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An evaporating landscape? Possible issues with the KKLT scenario

By Dr Thomas Van Riet, K.U. Leuven

What is this blog post about?

In 2003, in a seminal paper by Kachru, Kallosh, Linde and Trivedi (KKLT) (2000+ cites!), a scenario for constructing a landscape of de Sitter vacua in string theory with small cosmological constant was found. This paper was (and is) conceived as the first evidence that the string theory landscape contains a tremendous amount of de Sitter vacua (not just anti-de Sitter vacua) which could account for the observed dark energy.

The importance of this discovery cannot be underestimated since it fundamentally changed the way we think about how a fundamental, UV-complete theory of all interactions addresses apparent fine-tuning and naturalness problems we are faced with in high energy physics and cosmology. It changed the way we think string theory makes predictions about the low-energy world that we observe.

It is fair to say that, since the KKLT paper, the multiverse scenario and all of its related emotions have been discussed at full intensity, even been taken up by the media and it has sparked some (unsuccessful) attempts to classify string theory as non-scientific.

In this post I briefly outline the KKLT scenario and highlight certain aspects that are not often described in reviews but are crucial to the construction. Secondly I describe research done since 2009 that sheds doubts on the consistency of the KKLT scenario. I have tried to be as unbiased as possible. But near the end of this post I have taken the freedom to give a personal view on the matter.

The KKLT construction

The main problem of string phenomenology at the time of the KKLT paper was the so-called moduli-stabilisation problem. The string theory vacua that were constructed before the flux-revolution were vacua that, at the classical level, contained hundreds of massless scalars. Massless scalars are a problem for many reasons that I will not go into. Let us stick to the observed fact that they are not there. Obviously quantum corrections will induce a mass, but the expected masses would still be too low to be consistent with observations and various issues in cosmology. Hence we needed to get rid of the massless scalars. This is where fluxes come into the story since they provide a classical mass to many (but typically not all) moduli.

The above argument that masses due to quantum corrections are too low is not entirely solid. What is really the problem is that vacua supported solely by quantum corrections are not calculable. This is called the Dine-Seiberg problem and it roughly goes as follows: if quantum corrections are strong enough to create a meta-stable vacuum we necessarily are in the strong coupling regime and hence out of computational control. Fluxes evade the argument because they induce a classical piece of energy that can stabilize the coupling at a small value. Fluxes are used mainly as a tool for computational control, to stay within the supergravity approximation.

Step 1: fluxes and orientifolds

Step 1 in the KKLT scenario is to start from the classical IIB solution often referred to as GKP (1400+ cites), (see also this paper). What Giddings, Kachru and Polchinski did was to construct compactifications of IIB string theory (in the supergravity limit) down to 4-dimensional Minkowski space using fluxes and orientifolds. Orientifolds are specific boundary conditions for strings that are different from Dirichlet boundary conditions (which would be D-branes). The only thing that is required for understanding this post is to know that orientifolds are like D-branes but with negative tension and negative charge (anti D-brane charge). GKP understood that Minkowski solutions (SUSY and non-SUSY) can be build from balancing the negative energy of the orientifolds \(T_{{\rm O}p}\) against the positive energy of the 3-form fluxes \(F_3\) and \(H_3\):\[

V = H_3^2 + F_3^2 + T_{{\rm O}p} = 0

\] This scalar potential \(V\) is such that it does not depend on the sizes of the compact dimensions. Those sizes are then perceived as massless scalar fields in four dimensions. Many other moduli directions have gained a mass due to the fluxes and all those masses are positive such that the Minkowski space is classically stable.

The 3-form fluxes \(H_3\) and \(F_3\) carry D3 brane charges, as can be verified from the Bianchi identity for the five-form field strength \(F_5\)\[

\dd F_5 = H_3 \wedge F_3 + Q_3\delta

\] The delta-function on the right represent the D3/O3 branes that are really localised charge densities (points) in the internal dimensions, whereas the fluxes correspond to a smooth, spread out, charge distribution. Gauss' law tells us that a compact space cannot carry any charge and consequently the charges in the fluxes have opposite sign to the charges in the localised sources.

