4 Vacuum Solutions

The Einstein equations for a near-horizon geometry (7View Equation) in the absence of matter fields are equivalent to the following equation on H
Rab = 12hahb − ∇ (ahb) + Λ γab, (66 )
with the function F determined by
1 a 1 a F = 2h ha − 2∇ah + Λ, (67 )
see Eqs. (17View Equation), (18View Equation). In this section we will explore solutions to Eq. (66View Equation) in various dimensions. It is useful to note that the contracted Bianchi identity for the horizon metric is equivalent to
b b ∇aF − F ha − 2h ∇ [ahb] + ∇ ∇ [ahb] = 0. (68 )

4.1 Static: all dimensions

A complete classification is possible for Λ ≤ 0. Recall from Section 3.2.1, the staticity conditions for a near-horizon geometry are dh = 0 and dF = hF.

Theorem 4.1 ([42]). The only vacuum static near-horizon geometry for Λ ≤ 0 and compact H is given by ha ≡ 0, F = Λ and Rab = Λγab. For D = 4 this result is also valid for Λ > 0.

Proof: A simple proof of the first statement is as follows. Substituting the staticity conditions (50View Equation) into (67View Equation) gives

∇2ψ2 + 2Λψ2 = 2A , (69 ) 0
where ψ ≡ e− λ∕2. Irrespective of the topology of H one can argue that ψ is a globally-defined function (for simply connected H this is automatic, otherwise it can be shown by working in patches on H and exploiting the fact that λ in each patch is only defined up to an additive constant). For Λ = 0 it is then clear that compactness implies ψ must be a constant. For Λ < 0, if one assumes ψ is non-constant one can easily derive a contradiction by evaluating the above equation at the maximum and minimum of ψ (which must exist by compactness). Hence in either case h ≡ 0 a, which gives the claimed near-horizon data.

In four dimensions one can solve Eq. (66View Equation) in general without assuming compactness of H. For non-constant ψ and any Λ one obtains the near-horizon geometry (sending r → ψ2r)

dψ2 g = ψ2[A0r2dv2 + 2dvdr ] + ------+ P (ψ)dχ2, (70 ) P (ψ )
where P(ψ ) = A0 + βψ −1 − 1Λ ψ2 3. (This is an analytically-continued Schwarzschild with a cosmological constant.) This local form of the metric can be used to show that for Λ > 0 and ψ non-constant, there are also no smooth horizon metrics on a compact H.

4.2 Three dimensions

The classification of near-horizon geometries in D = 3 vacuum gravity with a cosmological constant can be completely solved. Although very simple, to the best of our knowledge this has not been presented before, so for completeness we include it here.

The main simplification comes from the fact that cross sections of the horizon H are one-dimensional, so the horizon equations are automatically ODEs. Furthermore, there is no intrinsic geometry on H and so the only choice concerns its global topology, which must be either ∼ 1 H = S or ∼ H = ℝ.

Theorem 4.2. Consider a near-horizon geometry with compact cross section H ∼= S1, which satisfies the vacuum Einstein equations including cosmological constant Λ. If Λ < 0 the near-horizon geometry is given by the quotient of AdS3 in Eq. (73View Equation). For Λ = 0 the only solution is the trivial flat geometry ℝ1,1 × S1. There are no solutions for Λ > 0.

Proof: We may choose a periodic coordinate x on H so the horizon metric is simply 2 γ = dx and the 1-form h = h(x)dx. Observe that h (x) must be a globally-defined function and hence must be a periodic function of x. Since the curvature and metric connection trivially vanish, the horizon equations (66View Equation) and (67View Equation) simplify to

