Unstoppable brane-flux decay of $\overline{\text{D6}}$ branes

We investigate $p$ $\overline{\text{D6}}$ branes inside a flux throat that carries $K \times M$ D6 charges with $K$ the 3-form flux quantum and $M$ the Romans mass. We find that within the calculable supergravity regime where $g_s p$ is large, the $\overline{\text{D6}}$ branes annihilate immediately against the fluxes despite the existence of a metastable state at small $p/M$ in the probe approximation. The crucial property that causes this naive conflict with effective field theory is a singularity in the 3-form flux, which we cut off at string scale. Our result explains the absence of regular solutions at finite temperature and suggests there should be a smooth time-dependent solution. We also discuss the qualitative differences between $\overline{\text{D6}}$ branes and $\overline{\text{D3}}$ branes, which makes it a priori not obvious to conclude the same instability for $\overline{\text{D3}}$ branes.


Introduction
A useful way to break supersymmetry in flux backgrounds is to insert branes that preserve different supercharges than the background fluxes and branes. Supersymmetry breaking by combining branes and antibranes is perturbatively unstable since the branes are mobile and can move towards each other and annihilate. The virtue of combining branes with fluxes that have opposite orientation is that there is no direct analog of brane/antibrane annihilation. Instead there can be brane-flux annihilation [1] which proceeds via first nucleating branes out of the fluxes which then annihilate with the antibranes. The question then is whether this type of supersymmetry breaking can be metastable. If so, then it would be of great practical use in constructing string models for dS vacua [2], inflation [3], near-extremal black hole micro-states [4,5] and holographic duals to dynamical supersymmetry-breaking [1,[6][7][8][9][10][11] (and see [12] for related work).
In [1] it was indeed argued that metastable states do exist for p D3 branes in the Klebanov-Strassler throat [13] with three-form fluxes carrying M × K D3 charges, where K and M are respectively the H 3 and F 3 flux quanta. Brane-flux annihilation proceeds via the Myers effect [14]: the p D3 branes polarise into an NS5 brane wrapping a two-cycle inside the 3-cycle (i.e. A-cycle) at the tip of the throat. The NS5 brane carries D3-charge that is a function of its position on the A-cycle. When the NS5 brane pinches off at the opposite side of the 3-cycle it induces M − p D3 charges instead of p D3 charges. Effectively, the motion over the 3-cycle corresponds to brane-flux annihilation: out of the background fluxes M D3 branes have materialised and at the same time K dropped by one unit. The M D3 branes then annihilate with p D3 branes to give a supersymmetric vacuum with M − p D3-branes. Kachru, Pearson and Verlinde (KPV) argued [1] that this supersymmetry breaking is metastable in the limit of small p/M . This limit is the same limit in which the characteristic size of 3-sphere at the tip R 2 S 3 ∼ g S M 2 s is much bigger then the characteristic size of the D3 horizon R 2 D3 ∼ g s p 2 s . Hence one might expect the D3 to only modify the geometry locally. KPV argued the metastability by computing the potential energy for the NS5 brane and finding it has a local minimum at a radius r ∼ (p/M ) √ g s M s . In the past years concerns have been raised in the literature about the validity of their result. First the NS5 potential was computed by using an action obtained from S-dualising the D5 action. The D5 action is valid at weak coupling and hence the obtained NS5 action can only be argued to be valid at strong coupling. An alternative analysis using the non-Abelian D3 action also suffers from the same problem since an S-dual version of the standard D3 action is required. Second, the computation of KPV is at the probe level and backreaction effects should therefore be subleading and tunably small. However many recent works have shown that the backreaction leads to a singularity which is difficult to interpret.
In this paper we study whether backreaction destroys the metastable state of D6 branes. For that we must first establish the metastability of D6-branes at the probe level which has not appeared in the literature before (see however [15]). We then indeed find that backreaction destabilises the D6 branes. We are unable to address the D3-brane stability conclusively but we comment on it.
In section 2 we recall the existing results on the backreaction of antibranes. We then describe the background solution in section 3 in which we place the D6-branes. In section 4 we show that D6-branes give rise to metastable states at the probe level when p/M is small (M the Romans mass, p is the antibrane charge). This state is however washed away when backreaction corrections are added (assuming g s p is large enough to allow for a supergravity description) as shown in section 5. In section 6 we comment on the case of branes of lower dimensionality, and discuss a possible loophole that could allow the interesting case of D3 branes to escape this instability. We conclude in section 7. All explicit computations can be found in the Appendices.