I want to stress the physics in equation the Bianchi identity. To a large extend one can think of the 3-form fluxes as a smeared configuration of actual D3 branes. Not only do they induce D3 charge, they also back-react on the metric because of their positive energy-momentum. We will see below that this is more than an analogy: the fluxes can even materialize into actual D3 branes.

This flux configuration is ‟BPS″, in the sense that various ingredients exert no force on each other: the orientifolds have negative tension such that the gravitational repulsion between fluxes and orientifolds exactly cancels the Coulomb attraction. This will become an issue once we insert SUSY-breaking anti-branes (see below).

Step 2: Quantum corrections

One of the major breakthroughs of the KKLT paper (which I am not criticizing here) is a rather explicit realization of how the aforementioned quantum corrections stabilize all scalar fields in a stable Anti-de Sitter minimum that is furthermore SUSY. As expected quantum corrections do give a mass to those scalar fields that were left massless at the classical level in the GKP solution. From that point of view it was not a surprise. The surprise was the simplicity, the level of explicitness, and most important, the fact that the quantum stabilization can be done in a regime where you can argue that other quantum corrections will not mess up the vacuum. Much of the original classical supergravity background is preserved by the quantum corrections since the stabilization occurs at weak coupling and large volume. Both coupling and volume are dynamical fields that need to be stabilized at self-consistent values, meaning small coupling and large (in string units) volume of the internal space. If this were not the case than one would be too far outside the classical regime for this quantum perturbation to be leading order.

So what KKLT showed is exactly how the Dine-Seiberg problem can be circumvented using fluxes. But, in my opinion, something even more important was done at this step in the KKLT paper. Prior to KKLT one could not have claimed on solid grounds that string theory allows solutions that are perceived to an observer as four-dimensional. Probably the most crude phenomenological demand on a string theory vacuum remained questionable. Of course flux compactifications were known, for example the celebrated Freund-Rubin vacua like \(AdS_5\times S^5\) which were crucial for developing holography. But such vacua are not lower-dimensional in any phenomenological way. If we were to throw you inside the \(AdS_5\times S^5\) you would not see a five-dimensional space, but you would observe all ten dimensions.

KKLT had thus found the first vacua with all moduli fixed that have a Kaluza-Klein scale that is hierarchically smaller than the length-scale of the AdS vacuum. In other words, the cosmological constant in KKLT is really tiny.

But the cosmological constant was negative and the vacuum of KKLT was SUSY. This is where KKLT came with the second, and most vulnerable, insight of their paper: the anti-brane uplifting.

Step 3: Uplifting with anti-D3 branes

Let us go back to the Bianchi identity equation and the physics it entails. If one adds D3 branes to the KKLT background the cosmological constant does not change and SUSY remains unbroken. The reason is that D3 branes are both BPS with respect to the fluxes and the orientifold planes. Intuitively this is again clear from the no-force condition. D3 branes repel orientifolds gravitationally as strong as they attract them "electromagnetically" and vice versa for the fluxes (recall that the fluxes can be seen as a smooth D3 distribution). This also implies that D3 branes can be put at any position of the manifold without changing the vacuum energy: the energy in the tension of the branes gets cancelled by the decrease in fluxes required to cancel the tadpole condition (Gauss' law).