h′ = 1h2 + Λ, (71 ) 21 2 1 ′ F = 2h − 2h + Λ. (72 )
This system of ODEs can be explicitly integrated as we explain below. Instead, we will avoid this and employ a global argument on H. If Λ ≥ 0, integrate Eq. (71View Equation) over H to deduce that h ≡ 0 and Λ = 0, which gives the trivial flat near-horizon geometry 1,1 1 ℝ × S. For 2- Λ = − ℓ2 < 0 we argue as follows. Multiply Eq. (71View Equation) by h′ and integrate over H to find ∫ S1 h ′2 = 0. Hence h must be a constant and substituting into the horizon equations gives h = 2ℓ and F = 0 (without loss of generality we have chosen a sign for h). The near-horizon geometry is then
4r 4r2 ( 2r )2 g = 2dvdr + ---dvdx + dx2 = − --2 dv2 + 2dvdr + dx + ---dv . (73 ) ℓ ℓ ℓ
This metric is locally AdS3 and in the second equality we have written it as a fibration over AdS2.

It is worth remarking that the ODE (71View Equation) can be completely integrated without assuming compactness. For Λ < 0 this reveals a second solution h = − 2 tanh (x) ℓ ℓ and F = − 2-sech2(x ) ℓ2 ℓ, where we have set the integration constant to zero by translating the coordinate x. Upon changing 2(x) r → rcosh ℓ the resulting near-horizon geometry is:

( ) ( 2 ) g = cosh2 x- − r-dv2 + 2dvdr + dx2. (74 ) ℓ ℓ2
Again, this metric is locally AdS3. Unlike the previous case though, H cannot be taken to be compact and hence we must have ∼ H = ℝ. For Λ = 0 there is also a second solution given by 2 h = − x and 1 F = x2, although this is singular. For Λ > 0 there is a unique solution given by ( ) h = 2ℓ tan xℓ and ( ) F = 1ℓ2sec2 xℓ, although this is also singular.18

4.3 Four dimensions

The general solution to Eq. (66View Equation) is not known in this case. In view of the rigidity theorem it is natural to assume axisymmetry. If one assumes such a symmetry, the problem becomes of ODE type and it is possible to completely solve it. The result is summarised by the following theorem, first proved in [122Jump To The Next Citation Point, 163Jump To The Next Citation Point] for Λ = 0 and in [154Jump To The Next Citation Point] for Λ < 0.

Theorem 4.3 ([122, 163Jump To The Next Citation Point, 154Jump To The Next Citation Point]). Consider a spacetime containing a degenerate horizon, invariant under an ℝ × U(1) isometry, satisfying the vacuum Einstein equations including a cosmological constant. Any non-static near-horizon geometry, with compact cross section, is given by the near-horizon limit of the extremal Kerr or Kerr-(A)dS black hole.

Proof: We present a streamlined version of the proof in [154Jump To The Next Citation Point]. As described in Section 3.2.2, axisymmetry implies one can introduce coordinates on H so that

dx2 Bk (x) Γ ′ γabdxadxb = -----+ B (x)dϕ2, h = ------d ϕ − --dx. (75 ) B(x) Γ Γ
The xϕ component of Eq. (66View Equation) implies k(x) ≡ k is a constant. The x component of Eq. (68View Equation) then implies
2 F = A0-+ Bk--, (76 ) Γ Γ 2
where A 0 is a constant. Substituting this into Eq. (67View Equation) gives
1 2 Bk2 A0 = --∇ Γ − ---- + ΛΓ . (77 ) 2 2Γ
Now subtracting the xx component from the ϕϕ component of Eq. (66View Equation) gives
Γ ′2 k2 2Γ ′′ − ---− ---= 0. (78 ) Γ Γ
A non-static near-horizon geometry must have k ⁄= 0 and therefore from the above equation Γ is non-constant. Using this, one can write Eq. (77View Equation) as
( ) B-Γ- ′ 2(A0-−-Λ-Γ )Γ- Γ ′ = Γ ′2 . (79 )
The solution to the ODE for Γ is given by
k2 βx2 Γ = ---+ ---, (80 ) β 4
where β is a positive constant, which can then be used to solve the ODE for B:
B = P(x)-, (81 ) Γ
βΛx4-- 2 −1 2 4k2-( 2 −1) P (x) = − 12 + (A0 − 2 Λk β )x + c1x − β2 A0 − Λk β (82 )
and c1 is a constant. Changing affine parameter r → Γ (x)r in the full near-horizon geometry finally gives
2 2 Γ (x) 2 P (x) 2 g = Γ (x)[A0r dv + 2dvdr ] + P(x-)dx + Γ-(x-)(dϕ + krdv ) , (83 )
with Γ ,P determined above. Observe that this derivation is purely local and does not assume anything about the topology of H (unlike the derivation in [154Jump To The Next Citation Point], which assumed compactness). If k = 0 we recover the general static solution (70View Equation), hence let us now assume k ⁄= 0.

