersymmetric toroidal vacuum Rn−1,1×Td−n. To prove these results we have made an assumption that the translation isometries along the Rn−1,1 subspace of these backgrounds commute with all the odd generators of their Killing superalgebra. The necessity and justification of this assumption will be made clear below.To begin, we shall first describe the steps of the proof of our result which are common to all d=11 and d=10 theories and then specialize at the end to present the special features of the proof for each individual theory. Schematically, the fields of Rn−1,1×wMd−n backgrounds are(1)ds2=A2ds2(Rn−1,1)+ds2(Md−n),F=W∧dvolA(Rn−1,1)+Z, where A is the warp factor that depends only on the coordinates of the internal space Md−n, dvolA(Rn−1,1) denotes the volume form of Rn−1,1 evaluated in the warped metric and F denotes collectively all the k-form fluxes of the supergravity theories. We take ds2(Rn−1,1)=2dudv+dz2+δabdxadxb, where we have singled out a spatial coordinate z which will be useful later. W and Z are (k−n)− and k− forms on Md−n which depend only on the coordinates of Md−n. Clearly if n>k, W=0. Therefore these backgrounds are invariant under the Poincaré isometries of the Rn−1,1 subspace. It is known that there are no smooth compactifications with non-trivial fluxes of d=10 and d=11 supergravities [4,5], i.e. solutions for which all fields are smooth including the warp factor and Md−n is compact without boundary. However here we do not make these assumptions. Md−k is allowed to be non-compact and the fields may not be smooth.To continue following the description of Rn−1,1×wMd−n backgrounds in [6–8], where one can also find more details about our notation, we introduce a light-cone orthonormal frame(2)e+=du,e−=(dr−2rA−1dA),em=Adxm,ei=eIidyI, on the spacetime with ds2(Md−n)=δijeiej. Then the Killing spinors of the Rn−1,1×wMd−n backgrounds can be written as(3)ϵ=σ++uΓ+ΓzΞ(−)σ−+A∑mxmΓmΓzΞ(+)σ++σ−+rΓ−ΓzΞ(+)σ++A∑mxmΓmΓzΞ(−)σ−, where xm=(z,xa), all the gamma matrices are in the frame basis (2) and the spinors σ±, Γ±σ±=0, depend only on the coordinates of Md−n. The remaining independent KSEs are a restriction of the gravitino and algebraic KSEs of the supergravity theories on σ± which schematically can be written as(4)Di(±)σ±=0,A(±)σ±=0, respectively, and in addition an integrability condition(5)(Ξ(±))2σ±=0, where Ξ(±) is a Clifford algebra element that depends on the fields. Ξ(±) will not be given here and can be found in the references above. The latter arises as a consequence of integrating the gravitino KSE of the theories along the Rn−1,1 subspace.Notice that the (spacetime) Killing spinors (3) may depend on the coordinates of the Rn−1,1 subspace. Such a dependence arises whenever σ± is not in the kernel of Ξ(±). Of course σ± is required to lie in the kernel of (Ξ(±))2. To see why this dependence can arise for Rn−1,1×wMd−n backgrounds, notice that AdSn+1 in Poincaré coordinates can be written as a warped product of Rn−1,1×wR. Therefore all AdS backgrounds, warped or otherwise, can be interpreted as warped Minkowski space backgrounds. It is also known that the former admit Killing spinors that depend on all AdS coordinates including those of the Minkowski subspace. Therefore Rn−1,1×wMd−n may also admit Killing spinors that depend on the coordinates of Rn−1,1, see also [9] for a more detailed explanation.The assumption we have made that the commutator of the translations P along Rn−1,1 and the odd generators Q of the Killing superalgebra [10,11] must vanish, [P,Q]=0, is required for Killing spinors ϵ not to exhibit a dependence on the coordinates of Rn−1,1. Indeed, if