Backreaction effects and instabilities
An explicit study of antibrane supersymmetry breaking is hard due to the lack of explicit solutions in the cases of interest. The best understood solutions at the moment are based on antibranes smeared over the A-cycle which has been most studied for D3 branes in the KS throat [16] (see also [17]). The only case for which a quantitative analysis is possible that does not rely on smearing are D6-branes since those solutions are described by ODE's [18] 1 . The qualitative feature that these studies have shown is the presence of a singularity in the H 3 flux density. This property can be shown to be unavoidable even for localised branes [21,22]. Two prominent interpretations of this singularity have appeared in the literature in the past years.
• Singularities are actually to be expected [23] since near the D3-brane we have an AdS 5 ×S 5 perturbed by three-form fluxes. Analogous, albeit supersymmetric, examples of this type are known to give rise to singularities that are resolved by letting the 3branes polarise to (p, q)-5 branes [24]. For smeared D3-branes there is a growing body of work that indicates that certain expected polarisation channels seem absent [25][26][27] and others lead to tachyonic modes [28][29][30][31]. Analogous results for fully localized branes are not as strong and a recent paper found that polarised D3-branes can be made regular by fixing boundary values of some fields [32]. It still remains to show that a full solution with these boundary values exists.
• The 3-form singularity of Dp solutions has a simple interpretation [33] which can be found in the sign of the divergent (but integrable) charge density F 6−p ∧ H 3 . This sign is opposite to the sign of the antibrane charge. The reason is that the flux is attracted gravitationally and electromagnetically towards the antibranes. If the antibranes are replaced instead with branes the electromagnetic repulsion will exactly counterbalance the gravitational pull [34]. This entails an obvious instability since a clumping of fluxes carrying Dp charges will enhance the brane-flux annihilation. This can be seen at the level of the probe actions describing the Myers effect [27,35,36] or from the point of view of bubble nucleation [36]. If the flux clumps in an unbound way one necessarily crosses a critical value that causes immediate brane-flux decay. If this picture is correct there should exist a smooth time-dependent solution describing the clumping of flux, which reaches a critical value at which the antibranes decay and then the flux clumping should stop and the BPS state should be reached after the closed string radiation decouples. This picture is not inconsistent with the expectation that backreaction is small when p/M is small, but the essential property that makes this possible is that the local backreaction, which can be big, determines stability.
In this paper we demonstrate that the second viewpoint is correct for D6 branes at large g s p but still arbitrarily small p/M .

D6 brane solutions
In order to analyse brane-flux annihilation, we first require a background with fluxes that carry D6 charges without the presence of localized D6 branes. This is possible in massive IIA supergravity as can be seen from the Bianchi identity for F 2 : Here F 2 is the RR 2-form field strength, F 0 is Romans mass and H 3 is the NSNS 3-form field strength. Notice that the term F 0 H 3 appears on the same footing as the D6 source term N δ 6 . Clearly when F 0 = 0 this term acts as a smooth source for 6-brane charge. A general study of backgrounds with these ingredients was carried out in [18]. It includes the "massive D6" brane with flat worldvolume and non-compact transverse space [37] and D6 branes with AdS 7 worldvolume and compact transverse space [18,38]. We start with brane-flux decay in the first example but before doing so we emphasize why the D6-brane is special.

Why the D6 is special
D6 branes stand out against antibranes of other dimensionality for their simplicity. The most important one arises in the description of the backreaction of D6 branes, which is described by ODE's [18,33,38] without having to smear the branes. A second feature-crucial for this paper-is that the annihilation of the H 3 -flux does not proceed via a polarisation into a higher-dimensional object. All Dp branes with p < 6 polarise into an NS5 branes [15]. D6 branes instead polarise into KK5 branes, which are smaller in dimension. However the KK5 assumes a circular isometry transverse to its worldvolume, one can therefore think of KK5s as 5-branes that are smeared over a circle. This circular isometry direction will live inside the D6 worldvolume such that the KK5 brane looks like a 6-brane and its backreaction is identical to that of the D6 brane since it carries D6 charge and tension. This is related to the fact that D6 solution is the only antibrane solution for which the singularity in H 3 is consistent with the singularity of the metric. In other words: the backreaction of the diverging H 3 flux does not destroy the local D6-metric, whereas it would for p < 6. For instance for p = 3 one finds that the H 2 3 scalar blows up near the horizon, but the horizon of a 3-brane should be smooth AdS 5 × S 5 . Hence the 3-form fluxes necessarily destroy the local AdS 5 throat, which is well known from the Polchinski-Strassler (PS) model [24]. In the PS model the singularity at the would-be horizon can be turned into a physical singularity corresponding to (p, q) 5-branes. Something similar can be expected for the D3 solution and indeed reference [32] has shown that a local 5-brane background with good singularities it is at least not inconsistent with having ISD fluxes at the UV and a non-zero (generalised) ADM mass.