Now assume H is compact, so by axisymmetry one must have either 2 S or 2 T. Integrating Eq. (77View Equation) over H then shows that if Λ ≤ 0 then A0 < 0 and so the metric in square brackets is AdS2. The horizon metric extends to a smooth metric on H ∼= S2 if and only if c1 = 0. It can then be checked the near-horizon geometry is isometric to that of extremal Kerr for Λ = 0 or Kerr-AdS for Λ < 0 [154Jump To The Next Citation Point]. It is also easy to check that for Λ > 0 it corresponds to extremal Kerr-dS. In the non-static case, the horizon topology theorem excludes the possibility of ∼ 2 H = T for Λ ≥ 0. If Λ < 0 the non-static possibility with H ∼= T 2 can also be excluded [164Jump To The Next Citation Point].

It would be interesting to remove the assumption of axisymmetry in the above theorem. In [145] it is shown that regular non-axisymmetric linearised solutions of Eq. (66View Equation) about the extremal Kerr near-horizon geometry do not exist. This supports the conjecture that any smooth solution of Eq. (66View Equation) on ∼ 2 H = S must be axisymmetric and hence given by the above theorem.

4.4 Five dimensions

In this case there are several different symmetry assumptions one could make. Classifications are known for homogeneous horizons and horizons invariant under a U(1)2-rotational symmetry.

We may define a homogeneous near-horizon geometry to be one for which the Riemannian manifold (H, γab) is a homogeneous space whose transitive isometry group K also leaves the rest of the near-horizon data (F, ha) invariant. Since any near-horizon geometry (7View Equation) possesses the 2D symmetry generated by v → v + c and (v,r) → (λv, λ−1r) where λ ⁄= 0, it is clear that this definition guarantees the near-horizon geometry itself is a homogeneous spacetime. Conversely, if the near-horizon geometry is a homogeneous spacetime, then any cross section (H, γab) must be a homogeneous space under a subgroup K of the spacetime isometry group, which commutes with the 2D symmetry in the (v,r) plane (since H is a constant (v,r) submanifold). It follows that (F,ha ) must also be invariant under the isometry K, showing that our original definition is indeed equivalent to the near-horizon geometry being a homogeneous spacetime.

Homogeneous geometries can be straightforwardly classified without assuming compactness of H as follows.

Theorem 4.4 ([158Jump To The Next Citation Point]). Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to

( 1 2 ) 2 2 2 g = − 2k + Λ r dv + 2dvdr + (ˆω + krdv ) + ˆg, (84 )
where ˆω is a U (1)-connection over a 2D base space satisfying Ric(ˆg ) = ˆλ ˆg with ˆλ = 1k2 + 2Λ 2. The curvature of the connection is √ -------- dˆω = k2 + 2Λˆ𝜖, where ˆ𝜖 is the volume form of the 2D base, and k2 + 2Λ ≥ 0.

The proof uses the fact that homogeneity implies h must be a Killing field and then one reduces the problem onto the 2D orbit space. Observe that for k → 0 one recovers the static near-horizon geometries. For k ⁄= 0 and Λ ≥ 0 we see that ˆλ > 0 so that the 2D metric ˆg is a round S2 and the horizon metric is locally isometric to a homogeneously squashed S3. Hence we have:

Corollary 4.1. Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to the near-horizon limit of the extremal Myers–Perry black hole with SU (2) × U (1) rotational symmetry (i.e., equal angular momenta). For Λ > 0 one gets the same result with the Myers–Perry black hole replaced by its generalisation with a cosmological constant [129Jump To The Next Citation Point].

