Choices of spinor inner products on M-theory backgrounds

M-theory backgrounds in the form of unwarped compactifications with or without fluxes are considered. Bilinear forms of supergravity Killing spinors for different choices of spinor inner products are constructed on these backgrounds. The equations satisfied by the bilinear forms and their decompositions into product manifolds are obtained for different inner product choices. It is found that the AdS solutions can only appear for some special choices of spinor inner products on product manifolds. The reduction of bilinears of supergravity Killing spinors into the hidden symmetries of product manifolds which are Killing-Yano and closed conformal Killing-Yano forms for AdS solutions is shown. These hidden symmetries are lifted to eleven-dimensional backgrounds to find the hidden symmetries on them. The relation between the choices of spinor inner products, AdS solutions and hidden symmetries on M-theory backgrounds are investigated.

solutions reduce to special Killing-Yano (KY) or special closed conformal Killing-Yano (CCKY) forms depending on the choices of the spinor inner products. KY forms are antisymmetric generalizations of Killing vector fields to higher degree differential forms and CCKY forms are a subset of antisymmetric generalizations of conformal Killing vector fields to higher degree forms. These special forms are called the hidden symmetries of manifolds. We also obtain KY and CCKY forms of eleven-dimensional backgrounds by lifting the hidden symmetries on product manifolds. So, we determine the relations between hidden symmetries, AdS solutions and choices of spinor inner products by exhausting all possibilities for spinor inner product choices. This may be considered as a first step of a classification of backgrounds in terms of spinor inner products.
The paper is organized as follows. In Section II, we summarize the equations for the bosonic sector of elevendimensional supergravity. Section III deals with M 4 × M 7 type backgrounds. We find the decompositions of field equations and supergravity Killing spinor equation and construct the bilinear form equations for both types of spinor inner products with their decompositions onto product manifolds. In Section IV, the same steps are achieved for M 7 × M 4 type backgrounds. Section V includes the situation for other types of backgrounds. In Section VI, the relation between hidden symmetries and AdS solutions are summarized and the lifts of hidden symmetries to elevendimensional backgrounds are considered. Section VII concludes the paper. There are also three appendices containing the topics of inner product classes of spinor spaces, Clifford algebra conventions and Clifford bracket and KY forms.

II. ELEVEN-DIMENSIONAL SUPERGRAVITY
Let us consider an eleven-dimensional Lorentzian spin manifold M 11 , with a metric g and a closed 4-form F . F is called the flux 4-form and the bosonic sector of the eleven-dimensional supergravity theory defined on M 11 is given by the following action where κ 11 is the eleven-dimensional gravitational coupling constant, capital letter indices take values A, B = 0, 1, 2, ..., 9, 10 and * 11 is the eleven-dimensional Hodge star operator. R AB are the curvature 2-forms, e A are coframe basis and A is the 3-form potential of the flux 4-form F = dA. The first term in (1) corresponds to the gravitational term and second and third terms are Maxwell-like and Chern-Simons terms, respectively. The field equations of the eleven-dimensional bosonic supergravity results from the above action by considering the variations of e A and A as follows * 11 (i XB P A ) = 1 2 i XA F ∧ * 11 i XB F − 1 6 g AB F ∧ * 11 F (Einstein) (2) dF = 0 (Closure) (4) where i XA denotes the interior derivative or contraction operator with respect to the vector field X A , g AB are components of the metric and P A are the Ricci 1-forms defined from the curvature 2-forms as P A = i X B R BA . The last equation (4) is the integrability condition for the definition of the flux form F . Moreover, the variation of the gravitino field in the fermionic sector will also lead to a condition on the spinor ǫ which is the supersymmetry parameter and in the bosonic sector it gives the following supergravity Killing spinor equation where ∇ X A corresponds the spinor covariant derivative and . denotes the Clifford multiplication. The co-frame basis e A define a basis of the Clifford algebra bundle Cl 10,1 on M 11 with the following equality e A .e B + e B .e A = 2g AB (6) where g AB are the components of the inverse metric. The supersymmetry parameter ǫ is an element of the spinor bundle S which corresponds to R 32 on M 11 and hence ǫ is a Majorana spinor. In the following chapters, we consider various types of unwarped compactifications of supergravity backgrounds which are the solutions of the field equations (2)-(4) of type M = M d × M 11−d . By constructing the bilinear forms of supergravity Killing spinors defined in (5) in those backgrounds, we show that the reduction or non-reduction of those bilinear forms into KY and CCKY forms on product manifolds require the existence or non-existence of AdS or Minkowski type solutions with or without internal and external fluxes. Moreover, we determine the correspondences between the choices of spinor inner products on product manifolds and the types of possible supergravity backgrounds. This gives a classification of unwarped compactifications of supergravity backgrounds in terms of spinor inner products.
