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Do supersymmetric anti-de Sitter black rings exist?

We determine the most general near-horizon geometry of a supersymmetric, asymptotically anti-de Sitter, black hole solution of five-dimensional minimal gauged supergravity that admits two rotational symmetries. The near-horizon geometry is that of the supersymmetric, topologically spherical, black hole solution of Chong et al. This proves that regular supersymmetric anti-de Sitter black rings with two rotational symmetries do not exist in minimal supergravity. However, we do find a solution corresponding to the near-horizon geometry of a supersymmetric black ring held in equilibrium by a conical singularity, which suggests that nonsupersymmetric anti-de Sitter black rings may exist but cannot be"balanced"in the supersymmetric limit.


Introduction
Supersymmetric black hole solutions of five-dimensional gauged supergravity have been known for a few years [1,2,3,4,5]. According to the AdS/CFT correspondence [6], these black holes should correspond to 1/16 BPS states of N = 4 SU (N ) super Yang-Mills theory on R × S 3 or, equivalently, gauge-invariant local operators of this theory on R 4 . A puzzle arises [7] because BPS operators carry five independent conserved charges: their spins J 1 , J 2 and R-charges Q 1 , Q 2 , Q 3 (where (Q 1 , Q 2 , Q 3 ) is a weight vector of the SU (4) R-symmetry group), whereas the most general known asymptotically AdS 5 × S 5 supersymmetric black hole solution has only 4 independent conserved charges [5].
There are two ways that this puzzle could be resolved. (i) We already know the most general supersymmetric black hole solution. Although generic 1/16-BPS operators have 5 independent charges, only a subset of these operators gives a large enough entropy to correspond to a macroscopic event horizon and this subset has only 4 independent charges. (Maybe because of finite coupling effects in the CFT.) This possibility has been explored in [8]. (ii) There exists a 5-parameter supersymmetric black hole solution that remains to be discovered.
If (ii) is correct, there appears to be some tension with what is known about black holes in minimal gauged supergravity. One can truncate to this theory by setting Q 1 = Q 2 = Q 3 = Q. the S 2 always has a conical singularity. This suggests that AdS black rings may exist, but they cannot be "balanced" in the supersymmetric limit without the presence of external forces corresponding to the conical singularity. The conical singularity vanishes when one takes the limit of vanishing cosmological constant, and one recovers the AdS 3 × S 2 geometry of an asymptotically flat supersymmetric black ring [15].
Interestingly, when lifted to a solution of type IIB supergravity, our singular black ring nearhorizon geometry is locally isometric to a solution obtained in [17]. It was shown in [17] that this solution can be extended to a globally regular metric in ten dimensions, but it does not appear possible to reduce the resulting solution to five dimension so any interpretation in terms of five dimensional black holes is lost.
This paper is organized as follows. We start by deriving a general constraint on black holes with two rotational symmetries. Then we review the results of [1], presenting the equations relevant to our analysis and deriving some general results. We solve these equations subject to the assumption of two rotational symmetries. There are only two interesting solutions and we show that these correspond to the near-horizon geometry of a supersymmetric black ring with a conical singularity and the near-horizon geometry of the solution of [3] respectively. Section 3 concludes. Some details of the analysis are relegated to appendices.

.1 General constraints
The bosonic sector of minimal D = 5 gauged supergravity is Einstein-Maxwell theory with a negative cosmological constant and a Chern-Simons term for the Maxwell field F. The unique maximally supersymmetric solution of this theory is AdS 5 with vanishing gauge field [18]. We shall denote by ℓ the radius of this AdS 5 solution.
Consider an asymptotically AdS 5 solution of this theory (not necessarily supersymmetric) that is stationary and admits two rotational symmetries, i.e., there is a R × U (1) × U (1) isometry group. Let k, m 1 , m 2 denote the Killing fields that generate time translations and rotations respectively. We assume that these Killing fields commute and also leave the Maxwell field invariant. Now we can use a standard argument from the theory of stationary axisymmetric solutions [19]: the Bianchi identity and the fact that the Lie derivatives of F along the Killing fields vanish imply that m µ 1 m ν 2 F µν is constant. However, since the solution is asymptotically AdS 5 , we can find coordinates so that the asymptotic metric is where k = ∂/∂t, m i = ∂/∂φ i and 0 ≤ θ ≤ π/2. Therefore, in the asymptotic region, m 1 and m 2 vanish at θ = 0, π/2 respectively. It follows that The same argument shows that k µ m ν i F µν ≡ 0 although we shall not need this result.

