N=4 Near-Horizon Geometries in D=11 Supergravity

Extreme near-horizon geometries in D=11 supergravity preserving four supersymmetries are classified. It is shown that the Killing spinors fall into three possible orbits, corresponding to pairs of spinors defined on the spatial cross-sections of the horizon which have isotropy groups SU(3), G2, or SU(4). In each case, the conditions on the geometry and the 4-form flux are determined. The integrability conditions obtained from the Killing spinor equations are also investigated.


Introduction
Black hole uniqueness is of particular interest in the context of string theory. The strongest black hole uniqueness theorems are formulated for asymptotically flat solutions in four dimensions [1,2,3,4,5,6] . However, it is notable that the inclusion of a (negative) cosmological constant weakens these types of uniqueness theorems, even in four dimensions [7,8]. The situation is furthermore complicated in relation to black holes in string theory, which frequently arise in higher dimensional supergravities. In this case the notion of black hole uniqueness is also weaker. The first signal of this was the discovery of asymptotically flat black rings (both with, and without, supersymmetry) [9,10,11], for which the event horizon spatial topology is S 2 × S 1 . Examples of black rings can be constructed in such a way as to have the same asymptotic charges as black holes with S 3 event horizon topology. Additional evidence of non-uniqueness has also been more recently discovered in [12,13], where a very large class of extreme, supersymmetric, black hole solutions has been found. These are both asymptotically flat, and have near-horizon geometries which include the geometry of the near-horizon BMPV solution [14] as a possibility, but which in addition have non-trivial topology located outside of the horizon. The charges associated with such solutions include the mass, angular momenta and electric charge which are evaluated at infinity; but in addition to these, there are also magnetic charges supported on certain, but arbitrarily many, 2-cycles. The thermodynamic properties of such solutions have also been considered in [15]. So, there is a large family of black hole geometries in five dimensions, which includes the BMPV black hole as a special case, but for which there is, a priori, no upper bound on the number of independent parameters which can be used to construct the geometries in this family. This is in marked contrast to the classical uniqueness theorems in four dimensions.
Nevertheless, in spite of this increased complexity, there are uniqueness results established when there are sufficiently many commuting rotational isometries [16,17,18,19]. Uniqueness theorems for static and asymptotically flat higher dimensional black hole solutions have also been established [20]. It is however notable that, even given such a rich structure of higher dimensional black hole solutions, there are currently no known analytic examples of asymptotically de-Sitter, or anti-de-Sitter, black rings in higher dimensions with regular horizons, and there are also non-existence theorems for such objects in certain theories [21,22,23,24]. It would appear that additional matter fields are required to be included for such solutions to exist.
Motivated by this, it is natural to investigate the issue of black hole uniqueness for extreme and supersymmetric black holes in eleven-dimensional supergravity. We remark that, in contrast to a large number of four and five-dimensional supergravity theories, black hole solutions in D=11 supergravity need not be extreme even if supersymmetry is imposed. This is because the algebraic identities satisfied by the gauge-invariant spinor bilinears differ between D=11 supergravity and four and fivedimensional theories. Hence, in this work, extremality will be imposed as an extra assumption in addition to supersymmetry, which enables the near-horizon limit of the black hole to be considered.
Although higher dimensional black holes are not determined uniquely by their near-horizon geometry, investigating the geometric structures associated with such near-horizon solutions is a necessary first step in establishing a systematic classi-fication of these types of black holes. As we have mentioned above, even in the presence of supersymmetry, black hole uniqueness breaks down in higher dimensions. However, the conditions on the geometry following from supersymmetry can be used to put constraints on the possible solutions, and in some cases to eliminate certain geometries entirely [21]. More generally near-horizon geometries typically admit additional symmetries when compared to the bulk black hole geometry, and this makes the analysis more tractable.