We first consider the case that the eleven-dimensional supergravity background M 11 has the product structure M 11 = M 4 × M 7 where M 4 is a Lorentzian spin 4-manifold and M 7 is a Riemannian spin 7-manifold. The frame and co-frame basis indices appeared in the previous equations will split into two parts A = {a, α} with a = 0, 1, 2, 3 and α = 4, 5, ..., 9,10. The Clifford algebra basis e A will decompose as where e a are the Clifford algebra basis on M 4 , z 4 is the volume form on M 4 , 1 7 is the identity on M 7 and e α are the Clifford algebra basis on M 7 which are pure imaginary. By considering the equalities e a .e b + e b .e a = 2g ab , e α .e β + e β .e α = 2g αβ and the properties z 2 4 = −1 and z 4 anticommutes with all 1-forms on M 4 , one can obtain the defining relation (6) from (7). Here, the inverse metric is decomposed as g AB = {g ab , g αβ } and there is no warped product factor. Similarly, the flux 4-form F will decompose as where λ and µ are constants and φ is a 4-form on M 7 . The flux components on M 4 and M 7 are called external and internal fluxes respectively and the constants λ and µ determine the existence or non-existence of external and internal flux components. The supersymmetry parameter ǫ will be constructed from four-dimensional and seven-dimensional spinors ǫ 4 and ǫ 7 as For the product structure M 4 × M 7 , the field equations (2)-(4) will decompose into four-dimensional and sevendimensional equations. For the Maxwell equation (3) and the closure condition (4), we can use the decomposition of the flux 4-form F in (8). For any product structure M n = M p × M q , the Hodge star operator * n satisfies the following equality where α is a k-form on M p and β is a l-form on M q [20]. So, in our case, we have where we have used z 4 = * 4 1 4 , * 4 * 4 = −1 and z 7 = * 7 1 7 . Its exterior derivative gives and the right hand side of (3) is since we have dz 4 = 0, z 4 ∧ z 4 = 0 and φ ∧ φ = 0 because of the fact that φ ∧ φ is a 8-form on M 7 . On the other hand, the closure condition dF = 0 gives dφ = 0 and hence we obtain the following equalities from equations (3) and (4) These equalities define a weak G 2 structure on M 7 and φ corresponds to the coassociative 4-form on it. So, M 7 will correspond to a proper weak G 2 manifold, a Sasaki-Einstein manifold or a 3-Sasaki manifold [21]. Moreover, (14) means that φ is a CCKY 4-form on M 7 and hence it must be generated from a geometric Killing spinor [15]. Since z 4 is the volume form on M 4 , it corresponds to a KY form on M 4 and as a result, the flux 4-form F in (8) is generated by KY and CCKY forms on M 4 and M 7 for λ = 0 and µ = 0. Einstein field equations given in (2) can also be decomposed into M 4 and M 7 components. From the flux 4-form F in (8), one can find the terms on the right hand side of (2) with similar calculations to the above as follows F ∧ * 11 F = λ 2 + µ 2 g 4 (φ, φ) z 11 (15) i XA F ∧ * 11 i XB F = {−λ 2 g ab z 11 , µ 2 g 4 (φ, φ)g αβ z 11 } where g p denotes the metric on p-forms. Here, we have used the definition of Hodge star in terms of the p-form metric; for any p-forms α and β we have α ∧ * β = g p (α, β) * 1. Hence, we have The left hand side of (2) corresponds to i XB P A = {i X b P a , i X β P α } where P a and P α are Ricci 1-forms on M 4 and M 7 , respectively. So, the Einstein field equations decompose into M 4 and M 7 as follows This means that for λ = µ = 0, both M 4 and M 7 are Ricci-flat manifolds and for the special case of λ = 0 and µ = 0, M 4 is a negative curvature and M 7 is a positive curvature Einstein manifolds (since the basis 1-forms are pure imaginary on M 7 , the metric components g αβ = g(e α , e β ) will have an extra minus sign). We will also analyze the decomposition of supergravity Killing spinor equation (5) into product manifolds. From (9), the left hand side of (5) corresponds to and by using the decompositions in (7) and (8), the right hand side of (5) gives So, the supergravity Killing spinor equation can be written as where we have used that the volume form z 4 anticommutes with basis 1-forms on even dimensions that is z 4 .e a = −e a .z 4 and on a Lorentzian 4-manifold it satisfies (iz 4 ) 2 = 1, so we have iz 4 .