Metastable Nonextremal Antibranes

We find new and compelling evidence for the metastability of supersymmetry-breaking states in holographic backgrounds whose consistency has been the source of ongoing disagreements in the literature. As a concrete example, we analyze anti- D 3 branes at the tip of the Klebanov-Strassler throat. Using the blackfold formalism we examine how temperature affects the conjectured metastable state and determine whether and how the existing extremal results generalize when going beyond extremality. In the extremal limit we exactly recover the results of Kachru, Pearson, and Verlinde, in a regime of parameter space that was previously inaccessible. Away from extremality we uncover a metastable black Neveu-Schwarz five-brane (NS5) state that disappears near a geometric transition where black anti- D 3 branes and black NS5 branes become indistinguishable. This is remarkably consistent with complementary earlier results based on the analysis of regularity conditions of backreacted solutions. We therefore provide highly nontrivial evidence for the metastability of antibranes in noncompact throat geometries since we find a consistent picture over different regimes in parameter space.


INTRODUCTION
The study of anti-branes in warped throats has a long history, starting with the original papers [1,2] whose motivation was the holographic description of dynamical breaking of supersymmetry (SUSY).Since then, antibranes have become an indispensable tool to break SUSY in various contexts.The applications beyond holography include de Sitter [3], inflationary model building [4] and the construction of non-extremal black hole micro-states [5].For these applications additional complications due to the compactness of extra dimensions arise, which we will not address.Instead, we restrict to the study of anti-branes in the non-compact KS throat, motivated by holography and the potential use as a local description inside compact throat geometries.
D3's at the tip of the throat can decay because the surrounding 3-form fluxes induce delocalised D3 charges out of which D3's can nucleate and annihilate with the D3's.This process can be described in terms of brane polarisation in which the D3's "puff" into a spherical NS5 wrapping a contractible S 2 inside the S 3 -cycle [2].As the NS5 moves over the S 3 , it changes the sign of the D3 charges it carries, effectively mediating the brane-flux decay.To find a meta-stable state the NS5 needs to find a balance between the H 3 -flux force that wants to push the NS5 over the S 3 and the force of its own "weight" doing the opposite.In the probe limit, KPV found that such a balance of forces exists whenever the ratio p/M is small enough [2].Here p denotes the number of D3's and M the quantum of 3-form flux piercing the S 3 .
The existence of this KPV state has been refuted in various works starting with the investigations of [6].The problem, found at the time, arises when trying to go beyond the probe limit and investigate what happens once the branes backreact.In particular [6], and many subsequent works [7], found that the backreacted geometry had singular 3-form fluxes in such a way that it would cause immediate brane-flux decay [8].As a response, [9] argued that the singularity is renormalised in such a way that does not affect stability when p = 1, which is a case that is not amenable to a supergravity analysis.Refs.[10,11] argued that meta-stability can also be retained when p 1 since the observed singularities cannot be proven to exist once one backreacts spherical NS5's instead of point-like D3's.In fact, all proofs of unphysical singularities rested on an assumption which was in contradiction with KPV from the start, since D3 meta-stable states are really NS5 states.
Several studies [12][13][14][15] have investigated the effect of adding temperature to the D3's.Most of these works were motivated by the would-be singularity in the 3-form fluxes.Whether or not a singularity can be cloaked by a horizon that arises when moving away from extremality is believed to be an important criterium for deciding the fate of singularities [16].Although strong indications were found that one should not worry about singularities at all, [9][10][11], it remains an outstanding problem to understand what happens when the D3's are at finite temperature.Aside the importance for understanding phase transitions in the holographic dual, finite temperature allows us to test the conjectured stability of the KPV state, which is crucial for the many applications listed in the beginning.If the state would destabilise infinitesimally away from extremality, it would be a sign of being gapless, which is not wanted.In this letter we present additional evidence that this does not happen and provide new, previously inaccessible, quantitative results that support the picture of [10,11].