D6 branes in flat space
We start with a flux background geometry with a flat D6 worldvolume. The metric and dilaton 2 is that of a D6 brane However to carry the D6 charge in fluxes we include non-trivial H 3 flux and Romans mass given by Here˜ 3 is the unwarped Hodge operator on the transverse space such that˜ 3 1 = r 2 dr ∧Ω 2 = vol 3 and Ω 2 is the volume form on the unit 2-sphere. We denote the line element of 7D Minkowski space by ds 2 7 . Further on, once we discuss the KK5 brane it will turn out to be useful to pick out a specific direction ψ that plays the role of the isometry direction of the KK5. We then write ds 2 7 = ds 2 6 + dψ 2 with ψ a circle direction. The fluxes obey a Hodge-duality relation: where the Hodge-star includes warp factors. This condition is identical to the famous Imaginary Self-Dual (ISD) condition for fractional D3 brane backgrounds after performing three T-dualities along the Minkowski directions. One can furthermore verify that (3.4) ensures a no-force condition for probe D6 branes. D6 branes on the other hand do feel a force. The Bianchi identity for F 2 (with N D6-brane sources located at the origin r = 0) implies˜ 2 We use string frame throughout.
to which a spherically symmetric solution is where N is the number of D6-branes sitting at r = 0. We will put N to zero for now and consider a background without any explicit D6 sources. This solution is then regular around r = 0. The solution has a peculiar singularity for In order to examine the local behaviour of the singularity we expand around it: and find This is precisely the local singularity structure of an O6-plane (see appendix A). A discussion of this singularity can be found in [39] where it was argued that strings are well behaved in the singular background. Our interpretation of the singularity as an O6-plane confirms that analysis. We will not go into more details on the global analysis of the solution since the computation we perform is local to r = 0. This solution is not supersymmetric [37,40]. A supersymmetric solutions with the same ingredients do not preserve the Poincaré symmetries of the D6 worldvolume. The flux background is however extremal in the following sense • The expression of the metric and form fields is the one typical for extremal p-branes.
• There is a no-force condition for inserting D6 branes into the background whereas D6 branes are pulled towards the tip of the throat.
• It is T-dual to supersymmetry breaking using (3,0)-fluxes in fractional D3 backgrounds which is rather well understood and does not influence any of the details of brane-flux decay in that context [1].

Inserting D6 branes
To describe the same throat geometry but with a stack of D6's at the tip, we use the following general Ansatz preserving the worldvolume symmetries (possibly broken at finite temperature) and the transverse rotational symmetries [33] (in string frame): From combining the form field equations of motion one can deduce that Solutions of the above form will break the ISD condition (3.4) whenever λ = 1 and such backgrounds exert non-zero forces on probe D6 branes. The extremal flux background discussed earlier is recovered for the choices λ = 1, e −4A = e 4B = S and e φ = g s S −3/4 . It was shown in [18,33,41] that insisting that the metric has an D6 singularity at r = 0 and asymptotes to the ISD solution, we find that α goes to a non-zero constant near the D6-horizon 3 which we denote α 0 . This implies: since the combination e −7A+φ−f blows up at zero or non-zero temperature. At non-zero temperature this is solely due to e f going to zero near the horizon whereas the other fields remain finite. Whereas at zero temperature e f = 1 but e −7A+φ is combination that necessarily diverges near a 6-brane. This singularity is not a coordinate artefact since it appears in scalar quantities like the H 3 -flux density: One readily verifies that the singularity in the associated charge density M H is still integrable, since H 3 ∼ r −1 Vol 3 = rdrdΩ 2 near the source at r = 0. Another peculiar feature is the fact that the backreaction of the fluxes does not destroy the local 6-brane geometry (at zero temperature). This is different for the Dp solutions with p < 6 as explained in the Introduction.
The sign of the singularity in λ is such that it corresponds to a diverging D6 charge density dissolved in the fluxes as illustrated in figure 1.
Given the interpretation of this singularity as a diverging charge density one should be worried about the stability of this background against brane-flux annihilation. Heuristically one expects a large charge density dissolved in fluxes to increase the probability of localised and mobile D6 branes materialising out of the flux cloud that subsequently annihilate against the D6 branes. This is the topic of section 4.