ǫ 4 = ±ǫ 4 . The decompositions of supergravity Killing spinor equation on M 4 and M 7 have to be considered separately for the cases of existence or nonexistence of internal and external fluxes. For the fluxless case λ = µ = 0, we have ∇ X a ǫ 4 = 0 ∇ X α ǫ 7 = 0 (21) and this means that ǫ 4 and ǫ 7 are parallel spinors on M 4 and M 7 , respectively. This is consistent with the Ricciflatness property in (17) and (18). 7-dimensional Riemannian manifolds admitting parallel spinors correspond to G 2 holonomy manifolds [17]. 4-dimensional Lorentzian manifolds admitting parallel spinors can be Minkowski or plane-wave spacetimes. However, Ricci-flatness property restricts the case to the Minkowski spacetime. Then, this case corresponds to the solution Mink 4 × G 2 . For the existence of only the external flux λ = 0 and µ = 0, we have and this corresponds to the case that ǫ 4 and ǫ 7 are geometric Killing spinors on M 4 and M 7 respectively which is consistent with being Einstein manifolds from (17) and (18). The geometric Killing spinors on M 4 and M 7 are real and imaginary Killing spinors, respectively [27]. 7-dimensional Riemannian manifolds admitting imaginary Killing spinors correspond to weak G 2 manifolds. In the case of admitting one Killing spinor, it is a proper weak G 2 manifold. For the existence of two and three Killing spinors, it corresponds to Sasaki-Einstein and 3-Sasaki manifolds, respectively. If there are maximal number of Killing spinors, then M 7 is a round sphere S 7 . 4-dimensional Einstein manifolds with negative curvature admitting real Killing spinors correspond to AdS 4 spacetimes. Then, the solutions in that case corresponds to AdS 4 × S 7 and AdS 4 × weak G 2 . For the general case of λ = 0 and µ = 0, we have both nonzero internal and external fluxes. In the literature, the presence of internal fluxes generally implies the consideration of a warp factor in the metric to obtain consistent solutions of field equations [6][7][8][9][10]22]. However, there is also a possibility of a solution for the unwarped case if the internal flux component φ satisfies a specific condition. We know that the internal flux φ satisfies the CCKY form equations (14) which means that they are constructed from geometric Killing spinors. If φ satisfies the condition φ.ǫ 7 = ± 1 2 ǫ 7 , then the supergravity Killing spinor equation decomposes into the following equations Moreover, one can write the Clifford product of a 1-form e α with an arbitrary form ω in terms of the wedge product and interior derivative as follows where the automorphism η acts on a p-form ω as ηω = (−1) p ω. Then, we have By applying the interior derivative operator i X α to the equations (14), one can see that φ satisfies and from the definition of the Lie derivative L X α = di X α + i X α d on forms, one obtains If the following condition on φ is satisfied then the equation (24) is transformed into Now, if we choose the constant µ as µ = ± λ 5 , then the supergravity Killing spinor equation decomposes into the following equations from (23) and (30) which correspond to geometric Killing spinors on M 4 and M 7 . This is consistent with the condition φ.ǫ 7 = ± 1 2 ǫ 7 , since if φ is constructed from ǫ 7 as a bilinear 4-form, then it automatically satisifies this condition from Fierz identities [15]. So, the only restriction on φ to obtain geometric Killing spinors on product manifolds is the condition (28). Indeed, the equations satisfied by φ correspond to the case that M 7 is a weak G 2 manifold and φ is the coassociative 4-form defined on it. In that case, g 4 (φ, φ) in (17) and (18) is constant and g 3 (i Xα φ, i X β φ) is proportional to g αβ [23]. So, equations (17) and (18) imply that M 4 and M 7 are Einstein manifolds. Then, the case λ = 0 and µ = 0, for the special choice of µ = ± λ 5 , also corresponds to the solutions AdS 4 × S 7 and AdS 4 × weak G 2 . But, for the general case of λ = 0 and µ = 0, the supergravity Killing spinor equation (20) cannot be decomposed into M 4 and M 7 components and one cannot find a general solution. For the final case of λ = 0 and µ = 0 which corresponds to the existence of only the internal flux, the equations will be similar to the previous case. However, if we take λ = 0 in (17), (18), (31) and (32), then (17) and (18) imply that M 4 and M 7 are Einstein manifolds, but (31) and (32) imply that they must admit parallel spinors which is inconsistent. So, λ = 0 and µ = 0 case does not correspond to a solution.

