*By Dr Thomas Van Riet, K.U. Leuven, Belgium*

**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 should not be underestimated since it profoundly 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 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.

*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

**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\[**

*probe*\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

‟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 .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.″

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). Were 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?

Thanks, Thomas, for this neat review of the KKLT landscape building and its apparent flaw!

ReplyDeleteTo summarize the bug: you seem to say that KKLT assume that the anti-D3-branes are given by a singularity which prevents them from seeing that the singularity is actually resolved in the full theory and the resolution drives the anti-D3-branes to annihilate against the D3-branes, is that right?

So you believe that the controllable semirealistic AdS vacua are OK and their number is huge, even if we demand a rather low cosmological constant, but the controllable dS vacua can't be produced easily, right?

Do you believe that the "uncontrollable" dS vacua exist, or are numerous? I mean those at strong coupling where you don't care about the uncalculability suggested by Dine-Seiberg...

The difficulties with building dS vacua have been intensely studied - or at least discussed - since the observation of a positive c.c. in the late 1990s. I still think that the difficulty in getting a de Sitter may be "just" a calculational thing - dS vacua inevitably break SUSY and all controllable vacua we have rely on SUSY at least at some level. Do you agree?

On the other hand, I do believe that the smallness of the C.C. ultimately will depend on (broken) SUSY, too, although it doesn't seem sufficient in today's formalism. Those things are about the model building.

Then there are all the conceptual things about de Sitter in quantum gravity. We don't know what the observables should be. There should be thermal radiation coming from the cosmic horizon, so we should need many microstates of an "empty" de Sitter, just like we have microstates for black holes, but this goes totally against the field-theory-based definitions of string theory we've been using (where we always have a unique vacuum). Those are subtle and deep issues but I think that they're ultimately separable from the phenomenological ones.

Thank you Dr. Van Riet for your guest blog and thanks to

ReplyDeleteLubos for inviting you to write. It will take me a long time to understand your

blog. But, even with practically zero

knowledge of ST, many of us would be thrilled when someone comes up

with a bug free DS solution of ST. I have asked this question before but never

really understood the answer from experts how one can trust conclusions of ADS for

our real universe when the experiments clearly show positive CC. But since so

many experts believe in it, I have to admit that there must be something to

that! Again this may be due to my total ignorance of the field. Thanks again.

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Thanks

Mrs Rose Veronica

A simple reason why superstring theory is difficult to formulate in de Sitter space and easy to formulate in anti-de Sitter space is that AdS group SO(d-1,2) can be extended to a supergroup which contains the super-Poincare group, which is the superconformal group in d-1 dimension (OSp(N|2) for d=4 and SU(2,2|N) for d=5), while the dS group SO(d,1) can be only extended as OSp(m-p,p|2n) and this group does not have a super-Poincare subgroup.

ReplyDeleteDear Lubos,

ReplyDeleteLet me try to address your questions in the order you asked them:

- At the time of the KKLT paper there was no mentioning of any singularity. This was only found in 2009. If the solution is physical then then the singularity must be resolved in string theory. All attempts so far point against that. My point is that, once you give up that you look at a dS solution, you can resolve it. The only thing you need to assume is that the anti-branes decay perturbatively: instead of a meta-stable vacuum the system is "side of the hill". That is why I said it evaporates, the vacuum energy is positive but then decreases in time to end up negative in the SUSY vacuum. Quintessence? Not sure. If you mean with quintessence slowly decaying dark energy then that is possible if the decay is slow enough. I did not attempt to compute it. But this paper might be useful: http://arxiv.org/abs/hep-th/0403123

- I do not have a counter argument against 10^hundreds of AdS vacua, but I would not really call it 100% established either. I am not enough an expert on non-perturbative corrections for my thoughts on this to be relevant. I just find it strange that in AdS/CFT we do not seem to have such a landscape of CFT's. But I am probably too ignorant here.