For Λ < 0 we see that there are more possibilities depending on the sign of ˆλ. If ˆλ > 0 we again have a horizon geometry locally isometric to a homogeneous S3. If ˆλ = 0, we can write ˆg = dx2 + dy2 and the U (1)-connection ∘ ---- ˆω = 2|Λ|(xdy − ydx ) is non-trivial, so the cross sections H are the Nil group manifold with its standard homogeneous metric. For λˆ< 0, we can write |ˆλ|ˆg = (dx2 + dy2 )∕y2 and the connection √ -------- |ˆλ|ˆω = k2 + 2 Λdx ∕y, so the cross sections H are the SL (2,ℝ ) group manifold with its standard homogeneous metric, unless k2 = − 2Λ, which gives H = ℝ × ℍ2. Hence we have:

Corollary 4.2. For Λ < 0 any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to either the near-horizon limit of the extremal rotating black hole [129Jump To The Next Citation Point] with SU (2 ) × U (1) rotational symmetry, or a near-horizon geometry with: (i) H = Nil and its standard homogenous metric, (ii) H = SL (2,ℝ ) and its standard homogeneous metric or (iii) H = ℝ × ℍ2.

This is analogous to a classification first obtained for supersymmetric near-horizon geometries in gauged supergravity, see Proposition (5.3).

We now consider a weaker symmetry assumption, which allows for inhomogeneous horizons. A U (1)2-rotational isometry is natural in five dimensions and all known explicit black-hole solutions have this symmetry. The following classification theorem has been derived:

Theorem 4.5 ([152Jump To The Next Citation Point]). Consider a vacuum non-static near-horizon geometry with a U (1 )2-rotational isometry and a compact cross section H. It must be globally isometric to the near-horizon geometry of one of the following families of black-hole solutions:

  1. H ∼= S1 × S2: the 3-parameter boosted extremal Kerr string.
  2. H ∼= S3: the 2-parameter extremal Myers–Perry black hole or the 3-parameter ‘fast’ rotating extremal KK black hole [190Jump To The Next Citation Point].
  3. H ∼ L(p,q ) =: the Lens space quotients of the above H ∼ S3 = solutions.


  • The near-horizon geometry of the vacuum extremal black ring [187] is a 2-parameter subfamily of case 1, corresponding to a Kerr string with vanishing tension [162Jump To The Next Citation Point].
  • The near-horizon geometry of the ‘slowly’ rotating extremal KK black hole [190] is identical to that of the 2-parameter extremal Myers–Perry in case 2.
  • The H ∼= S3 cases can be written as a single 3-parameter family of near-horizon geometries [135Jump To The Next Citation Point].
  • The 3 H ∼= T case has been ruled out [131Jump To The Next Citation Point].

For Λ ⁄= 0, the analogous problem has not been solved. The only known solution in this case is the H ∼= S3 near-horizon geometry of the rotating black hole with a cosmological constant [129], which generalises the Myers–Perry black hole. It would be interesting to classify near-horizon geometries with 1 2 H ∼= S × S in this case since this would capture the near-horizon geometry of the yet-to-be-found asymptotically-AdS5 black ring. A perturbative attempt at constructing such a solution is discussed in [152].

4.5 Higher dimensions

For spacetime dimension D ≥ 6, so the horizon cross section dimH ≥ 4, the horizon equation (66View Equation) is far less constraining than in lower dimensions. Few general classification results are known, although several large families of vacuum near-horizon geometries have been constructed.

4.5.1 Weyl solutions

The only known classification result for vacuum D > 5 near-horizon geometries is for Λ = 0 solutions with U (1)D− 3-rotational symmetry. These generalise the D = 4 axisymmetric solutions and D = 5 solutions with 2 U(1)-symmetry discussed in Sections 4.3 and 4.4 respectively. By performing a detailed study of the orbit spaces H ∕U (1)D−3 it has been shown that the only possible topologies for H are: S2 × T D− 4, S3 × T D− 5, L (p,q) × T D− 5, and TD −2 [139Jump To The Next Citation Point].