In terms of D=11 supergravity, there has been some significant progress in understanding the properties of supersymmetric near-horizon geometries. In particular, in [25], it was proven that all supersymmetric near-horizon geometries in D=11 supergravity must admit an even number of supersymmetries, and furthermore all such solutions admit a SL(2, R) symmetry algebra. The geometric structure of near-horizon solutions admitting N = 2 supersymmetry was classified in [25,26]. Following on from this, analysis of the superalgebras associated with supersymmetric near-horizon geometries [27,28] was used to further elucidate the nature of the symmetry algebras of solutions with N > 2 supersymmetry. In conjunction with the homogeneity theorem in [29], together with a classification of all compact 9-dimensional Riemannian manifolds in [30], it was shown that there are no supersymmetric near horizon geometries with 16 < N < 32 supersymmetries. Any maximally supersymmetric N = 32 near-horizon geometry must furthermore be one of the maximally supersymmetric geometries determined in [31].
It therefore remains to consider supersymmetric near horizon geometries with an even number 4 ≤ N ≤ 16 of supersymmetries, and in this work we shall consider the case of N = 4. We remark that the cases of 6 ≤ N ≤ 16 supersymmetries are not readily amenable to analysis via the techniques used in this paper. This is because, in order to determine the conditions obtained from the Killing spinor equations, we first explicitly integrate these equations along two of the light-cone directions which are generic to all near-horizon solutions. The remaining content of the Killing spinor equations reduces to finding spinors whose components depend only on the 9 remaining co-ordinates associated with the 9-dimensional spatial cross-section of the event horizon, which are parallel with respect to certain 9-dimensional supercovariant derivatives including contributions from the 4-form flux. To extract conditions on the geometry and fluxes from requiring the existence of such parallel spinors, Spin(9) gauge transformations are used to simplify the form of these spinors. Such simplifications can be implemented in the cases of N = 2 and N = 4 supersymmetry. However, the majority of this gauge freedom is used up in simplifying the spinors for the N = 4 case, and little further useful simplification can be made to the extra spinors for N = 6 solutions. Nevertheless, it is clear that for 6 ≤ N ≤ 16, the additional supersymmetry will impose further conditions on the geometry, and it would be interesting to consider these cases in more detail in future work.
In this paper we determine the conditions on the geometry of the 9-dimensional spatial horizon section, and also the 4-form flux, of extreme supersymmetric black hole near-horizon geometries preserving N = 4 supersymmetry in D = 11 supergravity. To this end, we shall solve the horizon Killing spinor equations (KSE) using spinorial geometry techniques [32,33]. This method has been used extensively to construct classifications of supergravity solutions (e.g. [34]), as well as to investigate properties of highly supersymmetric solutions [35] and black hole solutions [26]; see also [36] and references therein for a comprehensive list of applications. The key features of the spinorial geometry approach are firstly that appropriate gauge transformations are utilized to write explicitly the spinors in the simplest possible canonical form. Using this, the conditions obtained from the KSE can then be evaluated directly. This approach to analysing the KSE is particularly useful when investigating the properties of solutions with more than the minimal possible amount of supersymmetry, as it is not necessary to consider the rather detailed analysis of Fierz identities which would otherwise be required.
We apply these techniques to the N = 4 near-horizon geometries by determining explicitly the simple canonical forms for the spinors. In particular, the conditions required for N = 4 supersymmetry are equivalent to requiring the existence of two Majorana spinors whose components depend only on the horizon section co-ordinates. These two spinors must be positive chirality, with respect to a certain light-cone chirality projection, and must also be orthonormal as a consequence of the global analysis of [25]. Using appropriate gauge transformations, we find that there are three different stabilizers, namely SU(3), G 2 or SU(4), for the two positive chirality spinors. The three cases are treated separately. In each case, using the spinorial geometry techniques, we find the constraints on the geometry and we express the fluxes in terms of the geometry in a covariant fashion using the gauge-invariant bilinears which are constructed from the spinors.