- Whether string theory has huge amount of dS vacua or a unique one? No clue. But I can say I have tried to find them and failed whenever I was sticking to a set of ingredients that evade Maldacena-Nunes but are not too in-explicit. We have constructed order 10^3 dS solutions in massive IIA string theory and all where tachyonic. Hence my sympathy for an "anti-de Sitter" conspiracy theory :) I would not be surprised if it can be shown that string theory has no de Sitter vacua at all. It would not falsify string theory either. A "quasi de Sitter" solution would still be fine. And maybe this is what Polyakov is trying to tell us since the 80's?

- No: the difficulty of finding dS cannot be blamed on SUSY breaking. We have zillions of SUSY breaking AdS vacua and Minkowski vacua that are under control (skew whiffed solutions in M-theory, the GKP vacua with fluxes that have (0,3) component,...). Again: nobody ever succeeded to cook up a simple dS. Although Eva Silverstein and collaborators have original and smart ideas about that, I have never been convinced truly.

There are basic smell tests that the anti-D3 brane argument does not pass. It is being said that an anti-D3 brane in a background field arbitrarily weaker than the string scale becomes unstable. But since the field is weak, any instability that develops must have been at worst massless in the absence of the field. The only massless mode on the D3 is the position, so at worst the D3 shifts a bit at the bottom of the throat.

ReplyDeleteFrom another point of view, one does not trust SUGRA at distances less than the string length. By general principles, the backreaction is weak (O(g)) outside the string length. At shorter distances, it is meaningless.

Finally, backreaction of any object should include a Schwarzschild-like 1/(1-GM/r) metric. Expanding in powers of M gives arbitrarily high powers of 1/r. So it is not surprising that in any system, one gets apparent bad divergences. But (referring back to the previous paragraph), what plays the role of GM here is below the string length.

The anti-D3 papers do not address these, but these need to if one expects others to pay attention.

Layman here. What does "quasi de Sitter" mean? And; you're saying there's no reason to believe there's a way the vacuum energy could resist going negative? (Otherwise that would intuitively give us the observed small but nonzero CC, right?)

ReplyDeleteMost of this obviously goes over my head, but it's interesting to get a sense of what the current state of constructing realistic vacua looks like. Thanks for writing this!

With quasi dS I mean, like in inflation, you have a space-time that has the approximate conformal symmetries of dS but not quite since they are broken by the slow roll parameters. I can indeed imagine that the decay process, could slow down for smaller cc.

ReplyDeleteWell, I didn't quite understand the first part but I assume an answer I could understand would be quite lengthy and/or require that I get off my ass and actually do some studying ;)

ReplyDeleteA slowed down but not quite stopped would, I assume, either be the quintessence Lubos mentioned or then a step-by-step decay with lengthening time between decays that would make us live in a long-lived one. Are theorists just really careful about claiming some mechanism could explain our 10^-123 CC or am I far too hasty in thinking a (by me) very vaguely understood scenario seems to provide an answer?

yeah!! that is the kind of (guest) post that makes this blog deserve its name! from there you can actually watch and learn something about the pretty face of Nature, which looks almost generically distorted and redshifted from other places lost in the landscape. thanks!

ReplyDeleteI (who am not an anti-science sour-ball) hunch that the deeper physicists are able to look into What Is going on, guided by their best or most thoroughly workable string/M-theory type mathematical framework, what they (you) will find is a 'multiverse motivating/driving' - by mathematics deeply reflected - discrepancy; IOW, an ultimate infinitely innovative imbalance and indeterminancy (of course related to 'quantum uncertainty').

ReplyDeletethomas, could you give some specific references of polyakovian wisdom, please? it reminds me what i have read from his "from quarks to strings" (arxiv:0812.0183). i quote:

ReplyDelete"Even more important is to find the gauge theory for dS. I conjectured that the large N gauge theories have a fixed point at the complex gauge coupling corresponding to the radius of convergence of the planar graphs. Presumably this point is described by a non-unitary CFT corresponding to the intrinsically unstable de Sitter space. This approach will hopefully resolve the puzzle of the CC and cosmic acceleration."

if i may ask, lubos, maybe a guest post by polyakov could be useful ;)

Please one day do a big post about Maxwell equations and their various formats. Are they quantum with uncertainty?