Warped AdS 7 with D6 branes
If one allows the D6 worldvolume to be AdS 7 instead of Mink 7 the space transversal to the brane can be made compact and conformal to S 3 [34]. The resulting solution was described qualitatively in [18,33]. A much deeper understanding of the solution as well as a better numerical control was reached in [38], where it was understood that the solution can even be supersymmetric, rectifying some statements in [33] 4 . Consequently the dual SCFT's were uncovered in [44] (see also [45]).
What counts for this paper, all details aside, is that there are various AdS 7 solutions. They can be classified according to the way the six-brane charges are divided over the compact manifold. The D6 charges are either due to explicit D6 branes sitting at the poles of the S 3 or due to spherical D8 branes wrapping contractible S 2 ' inside the S 3 as in figure 2. Those D8 branes can induce D6 charges when they have the right kind of worldvolume fluxes on them.
The presence of D8 branes carrying the D6 charges is a crucial difference with the Minkowski solutions treated above. In the latter case it can be shown that the D8 branes would never sit at a stable position and pinch off to become pure D6 branes [25]. Only in AdS 7 can they reach a stable position [42]. Crucially, when the charges are carried by spherical D8 branes they do not cause flux-clumping singularities [25].
The role of the D8 brane can be confusing so we spend a few more words on this. Fluxes can decay against the p D6-charges in two ways [15]: 1) either by having a D8 brane polarising and move all the way over the "B"-cycle which decreases the Romans mass or 2) by having a KK5 brane moving inside the D6 worldvolume which decreases the H 3 -flux. In this paper we focus on the second channel the D8 will be stuck at a fully stable position in D8 × D6 the case of the AdS 7 worldvolume or cannot polarise when the worldvolume is Minkowski [25]. The effect on the KK5 channel should then be such that the KK5 polarisation does not occur at all for AdS worldvolume 5 and, as we argue in this paper, the KK5 will polarise and not even come to stop for flat worldvolumes. This drastic qualitative difference is caused by having the D8 brane polarised in one case and not in the other [27]. Because if the D8 polarises it removes the 3-form singularity, which lies at the heart of the immediate brane-flux decay.
4 Brane-flux annihilation at probe level

Brane-flux decay and T-duality
This implies also that we cannot have a single KK5-brane without introducing non-trivial KK-charge. Since we are not interested in KK-charge we introduce a pair of KK5 and KK5 that share a NUT direction ψ but are otherwise separated on that circle. We illustrate the D6 decay process in figure 4.1: The spacetime domain wall that mediates the decay of H-flux and D6-charge is a NS5-brane wrapping the S 1 but traveling radially in Mink 6 .

The probe potential
To compute the probe potential for KK5 branes in massive IIA we could use the actions derived in [46]. It is however technically easier to work in the T-dual frame, and use that the probe potentials are invariant under T-duality [15]. The T-dual setup involves a pair of probe NS5-branes, with opposite charge in the geometry Mink 6 × S 1 × M 3 where the branes are localised on the circle and on M 3 . The pair of NS5-branes induce D5 charge p D6 between them as explained below. This calculation was already sketched out in [15] but we will go through the details here for completeness. We start by writing down the background T-dual to (3.2) along the ψ-direction, ds 2 = S −1/2 ds 2 6 + S 1/2 dψ 2 + dr 2 + r 2 dΩ 2 2 , The function S takes takes the same form as above After T-duality the coordinate ψ remain periodic with ψ ∼ ψ + s . Before continuing on to the probe computation we review how a NS5-NS5 pair can induce a D5-brane charge. Let the NS5-brane be located at ψ = ψ 0 and the NS5-brane be located at ψ = −ψ 0 . The probe action relevant for these branes is The upper sign is used for the NS5-brane and the lower is for the antibrane. The field G ± is a worldvolume field where p is a constant and C 0 is the local gauge potential for F 1 . The D5 charge induced by the brane pair can be inferred from the sum of the above WZ couplings, where we have used C 0 = (M/ s )ψ and that C 6 is independent of ψ. The induced D5-charge is therefore controlled by the combination p − 2M ψ 0 . Since ∆ψ ≡ 2ψ 0 is the separation of the brane pair we see that when pulled together to the point ψ = 0 they induce the charge of p D5 whereas on the other pole ψ 0 = s /2 the charge is p − M . This shows that not only does the brane pair carry D5-brane charge, but the charge depends on the relative seperation of the NS5-NS5 pair. The computation of the probe potential is spelled out in Appendices B and C. Including the interaction term in the full probe potential we find: The potential is displayed in figure 4.2 and shows that the hight of the barrier is controlled by M g s and becomes smaller as M is increased. One can verify that there is a metastable state for M g s large whenever p M < 0.25 . 5 Including backreaction