Anti-D3 branes instead break SUSY. Heuristically that is straightforward since the no-force condition is violated. The anti-D3 branes can be drawn towards the non-dynamical O-planes without harm since they cannot annihilate with each other. The fluxes, however, are another story that I will get to shortly. The energy added by the anti-branes is twice the anti-brane tension \(T_{\overline{D3}}\): the gain in energy due to the addition of fluxes, required to cancel off the extra anti-D3 charges, equals the tension of the anti-brane. Hence we get\[

V_{\rm NEW} = V_{\rm SUSY} + 2 T_{\overline{D3}}

\] At first it seems that this new potential can never have a de Sitter critical point since \(T_{\overline{D3}}\) is of the order of the string scale (which is a huge amount of energy) whereas \(V_{\rm SUSY}\) was supposed to be a very tiny cosmological constant. One can verify that the potential has a runaway structure towards infinite volume. What comes to the rescue is space-time warping. Mathematically warping means that the space-time metric has the following form\[

\dd s_{10}^2 = e^{2A} \dd s_4^2 + \dd s_6^2

\] where \(\dd s_4^2\) is the metric of four-dimensional space, \(\dd s_6^2\) the metric on the compact dimensions (conformal Calabi-Yau, in case you care) and \(\exp(2A)\) is the warp-factor, a function that depends on the internal coordinates. A generic compactification contains warped throats, regions of space where the function \(\exp(A)\) can become exponentially small. This is often depicted using phallus-like pictures of warped Calabi-Yau spaces, such as the one below (taken from the KPV paper (I will come to KPV in a minute)):

Consider some localized object with some non-zero energy, then that energy is significantly red-shifted in regions of high warping. For anti-branes the tension gets the following redshift factor\[

\exp(4A) T_{\overline{D3}}.

\] This can bring a string scale energy all the way down to the lowest energy scales in nature. The beauty of this idea is that this redshift occurs dynamically; an anti-brane feels literally a force towards that region since that is where its energy is minimized. So this redshift effect seems completely natural, one just needs a warped throat.

The KKLT scenario then continues by observing that with a tunable warping, a new critical point in the potential arises that is a meta-stable de Sitter vacuum as shown in the picture below.

This was verified by KKLT explicitly using a Calabi-Yau with a single Kähler modulus .

The reason for the name uplifting then becomes obvious; near the critical point of the potential it indeed seems as if the potential is lifted with a constant value to a de Sitter value. This lifting did not happen with a constant value but the dependence of the uplift term on the Kähler modulus is practically constant when compared to the sharp SUSY part of the potential.

I am glossing over many issues, such as the stability of the other directions, but all of this seems under control (the arguments are based on a parametric separation between the complex structure moduli masses and the masses of the Kähler moduli).

The KKLT scrutiny

The issues with the KKLT scenario that have been discussed in the last five years have to do with back-reaction. As mentioned earlier, the no-force condition becomes violated once we insert the anti-D3 branes. Given the physical interpretation of the 3-form fluxes as a cloud of D3 branes, you can guess what the qualitative behavior of the back-reaction is: the fluxes are drawn gravitationally and electromagnetically towards the anti-branes, leading to a local increase of the 3-form flux density near the anti-brane.

Although the above interpretation was not given, this effect was first found in 2009 independently by Bena, Grana and Halmagyi in Saclay (France) and by McGuirk, Shiu and Sumitomo in Madison (Wisconsin, USA). These authors constructed the supergravity solution that describes a back-reacting anti-brane. Clearly this is an impossible job, were it not for three simplifying assumptions:

  • They put the anti-brane inside the non-compact warped Klebanov-Strassler throat since that is the canonical example of a throat in which computations are doable. This geometry consists of a radial coordinate measuring the distance from the tip and five angles that span the manifold which is topologically \(S^2\times S^3\). The non-compactness implies that we can circumvent the use of the quantum corrections of KKLT to have a space-time solution in the first place. Non-compact geometries work differently from compact ones. For example, the energy of the space-time (ADM mass) does not need to effect the cosmological constant of the 4D part of the metric. Roughly, this is because there is no volume modulus that needs to be stabilized. In the end one should ‟glue″ the KS throat, at large distance from the tip, to a compact Calabi-Yau orientifold.

  • The second simplification was to smear the anti-D3 branes over the tip of the throat. This means that the solution describes anti-D3's homogeneously distributed over the tip. In practice this implies that the supergravity equations of motion become a (large) set of coupled ODE's.