Near-horizon limit
Given a supersymmetric black hole we can take a near-horizon limit as explained in [11] to obtain a supersymmetric near-horizon solution. We want to classify such solutions. Necessary and sufficient conditions for a near-horizon geometry to be a supersymmetric solution of this theory were worked out in [1] using the method introduced in [11]. One can introduce Gaussian null coordinates on the horizon so that the near-horizon metric is [11] where the horizon is at r = 0, ∂/∂v is Killing, ∆ is non-negative, and the metric g ab is the metric on a spatial cross-section of the horizon. We shall denote this 3-manifold as H. Supersymmetry implies the following [1]: • There exists a globally defined unit 1-form Z on H: where g ab is the inverse of g ab .
• The (near-horizon) Maxwell field is where ⋆ is the Hodge dual on H. Hence the Bianchi identity is • The following equations hold on H where h · Z ≡ g ab h a Z b . This implies that • The Ricci tensor of H is These equations are not the complete set derived in [1] but they are the only ones we will need here. We have not been able to solve these equations in full generality. However, we have obtained some general results that is it convenient to record here. The near-horizon geometry is static if, and only if, the Killing field V ≡ ∂/∂v is hypersurface orthogonal, i.e., V ∧ dV ≡ 0. The following lemma gives the conditions for this to occur: Lemma 1. The following conditions are equivalent: (a) the near-horizon geometry is static, (b) dh ≡ 0, (c) ∆ ≡ 0. Proof. Assume (a). Then the rab components of V ∧ dV ≡ 0 give dh ≡ 0 so (a) implies (b). Now assume (b). If ∆ is nonzero then equation (7) implies that Z = −(ℓ/6)(h + d∆/∆) and hence Z is closed. Equation (9) then implies that ∆Z ∧ ⋆Z = 0, hence Z = 0, in contradiction with the fact that Z has unit norm. Hence we must have ∆ = 0 everywhere. Hence (b) implies (c).
Our second lemma shows that ∆ cannot vanish anywhere if the near-horizon geometry is nonstatic.
Lemma 2. If ∆ vanishes at a point then ∆ vanishes everywhere. Proof. Take the divergence of (7) and use equations (6) and (8) to obtain Assume ∆ = 0 at p. Then, since ∆ ≥ 0, ∆ is at a (global) minimum so d∆ = 0 at p. It then follows from this equation that ∇ 2 ∆ = 0 at p. However, since this is a minimum, the eigenvalues of the Hessian of ∆ must be non-negative, and ∇ 2 ∆ is the sum of the eigenvalues so they must all vanish. Hence the Hessian vanishes at p: ∂ m ∂ n ∆ = 0 at p. Assume inductively that ∆ and its first 2n derivatives vanish at p. Then the Taylor expansion at p begins where x i are normal coordinates at p. Now ∆ has to be non-negative but this leading term changes sign under x → −x and so we must require it to vanish. Therefore the (2n + 1)th derivatives of ∆ must vanish at p. Now the expansion of ∆ is where the symmetric tensor M is the (2n + 2)th derivative of ∆. From our induction hypothesis and equation (11), we can obtain which implies M i 1 ...i 2n jj = 0.
This implies that the leading term in the expansion of ∆ is harmonic in R 3 . But we also need this term to be non-negative, which implies that it attains its minimum at the origin. Then from the maximum principle, it follows that this term must vanish everywhere, i.e., M = 0. So we've shown that the first (2n + 2) derivatives of ∆ vanish. It follows by induction, if ∆ = 0 at p then all derivatives of ∆ vanish at p hence ∆ ≡ 0 by analyticity 4 .
In summary, these lemmas reveal that a static near-horizon geometry has ∆ ≡ 0 and a non-static one has ∆ > 0. We end this section by noting that a static near-horizon geometry can arise from a non-static black hole. Indeed, this is what happens for supersymmetric black rings in ungauged supergravity [15]. We shall see that the same appears to be true in gauged supergravity.