ReplyDeleteThanks for the interesting comment. My knowledge of group theory is too limited to say one way or other.It will be really unfortunate if it is true that DS is in conflict with SUSY ST on group theory grounds, since experiments support DS. I would like to see opinion of some knowledgeable people such as Lubos and others on this issue.

ReplyDeleteCome on, Nik, you may find Maxwell's equations and millions of their aspects in every textbook about electromagnetism.

ReplyDeleteIn their original form, they're classical equations without uncertainty, with full determinism.

In the era of quantum mechanics, you may put a "hat" above all the fields, and then they become operators (mathematically encoding observables with uncertainties), and the Maxwell's equations become an example of the Heisenberg equations of motion for operators, away to formulate any quantum mechanical theory, and the theory with these operators obeying these equations is known as Quantum Electrodynamics, the oldest and "practically most important" part of the Standard Model, a part that's been around since 1930 or so.

I do not follow this discussion. Nature is observed as non-SUSY. So if dS breaks SUSY that is no problem. Second, dS is not inconsistent with SUSY in a mathematical sense. There exist de Sitter superalgebras. That is no problem. Even more: you can write down supergravity models in which SUSY dS are the natural vacua. So where is the catch? Well there is no unitary representation. In other words: the SUGRA models all contain ghosts. So in that sense SUSY dS is not sensible, in the physics sense. But, again, mathematically SUSY dS exists. Ok, up to the issue that the Killing spinors are not globally defined. But that is no algebraic problem, rather a topological problem.

ReplyDeleteThe anti-D3 papers do discuss this issue. We are aware of the fact that SUGRA solutions cannot be trusted at small length scales.

ReplyDeleteHere is Polyakovian wisdom for you:

ReplyDeletehttp://arxiv.org/abs/1209.4135

http://arxiv.org/abs/0709.2899

Can you give a reference? And am I correct in understanding that you would say that a single anti-D3 in a weak background field is already problematic?

ReplyDeleteOther papers by him, perhaps some of them are also relevant:

ReplyDeletehttp://inspirehep.net/search?p=author%3AA.M.Polyakov.1%20AND%20collection%3Aciteable

There are some new results about 4d supersymmetric field theories in dS in this paper http://arxiv.org/abs/arXiv:1403.5038

ReplyDeleteEven tbough I would be lying when claiming to understand all details, this great gust post was very exciting to read!

ReplyDeleteThanks very much for sharing this live insight into what is shaking (as Sheldon says) in cool cutting edge physics :-)

Lets hope that the final getting rid of this at present disturbing singularity will lead to great new insights and disciveries

Cheers

Her's a real women in STEM: http://www.smbc-comics.com/?id=3548#comic

ReplyDeleteWell, Thomas Hertog is my colleague in Leuven and I am planning to understand his paper. It definitely seems intriguing and could be related to the discussion of the existence of dS vacua in string theory.

ReplyDeleteLubos - One must be careful when collapsing the state vector: http://www.smbc-comics.com/?id=3544#comic

ReplyDeletehttp://checkmyworking.com/2012/01/how-to-get-beautifully-typeset-maths-on-your-blog/#disqus

ReplyDeleteDear Michael, the page you linked to contains nothing new or interesting, right?

ReplyDeletePff lol!

ReplyDeleteInteresting indeed. I do have a French/parisian way of seeing that side of the truck driver...I prefer your way though ;-)

Right, sorry for wasting your time, it was the reverting to an older version I naively thought might be of interest, but I just found its not even possible anymore. Will be more careful in the future.

ReplyDeleteSorry for being late. This mudslinging at KKLT discussion does upset me quite a bit, and likely has contributed to Shamit now working on moonshine. What the quoted papers show, as you say many times in your post, is basically that there is no 10-dimensional supergravity solution corresponding to KKLT. But KKLT never claimed such a thing! If you read the paper carefully, or travel back in time to talk to K or T, you will see that they fully understood this point. The most important ingredient that is actually new in KKLT was the instanton that stabilizes the Kahler modulus. This effect has no 10-d description, and only exists after compactification. KKLT fully realized that 10-d SUGRA is unreliable, and since full string theory in RR backgrounds is too hard, they turned to the second best thing -- effective field theory. They included all the light fields, wrote down all the terms allowed by symmetries, calculated the bosonic potential and looked for critical points. One of them turned out to be dS. They (and others) estimated the size of the higher-order corrections, establishing self-consistency. They (and others) analyzed possible decay modes, showing their vacuum is long-lived. Anybody who wants to cast doubt on KKLT, first has to show why the EFT breaks down. Respectfully, why is this so difficult to understand?