The paper is organized as follows. In section 2 we present the near-horizon fields of D = 11 supergravity and write the field equations, the Bianchi identities and the Killing spinor equations in terms of these near-horizon fields. In particular, the conditions required for supersymmetry reduce to finding spinors which depend only on the co-ordinates of the spatial horizon section, and are parallel with respect to certain supercovariant derivatives defined on the horizon sections. We also review some global and Killing superalgebra analysis. In section 3 we present the solution of the Killing spinor equations of near-horizon geometries preserving N = 2 supersymmetries, fully expressing it in terms of the Spin(7) gauge-invariant bilinears constructed out of the spinors. In section 4 we solve the Killing spinor equations of near-horizon geometries preserving N = 4 supersymmetries, splitting the analysis in three distinct cases, SU(3), G 2 or SU(4), corresponding to the isotropy groups of the spinors associated with the N = 4 solutions. In section 5 we analyse the integrability conditions of near-horizon geometries in D = 11 supergravity. Using a purely local argument, we shall show that all the components of the Einstein equations are implied by the 11-dimensional KSE, the 4-form field equations and the Bianchi identites. The paper concludes with four appendices, where we have gathered some of the lengthier formulae, namely the N = 4 linear systems associated to the KSE, the expression of the 4-form in terms of the SU(3) gauge-invariant bilinears, and various G 2 and Spin (7) identities.
Using Gaussian null coordinates [38,39], the near horizon metric and the 4-form flux 1 of D = 11 supergravity can be written [25,26,37] in the following form ds 2 = 2e + e − + δ ij e i e j = 2du(dr + rh − 1 2 r 2 ∆du) + ds 2 (S) , where the vielbein {e + , e − , e i }, with i = 1, 2, 3, 4, 6, 7, 8, 9, ♯ is given by and is the metric of the horizon spatial cross-section S given by r = u = 0. In these expressions, h = h i e i , ∆, e i , Y and X depend only on the co-ordinates y of S. Furthermore, we assume that S is a compact 9-dimensional manifold without boundary, and that ∆, h, Y and X are globally defined and smooth 0-, 1-, 2-and 4-forms on S, respectively.

Bianchi identities and field equations
The Bianchi identities and field equations of D = 11 supergravity [40] can be decomposed along the lightcone directions and those of the horizon section S. For the Bianchi identities dF = 0, such a decomposition yields The field equation of the 4-form flux F reads where ⋆ 11 is the Hodge star on 11-dimensional spacetime. Equation (2.5) can be written as and where ⋆ 9 denotes the Hodge dual on S. The volume form is chosen as ǫ 11 = e + ∧e − ∧ǫ S , where ǫ S is the volume form of S. The Einstein equation is for which the independent components are the ij components (2.9) and the +− component Here∇ is the Levi-Civita connection of the metric ds 2 (S) andR ij is the Ricci tensor of S. The ++ and +i components of the Einstein equation hold as a consequence of (2.4), the 3-form field equations (2.6) and (2.7) and the components of the Einstein equation in (2.9) and (2.10). The −− and the −i components of the Einstein equations are also satisfied automatically for all near-horizon solutions (2.1). Thus, the conditions on ds 2 (S), ∆, h, Y and X simplify to (2.4), (2.6), (2.7), (2.9) and (2.10).

