Supersymmetric dS4 solutions in D=11 supergravity

Supersymmetric warped product dS4 solutions in D=11 supergravity are classified. The Killing spinor is associated with two possible stabilizer groups, SU(3) and G_2. We show that there are no solutions to the Killing Spinor equations in the G_2 stabilizer case. For the SU(3) stablilzer case, all of the conditions imposed from supersymmetry on the 4-form flux, and the geometry of the internal manifold, are determined in terms of SU(3) invariant spinor bilinears.


Introduction
De Sitter geometry is of particular interest in terms of string cosmology and also in the context of the holographic principle.De Sitter spacetime plays a central role in the understanding of our present universe.From the work of [1,2,3] it has been observed that our universe is asymptotically dS 4 , corresponding to a very small positive cosmological constant.However, the observed value of the cosmological constant differs by many orders of magnitude from the vacuum energy density value predicted by quantum field theory [4,5].Moreover, in the context of string cosmology there are also difficulties in obtaining de Sitter space via compactification from higher dimensions.In particular, there are no go-theorems proving that smooth warped de Sitter solutions with compact, without boundary, internal manifold cannot be found in ten-and eleven-dimensional supergravity [6,7,8].Issues relating to quantum gravity in de Sitter space have been investigated in [9].
In terms of holography, the AdS/CFT correspondence relates string theory in Anti-de Sitter (AdS) space to conformal field theories (CFT) defined on an appropriate boundary [10].This has been particularly useful in developing a deeper understanding of the the microscopic nature of the entropy-area law [11,12].In spite of the considerable insights produced via the holographic principle, there are still many open issues in this area.Building from the AdS 3 /CFT 2 correspondence proposed by Brown and Henneaux in [13], the relation between quantum gravity on de Sitter space and conformal field theory on a sphere, the so-called dS/CFT correspondence, was considered in [14,15,16].However, our understanding of the conjectured dS/CFT correspondence is less complete than for the case of AdS/CFT for a number of reasons.Firstly, in contrast to AdS, there is a lack of de Sitter space solutions in string theory (or in any quantum gravity theory) in which the conjecture can be tested.Also, there are subtle issues with defining the dual CFT on the past and future spheres I ± , relating to the causal structure of dS space.Nevertheless, the macroscopic entropyarea law applies to a very wide class of black holes, including asymptotically flat, asymptotically AdS, and also asymptotically dS cases.The universality of this law provides strong motivation for understanding de Sitter holography.
Motivated by this, it is of particular interest to systematically understand the different types of de Sitter solutions which are possible in D=10 and D=11 supergravity.Such a classification may provide interesting new applications of the dS/CFT correspondence.As it is possible to embed dS n inside both R 1,n and AdS n+1 as a warped product geometry [17], it follows that the maximally supersymmetric AdS 7 × S 4 solution, as well as R 1,10 , can both be regarded as examples of warped product dS 4 geometries.However, as we shall establish here, there is a much larger class of supersymmetric warped product dS 4 solutions in D = 11 supergravity than these two very special solutions, and this is also somewhat in contrast to the results of recent analysis of supersymmetric warped product dS n geometries for 5 ≤ n ≤ 10.
In terms of D = 11 supergravity, there has been recent progress in the classification of supersymmetric warped product dS n geometries for 5 ≤ n ≤ 10 [18].There are a number of different possibilities: • For 7 ≤ n ≤ 10, the geometry is the maximally supersymmetic R 1,10 solution with vanishing 4-form flux.
• For warped product dS 6 solutions, the solution is either the maximally supersymmetric AdS 7 ×S 4 solution, or R 1,6 ×N where N is a hyper-Kähler 4-manifold.
• The warped product dS 5 solutions are all examples of generalized M5-brane solutions for which the transverse space is R × N, where N is a hyper-Kähler 4-manifold.