ReplyDeleteDear Johannes, first, moonshine is pretty cool, isn't it?

ReplyDeleteMy temporary conclusions about the dS vacua are fixed... But surely if one admits that the existence of the dS vacua is an artifact of the 4D EFT approximations and probably fails in the full string theory, then the derivation of such dS vacua isn't too important or true, is it?

Meanwhile, back in the States, we have an evaporating country.

ReplyDeleteDear Lubos, yes of course moonshine is hot :) Perhaps it even sheds light on the landscape at night ;) What I am questioning is precisely that the 10-d supergravity analysis shows that the conclusion of the EFT is wrong. A perhaps useful analogy is with Gepner models: After including all the worldsheet instantons that you find at large volume, you predict the existence of small-volume solutions just from the point of view of the EFT. And indeed these solutions exist (as Gepner models), contradicting any supergravity analysis. Similarly, the KKLT solution simply does not exist in the supergravity approximation, in which the Kahler modulus is not stabilized.

ReplyDeleteThomas, has anybody tried to analyze the D3-brane annihilation instability from the point of view of EFT?

Dear Johannes, now I understand what you question. But if I read the same text by Thomas, I didn't have the feeling that he only discusses a description using 10D SUGRA.

ReplyDeleteIt's full of real objects in string theory - that have some description in terms of 4D EFT, 10D SUGRA, and full string theory, but they're *real objects* even if we don't define any description.

Those objects include the anti-D3-branes which are somewhere and which may be inflated to 5-branes, and the 10D string theory story - not just 10D SUGRA - seems compatible with the claim that they get annihilated against the D3-branes.

In your Gepner example, the 10D SUGRA clearly omits the string-scale alpha' corrections etc. that are essential for the stabilization of the volumes near the string-scale radii. Can you find a similar omission - of terms bringing a new parameteric scale to the discussion - in the analyses that Thomas is referring to? Just to be sure, if you can't, then the two situations are not analogous at all.

Dear Lubos, the new ingredient in KKLT is the (spacetime) instanton that stabilizes the Kahler modulus.

ReplyDeleteWhen we include too many stringy objects into a background, the 10d sugra becomes inappropriate. We might be wrong, and there is indeed an instability towards annihilation of the anti-D3 against the flux. But in that case, there should be an EFT description of this instability, in which you can analyze the effect together with the instanton.

Perhaps.

ReplyDeleteAlternatively, the effect that destabilizes the background may also be much stronger than the instanton and should be analyzed well before (and not together with) the instanton, right? Like a tachyonic field describing a location or a thickness of a 5-brane that is overlooked in the current EFT because it's nearly decoupled.

Dear Johannes, I disagree with this way of presenting things. Obviously there is no dS critical point from 10D SUGRA. And I never claimed so. You indeed need the instantons and they arise only in EFT after compactification. They give you AdS. I did not critize that part of the paper. What I did is to criticize the uplifting mechanism using an anti-D3 brane. I simply investigate whether an anti-D3 brane can consistently exist in a throat filled with ISD three-form flux. That is also what KPV analyses. You can nicely do this from a 10D point of view if you use a non-compact geometry. Then I investigate the OPEN string modes. These modes were NOT discussed in the KKLT paper (only in the earlier KPV paper) and they are supposed to be stabilised within SUGRA. . What we did is to ask ourselves the question of whether the open string modes remain stable upon backreaction. This question can be studied using SUGRA since the open string stability is a Myers effect. So within SUGRA a perfect valid undertaking!

ReplyDeleteIf this kind of scrutiny is considered upsetting, well...it is called research ;)

Dear Thomas, Thank you for your reply (and I apologize for the strong words in my first comment).

ReplyDeleteI agree.