Including the symmetries
Consider a supersymmetric, asymptotically anti-de Sitter black hole admitting two rotational Killing fields m 1 and m 2 . The near-horizon solution will inherit these symmetries. Hence we are interested in classifying near-horizon solutions for which there exist two commuting Killing vector fields m 1 , m 2 on H that preserve h, ∆ and the Maxwell field F. It follows from (5) that the Killing fields must also preserve Z. Furthermore, the near-horizon solution will also inherit the condition (2).
We can choose local coordinates x a = (ρ, x i ) so that ∂/∂x i are Killing, the metric on H is and ∆ and the components of h are functions only of ρ. It is convenient to allow the Killing fields ∂/∂x i to be arbitrary linear combinations of m 1 and m 2 , i.e., ∂/∂x i need not have closed orbits. We are then free to perform GL(2, R) transformations on the coordinates x i to simplify our analysis. We shall enforce the fact that orbits of m 1 and m 2 must close once we have determined a local solution.
It is convenient to parameterize the components of h as where Γ(ρ) is positive and a prime denotes a derivative with respect to ρ. The spatial components of the Maxwell field strength can be decomposed as where we have used equation (2) to deduce that the ij components must vanish. Comparing with equation (5) gives where ⋆ 2 denotes the Hodge dual with respect to the two-dimensional metric γ ij (with volume form η 2 oriented so that dρ ∧ η 2 is the volume form of H).
The ρi component of equation (10) gives hence we have Substituting the expression (19) for Z into equation (7) gives where, for a 1-form ω i (ρ)dx i , we define ω ′ = ω ′ i dx i . Equation (9) gives Solving equation (22) gives where ∆ 0 is a non-negative constant. There are two cases to consider depending on whether ∆ 0 > 0 or ∆ 0 = 0. These correspond to non-static and static near-horizon geometries respectively.

Non-static near-horizon geometry
Consider first the non-static case ∆ 0 > 0. In this case, the constants k i cannot both vanish for if they did then h would be closed and lemma 1 then gives ∆ = 0, contradicting ∆ 0 > 0. Equation (23) determines B, which can be plugged into (19) to obtain Substituting the expression for B into (24) yields and, since k i are constants, this can be integrated to give where C is a positive constant. Now consider equation (8). The ρρ component gives The ij component gives Let's deal first with the special case in which Γ is constant. These two equations reveal that this implies h = −(3/ℓ)Z (Z i cannot vanish when Γ ′ = 0 since this would imply that Z vanishes but Z has unit norm). All solutions with this property were obtained in [1]. There are three possibilities: corresponding to the metric on H being a homogeneous metric on the manifolds N il, SL(2, R) or S 3 respectively. Now assume Γ is non-constant. Multiply (31) by γ ij to get Since Γ ′ is nonzero, we can divide by Γ ′ and integrate to get where β is a positive constant. Multiplying (31) by k i k j leads to an equation which can be rearranged to read But from equation (30) we have where This implies that (34) can be rewritten as with solution where α 0 is a constant of integration. Hence we have where Note that P (Γ) ≥ 0 implies that k cannot vanish anywhere unless α 3 0 = ∆ 2 0 /C 2 . At this point, it is convenient to exploit the GL(2, R) symmetry to set k 1 = 1, k 2 = 0. Equation (29) then gives Plugging equation (39) into equation (35) and using the explicit form of Z (equation (27)) gives an ODE for γ 12 : d dΓ This can be integrated to give plus a constant times γ 11 which can be eliminated using the remaining GL(2, R) freedom to shift x 1 by a constant times x 2 . Finally we can get γ 22 from (33), using the freedom to rescale x 2 to set β = 1. The 2-metric γ ij is now fully determined in terms of Γ, so it is convenient to use Γ, instead of ρ as the 3rd coordinate on H. The full near-horizon solution is where with C and ∆ 0 positive constants and α 0 an arbitrary constant. It can be checked that the remaining equations of section 2.2 are all satisfied.
To summarize, we have determined all non-static near-horizon solutions admitting two commuting Killing fields. There are several cases: I If Γ is constant then the near-horizon solution is one of the solutions given in section 3.2 of [1], for which H is locally isometric to a homogeneous SL(2, R), N il or S 3 manifold. The latter case arises as the near-horizon limit of the asymptotically AdS black hole solutions constructed in [1].
