A non-Abelian Black Ring

We construct a supersymmetric black ring solution of SU(2) N=1, d=5 Super-Einstein-Yang-Mills (SEYM) theory by adding a distorted BPST instanton to an Abelian black ring solution of the same theory. The change cannot be observed from spatial infinity: neither the mass, nor the angular momenta or the values of the scalars at infinity differ from those of the Abelian ring. The entropy is, however, sensitive to the presence of the non-Abelian instanton, and it is smaller than that of the Abelian ring, in analogy to what happens in the supersymmetric coloured black holes recently constructed in the same theory and in N=2, d=4 SEYM. By taking the limit in which the two angular momenta become equal we derive a non-Abelian generalization of the BMPV rotating black-hole solution.


Introduction
The discovery of black rings by Emparan and Reall in Ref. [1] showed how two important properties of 4-dimensional asymptotically-flat black holes, uniqueness/no-hair and spherical topology of the event horizon (which, for the 5-dimensional black ring, is S 2 × S 1 ), could be violated in higher dimensions. 1 For a range of values of the conserved charges (mass, angular momenta) that may characterize an uncharged black ring, a different black-ring and a black-hole solutions are also possible. For charged black rings (the first of which was constructed in Ref. [5]) the non-uniqueness becomes infinite; for the same conserved electric charges one can construct black rings with regular horizons with magnetic dipole momenta taking continuous values in some interval [6]. Despite being innocuous to the conserved charges, these dipole momenta do contribute to the BH entropy. The construction of supersymmetric black-ring solutions in minimal [7] or matter-coupled N = 1, d = 5 supergravity [8,9,10,11,12] using the general classification of supersymmetric solutions of these theories started in Ref. [13] opened up the possibility of constructing very general families of black-ring solutions with various kinds of electric charges and moduli in which these issues could be studied.
The violation of the no-hair conjecture by non-Abelian fields in 4-dimensions is also a well-known but less stressed fact, perhaps because the first solutions in which this was observed [14,15,16], black-hole generalizations of the "Bartnik-McKinnon particle" [17] with asymptotically vanishing gauge charges, were purely numerical, which makes more difficult their study and understanding. 2 The first black-holes with non-Abelian hair (not related to the embedding of an Abelian field into a non-Abelian one through a singular gauge transformation) given in an analytical form were found using supersymmetry techniques in the context of N = 2, d = 4 Super-Einstein-Yang-Mills (SEYM) theory 3 in Refs. [20] and [21] using the general classification of the timelike supersymmetric solutions of these theories made in Ref. [22]. The black-hole solutions constructed in Ref. [21] include the field of an SU(2) coloured monopole found by Protogenov in [23] which also has asymptotically vanishing gauge charge. The monopole charge does contribute to the entropy, though. These black holes, which can be seen as the result of adding the coloured monopole to a standard black hole with Abelian charges, modifying the entropy but none of the asymptotic charges, were called coloured black holes and they seem to be ubiquitous [24].
The results of Ref. [22] have been used more recently to construct new single-center and two-center non-Abelian solutions of N = 2, d = 4 SEYM models that can be obtained by dimensional reduction of N = 1, d = 5 SEYM models 4 in Ref. [25]. One of the main goals of that exercise was to open the possibility for the construction of the first non-Abelian black-hole solutions in d = 5 by oxidation to d = 5 of those solutions, because the direct construction using the general classification of timelike supersymmetric solutions of Refs. [26,27] turns out to be too complicated. This can only be done for certain models of the lower dimensional theory. The oxidation itself turned out to be a non-trivial exercise if one wanted to construct solutions without spatial translation isometries (which would be black strings instead of black holes), but, as was shown in Ref. [28], one can use non-trivial cycles to perform the reduction and still preserve supersymmetry, basically using Kronheimer's mechanism [29]. Both kinds of black solutions (strings and holes) were recently constructed in Ref. [30].
The d = 5 non-Abelian black holes constructed there are, again, coloured black holes, with asymptotically vanishing gauge fields. They can be understood as the result of adding a BPST instanton to a black hole with Abelian charges, leaving the mass and electric charges unmodified. Just as in the 4-dimensional case, the non-Abelian field does contribute to the entropy. The BPST instanton field turns out to be related by dimensional redox to the coloured monopole at the heart of the 4-dimensional coloured black holes.