OUR APPROACH
From the NS5 viewpoint D3-NS5 polarisation is most naturally described in the supergravity regime where arXiv:1812.01067v1[hep-th] 3 Dec 2018 g s p 1 (see e.g.[17]).This involves a daunting task: solving the type IIB supergravity equations to find D3-NS5 bound state solutions wrapping an S 2 in the presence of the fluxes of the KS background.In this paper, we attack this problem using blackfold techniques [18][19][20].
In the blackfold formalism, problems of the above type are treated systematically by setting up a scheme of matched asymptotic expansions, where the solution is approximated in a far-zone by the background solution of interest (here the KS background) and in a near-zone by a uniform flat-space p-brane solution (here the D3-NS5 bound state).This scheme is possible when the characteristic length scales of the near-zone solution, denoted collectively by r b , are hierarchically smaller than the characteristic length scales R of worldvolume inhomogeneities, and the characteristic length scales L of the background.The regime r b R, L is a regime of longwavelength expansions.For a general discussion of blackfolds and the approximations they entail we refer the reader to [20] and references therein.The specifics of the regimes of interest in our context are summarised below.
A key component of the above analysis reformulates part of the supergravity equations as an effective worldvolume theory.In the case at hand, this is a supergravityderived 6d effective theory that describes the longwavelength properties of NS5 branes with dissolved 3brane charge.In what follows, we focus exclusively on the leading-order equations of the 6d theory (blackfold equations).At the very least, these equations pose necessary conditions for the existence of the long-sought supergravity solution in the above regime.

FORCED BLACKFOLD EQUATIONS
The leading order blackfold equations for black branes in the presence of general external fluxes in (super)gravity were obtained in [20].We consider D3-NS5 branes at the tip of the KS throat, where in appropriate units the string frame metric is [21] Here, M 0123 is Minkowski space in the directions 0123, dΩ 2 n is the metric element of the round unit n-sphere, b 2 0 0.93266 and s the string scale.There is also a RR field strength F 3 = dC 2 = 2M 3 s Vol(S 3 ) across the S 3 and a 7-form flux The dilaton is constant and the self-dual 5-form field strength vanishes at the tip.The boosted D3-NS5 bound state solution in flat space, which forms the seed of our blackfold expansion, is well known (see e.g.[22,23]).The following thermodynamic data of this solution (in the Einstein frame) are needed for our purposes: the energy-momentum tensor and the charge currents where and γ ab is the induced metric on the fivebrane worldvolume (parameterised by σ a with a, b, . . .= 0, 1, . . ., 5).The Hodge dual of γ ab is * and ĥab is a projector onto the directions of the dissolved D3-brane charge inside the fivebrane.General boosts/rotations of the D3-NS5 bound state solution are expressed in terms of the velocity timelike unit vector u a and the spacelike orthonormal vectors v a , w a .In terms of these vectors ĥab = γ ab − v a v b − w a w b .The electric currents J 4 , j 6 express the D3, NS5 currents of the solution while J 2 is a consequence of the non-zero C 2 -field of the solution.r 0 is the Schwarzschild radius.In the extremal limit, r 0 → 0, α → ∞ while the combination r 2 0 e 2α is kept fixed.The parameter θ controls how much D3 brane charge is dissolved inside the NS5 brane.
The general effective blackfold equations of the D3-NS5 brane in the presence of 3-form NSNS/RR fluxes, constant dilaton and vanishing 1-, 5-form RR fluxes are [20] Note that the index µ in ( 7) is an index in the tendimensional ambient KS background metric.Similarly, in ( 8), ( 9) is the Hodge dual for the 10d KS metric.The differential equations ( 7)-( 9) should be viewed as a system of dynamical equations for the unknown degrees of freedom of a 6d effective theory.These degrees of freedom include the transverse scalars (that express γ ab ), the functions r 0 , α, θ and the orthonormal vectors u a , v a , w a .
The last equation in ( 9) is equivalent to the statement that the NS5-brane charge density Q 5 = Cr 2 0 sinh α cosh α cos θ is a constant of motion proportional to the number N 5 of NS5 branes.For fivebrane configurations wrapping an S 2 inside the S 3 in (1), which will be considered below, the first equation in (9) similarly implies that the modified 3-brane charge ) is also a constant of motion proportional to the induced D3-brane charge of the bound state.We denote Q 3 /(4πQ 5 ) = πpg s 2 s , using p from now on as the ratio of the number of anti-D3s to NS5s.