  • These two papers solved the ODE's approximately: They treated the anti-brane SUSY breaking as small and expanded the solution in terms of a SUSY-breaking parameter, keeping the first terms in the expansion.
Regardless of these assumptions it was an impressive task to solve the ODE's. In this task the Saclay paper was the more careful one in connecting the solution at small radius to the solution at large radius. In any case these two papers found the same result, which was unexpected at the time: The 3-form flux density became divergent at the tip of the throat. More precisely, the following scalar quantity diverges at the tip:\[

H_3^2 \to \infty.

\] (I am ignoring the string coupling in all equations.) Diverging fluxes near brane sources are rather mundane (a classical electron has a diverging electric field near its position). But the real reason for the worry is that this singularity is not in the field sourced by the brane (since that should be the \(F_5\)-field strength and it indeed blows up as well).

In light of the physical picture I outlined above, this divergence is not that strange to understand. The D3 charges in the fluxes are being pulled towards the anti-D3 branes where they pile up. The sign of the divergence in the 3-form fluxes is indeed that of a D3 charge density and not anti-D3 charge density.

Whenever a supergravity solution has a singularity one has to accept that one is outside of the supergravity approximation and full-blown string theory might be necessary to understand it. And I agree with that. But still singularities can — and should — be interpreted and the interpretation might be sufficient to know or expect that stringy corrections will resolve it.

So what was the attitude of the community when these papers came out? As I recall it, the majority of string cosmologists are not easily woken up and the attitude of the majority of experts that took the time to form an opinion, believed that the three assumptions above (especially the last two) were the reason for this. To cut a long story short (and painfully not mention my own work on showing this was wrong) it is now proven that the same singularity is still there when the assumptions are undone. The full proof was presented in a paper that gets too little love.

So what was the reaction of the few experts that still cared to follow this? They turned to an earlier suggestion by Dymarsky and Maldacena that the real KKLT solution is not described by anti-D3 branes at the tip of the throat but by spherical 5-branes, that carry anti-D3 charges (a.k.a. the Myers effect). This then would resolve the singularity they argued (hoped?). In fact, a careful physicist could have predicted some singularity based on the analogy with other string theory models of 3 branes and 3-form fluxes. Such solutions often come with singularities that are only resolved when the 3-branes are being polarised. But such singularities can be of any form. The fact that it so nicely corresponds to a diverging D3 charge density should not be ignored — and it too often is.

So, again, I agree that the KKLT solution should really contain 5-branes instead of 3-branes and I will discuss this below. But before I do, let me mention a very solid argument of why also this seems not to help.

If indeed the anti-D3 branes "puff″ into fuzzy spherical 5-branes leading to a smooth supergravity solution then one should be able to "heat up″ the solution. Putting gravity solutions at finite temperature means adding an extra warp-factor in front of the time-component in the metric that creates an event horizon at a finite distance. In a well-known paper by Gubser it was argued that this provides us with a classification of acceptable singularities in supergravity. If a singularity can be cloaked by a horizon by adding sufficient temperature it has a chance of being resolved by string theory. The logic behind this is simple but really smart: if there is some stringy physics that resolves a sugra singularity one can still heat up the branes that live at the singularity. One can then add so much temperature that the horizon literally becomes parsecs in length such that the region at and outside the horizon become amendable to classical sugra and it should be smooth. Here is the suprise: that doesn't work. In a recent paper, the techniques of arXiv:1301.5647 were extended to include finite temperature and what happened is that the diverging flux density simply tracks the horizon, it does not want to fall inside. The metric Ansatz that was used to derive this no-go theorem is compatible with spherical 5-branes inside the horizon. So it seems difficult to evade this no-go theorem.

The reaction sofar on this from the community, apart from a confused referee report, is silence.