It is natural to try to see if black-rings also admit the addition non-Abelian instanton fields and the effect this addition may have on the mass and entropy. In this paper we are going to construct and study a regular supersymmetric black-ring solution of N = 1, d = 5 SEYM with a distorted BPST instanton. We start by reviewing in Section 1 the recipe that we are going to use to construct timelike supersymmetric solutions, which was obtained in Ref. [30]. In Section 2 we will carry out the construction of the solution after which we will study its regularity and we will compute its essential properties. In Section 3 we will study the limit in which the black ring becomes a non-Abelian rotating black hole. Our conclusions are in Section 4.

The recipe to construct solutions
In Ref. [30] we have found a procedure to construct systematically timelike supersymmetric solutions admitting an additional spacelike isometry (with adapted coordinate z) of any N = 1, d = 5 Super-Einstein-Yang-Mills (SEYM) characterized by the tensor C I JK and the structure constants f I J K : 5 4 Again, these are the simplest, but not unique N = 1 (minimal) supersymmetrizations of the d = 5 Einstein-Yang-Mills theory and the supersymmetrization requires the addition of, at least, scalars. They also go by the name of non-Abelian-gauged N = 1, d = 5 coupled to vector supermultiplets. 5 Our conventions are those of Refs. [31,26] and are based on Ref. [32]. The supersymmetric solutions of the most general N = 1, d = 5 supergravity theory including vector supermultiplets and hypermultiplets and generic gaugings were characterized in Ref. [26]. The inclusion of tensor supermultiplets was considered in Ref. [27].
1. Find a set of tand z-independent functions M, H, Φ I , L I and 1-forms ω, A I , χ in E 3 satisfying the equations (defined in E 3 as well) The 2. Using them, reconstruct the solution's 5-dimensional spacetime fields as follows: (a) The scalars can be found from this equation for the quotients because there is always a parametrization of the scalar manifold such that With the above equation for the quotients h I (φ)/f one can also determine the functionf . For the special case of symmetric scalar manifolds, it is given by 6 6 In this expression, The metric has the form wheref has been determined above, the 1-formω is given by 7 and where the 4-dimensional Euclidean metric dŝ 2 is given by 8 The vector fields and their corresponding field strengths are given by where the vector fieldsÂ I , defined on the 4-dimensional Euclidean space dŝ 2 , and their field strengths are given bŷ whereD (resp.D) is the exterior gauge-covariant derivative with respect to the connectionÂ I (resp.Ȃ I ).
In Ref. [30] we used this recipe to construct black-hole solutions with non-Abelian gauge and scalar fields for the SU(2)-gauged ST [2,5] model. 9 This model has 4 vector 7 The unhatted ω is the one occurring in Eq. (1.5). 8 With H and χ related by Eq. (1.2), this is a hyperKähler metric admitting a triholomorphic Killing vector, also known as Gibbons-Hawking metric [34,35]. We will also denote the compact coordinate z by ϕ. It will be assumed to take values in [0, 4π). 9 Actually, this is the name of the model of N = 2, d = 4 supergravity one obtains by dimensional reduction. multiplets and, hence, 4 scalar fields that parametrize the symmetric space SO(1, 3)/SO(3). It is defined by a tensor C I JK with the following non-vanishing components The directions to be gauged are the last three, which we will denote by indices α, β, . . . = 2, 3, 4. the ungauged directions will be denoted by indices i, j, . . . = 0, 1.
Being a symmetric space, we can use Eq. (1.8) to write the metric functionf as a function of the building blocks H, L I , Φ I : Now, in order to find solutions of this model, we just need to find building blocks that satisfy Eqs. (1.1)-(1.5). In the next section we will just do this to find a solution that describes a black ring.