It is clear from this list that the possible warped product dS n geometries for 5 ≤ n ≤ 10 is very highly constrained.In addition, a similar recent analysis of warped product dS n solutions in heterotic supergravity [19], including first order α ′ corrections, has also produced a rather restricted class of such solutions.In this case, for n ≥ 3, the geometry is R 1,n × M 9−n , where M 9−n is a (9 − n)-dimensional manifold.The dilaton depends only on the co-ordinates of M 9−n , and all p-form fields have components only along the M 9−n directions.The heterotic warped product dS 2 solutions are the direct product AdS 3 × M 7 solutions which have been classified in [20].Compared to these types of solutions, the conditions on supersymmetric warped product dS 4 solutions in D = 11 supergravity are rather weaker.
Motivated by these results, in this paper we classify the warped product dS 4 solutions in D = 11 supergravity.We find, on integrating the Killing spinor equations along the dS 4 directions, that all of the necessary and sufficient conditions for supersymmetry are encoded in a single gravitino-type equation, which is satisfied by a spinor ψ + whose components depend only on the co-ordinates of the internal space.We analyse the solutions of this equation using spinorial geometry techniques.This technique was introduced in [21] and consists of writing the Killing spinors in terms of multi-differential forms and, utilizing the gauge-covariance of the KSE, gauge transformations are then used to write the spinors in one of several simple canonical forms.The main outcome of this approach is a linear system which imposes conditions on the spin connection and the fluxes of the theory.This in turn can be used to obtain conditions on the geometry which are necessary and sufficient for supersymmetry.These techniques have been applied to classify a wide variety of supergravity solutions [22].
In the case of warped product dS 4 solutions, we state explicitly the Spin(7) gauge transformations which are used to write the spinor ψ + in canonical forms with stabilizer subgroups SU(3) and G 2 .We then solve the linear system obtained from the Killing spinor equations.In particular, we show that the linear system implies that there are no Killing spinors for which the stabilizer of ψ + is G 2 .For the case of SU(3) stabilizer subgroup, the Killing spinor equations determine all components of the 4-form flux in terms of the geometry of the internal manifold, and we present the geometric conditions and the components of the flux, written in a SU(3) covariant fashion.On considering these conditions, we note that the warped product dS 4 geometries are manifestly less restricted in terms of the geometric structure and the 4-form flux in comparison to the warped product dS n solutions for 5 ≤ n ≤ 10.Our analysis does not utilize the global techniques developed for the investigation of supersymmetric black holes [23]; we consider only local properties of the Killing spinor equations.This avoids the no-go theorems which exclude warped product dS n solutions when the warp product and 4-form flux are smooth, and the internal manifold is smooth and compact without boundary.
The plan of this paper is as follows.In Section 2 we summarize the bosonic field equations, Bianchi identities, and Killing spinor equations of D = 11 supergravity, we also describe the ansatz for the warped product dS 4 solutions, and present the reduction of the bosonic conditions to the internal manifold.In Section 3, we derive several integrability conditions from the Killing spinor equations, and we demonstrate how some of these integrability conditions can be derived from others.In Section 4, we explicitly integrate up the Killing spinor equations along the dS 4 directions, and show how the Killing spinor equations reduce to a single gravitino-type equation for a spinor ψ + which depends only on the internal manifold co-ordinates.We also prove that the supersymmetric dS 4 warped product solutions preserve N = 8n supersymmetries for n = 1, 2, 3, 4. In Section 5 we utilize spinorial geometry techniques, and prove that the spinor ψ + can be written in a particularly simple canonical form on applying appropriate Spin(7) gauge transformations.Furthermore, we prove that such a spinor has stabilizer subgroup which is either SU(3) or G 2 ; in the SU(3) case we also consider several possible special sub-cases.In Section 6, we present the SU(3) covariant conditions on the flux and geometry, obtained from the gravitino-type equation in the case of SU(3) stabilizer.In Section 7 we also prove that there are no supersymmetric warped product dS 4 solutions for which the stabilizer subgroup of ψ + is G 2 .We present our conclusions in Section 8.In Appendix A, we list some conventions.In Appendix B we present some properties of the explicit representation of the Clifford algebras in terms of differential forms, as utilized in the spinorial geometry method.In Appendix C we list some equations which are used in the process of integrating up the Killing spinor equation along the dS 4 directions in Section 4. In Appendices D and E we list the linear system of equations obtained from the gravitino-type equation for the cases of ψ + with SU(3) and G 2 stabilizer respectively.