II If Γ is non-constant then the near-horizon solution is (44).

Static near-horizon geometry
We now turn to the static case. This is analyzed in Appendix A, where it is proved that the near-horizon solution must be one of the following: III If Γ is constant then the near-horizon solution is the solution given in section 3.2 of [1] for which H is locally isometric to R × H 2 . This is the near-horizon limit of the supersymmetric black "string" of [20].
IV A solution (88) that can be obtained by taking the limit ∆ 0 → 0 (with other constants and the coordinates fixed) of the non-static solution (44). (This amounts to simply setting ∆ 0 = 0 in the solution (44).) V A solution (81) that can be obtained from (44) by setting VI The AdS 5 solution obtained in section 3.2 of [1], for which H is locally isometric to hyperbolic space.
The near-horizon geometry of the supersymmetric black holes of [3] is non-static with nonconstant Γ and hence must be described by (II) which we shall prove below. We shall also see that solution (IV) describes the near-horizon geometry of a supersymmetric black ring but suffers from a conical singularity.

Global considerations
So far, the discussion has been local. However, we are interested mainly in solutions for which H is compact since this is true of any solution arising as the near-horizon limit of a black hole. A compact 3-manifold admitting a U (1) × U (1) symmetry must be homeomorphic to T 3 , S 1 × S 2 , S 3 or a lens space (see e.g. [21]). This excludes the solutions (I) and (III) with constant Γ except for the solution of [1] with H locally isometry to S 3 . The known black hole solutions with this near-horizon geometry [1] arise as a special case of those of [3], and since we shall show that the near-horizon limit of the latter solutions is described by (44), it follows that the S 3 solution with constant Γ can be obtained as a limit of (44), so we shall not consider this solution further.
The solution (VI) is excluded because H cannot be compactified without breaking the rotational symmetries. So consider the other solutions with non-constant Γ. Note that γ 11 is a scalar invariant: it is the norm of ∂/∂x 1 (which is some linear combination of m 1 and m 2 ). Furthermore, for all of the solutions with nonconstant Γ, γ 11 is a monotonic function of Γ when Γ > 0. This implies that Γ can be expressed uniquely in terms of the invariant γ 11 and is therefore globally defined.
If H is compact and Γ nonconstant then Γ must achieve a distinct minimum and maximum on H. Hence dΓ must vanish at two distinct (positive) values of Γ. By calculating (dΓ) 2 , one finds that this is not possible for solution (V) above so this solution is necessarily noncompact. This leaves solutions (II) and (IV). Hence any compact solution with nonconstant Γ must be described by the line element (44), whether static (∆ 0 = 0) or not (∆ 0 > 0).
For this solution we have (dΓ) 2 = 4P (Γ)/(ℓ 2 Γ), so P (Γ) must be non-negative. Compactness implies that P (Γ) must have two distinct positive roots Γ 0 , Γ 1 with P (Γ) positive for Γ 0 < Γ < Γ 1 . It is not hard to see that this is only possible if P (Γ) has a third root Γ 2 > Γ 1 . (We can't have Γ 2 = Γ 1 since this would make the proper distance to Γ = Γ 1 infinite, i.e., H would not be compact.) So, for a compact horizon, P (Γ) must have the form sketched in figure 1. This imposes restrictions on the parameters of the solution.
At first sight, it appears that (44) is a 3-parameter solution. However, it is not hard to see that Γ is defined only up to multiplication by a positive constant. This freedom can be used to eliminate 2.7 ∆ 0 = 0: unbalanced black ring The metric of H is where we have rescaled x 2 . The metric has conical singularities at Γ = Γ 0 and Γ = Γ 1 , where ∂/∂x 2 vanishes. If one could remove these (by identifying x 2 with a suitable period) then one would be left with a smooth geometry of topology S 1 × S 2 with the S 1 parameterized by x 1 and the S 2 by Γ and x 2 . (Note that in this case we have m i ∝ ∂/∂x i .) Thus it would describe the horizon of a supersymmetric black ring. The necessary and sufficient condition for P (Γ) to have distinct real positive roots is It is convenient to define a parameter b by where 0 < b < 1. This defines b uniquely since the function on the RHS decreases monotonically from 1/27 at b = 0 to 0 at b = 1. We can now find explicit expressions for the roots: The coordinate x 2 can be periodically identified to eliminate a conical singularity at Γ = Γ 0 . The necessary and sufficient condition for the absence of a conical singularity of Γ = Γ 1 is then which implies b = 1. This is not allowed since b < 1 and hence one can never eliminate conical singularities from this metric 5 . So, at best, this solution describes the near-horizon geometry of an unbalanced supersymmetric black ring: the conical singularity provides the force required to hold the ring in equilibrium.