RECOVERING KPV AT EXTREMALITY
Consider the extremal limit of the equations ( 7)-( 9).We focus on a possibly time-dependent configuration where the fivebrane bound state at r = 0 wraps an S 2 inside the S 3 of the KS background.Writing the S 3 metric as dΩ 2  3 = dψ 2 + sin 2 ψ dω 2 + sin 2 ωdϕ 2 we turn on the single transverse scalar Then, the full set of equations ( 7)-( 9) reduces to After eliminating tan θ, it is trivial to check that the resulting equation for ψ coincides with the Euler-Lagrange equations of the DBI action obtained in [2] by S-duality of the D5 brane.In this manner, we recover the equations of motion of the KPV effective action from supergravity.A more general relation between the extremal blackfold equations and DBI equations can be derived along the lines of [24] (see also [25]).
The maximum value p * that allows a meta-stable vacuum has the following meaning in the blackfold language.Since the NS5 branes at extremality have a non-vanishing Hagedorn local temperature | cos θ|, one can show that p * is precisely the point where this temperature takes its maximum possible value.That occurs when cos θ = 1, i.e. when the 3-brane charge is depleted.

NON-EXTREMAL STATIC CONFIGURATIONS
It is straightforward to repeat the above exercise for non-extremal configurations at finite α.We continue to focus on the same ansatz for the worldvolume fields, now restricting to time-independent profiles.The nonextremal, static version of equation ( 10) is exactly the same as before.On the other hand, when expressed in terms of α eq. ( 11) becomes This equation can be written alternatively in terms of the local temperature T = 1 2πr0 cosh α , or the local entropy density s = 2πCr 3 0 cosh α, or any other quantity that characterises the deviation from extremality.

NON-EXTREMAL THERMODYNAMIC POTENTIALS
Eqs. ( 10), ( 12) can be obtained by extremising an appropriate effective potential, which can be derived from thermodynamic considerations (see e.g.[26,27]).Since the local temperature of the NS5 brane does not vanish at extremality, it is appropriate to choose an effective potential that holds some other quantity fixed, e.g. the global entropy S = B5 √ −γs/k = 8π 2 (g s M b 2 0 ) 5/2 Cr 3 0 cosh α sin 2 ψ, where B 5 is the spatial part of the worldvolume M 6 , s is the local entropy density defined above and k the Killing vector in the direction of the velocity vector u.The potential at fixed global entropy is  3)) and P[B 6 ] the pullback of the background B 6 -field [28].V S [ψ] is the total energy in the system and can be obtained from a Legendre transform of the Euclidean onshell action of the D3-NS5 bound state.

RESULTS FOR NON-EXTREMAL CONFIGURATIONS
Consider first the regime p/M < p * /M 0.08, where the extremal solutions have a meta-stable vacuum.In Fig. 1 we show how the effective potential V S changes as we vary the entropy S for a fixed value of p/M .We observe two interesting new features.Firstly, as soon as S is turned on, a new unstable vacuum emerges (black dots on the right plot of Fig. 1) near the North pole, ψ 0. For sufficiently small values of S there are three extrema: two unstable and one meta-stable.Secondly, as we increase S further the new unstable extremum comes closer to the meta-stable vacuum and the two merge at a critical value of the entropy S * , which is a function of p/M .Above this value the meta-stable vacuum is lost.The new unstable state represents a fat black NS5 with a highly pinched R 3 × S 5 horizon geometry that resembles a black D3.Instead, the meta-stable state starts life near extremality as a thin black NS5 with R 3 × S 2 × S 3 horizon topology.At the merger the meta-stable black NS5 turns effectively into a black D3.The picture of a merger driven by horizon geometry is reinforced by the following observation.
A quantitative measure of the 'fatness' of a black NS5 wrapping an S 2 is provided by the ratio d = 2 √ pr 0 /( √ M sin(ψ)) [29], where r0 ≡ √ Cr 0 / √ Q 5 is dimensionless.The ratio d, which is a natural function of p/M and the equilibrium ψ, compares the scale 2 √ g s pr 0 s = 2(g s pN −1 5 ) 1/2 r 0 associated to the Schwarzschild radius and the scale of the S 2 wrapped by the NS5 worldvolume √ g s M s sin(ψ).As an illustration, on the left plot in Fig. 2 we see how d behaves in the unstable branch (blue color) and the meta-stable branch (orange color).Recall that α → ∞ represents ex-tremality.The unstable branch has visibly higher values of d, expressing the dominance of the Schwarzschild radius.The meta-stable branch captures a thin black NS5 with small values of d.The merger occurs at a value of d notably close to 1.
On the right plot of Fig. 2 we present numerical data that exhibit how the ratio d at the merger point behaves as a function of p/M .Remarkably, these data show that the ratio remains effectively constant, near the value 0.89 over a significant range.It deviates slightly from this value in the vicinity of the upper bound of p/M , where effects from the second unstable state (already visible in KPV [2]) become important.The characteristically weak dependence of d on p/M is a clear signal that the properties of the merger point are closely tied to the properties of the horizon geometry.
Finally, by increasing p/M further, above the critical value p * /M 0.08, we observe the complete loss of the meta-stable vacuum exactly as in the extremal KPV analysis [2].The unstable vacuum in the vicinity of the North pole, however, remains, even above p * /M , and constitutes the single vacuum of the non-extremal static blackfold equations.