ReplyDeleteI agree with you, Brussels isn't the originator. But Brussels plays an essential role because the international organizations don't have any direct sovereignty over the nation states.

ReplyDelete"

ReplyDeleteBut Brussels plays an essential role because the international organizations don't have any direct sovereignty over the nation states."Ah well now, that's the crunch for any nation stupid enough to belong to the EU isn't it? :)

Actually, as an aside, I suppose one ought to distinguish between the euro-area and the rest of the EU here. The nations signed up to the euro are completely stuffed — they've already been overrun by the monster; their sovereignty is no more; they're now vassal states.

For the others, one might debate the sovereignty issue. Our politicos will make out that we're "of course" still a sovereign nation, that the Lisbon Treaty (the EU constitution repackaged) was "just a tidying up exercise" (as our liars claimed) but we're deep in the same shit too, just not quite as much — we can still breathe through our nostrils, just.

But the point is that none of this need be this way. (Again, as another aside when we joined what was called the Common Market in 1973. It was sold as being purely about trade, nothing to do with sovereignty yak yak .... All lies of course.) There's no reason this whole business can't be solely about trade. No political union is needed for that.

Moreover, given the extent of today's "global markets" the trade context is wholly different from those earlier days. In that regard the EU is a complete anachronism. Of course, it's far worse than that but that's another matter.

I think the main point though is that the game is global and the EU gets its regulations faxed to it now too. We simply don't need the leech. But then we never did.

Anyway, my point was not to argue the cons and cons of the EU but to mention a fact which is relatively new to me, and I imagine almost everyone else, namely that the bulk of regulation nowadays does not originate in the EU.

"

Not a lot of people know that," as Michael Caine used to say. :)I have to disagree with this statement. Singularities are indeed often short distance problems. It does not mean you have to accept them since you think that the UV complete theory will resolve them. The proper question is not: "why would the UV complete theory not resolve them?". The question is: "why would the UV complete theory resolve them?" Please do not turn it around. The GR community has lots of experience with singularities and they can tell you that singularities occur throughout if you are careless. The typical example is exactly that of forcing an unstable, inherently time-dependent system, into a static Ansatz.

ReplyDeleteSecondly if you read the whole post you will notice that I discuss a trick by Gubser to investigate indirectly whether string theory will resolve the singularity. This you do by heating up the system so much that its temperature horizon lives at large scales. Now guess what: the H^2 singularity refuses to fall inside the singularity. It tracks the position of the horizon. This is actually perfect in line with the interpretation Danielsson, Blaback and me gave to singularity. Can you tell me whether you disagree with this statement?

Thirdly: I was explaining that such singularities are rather mundane in systems of 3-branes and 3-form fluxes. People (actually not just any people, it were Polchinski and Strassler) have found that brane polarisation typically resolves the singularities. But brane polarisation can be seen within SUGRA. So SUGRA is smart enough in many cases to resolve the singularity itself (this is the fuzzball philosophy). Well, here is the issue with that: I can cook up systems of anti-branes in 3-form flux in which exactly the H^2 singularity appears but is then resolved by polarisation. But whenever I do that computation for a model that is related to KPV or KKLT, it simply does not work.

Dear Ramanujan, I think that I agree with Thomas that your opinion about the fate of a singularity is way too fast.

ReplyDeleteSingularities aren't not "automatic inconsistencies" that are simply cured by string theory in a universal way.

First, singularities in the geometry may be a part of fully consistent backgrounds, like orbifold or conifold singularities. Some probes see the geometry to be *exactly* the singular geometry that classical GR would suggest, and one may show that the full string theory on that background is consistent, anyway.

Second, the message that the UV-complete theory offers as the "resolution" of a singularity isn't universal. Some singularities are "resolved" in your sense, that the background is qualitatively the same, equally stable as the singular one seems to be, but free of any problems one would associate with singularities.

But some singularities are "resolved" by the UV-complete theory by noticing that the background isn't really a solution because some extra condition - a part of the "stringy equations of motion" - isn't obeyed "at" the singularity. Or they are refined by becoming unstable. For example, see APS (Adams Polchinski Silverstein) who show that the orbifold singularities in the bosonic string become unstable in string theory - with respect to "smearing" the singularity.