Bosonic field equations and KSE
In this section, we summarize the bosonic field equations and Killing spinor equations (KSE) of D = 11 supergravity, and describe the ansatz for the warped product dS 4 geometries.The bosonic fields of D = 11 supergravity consist of a metric g, and a 3-form gauge potential A with 4-form field strength F = dA.The action for the bosonic fields is given by where κ 2 is proportional to the gravitational coupling constant.The equations of motion are thus given by the first equation in (2.2) becomes The supercovariant derivative D A is defined as (2.5) Bosonic solutions to the equations of motion that preserve at least one supersymmetry are those that admit at least one non-vanishing Killing spinor ǫ, which satisfies In order to analyse supersymmetric warped product dS 4 solutions, we shall split the D = 11 spacetime in a 4+7 fashion ds 2 = dS 4 × w M 7 , where × w denotes a warped product of dS 4 with an internal manifold M 7 .In terms of the D = 11 frame, capital latin letters such as A,B denote D = 11 frame indices.These D=11 frame indices are split in a 4+7 fashion as follows: we use greek letters for dS 4 frame directions, and latin letters from the middle of the alphabet and onwards for M 7 .Latin letters from the beginning of the alphabet denote M 7 spacetime indices.M 7 is equipped with local co-ordinates y a , whereas dS 4 is equipped with local co-ordinates x µ .For further details about the conventions used are set out in Appendix A.
The warped dS 4 product metric g is where the vielbein frame is defined as The conformal factor A and the vielbein e j a depend only on y a co-ordinates.The scalar K is constant and greater than zero.
We require that the field strength F must be invariant under the isometries of dS 4 , hence it decomposes as follows: where c is a constant due to the Bianchi identity and X is a closed 4-form on M 7 depending only on y a co-ordinates.The gauge field equation (2.2) is equivalent to (2.11) It will be convenient to state the non-vanishing components of the spin-connection, and curvature components.The non-vanishing spin-connection components are where on the LHS Greek indices are frame indices on dS 4 , and on the RHS they are co-ordinate indices on dS 4 .∇ i denotes the Levi-Civita connection on M 7 .
The non-vanishing Riemann tensor components are where on the LHS Greek indices are frame indices on dS 4 , and on the RHS they are co-ordinate indices on dS 4 .The Ricci curvature tensor components are where on the LHS Greek indices are frame indices on dS 4 , and on the RHS they are co-ordinate indices on dS 4 .The (µν)-component of the Einstein equations of motion (2.4), imply that From the (ij)-component of the Einstein equation of motion (2.4) and the third equation in (2.14), one finds On taking the trace of (2.16), and using (2.11) and (2.15), we obtain

Integrability Conditions from the KSE
In this section, we begin the analysis of the KSE by computing the integrability conditions associated with (2.6).These results will be particularly useful when we explicitly integrate up the KSE along the dS 4 directions in the next section.From (2.5), we find and where We remark that (3.2) is equivalent to where ∇ i denotes the Levi-Civita connection on M 7 .
We use these expressions to derive several integrability conditions.First, from the integrability condition on dS 4 spacetime we get On the other hand, from the integrability condition with one direction on dS 4 and the other on M 7 , i.e.
So far, we have analyzed the integrability conditions involving the dS 4 part of the covariant derivative (3.1).The integrability condition on M 7 given by (3.10) In fact, (3.8) is implied by (3.10).To see this, contract (3.10) with Γ j and use the Einstein equation (2.16), the Bianchi identity, R l[ijk] = 0, and the condition dX = 0, as well as the gauge field equations (2.11).In particular: • The condition dX = 0 is used to derive: • The gauge field equation (2.11) implies that from and from this condition, it follows that • The gauge field equation (2.11) also implies that and from this condition, it follows that Hence, it follows that the integrability conditions (3.6) and (3.8) are both implied by (3.10), which is derived from the integrability condition of (3.4).
