### Supercritical

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:

- A flat background, with a linearly varying dilaton (varying along a
*timelike*direction). - 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 AdS_{d} backgrounds, the observables are correlation functions of a conformal field theory on the $d-1$ dimensional conformal boundary. However, there *are* no nontrivial field theories for dimensions greater than 6. So, for AdS_{d}, $d\gt 7$, 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 $\text{AdS}_n\times M^{d-n}$, for $n\leq 7$. So long as the isometry group of $M$ is one of those allowed by Nahm’s classification, then nothing I’ve said precludes such a background.

## 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.