Construction of the Solution
Inspired by Refs. [11,8], we choose a point x 0 ≡ 0, 0, −R 2 /4 in E 3 and a harmonic function N with a pole at that point, in terms of which we can write the non-vanishing building blocks in the ungauged directions as (2.2) These functions contain the integration constants q i , Q i and λ i . The first two can be interpreted as charges. The latter, whose value will be restricted by requirements such as the normalization of the metric at infinity, are moduli. Eq. (1.1) is satisfied automatically. Eq. (1.2) is satisfied with where r, θ ∈ (0, π) and ψ ∈ [0, 2π) are spherical coordinates centered at r = | x| = 0 with the definitions and orientation where r n , θ n ∈ (0, π) and ψ n ∈ [0, 2π) are spherical coordinates centered at r n = | x n | = 0 with the definitions This solution, originally found by Protogenov in Ref. [23], describes a magnetic colored monopole placed at r n = 0. It is singular at r n = 0 as a field configuration in E 3 , but this behaviour can change when we analyze the whole picture. In fact, we showed in Ref. [28] that the monopole field gives rise to a BPST instanton in E 4 through (1.14), and we used this result in Ref. [30] to construct a regular black hole of the same supergravity theory we consider in this work.
In the present case we obtain a different instanton field configuration from (1.14), which we call distorted BPST, because the pole of the harmonic function H is placed in a different point (r = 0) than that of the coloured monopole (r n = 0). This distorted BPST is singular at r n = 0, which might turn the black ring solution ill-defined. Happily this is not the case. The complete vector field contains the instanton plus an additional term, see (1.13), where the latter cancels precisely this divergence at that critical point Observe that in the ungauged case the Φ α s would have been harmonic functions −q α N/(4 √ 2) and the combinations C ijk q j q k should have been replaced by C iJK q J q K . Here the asymptotic behaviour of the non-Abelian gauge field indicates that the "non-Abelian q α s" do not contribute in the same way the q i s do. However, they have a similar near-horizon behaviour.
The above functions define completely the solution. In what follows we are going to analyze its metric to show that it describes a regular black ring and to compute its main properties.

Analysis of the Solution
In this analysis it is convenient to use two set of coordinates: those centered at r = 0, (r, θ, ψ, defined in Eq. (2.4)) supplemented by the time coordinate t and the angular coordinate ϕ, and those centered at r n = 0 (r n , θ n , ψ n , defined in Eq. (2.6)) supplemented by the time coordinate t n and the angular coordinate ϕ n . The relations and a solution can be readily found assuming ω has only one non-vanishing component, ω ψ : 10 10 The expression coincides with that of [36] despite we have chosen x 0 to be on the negative x 3 axis. This is because the coordinate θ has also a relative sign with respect to the used in that reference.
Observe that, since L α = 0 the non-Abelian terms do not affect ω. However, they do affect the whole 5-dimensionalω given in Eq. (1.10) via ω 5 in Eq. (1.11): The last term in G has a non-Abelian origin. In the r → ∞ limit in which the metric tends to Minkowski's (so we have an asymptotically flat solution), though, it is subdominant and we do not expect it to contribute to the the angular momentum of the solution.
So far we have been working in coordinates in which the hyperKähler metric Eq. (1.12) is of the form dŝ 2 = r (dϕ + cos θdψ) 2 + 1 r dr 2 + r 2 dθ 2 + sin θ 2 dψ 2 , (2. 16) but, in order to compute mass and angular momentum, it is convenient to use a different coordinate system (also centered at x = 0) t, Θ, φ 1 , φ 2 , related to the former by (2.17) in which the complete 5-dimensional metric is of the form The independent components of the angular momentum are now obtained from the metric behaviour in the ρ → ∞ limit 11 11 We use units in which and, from the absence of contribution proportional to g, we see that they coincide with those of the Abelian black ring, as we expected.
Observe that these formulae allow us to identify Before we move to study the possible presence of an event horizon, let us point out that the solution does not contain any Dirac-Misner strings. 12 Indeed, the g tφ 1 (resp. g tφ 2 ) metric component vanishes when the coordinate φ 1 (resp. φ 2 ) is not well defined, which happens at Θ = π/2 (Θ = 0).