An explicit classification of the possible near-horizon geometries (for the non-toroidal case) was derived in [135Jump To The Next Citation Point], see their Theorem 1. Using their theorem it is easy to show that the most general solution with ∼ 2 D −4 H = S × T is in fact isometric to the near-horizon geometry of a boosted extremal Kerr-membrane (i.e., perform a general boost of Kerr× ℝD −4 along the {t} × ℝD −4 coordinates and then compactify ℝD − 4 → T D−4). Non-static near-horizon geometries with H ∼= TD −2 have been ruled out [131] (including a cosmological constant).

4.5.2 Myers–Perry metrics

The Myers–Perry (MP) black-hole solutions [183] generically have isometry groups ℝ × U(1)s where s = ⌊ D−21⌋. Observe that if D > 5 then s < D − 3 and hence these solutions fall outside the classification discussed in Section 4.5.1. They are parameterised by their mass parameter μ and angular momentum parameters ai for i = 1,...s. The topology of the horizon cross section ∼ D−2 H = S. A generalisation of these metrics with non-zero cosmological constant has been found [99Jump To The Next Citation Point]. We will focus on the Λ = 0 case, although analogous results hold for the Λ ⁄= 0 solutions.

The location of the horizon is determined by the largest positive number r + such that in odd and even dimensions 2 Π (r+) − μr+ = 0 and Π (r+) − μr+ = 0, respectively, where

s ∏ 2 2 Π (r) = (r + a i). (85 ) i=1
The extremal limit of these black holes in odd and even dimensions is given by Π ′(r+ ) = 2μr+ and Π ′(r+ ) = μ, respectively. These conditions hold only when the black hole is spinning in all the two planes available, i.e., we need ai ⁄= 0 for all i = 1,⋅⋅⋅,s. Without loss of generality we will henceforth assume ai > 0 and use the extremality condition to eliminate the mass parameter μ. The near-horizon geometry of the extremal MP black holes can be written in a unified form [79Jump To The Next Citation Point]:
( ′′ ) gMP = F+ − Π--(r+)r2dv2 + 2dvdr + γμμ d μidμj 2Π (r+) ij ( 2r+ai ) ( 2r+aj ) + γij dϕi + --2----2-2rdv dϕj + --2----2-2rdv , (86 ) (r+ + ai) (r+ + aj)
∑ s a2μ2 1 F+ = 1 − -2-i-i2-, γij = (r2+ + a2i)μ2iδij + ---aiμ2iajμ2j, (87 ) i=1 r+ + ai F+
and in odd and even dimensions
∑ s ∑s γμiμjdμidμj = (r2+ + a2i)dμ2i, γμiμjd μidμj = r2+d α2 + (r2+ + a2i)d μ2i, (88 ) i=1 i=1
respectively. The direction cosines μi and α take values in the range 0 ≤ μi ≤ 1 with − 1 ≤ α ≤ 1 and in odd and even dimensions satisfy
s s ∑ 2 ∑ 2 2 μi = 1, μ i + α = 1, (89 ) i=1 i=1
respectively. The generalisation of these near-horizon geometries for Λ ⁄= 0 was given in [165]. It is worth noting that if subsets of the angular momentum parameters a i are set equal, the rotational symmetry enhances to a non-Abelian unitary group.

Since these are vacuum solutions one can trivially add flat directions to generate new solutions. For example, by adding one flat direction one can generate a boosted MP string, whose near-horizon geometries have H ∼= S1 × SD −3 topology. Interestingly, for odd dimensions D the resulting geometry has ⌊ D−1⌋ 2 commuting rotational isometries. For this reason, it was conjectured that a special case of this is also the near-horizon geometry of yet-to-be-found asymptotically-flat black rings (as is known to be the case in five dimensions) [79].