In this section, we will explicitly integrate the KSE along the dS 4 directions.In this analysis, we shall show that the KSE reduce to a single gravitino-type KSE acting on a spinor ψ which is independent of the dS 4 co-ordinates.To begin, we shall define a spinor Φ, as follows: where a is a constant to be fixed.We have chosen the relative coefficients between / X and / dA in (4.1) motivated by the first two terms in (3.8).We shall show that one can choose the constant a, as well as other constants k 1 , k 2 , q 1 , q 2 , q 3 , q 4 , q 5 such that Details of this calculation are presented in Appendix C. One finds that Given this choice of constants, the spinor Φ is which satisfies the following equations Equations (4.5) and (4.6), will be particularly useful in the process of integrating up the KSE along the dS 4 directions.By using (3.6), (4.5) becomes By using the definition of Φ (4.4), one can rewrite ∂ ∂x µ ǫ as Applying a second derivative ∂ ∂x ν to (4.8), using (4.7) and finally exploiting (4.8) to cancel R −1 Γ µ Φ terms, one gets a second order differential equation for ǫ, namely On defining η by it is straightforward to see that (4.9) is equivalent to and hence this equation can be integrated to find where ψ, τ λ with λ = 0, 1, 2, 3 are Majorana spinors which do not depend on the x µ co-ordinates.Given this expression for ǫ, i.e.
we substitute it into the KSEs (3.1) and (3.2).As the spinors ψ, τ λ are independent of the dS 4 co-ordinates, on expanding (3.1) and (3.2) order-by-order in x α , we find various conditions.In particular, from the KSE along the dS 4 directions (3.1), the vanishing of x−independent terms imply that the Majorana spinors τ µ are given in terms of ψ, as follows: The vanishing of the terms that are linear in x µ in (3.1) imply and we remark that this condition is equivalent to the integrability condition (3.6), but with ǫ replaced with ψ.The terms in (3.1) which are quadratic in x µ vanish identically; this then exhausts the content of (3.1).
Next we consider the KSE along the seven-dimensional internal directions, (3.2).Again, we substitute in (4.13) and expand order-by-order in dS 4 co-ordinates.The vanishing of x−independent terms gives The above equation (4.16) implies that ψ satisfies a gravitino KSE along the internal directions, which is identical to the condition (4.16) but with ǫ replaced with ψ.
From the terms in (3.2) which are linear in x µ we obtain which is identical to the integrability condition (3.8), with ǫ replaced by ψ.This then exhausts the content of (3.2).Hence, we have shown that the spinor ǫ is given by where The Majorana spinor ψ is independent of the dS 4 co-ordinates, and satisfies (4.16).Furthermore, ψ must also satisfy the algebraic conditions (4.17) and (4.15).However, as we have shown in the previous section, the integrability conditions of (4.16), together with the bosonic field equations and Bianchi identities, imply that (4.17) holds.Furthermore, we have shown that (4.17) also implies (4.15).Hence, the necessary and sufficient conditions for supersymmetry are encoded in (4.16).

Counting the supersymmetries
Having determined that the necessary and sufficient conditions for supersymmetry are given by (4.16), we shall now count the number of solutions to this equation.In particular, if ψ satisfies (4.16), then so does Γ µν ψ.We choose a null basis for the Majorana representation of Spin(10,1) and take the dS 4 frame directions to correspond with the +, −, 1, 1 directions, see Appendix B. The frame directions associated with the internal manifold M 7 correspond to the 2, 3, 4, 2, 3, 4, # directions.With these conventions for the de Sitter and internal manifold frames, we define lightcone projection operators as As the projection operator P ± commutes with the supercovariant derivative (4.16), we then decompose the spinor ψ using the lightcone projectors and we define ψ ± to be Without loss of generality, utilizing these projection operators, any supersymmetric solution must admit a positive chirality solution ψ + to (4. 16).Given such a ψ + spinor, we can then define ψ+ is an additional positive chirality solution to (4.16), and {ψ − , ψ− } are two negative chirality solutions to (4.16).{ψ + , ψ+ , ψ − , ψ− } are linearly independent, as by construction they are mutually orthogonal with respect to the Dirac inner product • , • .It would therefore appear, a priori, that the number of supersymmetries is 4n.However, there are, in fact further additional spinors.To see this, note that (4.6) implies that is also a positive chirality solution of (4.16).Furthermore, it can be shown that {ψ + , ψ+ , ψ} are linearly independent.To see this, suppose that for real constants c 1 , c 2 .Acting on both sides of this condition with the operator , and utilizing the integrability condition (4.15) to simplify the LHS, we find where we have also used (4.24) to simplify the RHS.It is clear that this admits no solution, as K > 0. Hence, we find that we can construct four linearly independent positive chirality spinors which solve (4.16), corresponding to {ψ + , ψ+ , ψ+ , ψ+ }, where ψ+ ≡ iΓ 1 1 ψ.There are also four corresponding negative chirality spinors given by Hence we have constructed 8 linearly independent solutions to (4.16), and it follows that the number of supersymmetries for warped product dS 4 solutions is 8n, n = 1, 2, 3, 4. We remark that the existence of the additional spinors ψ± , ψ± is somewhat analogous to results found in the analysis of near-horizon geometries of supersymmetric extremal black holes [23] and also for warped product AdS solutions [24].In these cases, given a Killing spinor, one also finds that additional Killing spinors can be generated by the action of certain algebraic operators constructed out of the fluxes of the theory.
