Non-perturbative decay of Non-Abelian hair

We construct a solution of Heterotic supergravity which interpolates between two different AdS$_{3}\times S^{3}\times T^{4}$ geometries corresponding to the near-horizon limits of two 5-dimensional black holes, only one of which has non-Abelian hair. This solution can be used to estimate the amplitude of probability of the non-perturbative decay of the gauge 5-brane responsible for the non-Abelian hair into eight solitonic 5-branes by evaluating its Euclidean action. The Wick rotation of this solution poses several problems which we argue can be overcome by using a non-extremal off-shell (NEOS) deformation of the solution. This NEOS field configuration can be Wick rotated straight away and its Euclidean action can be computed for any value of the deformation parameter. The Euclidean result can then be anti-Wick-rotated and its extremal limit gives the Euclidean action of the original solution, which turns out to be one half of the difference between the entropies of the 5-dimensional black holes.


Introduction
During the last few years, a number of solutions of 4-and 5-dimensional Super-Einstein-Yang-Mills theories 1 describing extremal black holes, strings and rings with different kinds of non-Abelian hair have been obtained in completely analytic form [1,2,3,4,5,6,7]. The naive form of their masses and entropies is puzzling, since the non-Abelian hair falls too fast at infinity to contribute to the mass but it is relevant at the horizon and contributes to the entropy. Thus, there seemed to be an infinite number of black holes with the same conserved charges at infinity but completely different entropies.
In order to clarify the situation, the embedding of a specially simple 5-dimensional black hole with non-Abelian hair in 10-dimensional Heterotic Supergravity was recently studied in Ref. [8]. This embedding leads to the identification of the physical parameters of the 5-dimensional solution (Abelian charges and moduli) with the numbers of certain branes of Heterotic String Theory, namely fundamental strings (F1s), solitonic 5-branes (S5) and wave momentum in a compact direction (W). Furthermore, it was found that a single gauge 5-brane [9] is responsible for all the black hole's non-Abelian hair [10] and that this gauge 5-brane contributes to the same 5-dimensional charge as the S5-branes with 8 units. Thus, this 5-dimensional charge, which occurs in the mass formula, should be split into two different charges, Abelian and non-Abelian, both of which contribute to the mass. In this way, all the branes of the solution contribute to the mass, as expected, and the non-Abelian hair puzzle is solved by the correct stringy identification of the charges.
The solution to this puzzle poses new questions. Many of the non-Abelian (single) black hole and string solutions we have constructed have very interesting near-horizon geometries of a new kind that we have called dumbbell solutions in Ref. [11] because they interpolate between two Bertotti-Robinson-like spaces AdS n × S m [12,13] with different radii. They were first noticed in Ref. [14], but they arise for several values of n and m in near-horizon limits of non-Abelian black holes and black strings. In particular, the near-horizon geometry of the simple non-Abelian 5-dimensional black hole studied in Ref. [8] interpolates between two AdS 2 × S 3 geometries with different radii which are found in two different limits, ρ → 0, ∞ of the radial coordinate. One of them contains the contribution of the non-Abelian hair (i.e. the contribution of the gauge 5-brane) while the other does not and it is just the AdS 2 × S 3 one would obtain as the near-horizon geometry of a 3-charge Abelian black hole.
The existence of this solution suggests its potential use to study the quantum transition between one AdS 2 × S 3 vacuum and the other by Euclidean path integral methods, 2 if a suitable instanton associated to this Lorentzian solution can be found. As a matter of fact, it is easier to work with and interpret the corresponding 10-dimensional 1 These are the minimal supersymmetrizations of the Einstein-Yang-Mills theories that admit supersymmetric black-hole solutions. Therefore, they are gauged supergravities with non-Abelian gauge groups and, due to the last condition, they must have at least 8 supercharges.
2 See, for instance, the collection of reprints [15].
solution of Heterotic Supergravity that one obtains by oxidizing the 5-dimensional dumbbell solution. In particular, as we are going to argue, in 10-dimensional language, the transition between the two vacua can be interpreted as a transition between a configuration that includes a gauge 5-brane and another in which there is no gauge 5-brane but there are 8 additional S5-branes or, in other words, the decay of a gauge 5-brane into 8 S5-branes (whose overall charge is the same).