4.5.3 Exotic topology horizons

Despite the absence of explicit D > 5 black-hole solutions, a number of solutions to Eq. (66View Equation) are known. It is an open problem as to whether there are corresponding black-hole solutions to these near-horizon geometries.

All the constructions given below employ the following data. Let K be a compact Fano Kähler–Einstein manifold19 of complex dimension q − 1 and a ∈ H2(K, ℤ ) is the indivisible class given by c1(K ) = Ia with I ∈ ℕ (the Fano index I and satisfies I ≤ q with equality iff q−1 K = ℂ ℙ). The Kähler–Einstein metric ¯g on K is normalised as Ric(¯g) = 2q¯g and we denote its isometry group by G. The simplest example occurs for q = 2, in which case K = ℂℙ1 ∼ S2 = with 1 2 2 2 ¯g = 4(d𝜃 + sin 𝜃dχ ).

In even dimensions greater than four, an infinite class of near-horizon geometries is revealed by the following result.

Proposition 4.1 ([156Jump To The Next Citation Point]). Let m ∈ ℤ and Pm be the principal 1 S-bundle over any Fano Kähler–Einstein manifold K, specified by the characteristic class ma. For each m > I there exists a 1-parameter family of smooth solutions to Eq. (66View Equation) on the associated S2-bundles H ∼= Pm ×S1 S2.

The dim H = 2q ansatz used for the near-horizon data is the U (1 ) × G invariant form

2 γ dxadxb = dx---+ B (x)ω ⊗ ω + A(x )2¯g, (90 ) ab B(x ) kB (x)ω Γ ′(x) h = --------− ----- dx, (91 ) Γ Γ
where ω is a U (1)-connection over K with curvature 2πma. The solutions depend on one continuous parameter L > 0 and the integer m > I. The continuous parameter corresponds to the angular momentum J [∂ ] ϕ where ∂ ϕ generates the U(1)-isometry in the S2-fibre. The various functions are given by 2 2 2 A (x) = L (1 − x ), 2 Γ (x) = ξ + x, √ -- k = ±2 ξ and B (x) = P (x)∕(A (x )2q−2Γ (x)) where P is a polynomial in x2 and smoothness fixes ξ to be a function of m.

The simplest example is the q = 2, Λ = 0 solution, for which the near-horizon data takes the explicit form

( 2) L2 (ξ + x2)(1 − x2)dx2 L2 (4 − m2x2 ) ξm − 43xm2- ( )2 γabdxadxb = -----m------(------4x2) + ------------2-------2----- dϕ + 12 cos 𝜃dχ (4 − m2x2 ) ξm − 3m2-- (ξm + x )(1 − x ) + 1L2 (1 − x2 )(d 𝜃2 + sin2𝜃d χ2), (92 ) 4√ --- ( ) 2 ξmL2 (4 − m2x2 ) ξm − 4x22- ( ) hadxa = ± --------------------------3m--- d ϕ + 1cos 𝜃dχ − --2x---dx, (93 ) (ξm + x2)2(1 − x2 ) 2 ξm + x2
( 4 ) 4- 3 −-m2- ξm = 3 4 + m2 , (94 )
and m > 2. The coordinate ranges are − 2∕m ≤ x ≤ 2∕m, 0 ≤ 𝜃 ≤ π, ϕ ∼ ϕ + 2π∕m, χ ∼ χ + 2π. Cross sections of the horizon H, are homeomorphic to S2 × S2 if m is even, or the non-trivial bundle S2 &tidle;×S2 ∼= ℂ ℙ2# ℂℙ2- if m is odd. For Λ ⁄= 0 the solutions are analogous.

For q > 2 the Fano base K is higher dimensional and there are more choices available. The topology of the total space is always a non-trivial S2-bundle over K and in fact different m give different topologies, so there are an infinite number of horizon topologies allowed. Furthermore, one can choose K to have no continuous isometries giving examples of near-horizon geometries with a single U (1)-rotational isometry. Hence, if there are black holes corresponding to these horizon geometries they would saturate the lower bound in the rigidity theorem.