A. Bilinear forms
Now, we will construct bilinear forms of supergravity Killing spinors by using the defining equation (5). The spinor bilinear of a spinor ǫ is defined in terms of the spinor inner product ( , ) and co-frame basis as a sum of different degree differential forms as follows ǫǫ = (ǫ, ǫ) + (ǫ, e a .ǫ)e a + (ǫ, e ba .ǫ)e ab + ... + (ǫ, e ap...a2a1 .ǫ)e a1a2...ap + ... + (−1) ⌊n/2⌋ (ǫ, z.ǫ)z where e a1a2...ap = e a1 ∧ e a2 ∧ ... ∧ e ap and z is the volume form. The bilinear p-form of the spinor ǫ is defined as the p-form component of the spinor bilinear However, we can consider two different spinor inner products on the spinor bundle of M 11 . We have the Clifford algebra Cl 10,1 and its even subalgebra that is isomorphic to Cl 0 10,1 ∼ = Cl 1,9 . So, the spinor space is isomorphic to R 32 and we have Majorana spinors with the spinor inner product choices R-skew with ξη involution or R-symmetric with ξ involution, where ξ is acting on a p-form ω as ω ξ = (−1) ⌊p/2⌋ ω with ⌊ ⌋ is the floor function. The details of the spinor inner product classes in all dimensions can be found in Appendix A. In the literature, only the first choice of the inner product is considered and the investigations are based on this choice. We will consider both choices separately and analyze the decomposition of bilinear forms on product manifolds in both cases.

R-skew ξη inner product
First, we choose the spinor inner product as R-skew with ξη involution and find the decomposition of bilinear forms on product manifolds. The bilinear forms constructed from a spinor are elements of S ⊗ S * where S is the spinor space and S * is the dual spinor space. Since the connection ∇ is compatible with the spinor inner product ( , ) and preserves the degree of a form, it is also compatible with the projection operation ( ) p on p-form bilinears and we can write for a supergravity Killing spinor ǫ as where we have used (5). For any spinor ψ, the dual spinor ψ can be written in terms of the involution operation J as ψ = ψ J . Since we have J = ξη, for any Clifford form ω and spinor ψ, we have ω.ψ = (ω.ψ) ξη = ψ ξη .ω ξη = ψ.ω ξη . Then, we can write where we have used F ξη = F and X ξη = − X. By using this equality in (34), we obtain If we add and subtract the term 1 6 ǫǫ.(F. X − X.F ) p to the right hand side, we find So, the bilinear form equation of supergravity Killing spinor ǫ which is also called the supergravity Killing form equation can be written as where [ , ] Cl denotes the Clifford bracket. Since we can write and so the supergravity Killing form equation (38) turns into The only non-zero bilinear forms of a spinor on an eleven-dimensional Lorentzian manifold are 1-, 2-, 5-, 6-, 9-and 10-forms as can be seen from Table XVII in Appendix A. So, the spinor bilinear of the supergravity Killing spinor ǫ is We can find the equations satisfied by all of the bilinear forms by considering the definition of the Clifford bracket and projection operation given in (B9). For p = 1, we have the following equation for the bilinear 1-form (ǫǫ) 1 from (41) where we have used the definition of the contracted wedge product given in (B8). If we use the definitions of the exterior derivative and coderivative in terms of the covariant derivative as d = e A ∧ ∇ XA and δ = −i X A ∇ XA for zero torsion, we obtain By comparing the equations (43) and (44), one can easily see that (ǫǫ) 1 satisfies the equation and hence (ǫǫ) 1 is a KY 1-form. Consequently, the vector field which is metric dual to the 1-form (ǫǫ) 1 is a Killing vector field. The definition and properties of KY forms can be found in Appendix C. For p = 2, the bilinear form equation (41) gives and the exterior and co-derivatives are So, (ǫǫ) 2 does not satisfy the KY equation. For p = 5, the bilinear form equation gives where the terms on the right hand side can also be written in a more explicit way by using the identity Similarly, for p = 6, we have and 5-and 6-form bilinears also do not satisfy the KY form equation. For the case of p = 9, (41) gives and For p = 10, we have and So, except the 1-form bilinear, all the higher degree bilinear forms of supergravity Killing spinors do not correspond to KY forms and satisfy different types of equations. Now, we can consider the decomposition of bilinear forms onto product manifolds M 4 and M 7 . Since the supergravity Killing spinor ǫ decomposes as in (9), the spinor bilinears decompose as ǫǫ = {ǫ 4 ǫ 4 , ǫ 7 ǫ 7 }. By considering the definitions ǫǫ (4) := ǫ 4 ǫ 4 and ǫǫ (7) := ǫ 7 ǫ 7 , the p-form bilinears on product manifolds correspond to Since, the degree of differential forms cannot be greater than the volume form, from (42) we have ǫǫ (4) = (ǫǫ (4) ) 1 + (ǫǫ (4) ) 2 ǫǫ (7) = (ǫǫ (7) ) 1 + (ǫǫ (7) ) 2 + (ǫǫ (7) ) 5 + (ǫǫ (7) ) 6 .
