Supersymmetric $dS_n$ solutions for $n \geq 5$ in $D=11$ supergravity

We determine the necessary and sufficient conditions for warped product $dS_n$ solutions, $5 \leq n \leq 10$, to preserve supersymmetry in $D=11$ supergravity, without assuming factorization of the Killing spinors. We prove that for $7 \leq n \leq 10$, all such solutions are flat, with vanishing 4-form. We also show that the only warped product $dS_6$ solutions are either the maximally supersymmetric $AdS_7 \times S^4$ solution, or $\mathbb{R}^{1,6} \times N_4$ where $N_4$ is hyperKahler, with vanishing 4-form. Supersymmetric warped product $dS_5$ solutions are then classified; it is shown that all such solutions are generalized M5-brane configurations, for which the transverse space is $\mathbb{R} \times N_4$, and $N_4$ is a hyperKahler manifold. If the 4-form is covariantly constant, then $N_4$ admits a hyperKahler potential.


Introduction
De Sitter geometry is of significant interest in the context of string cosmology, and in terms of holography. In the case of string cosmology, there have been numerous different approaches for investigating possible viable models. It has been known for some time that there are strict no-go theorems which imply that compactifications from regular warped product de-Sitter solutions are excluded [1,2,3]. Furthermore, additional phenomenological requirements, such as requiring the de-Sitter minima to be stable, and that the resulting models are compatible with slow roll inflation, also need to be satisfied [4,5]. Taking such requirements into account, there has been significant progress in understanding how to construct viable models in string cosmology, including systematic analysis of viable models from the perspective of N = 1, D = 4 supergravity [6], as well as the derivation of models from novel G 2 geometric structures in D = 11 [7,8] and SU(3) structures from type IIA supergravity [9,10,11]. The latter SU(3) structures were obtained by taking a supersymmetric class of IIA AdS 4 warped product solutions [12,13,14] and modifying the ansatz in such a way as to break the supersymmetry and admit dS 4 solutions. Notable further interest in de Sitter geometry has arisen from considerations of holography and entropy. The original gauge-gravity duality [15] formulated in terms of a conformal field theory dual to string theory in an AdS space has been generalized in numerous ways. In terms of de Sitter space, several proposed holographic dualities have been developed and applied, including [16,17,18,19].
Motivated by the importance of de Sitter solutions in string theory, in this work we initiate a systematic classification of supersymmetric de Sitter solutions in D = 11 supergravity. In particular, we classify all supersymmetric warped product dS n solutions for 5 ≤ n ≤ 10, for which the 4-form flux is invariant with respect to the isometries of dS n . We shall leave the classification of warped product dS 2,3,4 solutions to future work. As observed, there are strict no-go theorems which hold for such geometries. Hence, we shall not make any assumptions on smoothness of the warp factor or 4-form flux, nor do we assume that the internal manifold is smooth or compact; the analysis will be done entirely locally. In addition, we do not make any assumptions regarding factorization of the Killing spinors, as assuming such factorization may produce an erroneous counting of supersymmetries, as observed in [20]. The results of this paper therefore extend the classification constructed in [21] for supersymmetric AdS and flat warped-product solutions in D = 11 supergravity to include warped product dS n solutions 5 ≤ n ≤ 10. Classifications of warped product AdS and flat geometries have also been constructed for type II supergravity [22,23], and for warped product AdS geometries in heterotic supergravity [24].
In order to carry out the analysis of the conditions on the geometry and fluxes obtained from the Killing Spinor Equations (KSEs), we shall utilize spinorial geometry techniques, [25,26]. In this method, the Killing spinor is written as a differential multi-form, and gauge transformations are used to simplify the structure of this multiform into one (or more) simple canonical forms. This enables the KSEs to be written in terms of a linear system in the fluxes and spin connection, which can then be solved to extract the geometric conditions and the expression for the 4-form flux.
The plan of this paper is as follows. In section 2 we summarize the conditions on the bosonic fields associated with such warped product solutions obtained from the field equations and the Bianchi identities. We also analyse the KSEs and produce some results common to all of the dS n backgrounds. In Section 3 we use this to show that all warped product dS n backgrounds for 7 ≤ n ≤ 10 are flat, with vanishing 4form. In Section 4 we analyse the supersymmetric warped product dS 6 backgrounds, and show that these are either the maximally supersymmetric AdS 7 × S 4 solution, or R 1,6 × N 4 where N 4 is hyperKähler, with vanishing 4-form. In Section 5, we analyse the supersymmetric warped product dS 5 solutions, and we prove that these are warped products R 1,5 × w N 5 where N 5 is conformal to R×N 4 , and N 4 is a hyperKähler manifold. Such solutions are generalized M5-brane geometries, for which the transverse space is R × N 4 . It is also shown that if the 4-form is covariantly constant, then N 4 admits a hyperKähler potential. Some brief conclusions are presented in Section 6. There are also several Appendices. Appendix A contains a summary of curvature components associated with the warped product dS solutions. Appendix B contains further details of how the KSEs are integrated up along the de-Sitter directions, and then reduced to a gravitino KSE on the internal manifold. Some expressions for integrability conditions obtained from the KSEs are also presented. Appendices C and D contain more details of the analysis of the linear system obtained from the KSEs for dS 6 and dS 5 backgrounds, respectively. Appendix E contains more detail on the analysis of the dS 5 solutions in the special case when the 4-form is parallel.