Effective field theory approaches
A probe result should be thought off as a leading-order contribution in a perturbative series. If the series would be an expansion in p/M one can trust the leading order result in the limit of vanishing p/M . For the special case p = 1 some initial steps towards a direct effective field theory approach were taken in [47] based on [48]. An alternative description (see for instance [24]) is the one we take in this paper where p/M is small but g s p is nevertheless big. An effective field theory description of any kind will have to deal with the singularities typical to brane solutions, such as the singular warpfactors and diverging gauge fields sourced by the brane. Just as in standard electrodynamics one does not expect diverging field strengths to cause any harm as this corresponds to an infinite self-energy that is effectively subtracted off in the probe action. For diverging warpfactor, the same story applies.
The diverging H 3 field on the other hand is not as clear since it is not sourced by the 6-brane directly. It is tempting to think of it as the diverging field strength corresponding to the NS5/D6 boundstate since NS5-branes do source singular H 3 flux 6 . However it has been found in [32,49] that this cannot be, at least not within the Ansatz presented in this paper, since then the divergence in H 3 is not of the right kind to correspond to a NS5 source. As mentioned earlier the singularity really corresponds to a divergence in the flux clumping, which effectively makes the D6 charge dissolved in flux diverge. Our approach to dealing with this singularity consists of pealing off a small amount n p of branes and consider them to be a probes in the backround of p backreacting branes. In that way the singularities in the warpfactor and the gauge fields, generated by the backreacting stack (see picture 5, cancel each other since the singular contribution in the WZ term is of the same kind as the singular contribution in the DBI term. So this approach uses that string theory branes, even when non-supersymmetric, are locally behaving like supersymmetric branes where tension equals charge. If the probe is unstable one then takes it as strong evidence that the stack is unstable. This second approach effectively eliminates the diverging contributions of the geometry and the gauge field that couples to the brane, but in our case, it does not take care of the singularity in H 3 (and hence λ). Since the singularity in λ occurs at a place where the space-time metric is not to be trusted due to high curvature one has to be careful to conclude that the infinite flux-clumping destabilizes the system due to direct brane-flux decay. We therefore cut off the solution a string-length away from the singularities and investigate how large the flux-clumping is. In this section we find that the flux-clumping λ is proportional to g s p. We also show that for large g s p the system is destabilised independent of the ratio p/M being very small.
Another systematic way to approach the effective field theory would be to consider high-order corrections in the blackfold approach [50,51], which we hope to adress in the 6 Similarly for KK5-branes in massive IIA.

Flux-clumping at the cut-off
To evaluate the flux-clumping at a finite distance r = s from the antibranes, we need to evaluate λ there. Here we work directly with the D6 solution but later when we consider the T-dual background, the expression for λ remains the same. Near a stack of extremal antibranes we have e −4A = e 4B = h 6 , e φ = g s h Since λ = αe φ−7A , we can calculate λ once we know the value of α near the cut-of. The latter we obtain from the Smarr-like relations found in [21,32,49] (see also [52]), that relate the energy E (above the supersymmetric vacuum), to the values of the gauge fields at the sources: such that E = µ 6 pα(r = 0). To find α(0) we therefore need to know E. Following the reasoning of [6] we have (the background has e A = 1 at the tip) From the Smarr relation (5.3) we then find Concerning λ we have λ(r = l s ) = g 1/4 s π 2 pα(r = l s ). We take α(l s ) = α(0) and find λ ∼ g s p, consistent with the estimates in [47]. One can only consistently take α(l s ) = α(0) if α (0) is sufficiently small but this is easy to see. At the brane source we must have F 2 = Q Ω 2 and this provides an expression for α (0). Near the source we have we have: such that