In this section we shall use Spin(7) gauge transformations to bring the spinor ψ + to one of several simple canonical forms.We will describe the gauge transformations used to do this explicitly.
ψ + = w1 + we 1234 + λ 1 e 1 + λ1 e 234 + λ j e j − 1 3! ( * λ) l 1 l 2 l 3 e l 1 l 2 l 3 + Ωe 12 − Ωe 34 .(5.2) To proceed further, we define T 1 , T 2 , T 3 as It is straightforward to verify that T i with i = 1, 2, 3, which satisfy the algebra of the imaginary unit quaternions, preserve the span of the following basis elements and we remark that the Spin(7) gauge transformation generated by the T i is of the form p 4 id + p i T i where (p 1 , p 2 , p 3 , p 4 ) ∈ S 3 .Then one can carry out a SO(2) gauge transformation generated by T 3 to set w ∈ R.So far, the spinor ψ + can be written as A SU(3) gauge transformation generated by i(Γ 2 2− 1 2 Γ 3 3− 1 2 Γ 4 4), which leaves {1, e 1234 } invariant, is then used to set Ω ∈ R, so ψ + = w(1 + e 1234 ) + λ 1 e 1 + λ1 e 234 + λ j e j − 1 3! ( * λ) l 1 l 2 l 3 e l 1 l 2 l 3 + Ω(e 12 − e 34 ) .(5.6) We next exploit a SO(2) transformation generated by T 1 , acting on v 1 and v 3 to put Ω = 0.Then, we make a further SU(3) gauge transformation along the 2, 3, 4 directions to set λ 3 = λ 4 = 0 with λ 2 ∈ R, i.e. (5.7) In order to simplify further the spinor ψ + , we shall introduce additional Spin(7) generators L 1 , L 2 , L 3 given by (5.8) The L j also satisfy the algebra of the imaginary unit quaternions, and commute with the T i , and the Spin( 7) gauge transformation generated by the L j is of the form q 4 id + q j L j where (q 1 , q 2 , q 3 , q 4 ) ∈ S 3 .We shall then consider a generic gauge transformation generated by the T i and L j of the acting on the spinor (5.7) of the form p 4 id + p i T i q 4 id + q j L j ψ + . (5.9) We set q 2 = q 3 = 0 and q 1 = sin σ, q 4 = cos σ, such that and where the constant ℓ is chosen such that (p 1 , p 2 , p 3 , p 4 ) ∈ S 3 .With this choice of parameters, the gauge transformation given in (5.9) can be used to set λ 2 = 0 in (5.7), so the simplest canonical form for the spinor ψ + is given by (5.12)

Stabilizer Group of ψ +
It is useful to consider the stabilizer subgroup of Spin( 7) which leaves ψ + invariant.In particular, we must determine the generators f ij Γ ij , where f ij ∈ R are antisymmetric in i, j, and satisfy The conditions obtained from (5.13) are Depending on w and λ, there are two possible different stabilizer subgroups: (a) if w 2 −|λ| 2 = 0 then (5.14) implies that f αβ = 0 and f ♯α = 0, hence the stabilizer is SU(3).