In spite of its simplicity (as compared to the 5-dimensional one), it is very difficult to Wick-rotate the 10-dimensional solution to obtain the instanton whose Euclidean action we need to evaluate. We are going to argue that the most serious difficulties stem from the extremality of the solution. Indeed, the direct evaluation of the Euclidean action of extremal black holes has well-known problems [16,17] that do not arise when one deals with the non-extremal solutions and finds their physical quantities of interest (Hawking temperature and Bekenstein-Hawking entropy) taking then the extremal limit of these quantities.
Here we propose to use a non-extremal deformation of the solution which is not a solution: a non-extremal off-shell (NEOS) deformation since all one needs is that the NEOS configurations interpolate between the same vacua as the original solution (so they contribute to the path integral for the same process), that they can be Wick-rotated and that they have finite Euclidean action to take, afterwards, the extremal limit. There is no systematic prescription to construct the NEOS configuration, but we manage to construct a one-parameter family with just the right properties and we evaluate its Euclidean action finding a result that we interpret physically as the amplitude of probability of decay of a gauge 5-brane into 8 S5-branes in a background containing a number of other branes. The value of the Euclidean action turns out to be one half of the difference of the entropies of the non-Abelian and Abelian black holes with those branes.
This paper is organized as follows: in Section 2 we describe the solutions we are going to work with. In Section 3 we compute the above-mentioned amplitude, setting up the calculation in Section 3.1, rewriting it in Section 3.2 to make the Wick rotation easier, introducing the NEOS configuration in Section 3.3 and computing its Euclidean action in Section 3.4. In Section 4 we discuss our results.

Black holes with non-Abelian hair
In this work we are going to study solutions of 10-dimensional Heterotic Supergravity with just one set of SU(2) gauge fields. Its action, in the string frame, is given by where the 2-and 3-form field strengths F A and H are defined as The string coupling constant g s is equal to the vacuum expectation value of the exponential of the dilaton g s =<e φ >. In asymptotically-flat solutions, this should also be the value of the dilaton at infinity, and, therefore, in these solutions This action is part of the low-energy effective field theory action of any of the two Heterotic Superstrings at first order in α since SU(2) is contained in both of their gauge groups. From the supersymmetry point of view, this action is complete, i.e. the bosonic part of a complete locally supersymmetric action. There is, however, another term which enters the action at first order in α , proportional to R 2 − where R − is the Lorentz curvature 2-form of one of the torsionful spin connections Ω ab ± µ = ω ab µ ± 1 2 H ab µ . Also the Bianchi identity of the 3-form field strength H has another term to first order in α , proportional to Tr (R − ∧ R − ). Introducing these terms alone would break the supersymmetric completeness of the action (a quartic term would be required to restore it [18]) and this is the reason why we are not including them because we rely on supersymmetric solution-generating techniques to obtain solutions. However, in order to consider the solutions of this action as legitimate solutions of the full Heterotic String effective action to first order in α expansion, one has to show that the terms quadratic in the curvature, evaluated on the solutions, are much smaller than those we have kept. In the solutions that we are going to consider (Eq. (2.10)) the Tr (R − ∧ R − ) and other R 2 − terms are of higher order in α . 3 In Ref. [8] we obtained the following solution of SU(2) Heterotic Supergravity: where dΩ 2 (3) is the metric of the unit, round S 3 , v A L are the 3 left-invariant Maurer-Cartan 1-forms of SU(2), the coordinates y i , i = 6, 7, 8, 9 parametrize a T 4 and the Z functions are given bỹ where, in their turn, κ is the size parameter of a SU(2) BPST instanton and the charges Q 0 , Q − and Q + are related, respectively, to the number of solitonic five-branes, N S5 , the number of fundamental strings, N F1 , and the number of units of momentum flowing along the compact direction u of radius R z , N W , bỹ Apart from the S5, F1 and W ("Abelian") constituents, there is a single gauge 5-brane N G5 = 1 sourced by the SU(2) instanton [9].
When compactified on T 4 × S 1 , this solution describes a five-dimensional black hole with non-Abelian hair, whose entropy and mass read The objects we have referred to as "Abelian" source Abelian 1-forms in the 5dimensional theory and their charges contribute both to the mass and entropy. The non-Abelian gauge 5-brane manifests itself as a globally regular gravitating instanton [10] which contributes to the mass as 8 solitonic 5-branes would, but does not contribute to the entropy at all. This makes this solution less thermodynamically favored than another one with N S5 = N S5 + 8 solitonic 5-branes and no gauge 5-branes, which would have exactly the same mass, the same Abelian charges and moduli at infinity but larger entropy S = 2π (N S5 + 8)N F1 N W > S, suggesting that the spontaneous decay of a gauge 5-brane into 8 solitonic 5-branes is thermodynamically possible.