Note that b → 1 as ℓ → ∞, which suggests that the conical singularities can be eliminated as the cosmological constant is turned off. This is indeed true: in this limit, our ∆ 0 = 0 solution reduces to the AdS 3 × S 2 near-horizon geometry of the supersymmetric black ring of [15]. 6 It is interesting to consider the five-dimensional near-horizon geometry, which, after a coordinate change r = ΓR, is it can be checked that the expression in the square brackets is the line element of AdS 3 . Hence the five-dimensional near-horizon geometry is a warped product of AdS 3 and a deformed S 2 with a conical singularity at one pole. Our solution can be oxidized on S 5 to give a solution of type IIB supergravity [22]. Viewing S 5 as an S 1 bundle over CP 2 , the ten-dimensional solution is a warped product of AdS 3 with a 7-manifold M 7 , which is an S 1 bundle over S 2 × CP 2 where the S 2 is our singular S 2 . In fact, this ten-dimensional solution has been encountered before in a general exploration of such warped products [17]. Furthermore, it has been shown that the solution can be made into a globally regular metric if one takes M 7 to have a different topology, namely an S 1 bundle over a 6-manifold B 6 , where B 6 is an S 2 fibration over CP 2 [17]. The S 1 and S 2 here are not the same as for the uplift of our solution: if our Kaluza-Klein S 1 is parameterized by a coordinate ψ (and the S 2 by Γ and x 2 ) then the regular solution has S 1 parameterized by x 2 and S 2 parameterized by Γ and ψ.
So although our near-horizon geometry is necessarily singular in five dimensions, it can be made regular if lifted to ten dimensions. However, the resulting solution can no longer be reduced to five dimensions so one loses any interpretation in terms of five-dimensional black holes. It is natural to wonder whether the regular ten-dimensional solution might describe the near-horizon geometry of a supersymmetric, asymptotically AdS 5 × S 5 , black hole with horizon topology S 1 × M 7 .
We shall treat this case here. The special case α 0 = Γ 0 is treated separately in Appendix B. It turns out that it can be obtained as a limit of the generic case with no further complications. The range of Γ is Γ 0 ≤ Γ ≤ Γ 1 . The 2-metric γ ij is non-degenerate within this range but degenerates at the endpoints, where B(Γ) vanishes. This implies that the Killing field ω(Γ i )∂/∂x 1 − ∂/∂x 2 vanishes at Γ = Γ i , i = 0, 1. Now, using the fact that Γ i is a root of P (Γ), we have which implies that ω(Γ 0 ) = ω(Γ 1 ). Hence the Killing field that vanishes at Γ = Γ 0 is distinct from the one that vanishes at Γ = Γ 1 . In order to avoid conical singularities at Γ = Γ 0 , Γ 1 we must assume that these two Killing fields have closed orbits, in other words they generate rotational symmetries. Hence our two rotational Killing fields m i must be proportional to these two Killing fields: for some non-zero constants d 1 , d 2 . Introducing adapted coordinates φ i so that m i = ∂/∂φ i we have The condition φ i ∼ φ i + 2π fixes the constants d i up to signs: in order to avoid conical singularities one must take The solution is now globally regular. It is clear that H has S 3 topology, 7 with m 1 vanishing at Γ = Γ 0 and m 2 vanishing at Γ = Γ 1 . In Appendix B it is shown that the coordinate change from (x 1 , x 2 ) to (φ 1 , φ 2 ) is also valid for the special case α 0 = Γ 0 (although d 1 → 0 as α 0 → Γ 0 , the product d 1 ω(Γ 0 ) remains nonzero in this limit). Hence we do not need to treat this case separately any longer.