Bosonic field equations and KSE
The bosonic field equations of D = 11 supergravity [27] consist of the Einstein equations and the gauge field equations where A, B, . . . are 11-dimensional indices and F is the 4-form flux. Moreover, the Bianchi identities read The Killing spinor equations (KSE) of D = 11 supergravity [27] are where ∇ is the D = 11 Levi-Civita connection and ǫ is a Majorana spinor. In our work we study warped product dS n backgrounds in D = 11 supergravity, with n ≥ 5. The metric on the D = 11 spacetime M 11 is where A is a function of the co-ordinates of the Riemannian manifold M 11−n and is the metric tensor of n-dimensional de Sitter spacetime, with |x| 2 = η µν x µ x ν and k = 1 ℓ 2 . We introduce on M 11 the co-frame where a = n, n + 1, . . . , ♯; y α are the co-ordinates on M 11−n and In terms of the co-frame (2.7), the metric tensor (2.5) reads where ds 2 (M 11−n ) = δ ab e a e b . (2.10) Requiring the 4-form F to be invariant under the isometry group O(n, 1) of dS n , we have where X is a 4-form on M 11−n . In the following we reduce the bosonic field equations (2. where ∇ denotes the Levi-Civita connection on M 11−n and R ab is the Ricci tensor on M 11−n . Some details about the computation of the Ricci tensor of (2.9) are presented in Appendix A. Moreover, the gauge field equations (2.2) can be decomposed as whered is the exterior derivative on M 11−n . Eventually, we reduce the Bianchi identities (2.3) on M 11−n , obtainingd Now let us perform the reduction of the KSE (2.4) on M 11−n . The A = µ component of (2.4) is given by whose solution is given by where ψ and τ µ are Majorana spinors which depend only on the co-ordinates on M 11−n . Substituting (2.19) into (2.16), we obtain the following conditions Substituting (2.17) into (2.21), we get 2 The A = a component of the KSE (2.4) is given by (2.25) Substituting (2.24) into (2.25), we find ∇ a ψ = σ a ψ (2.26) 1 If ω is a p-form on M 11−n then / ω = ω a1...ap Γ a1...ap 2 Notice that Γ a1a2...a8 = 0 for n ≥ 5. and ∇ a C + Cσ a − σ a C ψ = 0 (2.27) where and Hence, the reduction of the KSEs (2.4) on M 11−n produces a gravitino KSE (2.26) on M 11−n , supplemented by (2.21) and (2.27), which are both quadratic in X. Equations (2.21) and (2.27) arise as integrability conditions of (2.26), implementing the bosonic field equations (2.12)-(2.14) and the Bianchi identities (2.15). Some details about the reduction of the KSE along M 11−n and the integrability conditions of (2.26) are presented in Appendix B.