However, the decay of a gauge 5-brane into 8 solitonic 5-branes can never take place perturbatively, as the SU(2) instanton is protected by topology and it can only occur non-perturbatively, by quantum tunneling.
In order to study this decay, it is convenient to consider a related solution, obtained by removing the 1's from the functions Z 0± , which can be seen as the near-horizon limit of the above solution. This solution reads explicitly where R 2 is the function In the absence of non-Abelian fields, this solution would just be AdS 3 × S 3 × T 4 , globally. This is the near-horizon geometry of the S5-F1-W brane configuration. However, the above solution, with the non-Abelian fields switched on, interpolates between two AdS 3 × S 3 × T 4 geometries of different radii: 4 • In the ρ → 0 limit the squared radius of the AdS 3 × S 3 factor 5 is R 2 0 =Q 0 . • In the ρ → ∞ limit the squared radius is R 2 Furthermore, the gauge fields are also different in these two limits: which is a pure gauge configuration. 4 The metric of the AdS 3 factor appears in a somewhat unconventional form but it can be checked that the Riemann curvature tensor corresponds to an AdS 3 space of radius R 0,∞ . 5 The radius of the AdS 3 and S 3 factors are equal, and we refer to this common value as the radius of the product geometry.
In order to compare the two limits, we must gauge-transform A A ∞ so that it also vanishes identically, A A ∞ = 0. After this gauge transformation, the 2-form B, which transforms simultaneously via Nicolai-Townsend transformations due to the presence of the Chern-Simons 3-form, takes the form which, on account of the first of Eqs. (2.7), tells us that the asymptotic geometry contains N S5 = N S5 + 8 S5-branes. We conclude that the complete solution Eq. (2.10) can be interpreted as an interpolation between the near-horizon geometries of a configuration with N S5 S5-branes and N G5 = 1 gauge 5-brane and another configuration with N S5 = N S5 + 8 S5-branes and N G5 = 0 gauge 5-branes.

Setting up the calculation
According to the Euclidean path integral approach, given an initial and a final state at fixed Euclidean times, the transition probability amplitude between them is given by where is the integral is taken over all the Euclidean field configurations Ψ which satisfy the boundary conditions associated to the given initial and final states and S E [Ψ] is their Euclidean action. This probability can be well approximated by for a classical solution Ψ 0 with the given boundary conditions and finite Euclidean action, i.e. an instanton. In some cases (when the initial and final states are vacua) this probability can be interpreted as the decay rate of a metastable vacuum into a more stable one. The simplest prescription to obtain a Euclidean solution is to Wick-rotate (t = −iτ) a Lorentzian one. However, when applied to non-trivial field configurations (non-static metrics, for example) this naive prescription fails to give real solutions of Euclidean signature unless some parameters of the solution are analytically continued into the complex domain (see, for instance, the seminal Ref. [21]). These Euclidean solutions can be thought of as real sections of a complexified solution obtained by analytical continuation, but their existence is by no means guaranteed and, in general, finding a real solution of Euclidean signature associated to a Lorentzian one is a well-known and complicated problem. 6 Here, we would like to find a real Euclidean solution associated to the Lorentzian dumbbell solution described in Eqs. (2.10) of the previous section. Such a solution, if of finite Euclidean action, could be interpreted as an instanton interpolating between the two vacua N S5 , N G5 = 1, N F1 , N W and N S5 + 8, N G5 = 0, N F1 , N W and the (minus) exponential of its Euclidean action would give the probability of decay from one vacuum to the other. Predictably, after the preceeding discussion, in the search for this real Euclidean solution we are going to meet several problems that we are going to try to solve.
The first problem arises in the Wick rotation of the Kalb-Ramond 2-form B, which makes the electric part purely imaginary. This is usually dealt with by Wick-rotating the "electric charge," (Q − , here) as well, but this would prove fatal in this case because it would make the dilaton and several components of the metric imaginary.
As we are going to explain, the root of this problem may lie in the extremality of our solution. Let us compare the Wick rotation of the solution at hands with that of a more familiar solution: an electrically-charged Reissner-Norström black hole.