It is worth noting that the local form of the above class of near-horizon metrics includes as a special case that of the extremal MP metrics H ∼= S2q with equal angular momenta (for m = I). The above class of horizon geometries are of the same form as the Einstein metrics on complex line bundles [186], which in four dimensions corresponds to the Page metric [185Jump To The Next Citation Point], although we may of course set Λ ≤ 0.

Similar constructions of increasing complexity can be made in odd dimensions, again revealing an infinite class of near-horizon geometries.

Proposition 4.2. Let m ∈ ℤ and Pm be the principal S1-bundle over K specified by the characteristic class ma. There exists a 1-parameter family of Sasakian solutions to Eq. (66View Equation) on ∼ H = Pm.

As a simple example consider K = ℂ ℙ1 × ℂ ℙ1. This leads to an explicit homogeneous near-horizon geometry with

( ) a b k2-+-2Λ- 2 γabdx dx = 2 λ2 (dψ + cos𝜃1dϕ1 + cos 𝜃2dϕ2) ( ) + 1- d 𝜃21 + sin2𝜃1d ϕ21 + d𝜃22 + sin2 𝜃2dϕ22 ∘ -------λ- a 2λ2 ∂ h ∂a = k -2---------, (95 ) k + 2Λ ∂ ψ
where for convenience we have written h is a vector field, k is a constant and λ = (k2 + 6Λ )∕4. Regularity requires that the Chern number m of the U (1)-fibration over each S2 to be the same and the period Δ ψ = 2 π∕m. The total space is a Lens space 3 S ∕ℤm-bundle over 2 S and is topologically ∼ 3 2 H = S × S. For k = 0 and Λ > 0 this corresponds to a Sasaki–Einstein metric on 3 2 S × S sometimes known as T 1,1.

The above proposition can be generalised as follows.

Proposition 4.3 ([158]). Given any Fano Kähler–Einstein manifold K of complex dimension q − 1 and coprime p ,p ∈ ℕ 1 2 satisfying 1 < Ip ∕p < 2 1 2, there exists a 1-parameter family of solutions to Eq. (66View Equation) where H is a compact Sasakian (2q + 1)-manifold.

These examples have U (1)2 × G symmetry, although possess only one independent angular momentum along the T2-fibres. These are deformations of the Sasaki–Einstein Y p,q manifolds [94].

There also exist a more general class of non-Sasakian horizons in odd dimensions.

Proposition 4.4 ([157Jump To The Next Citation Point]). Let Pm1,m2 be the principal T 2-bundle over any Fano Kähler–Einstein manifold K, specified by the characteristic classes (m a,m a) 1 2 where m ,m ∈ ℤ 1 2. For a countably infinite set of non-zero integers (m1, m2, j,k), there exists a two-parameter family of smooth solutions to Eq. (66View Equation) on the associated Lens space bundles H ∼= Pm1,m2 ×T 2 L (j,k).

The dim H = 2q + 1 form of the near-horizon data in the previous two propositions is the U (1)2 × G invariant form

a b dx2 i j 2 γabdx dx = ------+ Bij(x)ω ω + A (x) ¯g, (96 ) det Bj i ′ h dxa = Bijk--ω-− Γ-(x)dx, a Γ Γ
where ωi is a principal T 2-connection over K whose curvature is 2πm a i. The explicit functions 2 A (x ) ,Γ (x) are linear in x and Bij(x) are ratios of various polynomials in x. Generically these solutions possess two independent angular momenta along the T2-fibres. The Sasakian horizon geometries of Proposition 4.3 arise as a special case with m1 + m2 = Ij and possess only one independent angular momentum. The base K = ℂℙ1 gives horizon topologies ∼ 3 2 H = S × S or ∼ 3&tidle; 2 H = S ×S depending on whether m1 + m2 is even or odd respectively.

It is worth noting that the local form of this class of near-horizon metrics includes as a special case that of the extremal MP metrics H ∼= S2q+1 with all but one equal angular momenta. The above class of horizon geometries are of the same form as the Einstein metrics found in [37, 157Jump To The Next Citation Point].

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