the Killing spinors have a dependence on the Minkowski subspace coordinates, then the commutator [P,Q] of the Killing superalgebra will not vanish. This can be verified with an explicit computation of the spinorial Lie derivative of ϵ in (3) along the translations of Rn−1,1. Although this may seem as a technical assumption, it also has a physical significance in the context of flux compactifications. Typically the reduced theory is invariant under the Killing superalgebra of the compactification vacuum. So for the reduced theory to exhibit at most super-Poincaré invariance, one must set [P,Q]=0 for all P and Q generators. This physical justification applies only to compactification vacua but we shall take it to be valid for all backgrounds that we shall investigate below. Of course such an assumption excludes all AdS solutions of supergravity theories re-interpreted as warped Minkowski backgrounds. Therefore from now on we shall take Ξ(±)σ±=0 and so all Killing spinors ϵ will not depend on the coordinates of the Rn−1,1 subspace.Before we proceed further, let us describe the Killing spinors of Rn−1,1×wMd−n backgrounds in more detail. It turns out that if σ+ is a Killing spinor, then σ−≔AΓ−zσ+ is also a Killing spinor. Similarly if σ− is a Killing spinor, then σ+≔A−1Γ+zσ− is also a Killing spinor. Furthermore if σ+ is a Killing spinor, then σ+′≔Γmnσ+ are also Killing spinors for every m,n. Therefore the Killing spinors form multiplets under these Clifford algebra operations. The counting of Killing spinors of a background proceeds with identifying the linearly independent Killing spinors in each multiplet and then counting the number of different multiplets that can occur [6–8]. As all Killing spinors are generated from σ+ Killing spinors, we shall express all key formulae in terms of the latter.The 1-form bilinears that are associated with spacetime Killing vectors ϵr, r=1,…,N, that also leave all other fields invariant are(6)X(ϵr,ϵs)≔〈(Γ+−Γ−)ϵr,ΓAϵs〉seA, where in d=11 and IIA supergravity theories 〈(Γ+−Γ−)⋅,⋅〉s is the Dirac inner product restricted on the Majorana representation of Spin(10,1) and Spin(9,1), respectively, while in IIB it is the real part of the Dirac inner product. Note that X(ϵr,ϵs)=X(ϵs,ϵr). In particular, one finds that(7)X(σ−r,σ−s)=2A2〈σ+r,ΓzΓAΓ−Γzσ+s〉seA,X(σ−r,σ+s)=2A〈σ+r,ΓzΓAσ+s〉seA,X(σ+r,σ+s)=−〈σ+r,Γ+ΓAσ+s〉seA. Clearly the last 1-form bilinear is X(σ+r,σ+s)=−2〈σ+r,σ+s〉se−. The requirement that X(σ+r,σ+s) is Killing implies that 〈σ+r,σ+s〉s are constants. In particular, one can choose without loss of generality that 〈σ+r,σ+s〉s=(1/2)δrs. Then first 1-form bilinear in (7) is X(σ−r,σ−s)=2A2δrse+.Next consider the middle 1-form bilinear in (7). If σ+r is in the same multiplet as σ+s, i.e. σ+s=Γzaσ+r, then X(σ−r,σ+s)=−δrsAea. On the other hand if σ+s=σ+r, then X(σ−r,σ+r)=Aez. Thus all these bilinears generate the translations in Rn−1,1. However, if σ+r and σ+s are not in the same multiplet, then the bilinear(8)X˜rs≔X(σ−r,σ+s)=2A〈σ+r,ΓzΓiσ+s〉sei, will generate the isometries of the internal space. The Killing condition of X˜ implies that(9)X˜rsi∂iA=0. As the X˜ isometries commute with the translations, the even part, g0, of the Killing superalgebra decomposes as g0=p0⊕t0, where p0 is the Lie algebra of translations in Rn−1,1 and t0 is the Lie algebra of isometries in the internal space Md−n.So far we have not used the assumption that the backgrounds preserve N>16 supersymmetries. If this is the case, the Killing vectors generated by g0 span the tangent space of the spacetime at each point. This is a consequence of the homogeneity theorem proven for