In the SU(3) stabilized case it is particularly useful to consider the complex SU(3) invariant spinor bilinear scalar ψ + , Γ1ψ + = 2 √ 2wλ.There are various different cases, corresponding to whether this scalar vanishes, or it does not vanish: In fact, it is straightforward to see that the spinors associated with cases (i) and (ii) above are related by a P in (7) transformation.To see this, consider the spinor from case (ii), (5.15) The Spin (7)  It therefore follows that the spinor ψ + in case (ii) is Spin (7) gauge-equivalent to a spinor which in turn is Pin-equivalent, with respect to Γ 234 ∈ P in (7), to the spinor in case (i).The effect of the Γ 234 transformation is to flip holomorphic with antiholomorphic directions and to reflect along the # direction, namely α → ᾱ , # → −# . (5.17) It is therefore sufficient to consider spinors ψ + corresponding to the G 2 stabilizer case, and the two SU(3) stabilizer cases (i), (iii).Having determined the stabilizers associated with these three canonical types of spinors, we next proceed to obtain a linear system of equations by substituting these expressions for ψ + into (4.16).The linear system consists of relations between the flux and spin-connection, which when covariantized with respect to the appropriate stabilizer group, give rise to conditions on the flux X and the geometry of the internal manifold M 7 .In the following sections, we shall present the covariant solution of the linear system for each of the stabilizer subgroups.

SU(3) Invariant Spinor
In this section, we solve the KSEs (4.16) when the stabilizer of ψ + is SU(3), corresponding to for w 2 − |λ| 2 = 0. We begin by considering the case for which both w and λ are non-vanishing.Furthermore, we will write λ = ρe iθ , where ρ > 0 and θ ∈ [0, 2π[ are two real spacetime functions.The associated linear system and the components of the flux are presented in Appendix D. The linear system is initially expressed non-covariantly in terms of SU(3)-components of the spin-connection and the fluxes, but, it can be rewritten in SU(3)-covariant form by using the SU(3) gauge invariant bilinears.In Appendix E we set out the main relations which are used to write the relations in a manifestly SU(3) covariant fashion, in terms of the following SU(3) invariant bilinears: The above forms are obtained from the following SU(3)-invariant spinor bilinears: where a, b, c = α, ᾱ, #.
In constructing the solution to the linear system (D.2)-(D.13), it is convenient to make use of the two Lee forms built from χ, and ω, which are After some computation, the SU(3)-covariant conditions are as follows (here i, j, k = #, α, ᾱ are frame indices on M 7 ): We also obtain a SU(3) invariant expression for the flux X.In general, any real 4-form on M 7 can be written as where • σ is a real two-form; • β is a complex one-form, and β is its complex conjugate; • Y is real 3-form; • X TT is the traceless (2,2)-part of the flux.
We remark that X TT is the only part of the flux that is not fixed by the linear system.However, a traceless (2, 2) 4-form in 6 dimensions vanishes identically.To see this, note that X TT is dual (in 6 dimensions) to a (1,1) 2-form R, R = * 6 X TT .Furthermore, by definition as the contribution from trace terms in the final term vanishes.Hence R vanishes identically, and so X TT = 0.It follows that the flux can be written as where all of these terms are fixed by the Killing spinor equations.In particular, the components of the real 2-form σ and of the complex 1-form β are given by The real 3-form Y has components where q and h are functions, and V is a real one-form, given by: The SU(3) covariant geometric conditions obtained from the linear system are: (dω) (3,0) = (dω) (0,3) = 0 (6.27) χijk (L ξ ω) jk = 0 (6.29) The flux X can be expressed as and where In this section, we shall consider the case when the stabilizer of ψ + is G 2 , corresponding to the case with w 2 = |λ| 2 .We shall show that this orbit admits no solutions to the Killing spinor equations and the bosonic field equations.To establish this result, we set λ ≡ e iζ w, where ζ is a real function.The geometric conditions we obtained by solving the linear system are: Furthermore, we find that all of the components of the flux X vanish, As X = 0, the integrability condition Γ j [∇ i , ∇ j ] ψ + from (3.8) implies that Multiplying (7.9) by Γ l , we find We next take the inner product of (7.10) with ψ + , noting that the anti-hermitian terms vanish identically as ψ + is Majorana.The hermitian part gives The symmetric part of (7.11) then gives and the antisymmetric part of (7.11) implies The 3-form spinor bilinear in (7.13) is proportional to the G 2 -invariant 3-form ϕ given by Hence (7.13) implies that which in turn implies that dA = 0, so A is constant.However, from the Einstein field equation (2.15) we obtain It is clear that this equation admits no solution in the case for which A is constant and X = 0, as the LHS is strictly positive.Therefore, we conclude that there are no supersymmetric warped product dS 4 solutions for which the spinor ψ + is G 2 invariant.