inner product 1 2 5 6 Moreover, depending on the spinor inner product choices on M 4 and M 7 , one can determine the properties of nonzero bilinears constructed out of ǫ 4 and ǫ 7 . M 4 is a Lorentzian manifold, the spinor space corresponds to C 2 ⊕ C 2 and the spinors are Dirac-Weyl spinors. M 7 is a Riemannian manifold, the spinor space corresponds to R 8 and the spinors are Majorana spinors. So, from Table XVII in Appendix A, we have the bilinears for the chosen inner products given in Table I. So, we have to consider four different cases separately in the decomposition of bilinear forms onto product manifolds. i) M 4 : C * -sym ξ and M 7 : R-skew ξ; By considering the nonzero bilinears in the above table and the decomposition of the 4-form flux F in (8), the 1-form bilinear equation (43) decomposes as and the second equality implies that µ = 0 while the first equality implies that λ is real (since (ǫǫ (4) ) 1 is real and (ǫǫ (4) ) 2 is pure imaginary from the above table). From the first equality, one can obtain Then, the reduction of the 1-form bilinear onto M 4 is also a KY 1-form Similarly, the 2-form bilinear equation (46) decomposes as From the first equality, we have and so (ǫǫ (4) ) 2 is a KY 2-form However, the second equality in (61) implies that (ǫǫ (7) ) 2 is a parallel form and hence must be constructed from the parallel spinor ǫ 7 . This implies from (22) that λ must also vanish λ = 0. Then, the bilinears on M 4 also correspond to parallel forms and ǫ 4 is also a parallel spinor. So, the choice of inner product forces the flux F to vanish and the decompositions of 5-form and 6-form bilinear equations also imply this. Then, as a result, the first inner product choice allows only Mink 4 × G 2 solutions. ii) M 4 : C * -sym ξ and M 7 : R-sym ξη; In this case, all bilinear forms on M 7 which appear in the bilinear form equations are automatically zero as can be seen from the above table. So, the seven-dimensional parts of the decompositions are trivial and this does not give a restriction on µ. The inner product choice for M 4 is the same as for the first case and hence the four-dimensional parts of the bilinears correspond to KY forms and moreover they correspond to special KY forms. By direct computation, one can see that In summary, the relation between spinor inner product choices and M 4 × M 7 solutions is as given in Table II. Note that AdS solutions can exist only for the inner product choices for which the supergravity Killing forms decompose into special KY forms on product manifolds.

R-sym ξ inner product
In the second case, we choose the spinor inner product on the eleven-dimensional Lorentzian manifold M 11 as R-sym with ξ involution and consider the decomposition of bilinear forms in that case. p-form bilinear equation is the same as in (34) However, in this case the involution is ξ and for a Clifford form ω and a spinor ψ, we have ω.ψ = ψ ξ .ω ξ = ψ.ω ξ . So, we can write where we have used F ξ = F and X ξ = X. By adding and subtracting the term 1 6 ǫǫ.(F. X − X.F ) p to (67), we find where [ , ] +Cl denotes the Clifford anticommutator which is defined in (B11) and (B12). In terms of wedge product and interior derivative, the supergravity Killing form equation can also be written from (40) as The nonzero bilinear forms for R-sym ξ inner product are 0-, 1-, 4-, 5-, 8-and 9-forms and the spinor bilinear of the supergravity Killing spinor ǫ corresponds to ǫǫ = (ǫǫ) 0 + (ǫǫ) 1 + (ǫǫ) 4 + (ǫǫ) 5 + (ǫǫ) 8 + (ǫǫ) 9 .
So, (ǫǫ (4) ) 0 is constant and we can write the exterior and coderivatives of bilinear forms as d(ǫǫ (4) and d(ǫǫ (4) Then, by comparing (85) with (86) and (87), one can see that they satisfy the CCKY equation Moreover, they correspond to special CCKY forms So, ǫ 4 is a geometric Killing spinor generating the supergravity Killing forms which correspond to special CCKY forms. All of the bilinear form equations on M 7 are trivial and hence we have all types of solutions for this inner product choice. Namely, AdS 4 × weak G 2 , AdS 4 × S 7 and Mink 4 × G 2 . ii) M 4 : C * -sym ξ and M 7 : R-sym ξη; In this case, the situation for M 4 is the same as in the previous case and hence (ǫǫ (4) ) 1 and (ǫǫ (4) ) 4 are special CCKY forms. For M 7 , we have the following equalities ∇ Xα (ǫǫ (7) ) 0 = 0 So, we have µ = 0 and 0-and 4-forms are parallel. Then, we have the solutions AdS 4 × weak G 2 , AdS 4 × S 7 for λ = 0 and µ = 0. For λ = µ = 0, we have Mink 4 × G 2 . iii) M 4 : C-skew ξη and M 7 : R-skew ξ; This case gives The case for M 4 is the same as the previous case and for M 7 it is the same with case ii. So, both λ and µ vanishes and we have Mink 4 × G 2 solution.