The corrected probe result at large p
We now compute the probe potential for n D6-branes near the tip geometry with the backreacting of p D6-branes included. By cutting of the divergent flux-clumping by the value found above: λ ∼ g s p we find a finite and well-defined probe potential. To carry out this computation we need the solution T-dual to the D6 solution. A straightforward computation gives the following metric for the smeared D5-branes at r = 0: Where as before, the equations of motion demand α = λe 7A−φ and the fields have the standard D-brane singularities describing smeared 5-branes at r → 0. We now follow the exact same procedure of Appendix B to compute the probe potential by placing a single NS5/NS5-pair in the above background. This calculation can be found in Appendix D. The result is a run-away potential: as shown in figure 6. We note that the interaction term has a different dependence on the dilaton and warpfactor such that the warpfactor and dilaton do not simply factor to the front. The factor e 3A−φ in front of the interaction term reduces to g −1 s close to the brane, consistent with the uncorrected probe potential.
Note that this potential receives large corrections as ψ grows. Essentially, corrections are subleading until the value of ψ is such that the drop in energy is twice the tension of the n D6-branes. The decay of the branes causes at the same time a materilisation of M − n D6-branes and they will quickly dominate over the p D6-branes such that the background geometry cannot be trusted. It is satisfactory that at the value for ψ where the induced 6-brane charge in the KK5/KK5 pair becomes zero, the drop in energy is two times the tension of the n D6-branes. For a stable configuration we would expect that by dialling down n/M and p/M a metastable state would appear for the n mobile antibranes. This is however not the case which indicates that the whole stack of branes may be unstable. Of course when p keeps decreasing and eventually reaches small values for which our analysis is not valid. This is the limit of our approach and a different analysis must be carried out to determine the stability of small number of antibranes.
6 Dp-branes with p < 6 In this section we discuss how simple dimensional analysis can be employed to study flux clumping around antibranes of different dimensions.
The flux-clumping for unpolarised Dp-branes can be inferred either via the supergravity techniques of [21,22] or the brane effective field theory techniques of [47]. Both give the same result, namely with r a local radial variable. Hence the flux clumping at the cut-off scale r = s always equals λ c ∼ g s p .
In the supergravity regime this is too large to ensure stability of the probe. Now the question arises how λ c changes when the branes polarise via the Myers effect as predicted by KPV [1]. The essential observation of this paper is that λ c does not change for D6-branes polarising into a KK5/KK5 pair, because the pair has the same dimensionality as the original branes. We expect similar conclusion for D5-branes and hence they should be unstable as well in the supergravity regime. However when p < 5 the Dp charge spreads out on a spherical NS5-brane of radius R such that one expects, from dimensional analysis, 3) The largest possible value for R is the size of the full A-cycle, which we take proportional to R 2 max ∼ g s M 2 s like in the KS throat. For D4-branes we then have that this minimal value of λ c is still too large. However for D3-branes the situation improves. The radius at which the clumping is only of order 1 is found to be: This is far off the radius one finds from the probe computation, which is linear in p [1]: In fact at the probe radius the clumping can be estimated to be too high and should destabilise the brane. This already shows that the probe computation at best only gives qualitative information but certainly not quantitative results, very much in contrast with supersymmetric vacua [53]. Interestingly in reference [32] a √ p dependence of the polarisation radius was inferred on very different grounds: namely by using the aforementioned boundary condition for polarised branes that can regularise λ without the need to cut-off. This potential match seems interesting and we leave it for future research.

Conclusions
We now summarise our results and then give an outlook on the metastability of general antibranes.