In the non-extremal regime there is no problem with the simultaneous Wick rotation of the time t = −iτ and the electric charge q = −iq E because in the standard coordinates in which the metric takes the form with r ± = M ± M 2 − Q 2 , the electric charge Q only occurs quadratically in the metric. There is no extremal limit of this Euclidean solution, though, as the Lorentzian extremality condition M 2 − Q 2 = 0 becomes M 2 + Q 2 E = 0 and the near-horizon limit is always E 2 × S 2 (corresponding to the Lorentzian Rindler×S 2 ) and not H 2 × S 2 (which would correspond to AdS 2 ×S 2 ). Of course, one can always take the extremal limit of the physical quantities computed in the non-extremal case after they are re-expressed in terms of the Lorentzian charges. This is how, typically, the entropy and (vanishing) temperature of extremal black holes are computed in the Euclidean approach because the direct Wick rotation and the computation of the Euclidean action of the extremal solution present very serious problems. 7 Let us start with the problems presented by the direct Wick rotation of the extremal solution.
First of all, if one tries to Wick-rotate directly the extremal Lorentzian solution in which r + = r − = M = ±Q one finds that one has to Wick-rotate the mass as well, losing the reality of the metric. In our case there seems to be no way to make all the Wick-rotated fields real (specially the metric, due to its complicated form) simultaneously no matter how we treat the parameters of the solution.
There is a way out in the context of the 4-dimensional Maxwell-Einstein theory: one can dualize the electric charge into a magnetic charge, which needs not be Wickrotated. Dyonic solutions such as Eqs. (2.10) can be more difficult to rotate into a purely magnetic solution but we can split the 2-form into its electric and a magnetic parts and dualize only the electric one obtaining two magnetic fields (a 2-form and a 6-form) which do not need to be Wick-rotated. We will explain how to do this in detail later but we can advance that this trick turns out to be only good enough to keep the solution real for Q + = 0. This strongly suggests that we should try to work with a non-extremal Euclidean solution and then take the extremal limit of the Lorentzian results.
On top of the problems related to the Wick rotation there is another problem that seems to affect extremal solutions only and which supports the need of working with non-extremal solutions. As shown in Refs. [16,17], a direct calculation of the entropy of the extremal Reissner-Nordström black hole within the Euclidean approach gives a result (S BH = 0) which differs from the extremal limit of the entropy of the nonextremal black holes, which is the same as the value obtained by counting microstates in the String Theory context [23,24,25]. The technical reason is the existence of an inner boundary in the extremal Euclidean solution which does not exist in the nonextremal family of solutions for any value of the physical parameters.
The need to work with a non-extremal solution raises another problem, because the non-extremal version of the black-hole solution Eqs. (2.5) is not known and it has been argued that it may not exist. We are going to circumvent this problem by constructing a 1-parameter non-extremal deformation of the solution Eqs. (2.10) which is not a solution of the equations of motion but interpolates between the same two vacua as the extremal solution. This non-extremal off-shell (NEOS) deformation can be understood as a mere regularization procedure or as a computation of the action over an off-shell family of field configurations that contribute to the path integral in Eq. (3.1) because they have the boundary conditions demanded in this case. The extremal limit is, at the same time an extremum of the action because it is a solution of the classical equations of motion and, clearly, it makes sense to compute the action over the complete family of configurations and then take the extremal limit.
In the rest of this section we are going to carry out the program explained above. First of all, we are going to dualize the electric part of the Kalb-Ramond 2-form into a magnetic 6-form. This has to be done in the action and in the solution simultaneously. Next, we will make a first attempt at the Wick rotation and we will see that for Q + = 0 we need the NEOS deformation. Finally, we will compute the Euclidean action for this family of field configurations, taking into account all the boundary terms.

Dual action and solution
In order to dualize the electric part of the Kalb-Ramond 2-form B we replace it by the sum of a pair of 2-forms B 1 + B 2 such that H 1 · H 2 = 0 and then dualize the second into a 6-formB 2 with 7-form field strengthH 2 = e −2φ H 2 such that H 1 ∧H 2 = 0. The resulting action is 8 (3.4) and any solution of this action satisfying the constraint H ∧H = 0 is a solution of the original Heterotic Supergravity with We can reexpress the solution Eqs. (2.10) as the following purely magnetic solution of the above action whose Wick rotation is potentially simpler: (3.6) In order to proceed with Wick rotation, we first need to identify an appropriate time coordinate. Let us first begin with the case in which there is no momentum, Q + = 0. In this case, we can make the change of variables v = (t − x)/ √ 2, u = (t + x)/ √ 2, which makes 2dudv = dt 2 − dx 2 . Then we substitute t = −iτ, and redefine with a global sign the metric to get a positive-definite Euclidean metric (the rest of fields are unaffected) This solution interpolates between two H 3 × S 3 geometries of radii R 0 and R ∞ and is a gravitational instanton which represents a tunneling history in which one H 3 × S 3 vacuum decays into another one of larger radius, or, according to the previous discussion, a history in which a gauge 5-brane decays into 8 solitonic 5-branes. We may just compute the Euclidean action for this solution which is real but, as we will see, it vanishes.