In summary, for the inner product choice of R-sym ξ on M 11 , the solutions that appear for different types of inner product choices on M 4 and M 7 can be given as in Table IV. Note that, when the AdS solutions exist for the relevant choices of spinor inner products, the supergravity Killing forms decompose into special CCKY forms on product manifolds.  where λ and µ are constants and φ is a 4-form on M 7 . Similarly, the supersymmetry parameter can be written as By decomposing the field equations similar to the case in Section III, the Maxwell-like field equations give This means that φ is a CCKY 4-form on M 7 and must be generated from a geometric Killing spinor. Since the volume form z 4 is also a KY 4-form, the flux form F is generated by KY and CCKY forms and hence by geometric Killing spinors for λ = 0 and µ = 0. The decomposition of Einstein field equations will give the following equalities on M 4 and M 7 respectively This means that for λ = µ = 0, both M 7 and M 4 are Ricci-flat manifolds and for the special case of λ = 0 and µ = 0, M 7 is a negative curvature and M 4 is a positive curvature Einstein manifolds. The decomposition of the supergravity Killing spinor equation into product manifolds can be found as follows where we have used that z 4 .e α = −e α .z 4 and z 2 4 = 1 for the Riemannian manifold M 4 , so we have z 4 .ǫ 4 = ±ǫ 4 . Then, for the fluxless case λ = µ = 0, we have two equations on product manifolds and this means that ǫ 7 and ǫ 4 are parallel spinors on M 7 and M 4 , respectively and both M 7 and M 4 are Ricciflat manifolds. 4-dimensional Riemannian manifolds admitting parallel spinors correspond to Calabi-Yau manifolds with SU (2) holonomy (and also hyperkähler manifolds with Sp(1) holonomy but they are equivalent to Calabi-Yau manifolds with SU (2) holonomy). 7-dimensional Ricci-flat Lorentzian manifolds admitting parallel spinors can be Minkowski spacetimes. So, fluxless case corresponds to Mink 7 × CY 2 . For the existence of only internal flux λ = 0 and µ = 0, we have the following equations and hence ǫ 7 and ǫ 4 correspond to geometric Killing spinors on M 7 and M 4 , respectively. So, M 7 and M 4 are Einstein manifolds. The only four dimensional Riemannian manifold admitting geometric Killing spinors is the four-sphere S 4 and the solution in this case corresponds to AdS 7 × S 4 . If both internal and external fluxes are non-zero λ = 0 and µ = 0 and φ satisfies φ.ǫ 7 = ± 1 2 ǫ 7 , then the supergravity Killing spinor equation decomposes into By doing similar calculations as in Section III, one can find that if φ satisfies the condition (L X a * 7 φ).ǫ 7 = µe a .ǫ 7 (103) then (100) transforms into a geometric Killing spinor equation and both ǫ 7 and ǫ 4 are geometric Killing spinors. However, this case does not give a new solution and also corresponds to AdS 7 × S 4 solution. For the case of λ = 0 and µ = 0, the field equations and Killing spinor equations give an inconsistency and hence this case does not correspond to a solution. The decomposition of bilinear forms of supergravity Killing spinors have to be investigated separately for different choices of spinor inner products. For the choice of spinor inner product R-skew ξη on M 11 , the supergravity Killing forms (ǫǫ) p = {(ǫǫ (7) ) p , (ǫǫ (4) ) p } satisfy (41) and non-zero bilinear forms correspond to ǫǫ (7) = (ǫǫ (7) ) 1 + (ǫǫ (7) ) 2 + (ǫǫ (7) ) 5 + (ǫǫ (7) ) 6 ǫǫ (4) = (ǫǫ (4) ) 1 + (ǫǫ (4) ) 2 on M 7 and M 4 , respectively. M 7 is a Lorentzian 7-manifold, so the spinor space is H 4 and the spinors are symplectic Majorana spinors. M 4 is a Riemannian 4-manifold, so the spinor space is H ⊕ H and the spinors are symplectic Majorana-Weyl spinors. From Table XVII in Appendix A, we have the bilinears for the chosen inner products given in Table V. We consider four different inner product choices in the decomposition of supergravity Killing forms. i) M 7 : H − -sym ξ and M 4 : H-swap ξ; In that case, the bilinear form equations for (ǫǫ) 1 , (ǫǫ) 2 , (ǫǫ) 5 , (ǫǫ) 6 , (ǫǫ) 9 and (ǫǫ) 10 on M 7 corresponds to Table VI: The relation between the choice of spinor inner products and M7 × M4 solutions for R-skew ξη inner product on M11.