But still let us go back to zero temperature since there is some beautiful physics taking place. I said earlier that the true KKLT solution should include 5-branes instead of anti-D3 branes. This was described prior to KKLT in a beautiful paper by Kachru, Pearson and Verlinde, called KPV (again the same letter ‛K′). The KPV paper is both the seed and the backbone of the KKLT paper and the follow-up papers, like KKLMMT, but for some obscure reason is less cited. KPV investigated the ‟open-string″ stability of probe anti-D3 branes placed at the tip of the KS throat. They realised that the 3-form fluxes can materialize into actual D3 branes that annihilate the anti-D3 branes which implies a decay to the SUSY vacuum. But they found that this materialization of the fluxes occurs non-perturbatively if the anti-brane charge \(p\) is small enough\[

\frac{p}{M} \ll 1.

\] In the above equation \(M\) denotes a 3-form flux quantum that sets the size of the tip of the KS throat. The beauty of this paper resides in the fact that they understood how the brane-flux annihilation takes place, but I necessarily have to gloss over this such that you cannot really understand it if you do not already know this. In any case, here it comes: the anti-D3 brane polarizes into a spherical NS5 brane wrapping a finite contractible 2-sphere inside the 3-sphere at the tip of the KS throat as in the below picture:

One can show that this NS5 brane carries \(p\) anti-D3 charges at the South pole and \(M-p\) D3 charges at the North pole. So if it is able to move over the equator from the South to the North pole, the SUSY-breaking state decays into the SUSY vacuum: recall that the fluxes have materialized into \(M\) D3 branes that annihilate with the \(p\) anti-D3 branes leaving M-p D3 branes behind in the SUSY vacuum. But what pushes the NS5 to the other side? That is exactly the 3-form flux \(H_3\). This part is easy to understand: an NS5 brane is magnetically charged with respect to the \(H_3\) field strength. In the probe limit KPV found that this force is small enough to create a classical barrier if \(p\) is small enough. So we get a meta-stable state, nice and very beautiful. But what would they have thought if they could have looked into the future to see that the same 3-form flux that pushes the NS5 brane diverges in the back-reacted solution? Not sure, but I cannot resist from quoting a sentence out of their paper
One forseeable quantitative difference, for example, is that the inclusion of the back-reaction of the NS5 brane might trigger the classical instabilities for smaller values of \(p/M\) than found above.
It should be clear that this brane-flux mechanism is suggesting a trivial way to resolve the singularity. The anti-brane is thrown into the throat and starts to attract the flux, which keeps on piling up until it becomes too strong causing the flux to annihilate with the anti-brane. Then the flux pile-up stops since there is no anti-brane anymore. At no point does this time-dependent process lead to a singular flux density. The singularity was just an artifact of forcing an intrinsically time-dependent process into a static Ansatz. This idea is explained in two papers: arXiv:1202.1132 and arXiv:1410.8476 .

I am often asked whether a probe computation can ever fail, apart from being slightly corrected? I am not sure, but what I do know is that KPV do not really have a trustworthy probe regime: for details explained in the KPV paper, they have to work in the strongly coupled regime and they furthermore have a spherical NS5 brane wrapping a cycle of stringy length scale, which is also worrisome.

Still one can argue that the NS5 brane back-reaction will be slightly different from the anti-D3 back-reaction exactly such as to resolve the divergence. I am sympathetic to this (if one ignores the trouble with the finite temperature, which one cannot ignore). However, again computations suggest this does not work. Here I will go even faster since this guest blog is getting lengthy.

This issue has been investigated in some papers such as arXiv:1212.4828, and there it was shown, under certain assumptions, that the polarisation does not occur in a way to resolve the divergence. Note that, like the finite temperature situation, the calculation could have worked in favor of the KKLT model, but it did not! At the moment I am working on brane models which have exactly the same 3-form singularity but are conceptually different since the 4D space is AdS and SUSY is not broken. In this circumstance the same singularity does get resolved that way. My point is that the intuition of how the singularity should get resolved does work in certain cases, but it does not work sofar for models relevant to KKLT.

What is the reaction of the community? Well they are cornered to say that it is the simplifications made in the derivation of the ‛no polarisation′ result that is causing troubles.