dS 5 backgrounds
For dS 5 backgrounds, we define the 2-form on M 6 G = ⋆ 6 X . (2.37) The bosonic field equations (2.12)-(2.14) and the Bianchi identities (2.15) yield Furthermore, the KSE (2.26) are given by is the highest rank Gamma matrix on M 6 . We remark that it is straightfoward to count the number of supersymmetries preserved by the warped product dS 5 backgrounds. In particular, if ψ satisfies (2.42) then so does Γ µν ψ. On taking a frame basis for the dS 5 directions given by (frame indices are chosen to be compatible with the spinorial geometry calculation in Section 5): , e ♯ , e 4 , e 9 } (2.44) one can without loss of generality assume that (2.42) admits a solution with positive lightcone chirality ψ = ψ + where Γ + ψ + = 0. It then follows that all satisfy (2.42). These 8 spinors are linearly independent. Hence it follows that warped product dS 5 solutions preserve N = 8, N = 16, N = 24 or N = 32 supersymmetries. The N = 32 supersymmetric solutions of D = 11 supergravity are fully classified [28], and are R 1,10 , AdS 7 × S 4 , AdS 4 × S 7 , and a maximally supersymmetric plane wave solution. The 4-forms of AdS 4 × S 7 and the maximally supersymmetric plane wave solution are not compatible with the 4-form for the warped product dS n solutions for n ≥ 5. As we shall demonstrate in Section 5, the maximally supersymmetric N = 32 warped product dS 5 solutions correspond to R 1,10 and AdS 7 × S 4 . We shall however concentrate on the analysis of the N = 8 solutions in this paper.
3 Warped Product dS n backgrounds (7 ≤ n ≤ 10) In this section we show that warped dS n backgrounds, with 7 ≤ n ≤ 10, have vanishing 11-dimensional Riemann tensor and vanishing 4-form flux F . Let us consider first n ≥ 8. For such values of n the 4-form X vanishes identically, thus F = 0. Hence, the KSE (2.26) simplify to Moreover, the integrability conditions (2.23) reduce to and Equation (3.8) implies that ∇A = 0, since k > 0. Hence locally we can adapt a frame such that where we have denoted by 7, 8, 9, ♯ the directions along M 4 . Inserting (3.10) into (3.9), it follows that w = 0, thus Using (3.11), the integrability conditions (3.8) reduce to k − ( ∇A) 2 = 0 (3.12) and the KSE (2.26) simplify to ∇ a ψ = 0, which in turn implies that is M 4 is Ricci flat. Moreover, inserting (3.11) and (3.13) into (2.13) we find SincedA is non-zero and covariantly constant on M 4 , then locally Decomposing (3.13) along (3.15), it follows that M 3 is Ricci flat, hence M 3 is flat, which in turn implies that M 4 has vanishing curvature tensor, i.e.
• Class II
where L = L(x, z), with ∂ x L = y −2 g 2 and θ is a closed 1-form which does not depend on x and y. Notice that (5.82) is equivalent to V = 16g 2 y 2 dL . where is a hyperKähler manifold independent of y and s (or x and y). The manifold N 4 is hyperKähler as a consequence of (5.69). In order to show that the metric on N 4 is independent of x and y, we find that as the Lie derivative of the hyperKähler forms with respect to V vanishes, this implies that for A ∈ su(2). An appropriate SU(2) gauge transformation can be utilised, which leaves the spinor Ψ invariant, in order to set A = 0, and in this gauge Similarly, one can also without loss of generality choose a gauge in which L Wê P = 0 .
In the following we rescale s = 4ŝ and then we drop the hat on s  This geometry corresponds to that of a generalized M5-brane configuration, whose transverse space is R × N 4 [30].