inner product 0 1 4 5 Hence, this inner product choice coresponds to AdS 7 × S 4 solution and the geometric Killing spinor ǫ 7 generates the flux component φ which is a special CCKY form and the geometric Killing spinor ǫ 4 generates the bilinear forms (ǫǫ (4) ) 1 and (ǫǫ (4) ) 2 which are special KY forms. If we choose µ = 0, then the solution reduces to Mink 7 × CY 2 case and the bilinear forms correspond to parallel forms.
iii The bilinear form equations in this choice are exactly the same as in case (ii) and we have λ = 0 and µ = 0. So, the geometric Killing spinor ǫ 7 generates the flux component φ and the geometric Killing spinor ǫ 4 generates the bilinear forms (ǫǫ (4) ) 1 and (ǫǫ (4) ) 2 which are special KY forms. So, this case corresponds to AdS 7 × S 4 and Mink 7 × CY 2 solutions.
As a result, for the inner product choice of R-skew ξη on M 11 , the relation between the solutions AdS 7 × S 4 and Mink 7 × CY 2 and the inner product choices on M 7 and M 4 can be described as in Table VI.
We consider four different choices for the decomposition of supergravity Killing forms. i) M 7 : H − -sym ξ and M 4 : H-swap ξ;  As a result, for the inner product choice of R-sym ξ on M 11 , the relation between the solutions AdS 7 × S 4 and Mink 7 × CY 2 and the inner product choices on M 7 and M 4 can be described as in Table VIII.
Note that in the presence of AdS solutions, supergravity Killing forms decompose into special CCKY forms. Remember that we only consider the unwarped product manifolds and in that case these types of backgrounds will not give interesting examples for the reduction of supergravity Killing form bilinears into KY and CCKY forms by choosing different spinor inner products. The reason for that is the fact that the AdS solutions can only appear for these types of backgrounds in the presence of a warp factor and in the unwarped case we do not have AdS solutions.
In M 5 × M 6 case, we have the following decompositions where ψ is a 4-form on M 5 , φ is a 4-form on M 6 and λ and µ are constants. However, for these choices, the consistent decompositions of Maxwell-like, Einstein and supergravity Killing spinor equations can only be possible for the fluxless case λ = 0 = µ. Hence, in that case the solution for all types of spinor inner products is Mink 5 × CY 3 .
For M 6 × M 5 case, the situation is similar and the only consistent decomposition corresponds to the fluxless case. However, in that case we do not have any solution since there are no five-dimensional compact Riemannian manifolds admitting parallel spinors.
In M 3 × M 8 , we have the following decompositions where φ is a 4-form on M 8 . Similarly, the only consistent decomposition is in the fluxless case µ = 0 and for all types of spinor inner products the only solution is Mink 3 × Spin (7). inner product on M11 background reduction of bilinears on M4

VI. REDUCTION AND LIFT OF KY AND CCKY FORMS
The existence of AdS solutions for the unwarped M 4 ×M 7 and M 7 ×M 4 type backgrounds is highly dependent on the choice of spinor inner products on product manifolds. As we have have seen in sections III and IV, only some special choices of spinor inner products allow the AdS solutions. Moreoever, we have shown that, for the AdS solutions, there is a relation between supergravity Killing forms on M 11 and the hidden symmetries on product manifolds. The type of hidden symmetries on product manifolds is also dependent on the choice of the spinor inner product on M 11 . For the choice of R-skew ξη inner product on M 11 , supergravity Killing forms reduce onto special KY 1-and 2-forms on M 4 . If one chooses R-sym ξ inner product on M 11 , then the supergravity Killing forms reduce onto special CCKY 1-and 4-forms on M 4 . These are correct for both M 4 × M 7 and M 7 × M 4 type backgrounds. The situation can be summarized as in Table IX. KY and CCKY forms on product manifolds which are reduced from the supergravity Killing forms on M 11 can also be lifted to hidden symmetries on M 11 . For any manifold M with a product structure M = M m × M n and metric one can construct KY and CCKY forms on M by using the KY and CCKY forms on M . For a KY p-form ω on M and a CCKY q-form ν on M , the following forms are KY p-forms and CCKY (m + q)-forms on M , respectively [25]. Here z M is the volume form on M . So, for the solutions AdS 4 × S 7 and AdS 4 × weak G 2 , the internal component of the flux which is the 4-form φ is a CCKY 4-form and the following form is a CCKY 8-form on M 11 . For the spinor inner product R-skew ξη on M 11 and the solution AdS 7 × S 4 , we have special KY forms (ǫǫ (4) ) 1 and (ǫǫ (4) ) 2 on S 4 . So, we have the following KY 1-and 2-forms on M 11 ω 1 = (ǫǫ (4) ) 1 However, these do not need to be special KY forms. For the spinor inner product R-sym ξ on M 11 and the solution AdS 7 × S 4 , we have the special CCKY forms (ǫǫ (4) ) 1 and (ǫǫ (4) ) 4 on S 4 . So, we have the following CCKY 8-and 11-forms on M 11 Again, these do not need to be special CCKY forms. As a result, supergravity Killing forms constructed out of supergravity Killing spinors induce KY and CCKY forms on AdS backgrounds of eleven-dimensional supergravity.