But wait a minute... could it perhaps be that at this point in time the burden of proof has shifted? Apparently not, and that, in my opinion, starts becoming very awkward.

It is true that there is still freedom for the singularity to be resolved through brane polarisation. There is just one issue with that: to be able to compute this in a supergravity regime requires to tune parameters out of the small \(p\) limit. Bena et. al. have pushed this idea recently in arXiv:1410.7776 and were so kind to assume the singularity gets resolved, but they found the vacuum is then necessarily tachyonic. It can be argued that this is obvious since they necessarily had to take the limit away from what KPV want for stability (remember \(p\ll M\)). But then again, the tachyon they find has nothing to do with a perturbative brane-flux annihilation. Once again a situation in which a honest-to-God computation could have turned into the favor of KKLT, it did not.

Here comes the bias of this post: were it not for a clear physical picture behind the singularity I might be finding myself in the position of being less surprised that there is a camp that is not too worried about the consistency of KKLT. But there is a clear picture with trivial intuition I already alluded to: the singularity, when left unresolved, indicates that the anti-brane is perturbatively unstable and once you realise that, the singularity is resolved by allowing the brane to decay. At least I hope the intuition behind this interpretation was clear. It simply uses that a higher charge density in fluxes (near the anti-D3) increases the probability for the fluxes to materialize into actual D3 branes that eat up the anti-branes. KPV told us exactly how this process occurs: the spherical NS5 brane should not feel a too strong force that pulls it towards the other side of the sphere. But that force is proportional to the density of the 3-form fluxes... and it diverges. End of story.

What now?

I guess that at some point these ‟anti-KKLT″ papers will stop being produced as their producers will run out of ideas for computations that probe the stability of the would-be KKLT vacuum. If the first evidence in favor of KKLT will be found in that endeavor, I can assure you that it will be published in that way. It just never happened thus far.

We are facing the following problem: to fully settle the discussion, computations outside the sugra regime have to be done (although I believe that the finite temperature argument suggests that this will not help). Where fluxes not invented to circumvent this? It seems that the anti-brane back-reaction brings us back to the Dine-Seiberg problem.

So we are left with a bunch of arguments against what is/was a beautiful idea for constructing dS vacua. The arguments against have an order of rigor higher than the original models. I guess we need an extra level of rigor on top from those that want to keep using the original KKLT model.

What about alternative de Sitter embeddings in string theory? Lots of hard work has been done there. Let me do injustice to it by summarizing it as follows: none of these models are convincing to me at least. They are borderline in the supergravity regime or we don't know whether it is trustworthy in supergravity (like with non-geometric fluxes). Very popular are F-term quantum corrections to the GKP vacuum which are used to stabilize the moduli in a dS vacuum. But none of this is from the full 10D point of view. Instead it is between 4D effective field theory and 10D. KKLT at least had a full 10-dimensional picture of uplifting and that is why it can be scrutinized.

It seems as if string theory is allergic to de Sitter vacua. Consider the following: any grad student can find an anti-de Sitter solution in string theory. Why not de Sitter? All claimed de Sitter solutions are always rather phenomenological in the sense that the cosmological constant is small compared with the KK scale. I guess we better first try to find unphysical dS vacua. Say a six-dimensional de Sitter solution with large cosmological constant. But we cannot, or nobody ever did this. Strange, right? Many say: "you just have to work harder". That ‛harder′ always implies ‛less explicit′ and then suddenly a landscape of de Sitter vacua opens up. I doubt that seriously, maybe it just means we are sweeping problems under the carpet of effective field theory?

I hope I have been able to convince you that the search for de Sitter vacua is tough if you want to do this truly top-down. The most popular construction method, the KKLT anti-brane uplifting, has a surprise: a singularity in the form of a diverging flux density. It sofar persistently survives all attempts to resolve it. This divergence is however resolved when you are willing to accept that the de Sitter vacuum is not meta-stable but instead a solution with decaying vacuum energy. Does string theory want to tell us something deep about quantum gravity?

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