REGIMES OF VALIDITY
The validity of the blackfold expansion requires a large separation of scales r b R, L. In the case at hand, the characteristic length scale r b is the largest scale among the energy density radius (r ε ∼ r 0 sinh α) and the scales associated to the NS5 and D3 charge respectively.The scale R is controlled by the size of the S 2 that the NS5 wraps, while the background scale L is larger and set by the size of the S 3 .We note that in all configurations of interest the two terms on the RHS of (10) (πp/M and −ψ + 1 2 sin(2ψ)) are either comparable or the πp/M term dominates.Then, from r (N S5) h R and r (D3) h R respectively, we obtain the conditions N 5 g s M sin 2 ψ and p/M √ g s M sin 2 ψ.Both fail at the North pole, but for sufficiently large M , they allow our calculations to be valid everywhere except for a small region around the North pole.In turn, the constraint r ε R leads to the requirement d √ g s p( √ N 5 sinh α) −1 .This restricts the regime of validity of the unstable (blue) branch on the left plot of Fig. 2 close to extremality, at large values of α [30].

DISCUSSION
Our analysis uncovered two features of black D3-NS5 branes at the tip of the KS throat: a new unstable state near the North pole, and the persistence of the meta-stable state when the horizon radius is not too large.For a critical value of the horizon radius the metastable state and the new unstable state merge.Above this critical value, the meta-stable state is lost.Remarkably, these two features coincide with what was anticipated in [10] based on the very complementary viewpoint of exact solutions for fully backreacted NS5 branes.Finding such solutions is clearly beyond reach, but nonetheless some properties of the would-be exact solution can be diagnosed using GR techniques.In particular, a nogo-theorem for physically acceptable solutions was constructed and the outcome was that the black NS5, if it exists, has a maximum horizon radius until it disappears.A black D3 was also escaping the nogo, but was deemed unphysical in [10], since it does not persist in the extremal limit.Note that the findings of [10,11] were not proofs of existence, they were nogo's.What we showed here is that exactly when the nogo's do not apply we find go's in the blackfold approach.We regard this as very strong evidence for the existence of the meta-stable states, since by now they have been argued in rather complementary ways.
conventions for which the warpfactor at the tip has been absorbed in a rescaling of the Minkowskian coordinates.That explains, in particular, the factor of 2 in the definition of d.
[30] In addition, since the NS5 brane has a running dilaton one may worry whether regions of spacetime with large values of string coupling e φ invalidate our analysis.We note that the running of the dilaton is capped off at the horizon for non-extremal solutions at the value e φ (r0) = gs sin 2 θ + cosh 2 α cos 2 θ.Hence, by suitably tuning the asymptotic value of gs we can achieve wide areas in parameter space where our solutions are everywhere weakly coupled.Admittedly, this tuning is not possible for extremal solutions.However, since it is understood how to treat the strong coupling singularity of NS5 branes in flat space, and since the constraint (blackfold) equations can be obtained in a far-zone analysis of the solution, where the string coupling is weak, we anticipate that a large dilaton in the bulk of the solution does not invalidate the conclusions of our analysis even at extremality.

FIG. 1 .
FIG.1.Plots of the effective potential at fixed entropy, VS, as a function of the angle ψ on S 3 .Both figures represent plots at p/M = 0.03.The right plot zooms into the region near the North pole of the S 3 .As we increase the entropy we encounter a critical value S * , where the meta-stable vacuum of KPV (blue dots on the right) merges with a new unstable vacuum (black dots on the left).

FIG. 2 . 2 2 0 + |Q 5 √ 1 +
FIG.2.Plots of the ratio d that expresses how 'fat' a non-extremal D3-NS5 bound state is.On the left plot we depict the dependence of d on the non-extremality parameter α for the unstable (blue) and meta-stable (orange) branches for p/M = 0.03.On the right plot, we depict d at the critical merger point as a function of p/M .