VII. CONCLUSION
We show that the choices of spinor inner products play a central role for the M-theory backgrounds corresponding to unwarped compactifications. Especially, the existence of AdS solutions depends on the choice of some special types p − q(mod 8) Clp,q 0, 2 R(2 n/2 ) 3, 7 C(2 (n−1)/2 ) 4, 6 H(2 (n−2)/2 ) 1 R(2 (n−1)/2 ) ⊕ R(2 (n−1)/2 ) 5 H(2 (n−3)/2 ) ⊕ H(2 (n−3)/2 ) Table X: Clifford algebras corresponding to p positive and q negative generators of spinor inner products on product manifolds. For AdS solutions, supergravity Killing forms which are bilinear forms of supergravity Killing spinors reduce onto the hidden symmetries on product manifolds. These hidden symmetries correspond to special KY and special CCKY forms. Moreover, this reduction gives rise to the lift of hidden symmetries onto eleven-dimensional backgrounds and we find KY and CCKY forms on M-theory backgrounds. The methods leading to the relations between AdS solutions, choices of spinor inner products and reduction to hidden symmetries can be seen as a first step of a classification procedure for general string and M-theory backgrounds in terms of spinor inner products. One can also investigate the situation for warped product compactifications of M-theory backgrounds. Obviously, the field and bilinear form equations will be different from the unwarped case since they will include the warp factor in that case. On the other hand, these investigations can also be extended into ten-dimensional string backgrounds and their dependence on the choices of spinor inner products can be determined. So, by finding the relations between solutions, spinor inner products and reduction of bilinear forms, possible classification schemes can be obtained in that way. The conformal field theory equivalent of the choice of spinor inner products can also be investigated in the framework of AdS/CF T correspondence. These may be considered as motivations for future investigations about the topic of the paper. p − q(mod 8) S type of spinors 0 R 2 (n−2)/2 ⊕ R 2 (n−2)/2 Majorana-Weyl 1, 7 R 2 (n−1)/2 Majorana 2, 6 C 2 (n−2)/2 ⊕ C 2 (n−2)/2 Dirac-Weyl 3, 5 H 2 (n−3)/2 Symplectic Majorana 4 H 2 (n−4)/2 ⊕ H 2 (n−4)/2 Symplectic Majorana-Weyl 1 R-sym 6 H − -sym 2 R-skew 7 H -sym 3 C-sym 8 R-swap 4 C-skew 9 H-swap 5 C * -sym 10 C-swap which are called D j -symmetric or D j -skew inner products respectively where j denotes the identity for D = R, identity or complex conjugation ( * ) for D = C, quaternionic conjugation ( ) or quaterninonic reversion ( ) for D = H. Moreover, for any Clifford form ω, we have the following property (ψ, ω.φ) = (ω J .ψ, φ) where J corresponds to ξ or ξη involutions on the Clifford algebra and . denotes the Clifford product which is defined as in (B1) and (B2). Here ξ denotes the anti-involution acting on any p-form ω as ω ξ = (−1) ⌊p/2⌋ ω and η is the inner automorphism acting as ω η = (−1) p ω. ⌊ ⌋ denotes the floor function which takes the integer part of the argument. So, we have three choices for an inner product; symmetry or anti-symmetry, the involution J and the induced involution j. From the detailed analysis of Clifford algebras, one can see that there are ten different types of inner products on real Clifford algebras as in Table XIII [24]. In the table, swap means that when the arguments in the inner product are reversed, their semi-spinor space is changed.The inner products induced on Clifford algebra representations in different dimensions can be listed as in Table XIV [24]. In the table, for each dimension, the first row corresponds to the inner product with ξ involution and the second row corresponds to the inner product with ξη involution and the numbers in the table corresponds to the inner product classes in the table XIII. The inner product classes k ⊕ k denotes kth inner product class on each semi-spinor space. The table repeats itself after dimension 7 with respect to mod 8. As the representation spaces of even subalgebras, the inner products on spinor spaces can also be obtained from Table XIV via the isomorphism Cl 0 p,q ∼ = Cl q,p−1 as in Table XV. On a spin manifold M , the possible inner product choices on the spinor bundle can be determined from the Table  XV. So, the manifolds that we consider throughout the text can have the spinor inner products given in Table XVI. For any spinor field ǫ, the choice of the inner product determines the properties of the bilinear forms constructed from ǫ. For a p-form ω, we have