In the same way, the fate of the singularity here may be the annihilation of the anti-D3-brane with the D3-brane-like flux. You didn't write down anything that is a valid proof that this can't happen. Some unspecified quantities you discussed may be O(g) which is small. But we're talking about the stability of a configuration. If there's some new 4D effective field F with the potential

V = - g^5 * F^2,

then it still has a maximum at F=0 and will lead to fast, perturbative instability, won't it? To figure out what the fate of a singularity is is a nontrivial task with many answers possible a priori. Analyses look like in the APS paper I mentioned, or other papers that Thomas mentioned. It's simply not true that string theory always says "forget about the singularity, it actually becomes a smooth place with no extra implications". This is the kind of naive thinking about singularities that the loop quantum gravity "scientists" may believe but it has nothing to do with the reality.

String theory may say many different things about the fate of a singularity and one must study it - and/or its suitable approximations - carefully to figure out which scenario is right.

Yes, de Sitter superalgebras exist, for example osp(d,1|N), but none of them contains a super-Poincare subalgebra, simply because the so(d,1) algebra does not have a Poincare subalgebra.

ReplyDeleteOn the other hand, the AdS algebra so(d-1,2) is the conformal algebra in d-1 dimensions, and hence has a (d-1)-dimensional Poincare subalgebra, iso(d-2,1).

Since a SUSY algebra is by definition a superalgebra which has a super-Poincare subalgebra, therefore a dS superalgebra cannot be considered as a SUSY algebra.

The tax credit in the USA for a Tesla is $45,000 not $7500. In Norway the credit is $110,000 USD.

ReplyDeleteI met one if my Slavic friends returning from California, few years ago. He couldn't understand what's the big deal. The women are ugly and fat and the ocean is cold. No Pamela Anderson anywhere in sight. I thought about Supertramp song when he said that. He liked food in some San Francisco restaurants but he said he can get even better in shitholes in Europe. He wouldn't even swap French Riviera for California. No way. Now, granted, Google and Apple and Microsoft (sort of ) are around there, good luck getting a great software development job in French Riviera. As for particle physcists Rovelli likes Marseilles. That's my rant.

ReplyDeleteLOL, I've spent one year in California in total - 1/2 in Santa Cruz, 1/2 in Santa Barbara. And visits to Berkeley, San Francisco, USC, Monterey, and a few places.

ReplyDeleteIt's nice but when it comes to big conclusions, I would probably agree with the Slavic friends of yours and with the non-Slavic non-particle non-physicist quasiphilosophical babblers such as Rovelli.

The ocean is cold and it makes the place non-ocean-like. I may have spent more time swimming in the Atlantic Ocean near Boston than in California.

The redwoods in Santa Cruz are impressive but from a broader viewpoint, the environment of the campus is just another uncivilized monocultural forest. The food in CA largely sucks.

So the only improvement over the testimonies above is that I did meet Pamela Anderson but I wasn't terribly impressed.

My sister has been living in the French Riviera for years and I would probably agree it's a more luxurious place overall.

This is a bit a discussion about language. A superalgebra is for me a SUSY algebra. Super=SUSY. I do not see the relevance of having a Poincare subgroup. Maybe you can enlighten me on this. (I assume you are not referring to the fact that we locally observe Poincare invariance? This is on the level of particle physics. My laptop is already not Poincare invariant anymore and the universe at large scales seems to have an approximate de Sitter isometries)

ReplyDeleteI just purchased it! All they have now is the Mens XL size in stock.

ReplyDeleteSo the guy is a bit aspergian - so what? So are lots of people in the

ReplyDeletesciences. Now what about all the women who trade on their scantily clad

and nekkid bodies? What's their excuse?

thanks, I was looking for this info...

ReplyDeleteunfortunately they are sold out again,

but they are reissung them soon :D

oh yes, and they drool over how cool Lena Dunham is with all her sexual predatory sh*t (it's all in her book, mind you).

ReplyDeleteThe link in this blog isn't to where Matt gets his shirts. It's www.ellyprizeman.com

ReplyDelete