We shall now show that the near-horizon solution we have obtained is locally isometric to that of the supersymmetric black hole solution of Chong et al [3]. It is convenient to use the roots Γ i as parameters, as opposed to (C 2 , α 0 , ∆ 0 ). These are related by Note that Γ i are not totally arbitrary positive numbers: they are constrained by ∆ 2 0 > 0. Use the scale transformations (46) on Γ to fix which can always be done (i.e. the corresponding K > 0) since 0 < Γ 0 < Γ 1 < Γ 2 . We can define two constants a, b so that the roots are parameterized as: where g ≡ 1/ℓ. We need Γ i to be positive and correctly ordered, which gives the restrictions Note that b may be negative. One can solve for C, α 0 and ∆ 0 using (58). One finds that so the constraint coming from ∆ 2 0 > 0 is, in the notation of [3], We now define a coordinate θ by It is easy to check that taking 0 ≤ θ ≤ π/2 does indeed uniquely parameterize the range Γ 0 ≤ Γ ≤ Γ 1 (although θ = 0 corresponds to Γ = Γ 1 and θ = π/2 to Γ = Γ 0 ). We then have where and For completeness, we give the expressions for the constants d i : We have checked that our 3-metric in the (θ, φ 1 , φ 2 ) coordinates agrees exactly with the horizon metric of Chong et al in their (θ, ψ, φ) coordinates 8 with φ 1 = ψ and φ 2 = φ.
It is interesting to note that the near-horizon geometry is considerably simpler in the coordinates (Γ, x 1 , x 2 ) than in the coordinates (θ, φ 1 , φ 2 ).

Discussion
In this paper we have determined the most general regular, supersymmetric, near-horizon geometry of an asymptotically AdS 5 black hole solution of minimal gauged supergravity which admits two rotational isometries. We found that the only such solution is the near-horizon geometry of the topologically spherical supersymmetric black hole discovered by Chong et al [3]. Hence if new supersymmetric black hole solutions exist then either they do not have two rotational symmetries, or they have the same near-horizon geometry as the known solutions.
Our result implies that exotic topologies for supersymmetric black holes (such as black rings) are not allowed, unless they possess fewer than two rotational symmetries. We should emphasise that no known black hole in five dimensions possesses fewer than two rotational symmetries, including the asymptotically flat supersymmetric black rings of ungauged supergravity [15]. It has been conjectured, however, that there could exist asymptotically flat black holes with only one rotational symmetry [11], essentially because this is all one is guaranteed in general [16].
Curiously we did find a near-horizon geometry which describes a black ring, however it necessarily possesses a conical singularity. Physically this can be interpreted as suggesting that supersymmetric black rings in AdS cannot be "balanced", i.e., rotation and electromagnetic repulsion is not enough to counterbalance gravitational attraction (which is enhanced in AdS). However, known unbalanced ring solutions (e.g. [9,10]) can be balanced by increasing the angular momentum, so it seems likely the same will be true here (the mass and/or charge would also have to change for consistency with the BPS inequality). Hence our singular near-horizon geometry may be interpreted as evidence in favour of the existence of regular non-supersymmetric anti-de Sitter black rings.
It is interesting to compare our result with the corresponding result for the ungauged theory [11]. In the latter theory, one obtains a complete classification assuming only supersymmetry and compactness of the horizon. The allowed near-horizon geometries all admit two rotational symmetries (in fact H is homogeneous), but this is an output, not an input. In the gauged theory, we had to input the rotational symmetries as an assumption. It would be nice if this assumption could be relaxed and the general solution of the equations of section 2.2 determined (for compact H).
In the ungauged theory it is possible to construct a global uniqueness argument for supersymmetric black holes of spherical topology [11]. It is obviously desirable to have a corresponding result for the gauged theory, i.e., prove that the solution of [3] is the only supersymmetric asymptotically AdS black hole solution, even assuming two rotational symmetries. However, the arguments of [11] rely heavily on features of the ungauged theory that do not extend to the gauged theory so new ideas would be required to do this.
In the ungauged theory, it is straightforward to extend the argument to allow for abelian vector multiplets [23] and the same is probably true in the gauged theory. So it seems likely that any supersymmetric, asymptotically AdS black hole solution of gauged supergravity coupled to abelian vector multiplets must have the same near-horizon geometry as the solution of [5] if it admits two rotational symmetries.
The ρρ component of (8) gives which implies that Γ ′ is not identically zero hence we can change variable from ρ to Γ and rewrite this equation as where and for convenience the integration constant has been chosen such that Γ 0 is the real root of P (Γ).