The solution may have an event horizon at x = x 0 , where the norm of the timelike Killing vector of the metric vanishes. In order to study the near horizon limit we need to use a different coordinate system because several components of the metric blow up there in the coordinates we have been using so far. Recall the expression for the metric in the original frame centered at x = 0 (2.23) We first go to the auxiliary frame centered at the horizon with spherical coordinates and take the r n → 0 limit. The functions that appear in the metric behave in this limit as follows where we have defined the constants v = C ijk q i q j q k − 16 These expression for the constants v and l resemble those of the Abelian case [11], with an additional non-Abelian term. The precise form of the constants k 1 , k 2 and k 3 in terms of the charges are messy. They do not occur in the calculation of any physical quantity, but they play a role in the near horizon analysis, 13 since they are responsible for the disappearance of O(r −1 n ) in the metric after we perform the following coordinate transformation, where the constants b 1 , b 2 and c 1 can be chosen such that all divergences in the metric in the r n → 0 limit disappear: With this choice we find in the r n → 0 limit that the horizon has the following metric with the topology S 1 × S 2 , so the solution is a black ring with non-Abelian hair, i.e. a non-Abelian black ring. Using this metric we can compute the area of the horizon 14 , 37) 13 We give their form here for the sake of completeness, (2.33) so the entropy of the non-Abelian black ring can be written in terms of the charges and angular momenta using the expressions for the constants v and l Eqs. (2.29) and (2.30) together with Eq. (2.22) as follows: Finally, we would like to compute the mass of the solution. We do so by comparing the asymptotic behavior of the metric component g tt with that of the Schwarzschild solution, g tt ∼ 1 − 8MG 3πρ 2 + · · · . We get The constants λ i can be expressed in terms of physical constants. If we define the physical scalars of the theory as φ x ≡ h x /h 0 we find that the only scalar with a nonvanishing asymptotic value is the Abelian one and this value is φ 1 ∞ = λ 1 /λ 0 . On the other hand, the asymptotic normalization of the metric Eq. (2.10) implied λ 0 λ 2 1 = 2/3 3 . Then, and M takes the form and depends only on the moduli and on the electric charges Q 0 , Q 1 while the q i , which correspond to magnetic dipole momenta do not contribute to it [6]. The non-Abelian field do not contribute, either. This expression looks identical to that of the non-Abelian black hole solution constructed in Ref. [30], but the charges Q 0 and Q 1 are not the same than the charges q 0 and q 1 that appear in the black-hole mass formula given in that reference. They are, actually, related by Q BR BR . This is just reflecting the fact that the conserved electrical charges in the black ring receive contributions from the magnetic dipole momenta via the Chern-Simons term in the action. This effect is commonly described as "charges dissolved in fluxes" [9].
This non-Abelian black-ring mass formula, is, however, identical to that of the Abelian black ring that one would obtain by removing the non-Abelian fields from this solution. In other words: the presence of non-Abelian fields is not observable at spatial infinity. They do contribute to the entropy, though, as in the black-hole case, their entropy being smaller than that of their Abelian siblings.
Observe that the non-Abelian term in ω 5 , which does not contribute to the angular momentum, does contribute to the r → 0 limit just as the Abelian terms: Let us study the near-horizon limit → 0. Using Eqs. (3.4) and (3.7), we find that the metric Eq. (1.9) behaves in this limit as where r = (Y 3 − Z 2 ) 1/2 ρ, dΠ 2 (2) = ρ 2 dt 2 − dρ 2 ρ 2 is the metric of the AdS 2 of unit radius and sin 2 α = Z 2 /Y 3 . This space is the near-horizon limit of the BMPV black hole [38], but, due to the non-Abelian contribution to Z (which can be understood as a sort of "near-horizon angular momentum"), now α does not vanish for vanishing asymptotic angular momentum J and we can have a stationary black hole with J = 0 whose nearhorizon limit is not AdS 2 ×S 3 . The converse is also possible: we can make α = Z = 0 for J = 16 √ 3 q 0 /g 2 and have a rotating black hole whose near-horizon limit is AdS 2 ×S 3 . The area of the horizon is (3.10)

Conclusions
The existence of black-hole and black-ring solutions with identical asymptotic behaviour but with non-Abelian hair that contributes to the entropy [21,25,24,30] challenges our understanding of black-hole hair and the microscopic interpretation of the black-hole/black-ring entropy, just as the Abelian hair discovered in Ref. [6] did. More research is necessary to gain a better understanding of these solutions. In particular, the stability of these supersymmetric non-Abelian solutions (which are entropically disfavored) needs to be addressed and their possible non-supersymmetric and nonextremal generalizations have to be constructed and studied. Work in these directions is in progress.