Solutions with Parallel 4-form
A special class of solutions arises when we take the 4-form F to be covariantly constant with respect to the Levi-Civita connection of the D = 11 solution. It is known that AdS 7 × S 4 lies within this class of solutions. Here we shall investigate the additional conditions obtained on the geometry when where ∇ (11) is the Levi-Civita connection on M 11 . Using the 11-dimensional spin connection (A.1) and (5.98), equation (5.105) is equivalent to where ∇ denotes the Levi Civita connection on M 6 . Implementing (5.95) and (5.99), equation (5.106) gives the following set of PDEs The Einstein equations (5.100) are implied by (5.107)-(5.109). Moreover, (5.106) implies that where c is constant. Implementing (5.110) into (5.107)-(5.109), we find Some details about the derivation of (5.111)-(5.113) are presented in Appendix E. We remark that in the case for which c = 0, corresponding to G = 0, these conditions imply that f is constant. For such solutions, the metric (5.97) simplifies to that of R 1,6 × N 4 where N 4 is a hyperKähler 4-manifold. On taking N 4 = R 4 one recovers the maximally supersymmetric flat R 1,10 solution. Otherwise, if c = 0, (5.113) implies that P 2c 2 is a hyperKähler potential for N 4 , in accordance with Proposition 5.6 of [31].

Conclusion
In this work we have classified the warped product dS n solutions of D = 11 supergravity for 5 ≤ n ≤ 10. We have found that: (i) For 7 ≤ n ≤ 10, all such solutions are flat, with vanishing 4-form.
(ii) In the case of warped product dS 6 solutions, these are either the maximally supersymmetric AdS 7 × S 4 solution, or R 1,6 × N 4 where N 4 is hyperKähler, with vanishing 4-form.
(iii) In the case of warped product dS 5 solutions, we prove that all such solutions are warped products R 1,5 × w N 5 , as given in (5.97), where N 5 is conformal to R × N 4 , and N 4 is a hyperKähler manifold. The geometry corresponds to a generalized M5-brane configuration, which is determined by the choice of the hyperKähler manifold, and a harmonic function on R×N 4 . We have also shown that if the 4-form F is covariantly constant, then the hyperKähler manifold N 4 admits a hyperKähler potential, and demonstrated how this potential is related to the warp factor.
In analysing the warped product supersymmetric dS 5 solutions, we have shown that such solutions preserve N = 8k supersymmetries (k = 1, 2, 3, 4). Our classification has determined the necessary and sufficient conditions for N = 8 supersymmetry; the N = 32 solutions are flat space and AdS 7 × S 4 . It would be interesting to determine the additional conditions imposed on the warp factor and on the hyperKähler manifold N 4 in order for there to be N = 16 or N = 24 enhanced supersymmetry, extending to the de-Sitter case the analysis for AdS 5 solutions with enhanced supersymmetry [33].

Appendix A Computation of the Ricci tensor
In this Appendix, we outline the computation of the Ricci tensor of (2.9). First of all, the non-vanishing components of the spin connection in the frame (2.7) are where Ω a,bc denotes the spin connection on M 11−n . Using (A.1), the non-vanishing components of the 11-dimensional Riemann tensor are where R abcd is the Riemann tensor on M 11−n . Implementing (A.2), we compute the 11-dimensional Ricci tensor, whose non-vanishing components are where R ab is the Ricci tensor on M 11−n .

Appendix B Reduction of the KSE and integrability conditions
In this Appendix, we provide some details about equations (2.18), (2.19), as well as the integrability conditions of (2.26). In the following, we denote ∂ µ = ∂ ∂x µ . Acting on (2.16) with ∂ ν , we find Taking the symmetric part of (B.1), we obtain where η is a Majorana spinor and F is a function. Substituting (B.4) into (2.18), we obtain Enforcing the vanishing of the coefficient of the ( which implies (the integration constant has been absorbed into F ) Inserting (B.7) in (B.5) we obtain ∂ µ ∂ ν η = 0 (B.8) whose solution is given by where ψ and τ µ are Majorana spinors depending only on the co-ordinates of M 11−n . Substituting (B.7) and (B.9) into (B.4), we get (2.19).
Let us now analyze the integrability conditions of (2.26), that is Appendix D dS 5 backgrounds: linear system (D.24) Moreover, the linear system associated to the algebraic condition (5.12), with Ψ given by (5.31), is given by (D. 30)