Conclusion
We have classified the supersymmetric warped product dS 4 × w M 7 solutions in D = 11 supergravity.To do this, we first integrated explicitly the gravitino equation along the dS 4 directions.This reduces the conditions imposed by supersymmetry to a gravitinotype equation on M 7 acting on a Majorana spinor ψ + , whose components depend only on the co-ordinates of M 7 .Using spinorial geometry techniques, the spinor ψ + was then simplified to two possible canonical forms by Spin(7) gauge transformations.These two canonical forms have stabilizer subgroups corresponding to G 2 and SU(3).
In the G 2 case, we show that there is no solution to the Killing spinor equations.For the SU(3) case we have determined the 4-form flux in terms of SU(3) invariant geometric structures on M 7 , as well as determining all of the conditions imposed on the geometry of M 7 .This work fully classifies the supersymmetric dS 4 warped product solutions with minimal N = 8 supersymmetry.It would be interesting to consider the N = 16 case, as well as the N = 24 and N = 32 cases.In particular, for the latter two cases of N = 24 and N = 32 supersymmetry, it is possible to find further conditions on such solutions utilizing the homogeneity theorem analysis constructed in [25].To proceed with this, suppose that there we have N linearly independent solutions {ψ r : r = 1, . . ., N} for N = 24 or N = 32 to the gravitino equation (4.16).We then consider the integrability condition (4.15), which implies This implies that and hence On defining vector fields Θ rs i = ψ r , Γ 0 Γ i ψ s , this implies cL Θ rs A = 0 .(8.4) For N = 24 and N = 32 solutions, it follows from the homogeneity theorem analysis of [25] that the Θ rs span pointwise the tangent space of M 7 , and hence If c = 0, then this implies that dA = 0.However, (7.17) implies that there are no solutions for which A is constant.Hence, for N = 24 or N = 32 solutions, we must take c = 0.This determines all possible N = 32 warped product dS 4 solutions.From [26], where all maximally supersymmetric solutions in D = 11 supergravity were determined, the maximally supersymmetric solutions are R 1,10 with F = 0; AdS 4 × S 7 with 4-form F proportional to the volume form of AdS 4 , AdS 7 × S 4 , with 4-form F proportional to the volume form of S 4 , and a maximally supersymmetric plane wave solution which has F = 0, but F 2 = 0.In terms of possible N = 32 warped product dS 4 solutions, the condition c = 0 implies that F 2 ≥ 0 with equality if and only if F = 0. Hence we exclude AdS 4 × S 7 and the maximally supersymmetric plane wave as N = 32 warped product dS 4 solutions.It follows that the N = 32 warped product dS 4 solutions are R 1,10 and AdS 7 × S 4 .In particular, it is possible to write both R 1,4 and AdS 7 as warped product dS 4 geometries [17].It would be interesting to further understand the possible N = 16 and N = 24 warped product dS 4 solutions, though the homogeneity theorem does not apply to the N = 16 solutions.D.1 Solution for λ = 0, w = 0 From the linear system (D.2)-(D.13),we find that the components of the flux are given by the following expressions

Appendix E Covariant relations
In this Appendix, we present the main relations used to covariantize the linear system.These expressions relate spin connection terms to SU(3)-covariant terms involving the the SU(3) invariant 1-forms ξ, ω and χ, their Lie derivatives with respect to ξ, and also the Lee forms W and Z:

. 6 )
These equations are similar, but not identical, to the original Killing spinor equations for ǫ (3.1)-(3.2).The differences are in terms of certain signs appearing in (4.5)-(4.6),which are flipped with respect to (3.1)-(3.2) -in (4.5) the second and the fourth term with respect to (3.1) and in (4.6) the first and the third term with respect to (3.2).