Summary
We have shown that at probe level D6-branes experience a classical barrier against annihilation with fluxes of the opposite charge on the condition that the Romans mass quantum M is much bigger then the antibrane charge p, thereby extending the results of [15]. This condition is qualitatively T-dual to the condition for the metastability of D3-branes [1], where Romans mass gets replaced with RR 3-form flux. We have then shown that backreaction effects washes away the classical barrier, confirming the arguments of [35] and [36].
The reason this could be firmly shown rests on 2 facts: 1. D6-branes are special in the sense that they brane-flux decay via mutating into a KK5/KK5 pair carrying D6-charge. The pair has effectively the same dimensionality and backreaction as localised D6-branes.
2. The backreaction of localised D6-branes is described by ODE's. Near the source the backreacted geometries are T-dual to Dk-branes with k < 6 smeared over the k − 6 directions inside the A-cycle filled with RR-flux. For such branes there was already strong evidence that they are unstable [27].
The vanishing of the classical barrier upon backreaction is due to the attraction of the fluxes towards the D6-branes, something that is not taken into account in the probe approximation. This increase in flux diverges close to the antibranes at the classical level. Hence we followed the procedure of [47,48], which amounts to cutting of the singularity at string scale. The main observation is that the cut-off value for the flux density is still too high and causes immediate brane-flux decay. Effectively the increased flux density causes an increase in the probability that actual D6-branes materialise out of the flux cloud which subsequently annihilate with the D6-branes. Our result explains the lack of smooth D6 solutions with finite temperature [41].
The reason for this apparent failure of the probe approximation could be due to the fact that the probe calculation is outside of the regime of validity of the NS5 (or KK5) brane action.
We have also investigated whether the same picture could still be valid for Dk-branes with k < 6. Using dimensional analysis, similar to [47] we argue that the same physics should hold for D4 and D5 branes but that D3-branes cannot be shown to be unstable using the method described in this paper. We do however find, that if a metastable state exists for D3-branes, the radius of the spherical NS5-brane the D3-branes expand into must scale differently than predicted by the probe calculation [1].

Outlook
It has been argued that the instability can be circumvented by considering small brane charges (e. g. p = 1) [47,54]. For small brane charges the cut-off value of the flux clumping is small and furthermore there is no polarisation potential that can be computed within the supergravity regime. Given that flux clumping is not severe for small p instability cannot be argued solely based on singular flux density. Instead brane effective field theory should be employed to argue for stability [47]. Stability would most likely imply a finite temperature solution with small p in supergravity. However for D6-branes the absence of a smooth finite temperature solution was demonstrated in [41]. It could of course be that the metastable state at p = 1 has a very small gap such that the minimal temperature needed for a classical horizon already destabilises the vacuum [54]. This potential loophole might become visible by computing finite temperature corrections to probe actions, something that can be done in the blackfold approach [50,51].
Despite the possible room for the existence of metastable D3-branes, we want to be open-minded and contemplate about the possibility that also D3-branes are unstable against the flux-clumping effect [35,36]. As long as numerical evidence showing that the boundary conditions for regular solutions described in [32] and [49] can be taken, does not exist, the option of an instability has to be taken seriously. There exists a heuristic interpretation of the "Smarr relation" for flux throats that supports this interpretation.
The generalised ADM mass M ADM of a flux throat with sources, measures the energy above the supersymmetric vacuum. It was shown that M ADM obeys a Smarr-like relation [32,49,55] where Q M (Q D ) stands for monopole (dipole) charge and Φ M (Φ D ) denotes the gauge field sources by the monopole (dipole) charge evaluated at the horizon in a specific gauge. The exact expression, on the nose, corresponds to the on-shell Wess-Zumino action for the would-be polarised brane (see [21] for a similar derivation using on-shell brane actions). This almost fits the standard expectation that the energy of the flux throat equals twice the antibrane tension since where the DBI-term and WZ-term exactly equal each other in the probe approximation. In the backreacted case it seems all energy is therefore carried by the WZ term alone. This comes about since the warpfactor redshifts the on-shell DBI to zero, and alternatively this can be seen as a flux clumping effect, which makes the WZ dominate over the DBI. If this interpretation is correct then a metastable state can never exist since metastability truly requires a competition between the DBI and WZ term. We do want to point out that this is not a proof since our interpretation of the Smarr relation as on-shell probe potential is heuristic. Finally, we have not touched upon the 3-form singularities in those (supersymmetric) AdS 7 solutions of [38] with pure D6-branes (ie not all 6-brane charges come from spherical D8-branes). The logic of this paper suggests that those AdS 7 backgrounds, despite supersymmetry, are unstable and not suitable candidates for holographic descriptions of certain 6D CFT's. The typical 6D CFT's classified by these massive IIA solutions are however described mainly in terms of spherical 8-branes [44], but it would be interesting to understand this better.