The NEOS deformation and its Euclideanization
This is closely related to the fact that the solution with Q + = 0 does not produce a black hole in five dimensions. Indeed, it is known that the on-shell Euclidean action for black hole solutions is related to the entropy of the black hole. For Q + = 0 the entropy of the would-be black hole vanishes so it is reasonable that the Euclidean action does so. It is necessary to have Q + = 0 in order to get a non-vanishing action. 9 When Q + = 0 things are more involved: after the change of variables there is a crossed term in the metric Q + dtdx that becomes imaginary after the Wick rotation unless we rotate Q + as well. But Q + occurs in more places in the metric, which would become complex.
As explained before, this problem can be solved by using a 1-parameter a NEOS deformation of the metric 10 where and R(ρ) has the same form as before. Even though the NEOS configuration (3.9) is not a solution in general, it shares with the original extremal solution several interesting properties. If we consider the case of pure gauge vector fields, so that R(ρ) is constant, it is a solution with the geometry AdS 3 × S 3 × T 4 , albeit in different coordinates. With non-trivial gauge fields the metric interpolates between two AdS 3 × S 3 × T 4 geometries with radii R 0 and R ∞ , so the NEOS configurations contribute to the path integral that describes the transition between these two vacua, and the a = Q + extremizes the Euclidean action.
Furthermore, for arbitrary values of a, this metric can be analytically continued to a Euclidean metric by making t = −iτ, a = iℵ. In general, it is not a solution, but when the "extremality condition" a 2 = Q 2 + is satisfied in the Lorentzian configuration, we recover the solution (3.8).
As discussed above, we are going to Wick-rotate the above NEOS configuration first, then we are going to compute its Euclidean action and at the end we are going to take the a 2 → Q 2 + limit in the result. First, let us massage a bit the metric by making the change of coordinates which is a Lorentz boost. This sets the metric in a diagonal form (3.12) In the ρ → 0 limit, the t − ρ sector of the metric is just a 1 + 2-dimensional Rindler spacetime and, therefore, it corresponds to a non-extremal horizon whose Hawking temperature we will compute later.
To complete the definition of this field configuration we have to determine the period of the coordinate y, which is assumed to be compact in the original solution we started with. By comparing the area of the horizon of this metric and of the original one we conclude that 11 Then, we perform Wick rotation: t = −iτ, a = iℵ and, after an overall change of sign, we get the Euclidean metric where now Note that the R = constant configurations (with pure gauge vector fields) are solutions (just as their Lorentzian partners) and have the geometry of H 3 × S 3 × T 4 with radius R.
In the ρ → 0 limit, redefining the radial coordinate ρ ≡ ρ 0 r 2 /R 2 0 , the metric takes the form 16) and, to avoid a conical singularity at r = 0, the Euclidean time τ must be periodic with period β so that the Hawking temperature (undoing the Wick rotation of the a parameter) is (3.17) and vanishes in the a → Q + limit.

Computation of the Euclidean action
The complete Euclidean action that we want to compute is given by where K is the trace of the extrinsic curvature of the boundary ∂M, c is a normalization constant which is fixed by the criterium that S E = 0 when evaluated on the vacuum (taken to be H 3 × S 3 × T 4 with radius R ∞ ) and (H ·B) µ ≡H µν 1 ···ν 6B ν 1 ···ν 6 . This surface term appears when we dualize the Kalb-Ramond 2-form B. For the normalization constant, we will use the usual prescription where K 0 is the extrinsic curvature of the boundary when it is embedded in the vacuum.
In order to evaluate the integrand of the action it is convenient to use the equations of motion, but here we have to be very careful because we are not dealing with a solution, and not all of them are satisfied. In particular, the equation of motion of the dilaton is not satisfied and we have to add a "source" term δ(ρ; ℵ) (3.20) which can be simply computed by plugging the fields in the l.h.s. We know, however, that δ = 0 for global H 3 × S 3 × T 4 and also that lim ρ→∞ δ(ρ; ℵ) = 0, because the configuration we are considering asymptotically tends to a solution. Indeed, δ decays so fast that the integral converges.