Independent Killing Spinor Equations
The Killing spinor equations (KSE) of 11-dimensional supergravity [40] are where ∇ is the Levi-Civita connection. The KSE can be integrated along the two lightcone directions, as all the bosonic fields are independent of the u co-ordinate, and the dependence on the r co-ordinate is known explicitly. The remaining content of the KSE can then be further decomposed along the directions corresponding to the spatial horizon section S. To begin, we write the Killing spinor ǫ as [26] Then after some computation, see [25,26,37], we find that and where φ ± = φ ± (y) do not depend on r or u and Using the field equations and Bianchi identities, the independent KSE are

Global and Superalgebra Analysis of Supersymmetry
For each of the "horizon gravitino KSEs" on S given in (2.16), one can associate a "horizon Dirac equation" as These Dirac equations, in addition to the Levi-Civita connection, also depend on the fluxes of the supergravity theory restricted on the horizon section S, and the properties of the horizon Dirac equations play an important role in counting the number of supersymmetries preserved by the solutions. In particular, generalized Lichnerowicz Theorems established in [25] (see also analogous results for type II supergravity theories in [41,42,43,36]) imply that and furthermore, via an index theory argument, we have The global analysis which is used to establish the generalized Lichnerowicz theorems, also implies, by a maximum principle argument, that φ + 2 is constant, and henceforth we shall take all such positive chirality spinors to satisfy φ + 2 = 1. The number of positive chirality spinors which are parallel with respect to ∇ (+) i is equal to the number of negative chirality spinors which are parallel with respect to ∇ (−) i , and for a supersymmetric near-horizon geometry preserving 2k supersymmetries, we take φ r ± spinors for r = 1, . . . k with, without loss of generality, φ r + , φ s + = δ rs , where , is a Spin(9)-invariant Dirac inner product.
There is furthermore an algebraic relation which can be used to construct φ r + spinors from φ r − spinors which are parallel with respect to ∇ and in particular, it has been shown that if φ r − = 0 then φ r + defined as above is non-vanishing. A further algebraic relation, based on a superalgebra computation [27,28], has also been constructed which relates φ r whereV is obtained from the horizon section isometries where Z rs is antisymmetric in the indices r, s. Moreover, the negative chirality spinors 26) and the positive chirality spinors φ r + satisfy The isometries V rs i on S are obtained from D = 11 spacetime isometries as set out in [27,28], and both the D = 11 metric and 4-form F are invariant with respect to these isometries [44]. Consequently, the isometriesV and Z rs leave invariant all of the near-horizon data: and similarlỹ Furthermore,V and Z rs commute The condition (2.23) does not however determine the φ r − spinors directly from the φ r + spinors becauseV depends implicitly on φ r − via (2.25). The vector fieldV may vanish, in such a case the near-horizon geometry is static, and is a warped product AdS 2 × w S [25]. Generically, a non-static N = 4 solution has non-vanishing commuting isometriesV and Z = Z 12 , and therefore admits a U(1) × U(1) symmetry on the horizon spatial section S. However, there are also special cases when this symmetry is reduced. Additional geometric conditions can be obtained from a detailed analysis of the Killing spinor equations, which we shall undertake in the following sections. We remark that it will be most straightforward to consider the conditions obtained by solving where r = 1 for N = 2 solutions, and r = 1, 2 for N = 4 solutions. This is because the constant norm condition φ r + , φ s + = δ rs , combined with a judicious application of Spin(9) gauge transformations, allows one to find very simple canonical forms for the φ r + spinors, and consequently the linear systems obtained from the KSE are rendered more tractable. Having solved the KSE for the positive chirality spinors, the global analysis guarantees that there must exist negative chirality spinors φ r − which are parallel with respect to ∇ (−) .