A O6 planes in flat space
The O6-plane metric is given by We expand this metric around the critical radius The result is

B Computing the probe potential
We now turn to calculating the probe potential. In order to obtain the expression for B 6 we make use of the Bianchi identity d H 7 − F 1 C 6 = 0, from which we infer The RR gauge potential C 6 can be calculated from the expression for F 7 = dC 6 as where the constant c explicitly shows the gauge freedom in the definition of C 6 . Using this we find The WZ terms in (4.3) can be evaluated to We could worry about the last term in the above expression which seems to break the gauge invariance of the potential. It is due to the fact that δB 6 = −C 0 δC 6 and The WZ Lagrangian is therefore not gauge invariant as we have written it down. One can introduce an extra worldvolume field to cure this problem. We will not pursue this direction since the term in question represents an overall shift in the potential and can be used to set our zero-point of the potential.
Turning to the DBI action, we simply evaluate Using the expression (4.2) for S and evaluating at the tip r = 0 we obtain the full probe action where we used that v g s M and disregarded terms of order (v/M g s ) 2 . Later we will add an interaction term which is of order v/M g s and is therefore leading order with respect to the terms we ignored. So far we have calculated the probe action of the NS5-brane located at ψ 0 , we also have to add to this the action of the NS5-brane located at −ψ 0 . This brane has opposite charge and opposite sign of p as explained above. The combined potential is then the same as above, but with an overall factor of 2: where ∆ψ = 2ψ 0 is the separation of the two branes. Here we have also fixed the zero-point energy by choosing c = 0 such that at ψ = 1/2 the potential vanishes.
Notice that for ∆ψ ≥ s p/M the potential vanishes, this is because we have not taken into account the electromagnetic and gravitational interaction between the two branes. For a brane-antibrane pair this is a subleading effect but is obviously important to obtain the full potential. We can compute the interaction potential between the two probe NS5-branes by placing one NS5-brane in the geometry and let it backreact. This computation is spelled out in the next Appendix C.

C The force between an NS5/KK5 pair
We can compute the interaction potential between the two probe NS5-branes by placing one NS5-brane in the geometry, let it backreact, and evaluate that backreaction in the probe brane action of the other NS5-brane. Hence, to find the force between an NS5/NS5 pair amounts to solving a differential equation equation for an harmonic function for a NS5-brane which we call K, in our throat geometry. We are only interested in the probe potential and thereby the interaction term close to r = 0 which simplifies our task immensely. The background metric at r = 0 simplifies to a cylinder: ds 2 → v −1 ds 2 6 + v(dψ 2 + dr 2 + r 2 dΩ 2 2 ) . (C.1) A single NS5-brane in this background deforms the metric and fields in the standard way: The form equations of motion reduce to where we have put the NS5-brane at the location r = 0 and ∆ψ measures the distance from the brane along the isometry direction. Furthermore we have put the number of NS5-branes to one and used that in our conventions µ 5 = 2π. In the region around r = 0 we have B 6 = π v 2 g 2 s (1 − cos(2π∆ψ/ s )) . (C.8) We obtain the interaction term by inserting this into the action of the other NS5-brane so that we get −µ 5 e −2φ √ −g + B 6 = −2 πµ 5 v 2 g 2 s (1 − cos(2π∆ψ/ s ))vol 6 . (C.9) The potential desribing the force starts rising until it reaches a maximum and then goes back in an identical matter. In reference [15] this potential was approximated to be piecewise linear, which is qualitatively similar. The reason the potential grows with ψ instead of going down is due to the throat geometry; the brane/antibrane forces are effectively confined to the circle dimension such that it mimics electrodynamics in 1 + 1 dimensions.

D Computing the corrected probe potential
To evaluate the probe potential of a single NS5/NS5-pair in the background described in section 5.3 we require the gauge field expressions: To calculate the interaction term we again follow the same procedure as in Appendix C. The metric is approximated by the cylinder metric evaluated at a given radius r = s ds 2 = e 2A ds 2 6 + e 2B e −2B−2A dψ 2 + dr 2 + r 2 dΩ 2 2 , (D.7) We rescale r to absorb the factors of e B−A asr = e B−A r. Using the method described in Appendix C we find the interaction potential (D.9) where we threw away an additive constant. If we were to taker to zero we would find that λ blows up in exactly the right way, see (3.11), to compensate for the vanishing prefactor, and hence we obtain finite result for the terms that include λ. All other terms, including the interaction term, vanish in this limit, which physically can be understood as an effect due to redshift. Since we cannot trust these expressions below the string scale, we need to evaluate them atr = s to investagate whether the redshift is large enough within the regime under our control. In paricular, we find that λ at its cut-off value is λ = g s p, which, indeed, is large.