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April 12, 2005


Luboš, on his blog, seems to have gotten into a bit of an argument with Eva Silverstein about the status of “supercritical strings.”

I hesitate to wade into the middle. But, while it’s clearly true that these supercritical strings have some nice properties, it’s also obscure whether one can really make them into sensible interacting theories.

Two types of backgrounds have been proposed:

  1. A flat background, with a linearly varying dilaton (varying along a timelike direction).
  2. An AdS background.

In flat space, on general grounds, the observables should be an S-matrix.

  • In critical string theory in flat space, the dilaton is a constant, and an S-matrix is defined (for sufficiently small dilaton VEV) for the scattering of perturbative string states into perturbative string states.
  • In 1+1 dimensional noncritical string theory, there is also a linear dilaton background (varying in a spacelike direction). And there’s a “Tachyon wall,” preventing strings from penetrating the region of strong coupling. So one has an S-matrix, of sorts, for string coming in from the weakly-coupled region, bouncing off the wall, and returning to the weakly-coupled region.

However, for supercritical strings, the theory is strongly-coupled either in the far future or in the far past. It’s not clear how one defines an S-matrix. It certainly isn’t for perturbative string states.

In AdSd backgrounds, the observables are correlation functions of a conformal field theory on the d1d-1 dimensional conformal boundary. However, there are no nontrivial field theories for dimensions greater than 6. So, for AdSd, d>7d\gt 7, the observables are, in some sense, all “trivial.”1

Those are the two paradigms that we know about for the observables of quantum gravity. Neither seems to lead to a satisfactory answer for supercritical strings. Perhaps there’s a third prescription for what the observables of quantum gravity are, which would yield a satisfactory answer in the case of supercritical strings.

But I’d like to hear what that is.

1 To be fair, there remains the possibility of taking a supercritical string on AdS n×M dn\text{AdS}_n\times M^{d-n}, for n7n\leq 7. So long as the isometry group of MM is one of those allowed by Nahm’s classification, then nothing I’ve said precludes such a background.

Posted by distler at April 12, 2005 12:00 AM

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Re: Supercritical

Hi Jacques:

As far as I understand, their theories are not spacetime SUSY. In that case I agree there are no known fixed point above six (actually four) dimensions, but I am not sure the possibility is excluded (Nahm classification is for superconformal theories). I agree that these fixed points are not very likely to exist, especially in families.

Posted by: Moshe Rozali on April 12, 2005 9:05 AM | Permalink | Reply to this


You’re right. Nahm’s classification was of superconformal algebras.

I was trying to be generous because, as you note, there are no nonsupersymmetric fixed points known above 4D. And supersymmetry (can) guarantee the existence of families of fixed points, allowing you to do stuff like extrapolate from weak coupling to strong.

Finding an isolated solution with n≥4 sounds even harder.

Posted by: Jacques Distler on April 12, 2005 9:26 AM | Permalink | PGP Sig | Reply to this

Re: Supercritical

Similar comments apply to the KKLT construction. They are naturally interested in 4-dimensional dS vacua, but nothing in their reasoning prevents us from getting non-supersymmetric AdS vacua, and thus lots and lots of non-SUSY fixed points. In the last step of their journey just break SUSY in a milder way, not generating a new metastable vacuum, surely that will only improve the relaibility of the analysis. If you believe this then the number of 4dim non-SUSY fixed points is at least as large as the number of meta-stable dS vaccua.

Posted by: Moshe Rozali on April 12, 2005 9:53 AM | Permalink | Reply to this


If you believe this then the number of 4dim non-SUSY fixed points is at least as large as the number of meta-stable dS vaccua.

You mean 3D non-SUSY fixed points (the boundary of AdS4 being 3 dimensional).

There are, likely, boatloads of 3D non-SUSY fixed points. And even more 2D non-SUSY fixed points.

Posted by: Jacques Distler on April 12, 2005 3:25 PM | Permalink | PGP Sig | Reply to this

Re: Supercritical

yes, of course,sorry, it makes it a little more plausible.

Posted by: Moshe Rozali on April 12, 2005 3:42 PM | Permalink | Reply to this

Re: Supercritical

Hi Jacques,

I hope all is well with you.
A couple of comments:

1) On the supercritical models:

In more general backgrounds than the timelike linear dilaton, the dilaton need not go to strong coupling in the past (given that it goes to weak toward the future). This is because there are other ingredients–e.g. orientifolds and fluxes–that provide enough independent forces to metastabilize it. We combined this with an asymmetric orientifold model to fix the rest of the moduli in the paper with Andy and Alex. Given the potential energy generated by these ingredients, the dilaton can start in a metastable minimum and tunnel out to the timelike linear dilaton phase in the future. (Or one can consider an AdS solution with the dilaton constant for all time.)

Of course these models are complicated and not very well explored yet, and conceivably there could be some concrete pathology ruling all of them out. (See the papers for a delineation of assumptions and issues involved–the main thing we identified to worry about is the potentially large number of RR species running in loops, though there are also lots of Chern-Simons couplings that may lift these RR fields.) But such a no go theorem would need to be proved rather than assumed in order to make a sharp prediction of the critical dimension from string theory.

2) On observables: this is of course a question you could ask similarly for metastable (A)dS models starting from either critical or supercritical limits, as Moshe also commented. While I agree we do not know fully yet how to formulate observables in cosmological backgrounds including metastable de Sitter with bubble decays, I don’t think our ignorance translates into a no go theorem. Instead I prefer to view it as a challenge motivated by the data and by the effective field theory arguments for these backgrounds.

In the AdS case, since the gravity side is the side with the effective weakly coupled description, it is not surprising that we found the gravity-side solutions before we know the CFT duals.

It is also clear that it would be silly to limit our considerations to backgrounds which have a currently known non-perturbative definition–I know you weren’t saying this but the argument has been made. Even for ridiculously symmetric systems like N=8 supergravity obtained from toroidal compactification of 11d supergravity, we do not yet have a non-perturbative formulation. (Recall that matrix theory and concrete examples of AdS/CFT both are rendered inapplicable by IR effects upon compactification to a hierarchically large 4d.) Again, to me this doesn’t suggest a no go argument but instead an interesting challenge.


Posted by: Eva Silverstein on April 12, 2005 5:47 PM | Permalink | Reply to this

Re: Supercritical

It is also clear that it would be silly to limit our considerations to backgrounds which have a currently known non-perturbative definition…

That would, indeed, be a silly position to take. And I’m certainly not saying that we shouldn’t explore these supercritical backgrounds.

On the other hand, we should be a little nervous (and perhaps somewhat skeptical) that we don’t know what the observables are in such a background, let alone how to calculate them. It may be, at the end of the day, that these backgrounds cannot be made sense of.

Now, having said that, there are other backgrounds, like … umh … cosmological (accelerating) backgrounds, where we don’t what the observables of quantum gravity are, either. There, at least, we have a pretty good reason for believing that a sensible definition exists. (And it may help that, in String Theory, at least, such backgrounds tend to be only metastable.)

But I understand your argument that — from a purely theoretical point of view — such backgrounds are no less mysterious that the supercritical ones.

Posted by: Jacques Distler on April 16, 2005 10:02 PM | Permalink | PGP Sig | Reply to this

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