N = 2 Solutions
In this section we revisit near-horizon geometries preserving exactly N = 2 supersymmetries. This has already been discussed in [26] working in a particular gauge for the φ 1 − spinor. However, our analysis here does not involve that gauge fixing procedure, as we wish to retain sufficient gauge freedom to simplify the additional positive chirality spinor which we will consider in the later analysis of N = 4 solutions. It is convenient to solve the KSE in the positive chirality sector since, as we have mentioned in the previous section, φ 1 + has constant norm, which enables us to write this spinor in a particularly simple canonical form. In particular, using the Spin(9) gauge invariance of the KSEs (3.1), namely where Γ ij are the generators of Spin (9), the spinor φ 1 + can be written in the following particularly simple canonical form which is stabilized by a Spin (7) isotropy subgroup of Spin(9) [37]. It will be particularly useful to demonstrate explicitly how this can be achieved, as it will illustrate how a similar analysis can be done to determine the common isotropy group of two spinors {φ 1 + , φ 2 + } in the next section, which is the primary focus of this work. So, in order to show (3.3), let us start by writing down the general expression for a positive (lightcone) chirality Majorana spinor [45] where in (3.4) α, β = 1, 2, 3, 4, ⋆ is the Hodge dual on R 4 and w, ψ and B are functions of the horizon coordinates y. Defining A αβ := B αβ − (⋆B) αβ , we can rewrite (3.4) as follows where χ p := A 1p , with p = 2, 3, 4. Consider first the su(3) transformation S pq Γ pq . Then Since the action of SU(3) on C 3 is simply transitive, without loss of generality we can impose χ 3 = χ 4 = 0. Thus (3.5) simplifies to with χ := χ 1 . Next, let us consider the gauge transformations  (2) on R 2 allows us to set ℜ(χ) = ℑ(χ) = 0, that is, χ = 0. Also, iΓ 11 is a so(2) transformation acting on {v 1 , v 2 }. Thus, we can set ℑ(w) = 0, that is w ∈ R.
In solving the KSE (4.1), the three cases outlined above must be treated separately.

G 2 Isotropy Group
In this section we solve the KSE (4.1) assuming cos θ = 0 (thus sin θ = 1), that is the second spinor (4.2) boils down to where K is given by the first equation in (4.15) and φ is the fundamental G 2 3-form, that is where ℜ( χ) and ω are defined by (4.14) and e 7 ≡ √ 2 ℑ(e 1 ) is the seventh direction of the G 2 manifold. We can also construct the 4-form We have found the following (covariant) geometric conditions (ĩ,j, ... are 7-dimensional real G 2 indices) (L e ♯ e ♯ )˜i − (L K K)˜i = 0 , (4.53) (L K e ♯ )˜i + (L e ♯ K)˜i = 0 , is the Lee form of φ on S. Equations (4.50), (4.51), (4.53), (4.54), (4.60) and (4.61) imply that the exterior derivatives of K and e ♯ on S can be written as follows We can use (4.63) and (4.64) to introduce local coordinates on S. In fact, equations (4.63) and (4.64) imply that K and e ♯ form a 2-dimensional differential ideal on S. Frobenius' theorem then ensures that locally we can choose a coordinate system [47,48] y˜i, y 8 := v 1 , y 9 := v 2 (4.65) such that for some functions f 1 , f 2 , g 1 and g 2 . The slices v 1 = const, v 2 = const are 2dimensional integral manifolds, and every slice can be equipped with a G 2 structure with fundamental form given by It is worth noticing that the (2 + 7)-splitting of S which emerges in the G 2 case, there exists also in the SU(3) case, although it is less transparent. In fact, the conditions (4.16), (4.17), (4.21) and (4.22) imply that K and W form a 2-dimensional ideal on S. However, in the SU(3) case we also have the 1-form bilinear U, but there are no simple conditions on dU, the only condition on it being (4.19). After having covariantized the geometric conditions, we are left to covariantize X. A somewhat long computation allows to express X in terms of the G 2 gauge invariant bilinears (4.47) as follows

SU (4) Isotropy Group
In this section we solve the KSE (4.1) assuming sin θ = 0, i.e. the second spinor (4.2) boils down to (4.69) The linear system associated to the second spinor is given by where ω and χ are defined by (3.32). We have found the following geometric conditions (ǐ,ǰ, . . . are 8-dimensional real SU(4) indices) (dω)ǰ 1ǰ2ǰ3 ℜ(χ)ǰ 1ǰ2ǰ3ǐ = 0 , (4.84) Moreover, the fluxes can be covariantized as follows where θ (9) ℜ(χ) := ⋆ 9 (ℜ(χ) ∧ ⋆ 9 dℜ(χ)) , (4.91) θ (9) ω := ⋆ 9 (ω ∧ ⋆ 9 dω) , (4.92) are the Lee forms of ℜ(χ) and ω on S respectively, and X TT (2,2) is the SU(4)-traceless (2,2) part of X, that is where we have denoted Enforcing the gauge field equation (2.5) and the Bianchi identities, (5.2) boils down to In the analysis of the integrability conditions, we shall not assume any of the results which follow from the global assumptions made in the previous sections. In particular, on integrating up the KSE along the lightcone directions, we find the following Killing spinors Two separate cases must be considered. The first case corresponds to φ − = 0. Consider where M, N = +, −, i. Setting u = 0 in (5.6), we obtain which is equivalent to The first term in (5.8) has positive (lightcone) chirality, while the second has negative (lightcone) chirality, thus (5.8) holds iff both terms vanish independently, which in turn imply that This means that all the components of E M N must vanish, apart from E ++ . However, as is shown in [26], in the near-horizon limit, E ++ = 0 is a consequence of the other bosonic field equations, thus indeed E M N = 0 for all M, N.