Using the fact that Z is a unit one-form one can deduce that: and thus Γ 0 > 0. The ij component of (8) reduces to We can use the GL(2, R) freedom to set ω 1 = 1 and ω 2 = 0, i.e. ω = dx 1 . Therefore γ 12 and γ 22 satisfy the same first order ODE which can be written as: Thus γ 12 and γ 22 are both equal to some constant times P (Γ)/Γ 2 . This means that we can use the remaining GL(2, R) freedom to set γ 12 = 0 and γ 22 = P (Γ)/Γ 2 . Multiplying (78) by γ ij implies where γ is the determinant of γ ij , and hence γ = C 2 P (Γ)/Γ where C is a positive constant. These results imply γ 11 = C 2 Γ. Note that ω i ω i = γ 11 and thus upon comparison with (77) we deduce The rest of the equations in section 2.2 are satisfied without further constraint. Using Γ instead of ρ as a coordinate allows one to write the metric on the horizon as: This metric is smooth for Γ > Γ 0 (recall necessarily we have Γ 0 > 0). There is a conical singularity at Γ = Γ 0 but this can be removed by appropriately identifying x 2 . The 3-manifold this defines is topologically R 2 × S 1 . By rescaling x 1 suitably and letting Γ 0 → 0 we recover case (I).
(III) Consider the case where k is not identically zero. Equation (23) implies that Γ −1 k i = const , and therefore one can again deduce from (25) that: where ω i are constants. Expanding ⋆ 2 B in the basis {k, ⋆ 2 k} and using (24) implies for some constant α 0 . Therefore Note that if Γ is a constant, h = αZ for some constant α. This case was considered in section 3.2 of [1], where it was proved that H must be locally isometric to R × H 2 . In the following we assume Γ is non-constant. From the fact that Z is a unit one-form one can show that: where and we have defined the constant C > 0 by C 2 = Γ −1 k i k i (since k i are constants by assumption). Let us use the GL(2, R) freedom available to set k 1 = 1 and k 2 = 0. Using the definition of the constant C then gives γ 11 = C 2 Γ. Also, since Γ −1 k i is a constant we deduce that γ 12 = cΓ for some constant c. Using the GL(2, R) transformations (which leave k i invariant), we can arrange to have γ 12 = 0. Then, the 22 component of equation (8) simplies to: which implies γ 22 = P (Γ)Γ −2 (the integration constant have been fixed using the remaining GL(2, R) freedom). Collecting these results gives the horizon geometry g ab dx a dx b = ℓ 2 ΓdΓ 2 4P (Γ) + C 2 Γ(dx 1 ) 2 + P (Γ) Γ 2 (dx 2 ) 2 with P (Γ) given by (86). The rest of the equations in section 2.2 are now satisfied without further constraint. This metric is analysed in section 2.7 of the main text.
B Special case α 0 = Γ 0 This special case needs to be considered separately. The metric on H still takes the form (53). For Γ 0 ≤ Γ ≤ Γ 1 the function A(Γ) now vanishes for Γ = Γ 0 and is positive otherwise. The function B(Γ) now vanishes only at one end point Γ = Γ 1 and is positive in the rest of the interval. Therefore the 2-metric γ ij is non-degenerate for Γ 0 < Γ < Γ 1 and degenerates at Γ = Γ 0 and Γ = Γ 1 . The Killing field ∂/∂x 1 vanishes at Γ = Γ 0 and the Killing field ω(Γ 1 )∂/∂x 1 − ∂/∂x 2 vanishes at Γ = Γ 1 . In order to avoid conical singularities at Γ = Γ 0 , Γ 1 these two Killing fields must have closed orbits. Thus these two Killing fields must be proportional to m i , say Now introduce adapted coordinates such that m i = ∂/∂φ i : The condition φ i ∼ φ i + 2π implies that in order to avoid the conical singularities we must take: This solution is now globally regular and has S 3 topology with m 1 vanishing at Γ = Γ 0 and m 2 vanishing at Γ = Γ 1 . The coordinate change (x 1 , x 2 ) → (φ 1 , φ 2 ) can be obtained from the α 0 = Γ 0 case studied in section 2.8 by taking the limit α 0 → Γ 0 .