Using the dilaton equation, the Euclidean action Eq. (3.18) takes a simpler form: (3.22) Next, we massage the first term in the integrand (3.23) The first term cancels identically theH ·B boundary terms and the second combines with theBFF term into a term proportional to the equation of motion ofB, 12 which happens to be identically satisfied, 13 so we get (3.25) Since the only non-compact coordinate is ρ, the boundary of M consists just of the asymptotic region ρ = ρ ∞ , where ρ ∞ is a regulator that must be taken to infinity eventually. In the Euclidean NEOS configuration that we are considering, with the Euclidean time compactified with the period β = 1/T H the region ρ = 0 is not a boundary.
In the limit ρ → ∞ so these terms decay too fast to contribute to the integral. Therefore, only the bulk term gives a non-zero contribution and the Euclidean action is given by where the 10-dimensional Newton constant is given in Eq. (2.4), V T 4 = (2πl s ) 4 is the volume of the T 4 , V S 3 = 2π 2 is the volume of the S 3 of unit radius, R y is the radius of the compact coordinate y given in Eq. (3.13), β = 1/T H is the period of the Euclidean time given implicitly in Eq. (3.17) and ∆(ℵ) is the integral defined in Eq. (3.21). Substituting in the above expression, we find that Note that this is a finite result for all the finite values of ρ 0 . These NEOS configurations are, therefore, instantons, even if they are not solutions. We now have to undo the Wick rotation ℵ = −ia, so that ρ 2 0 = 1 2 (Q 2 + − a 2 ) 1/2 and take the limit a → Q + . In this limit both ∆(a) and ρ 0 go to zero, but the limit of ∆(a)/ρ 2 0 turns out to be finite and takes the value 12 This term has the form where we have used the assumption that N S5 >> 8. This leads us to our final result for the Euclidean action associated to the Lorentzian solution interpolating between two brane configurations we started from which leads to the transition probability . (3.31)

Discussion
First of all, we observe that, quite remarkably, this result coincides with e −∆S BH , where ∆S BH is the change of Bekenstein-Hawking entropy in the process: which is what one can expect on general grounds: for a single black hole, the Euclidean action is proportional to the Bekenstein-Hawking entropy [21,26] 14 and the Lorentzian solution we started from was interpreted as connecting two different blackholes near-horizon geometries, so it is natural that the Euclidean action yields the difference between the entropies of two black holes. Notice that the sign in the exponent is opposite to what one would expect from statistical mechanics, where the decay rate would be estimated to be of the order of e +∆S . In our case, the entropy difference is positive, so the process is favored thermodynamically. However, the decay process involves topology change, which is highly suppressed. This way we can interpret the minus sign in e −∆S as the fact that topology constrains the decay so effectively that it succeeds in suppressing the process even though it involves an increase of entropy.
Given the composition of the string background corresponding to the black hole (F1s, NS5s etc), the result can also be interpreted as the decay rate of a gauge 5-brane which lives in a configuration of fundamental strings, momentum waves and solitonic 5-branes.
When the numbers of each kind of component are comparable, this probability is tiny and the gauge 5-brane, though unstable, is long lived. However, when the number of S5-branes is much larger than the number of strings and waves, N S5 > N F1 N W , the gauge 5-branes decay quickly into S5-branes. One may say that S5-branes are "hungry" for gauge 5-branes, and the larger their number the faster they will eat them.
The result suggests that the non-Abelian 5-dimensional black hole Eq. (2.5) whose near-horizon limit gives the Lorentzian "dumbbell" solution we started from Eq. (2.10) is non-perturbatively unstable. Furthermore, and possible related to this fact, it seems very difficult or, perhaps, it is impossible, to find non-extremal black holes with the same charges and non-Abelian fields. Clearly, more work is necessary to clarify the situation.
Finally, we should comment on the relation between the result obtained here and the method employed and Brill's work Ref. [27] in which he computed the Euclidean action of an instanton whose Lorentzian counterpart connects several (at least three) asymptotic AdS 2 × S 2 geometries and which has non-trivial topology. 15 These solutions can be used to study the non-perturbative splitting of a Reissner-Nordström black hole into smaller black holes. In that case the Wick rotation offers no special complications but the geometry of the instanton is very complicated and it is not clear how to deal with the inner boundaries identified in Refs. [16,17]. The solution studied here is topologically simpler (at least from the metric point of view), but we have argued that a non-extremal off-shell (NEOS) deformation had to be used to go to the Euclidean, compute the action and then go back to Lorentzian signature and take the extremal limit, avoiding many of the pitfalls one finds along the way.
We expect the NEOS technique developed here to be of further use in other contexts.