This equation boils down to
The first and the fourth terms in (5.11) have negative (lightcone) chirality, while the second and the third ones have positive (lightcone) chirality, thus (5.8) holds iff both terms vanish independently, which in turn imply that

Conclusion
We have classified the conditions imposed on the geometry and 4-form flux obtained from requiring that an extreme near-horizon geometry in D = 11 supergravity preserves N = 4 supersymmetry. This analysis is tractable because previous global analysis in [25] has been used to reduce the calculation to that of solving ∇ (+) i φ r + = 0, r = 1, 2 (6.1) and for N = 4 solutions, there is sufficient Spin(9) gauge freedom present to enable one to write {φ 1 + , φ 2 + } in one of three simple canonical forms, on exploiting the condition φ r + , φ s + = δ rs . In particular, the common isotropy group of {φ 1 + , φ 2 + } is one of three classes; SU(3), G 2 or SU(4). For each class, the conditions on the geometry and fluxes have been expressed in a fully gauge-invariant fashion, in terms of the gauge-invariant spinor bilinears corresponding to each of these cases.
There are a number of additional issues relevant to these N = 4 solutions which would be interesting to explore further. First, although we have mentioned in section 2.3 that a generic N = 4 near-horizon solution admits two commuting rotational isometries on S,V and Z = Z 12 , it remains to determine how these isometries relate to the geometric structures on S which we have derived in section 4. In practice, it is rather difficult to do this, because the isometriesV and Z are constructed from both φ + and φ − spinors. We have used the majority of the gauge freedom to simplify the canonical forms of the φ + spinors. In general, the φ − spinors are rather complicated in form, as in the absence of useful gauge transformations to simplify these, one must instead utilize (2.22) and (2.23) to relate φ + to φ − , and as a result there is a non-trivial appearance of various flux terms in the explicit expressions for the isometries.
Furthermore, it would be useful to see if there are any further conditions on the N = 4 spinors from global analysis. In particular, we have seen that for the SU(3) isotropy group case, there is a function θ which appears in the spinor φ 2 + . Although the manner in which this function appears explicitly is dependent on the gauge choice we have made, one can straightforwardly write cos 2θ in terms of gaugeinvariant bilinears, and hence cos 2θ is a globally well-defined and smooth function on S. In addition, the analysis of the KSE produces an expression for d(cos 2θ) in terms of various SU(3) invariant bilinears (4.43). As we already have a constant norm condition, φ r + , φ s + = δ rs which is obtained via a (global) maximum principle argument, [25], it is natural to enquire if θ can be shown to be constant via an analogous analysis. However, on computing the Laplacian of cos 2θ, there does not appear, a priori, to be any way of controlling the sign of the resulting terms in a way which is compatible with a maximum principle argument. We remark that there is also no immediate contradiction obtained if one assumes that θ is constant, with sin θ = 0, cos θ = 0; this would appear to be simply a special case of the more general analysis. It may also be the case that other properties of the geometry in the N = 4 solutions could in principle be further constrained via similar global analysis, and it would be interesting to explore this further.