d=11 and d=10 supergravity backgrounds in [12,13]. This states that all solutions of these theories that preserve more than 16 supersymmetries must be locally homogeneous. In this particular case because of the decomposition of g0, the Killing vector fields generated by t0 span the tangent space of Md−n at every point. As a result the condition (9) implies that A is constant. The main result of this note then follows as a consequence of the field equation of the warp factor and those of the rest of the scalar fields of these theories. So to complete the proof we shall state the relevant equations on a case by case basis.In d=11 supergravity, the 4-form field strength of the theory for Rn−1,1×wM11−n, n≥3, backgrounds can be expressed as(10)F=dvolA(Rn−1,1)∧W4−n+Z, with W4−n=0 for n>4, and the warp factor field equation is(11)∇˜2logA=−n(∂logA)2+13⋅(4−n)!(W4−n)2+1144Z2, where ∇˜ is the Levi-Civita connection on M11−n. The superscripts on the forms denote their degree whenever it is required for clarity. Clearly if A is constant, as it has been demonstrated above for N>16 backgrounds, then Wn−4=Z=0. So F=0 and thus all the fluxes vanish. In fact, this is also the case for n=2 provided that A is taken to be constant. As F=0 and A is constant, the gravitino KSE in (4) implies that all the Killing spinors σ± are parallel with respect to the Levi-Civita connection, ∇˜, on M11−n. In turn this gives that all the Killing vector fields X˜ in (8), which span the tangent space of M11−n, are also parallel with respect to ∇˜. Thus M11−n is locally isometric to R11−n. Therefore the backgrounds Rn−1,1×wM11−n are locally isometric to the maximally supersymmetric vacuum R10,1. Notice that the last step of the proof requires the use of the homogeneity theorem.In (massive) IIA supergravity, the 4-form F, 3-form H and 2-form G field strengths of the theory for Rn−1,1×wM10−n, n≥3, backgrounds can be written as(12)F=dvolA(Rn−1,1)∧W4−n+Z,H=dvolA(Rn−1,1)∧P3−n+Q,G=L, where W4−n vanishes for n>4 and similarly P3−n vanishes for n>3. The field equations for the warp factor A and dilaton field Φ, n>2, are(13)∇˜2logA=−n(∂logA)2+2∂ilogA∂iΦ+12(P3−n)2+14S2+18L2+196Z2+14(W4−n)2,∇˜2Φ=−n∂ilogA∂iΦ+2(dΦ)2−112Q2+12(P3−n)2+54S2+38L2+196Z2−14(W4−n)2, where S=eΦm and m is the cosmological constant of (massive) IIA supergravity. Clearly if both A and Φ are constant, which is the case for all N>16 backgrounds, then the above two field equations imply that all the form fluxes will vanish. Significantly, the cosmological constant must vanish as well. There are no Rn−1,1×wM10−n solutions in massive IIA supergravity that preserve N>16 supersymmetries. In IIA supergravity, an argument similar to the one presented above in d=11 supergravity reveals that M10−n is locally isometric to R10−n and so all N>16 Rn−1,1×wM10−n backgrounds are locally isometric to the maximally supersymmetric vacuum R10,1. It is not apparent that the theorem holds for n=2 even if A and Φ are taken to be constant.In IIB supergravity the self-dual real 5-form F and complex 3-form H field strengths of the theory for Rn−1,1×wM10−n, n≥3, backgrounds can be expressed as(14)F=dvolA(Rn−1,1)∧W5−n+⋆W5−n,H=dvolA(Rn−1,1)∧P3−n+Q, where W5−n vanishes for n>5 and P3−n vanishes for n>3. The field equation of the warp factor is(15)∇˜2logA=−n(∂logA)2+38|P3−n|2+148|Q|2+4(5−n)!(W5−n)2.Clearly if A is constant, which we have demonstrated that it holds for N>16 backgrounds, then F=H=0. Moreover the homogeneity theorem implies that the two scalar fields of IIB supergravity, the axion and the dilaton, are also constant. 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