## December 15, 2003

### Thanks to Jacques

#### Posted by Luboš

It seems that this string coffee table works, because you are able to see this sentence. Thanks to Jacques. So what do you Gentlemen think, for example, about the stability of two-dimensional superstrings?

Posted at December 15, 2003 6:33 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/265

### Stability

Why, just on Friday, there was a paper addressing that question. Those backgrounds sure look rock-solid stable to me.

Posted by: Jacques Distler on December 15, 2003 6:54 PM | Permalink | Reply to this

### Re: Stability

Hi Jacques,

First of all, thanks for setting up this system. It looks like it will be very addictive.

Second, in your paper, you construct models with no massless fields, which means the graviton is massive. What sort of geometry is making the graviton massive? Is it like a linear dilaton?

And perhaps it isn’t so surprising that we don’t have holography when there’s no gravity. What do you think?

Posted by: Arvind on December 15, 2003 10:10 PM | Permalink | Reply to this

### Re: Stability

As you know, there are no gravitons in 1+1 d. In fact, there’s no dilaton field either. Both the metric and the dilaton can locally be gauged-away.

The “novelty” of our backgrounds is that there are no other massless bosons. We have a theory in flat 1+1 dimensional space, with unbroken (8,8) or (24,0) supersymmetry, and no massless bosons.

No, this isn’t a linear-dilaton vacuum. It’s genuinely Poincaré-invariant.

Posted by: Jacques Distler on December 15, 2003 10:56 PM | Permalink | Reply to this

### Re: Stability

Yep, I should have put “gravitons” in quotes. I should probably ask rather about the vertex operator \del X \del X (OK, so now I have to figure out how to use MathML). I suppose this has some nonzero mass.

I am trying to see if there is really any application of holography here. If there is no graviton field, there are no
black holes, etc. So how can we apply the entropy law?

Posted by: Arvind on December 15, 2003 11:16 PM | Permalink | Reply to this

### Re: Stability

You don’t “do MathML”; you just type LaTeX-like expressions and rely on the plugin to do the conversion. So $\partial X \partial X$ becomes $\partial X\partial X$ automagically.

It’s not true that just because there are no gravitons there are no blackholes. There are no gravitons in 2+1 d, but there are certainly blackholes. And the same is true in 1+1 d. In any case, the Goheer et al paper was was about the entropy of Rindler space.

As to vertex operators, as you well-know, BRST-invariance kills 4 states at the massless level (two are not annihilated by $Q$, two are $Q$-exact). That gets rid of all three components of the metric field and gets rid of the dilaton field in 1+1 d.

Posted by: Jacques Distler on December 16, 2003 12:11 AM | Permalink | Reply to this

### Re: Stability

As you can probably tell, I’m still just trying to see if one should just drop this whole holography idea, or if there’s some loophole. I admit I’m not seeing one yet.

The only thing which looks to me a possibility is that the gravitational sector is somehow frozen out in the theory you are considering. It seems a possibility, since the graviton is massive, so we don’t have fluctuations in the geometry. I don’t know if this solves the problem that Goheer et al. had.

BTW, the connection from your theory to Rindler space is by considering an accelerated observer, right?

Posted by: Arvind on December 16, 2003 1:34 AM | Permalink | Reply to this

### Re: Stability

Well, I wouldn’t want to abandon holography just yet. It’s still a neat idea. On the other hand, Marolf and Sorkin, hep-th/0309218, have a nice argument against the derivation of holography from the generalized second law. I’ve always been a bit of a holography skeptic, myself.

It’s not terribly difficult to find a way to rescue holography from our paper – it wasn’t really intended to be an attack on it, after all, just on the argument in Goheer et al. The simplest (but not the only) way out is to just think of holography as a semi-classical imprint (to steal Bousso’s term) of the underlying quantum theory, whatever that may be. Thus, if we can’t work semi-classically, we can’t apply holography.

And yes, the Rindler coordinates just arise because the observer is accelerating.

Anyways, enough of this holography stuff. Who’s made it through Witten’s latest opus yet?

Posted by: Aaron Bergman on December 16, 2003 2:47 AM | Permalink | Reply to this

### Re: Stability

Ah yes, Witten’s opus. Looks like it’s going to be a huge thing. Hundreds of followups next week, I bet. Looks like there are a lot of open questions.

Posted by: Arvind on December 16, 2003 4:30 AM | Permalink | Reply to this

### Witten and twistors

Freddie Cachazo has prepared us for the basics, so we need some elementary stuff about the MHV amplitudes and the twistors. I am sort of amazed how similar it smells to the project on the mysterious duality - a similar way to describe not N=4 Yang-Mills, but M-theory on tori (including the U-dualities). Witten might be eventually faster to make even this work. :-) Of course, I think that the del Pezzo case is much more ambitious than the N=4 case.

Posted by: Luboš Motl on December 16, 2003 4:53 AM | Permalink | Reply to this

### Re: Witten and twistors

Hi Lubos,

Do you want to explain what you’re talking about? Mysterious duality? del Pezzos?

Posted by: Arvind on December 16, 2003 8:25 PM | Permalink | Reply to this

### Re: Witten and twistors

“A mysterious duality” is a leading contender to initiate the third superstring revolution. See the paper http://arXiv.org/abs/hep-th/0111068 by Iqbal, Neitzke, and Vafa.

Del Pezzo surfaces are 2-complex-dimensional manifolds: ${\mathrm{CP}}^{2}$ and its variations where $k=1,2,3,4,5,6,7,8$ points are resolved, and a special type of del Pezzo is ${\mathrm{CP}}^{1}$ cross ${\mathrm{CP}}^{1}$. These ten del Pezzo are in one-to-one correspondence with maximally supersymmetric string/M-theoretical backgrounds: ${\mathrm{CP}}^{2}$ is M-theory in 11 dimensions and ${\mathrm{CP}}^{1}$ cross ${\mathrm{CP}}^{1}$ is type IIB in ten dimensions.

The physical origin behind this duality relating a crazy geometry with M-theory might be a sort of string theory defined on the del Pezzo (or its supergeneralization) as the target space - a type of string theory that could give you physics of M-theory much like Witten’s twistors give you the amplitudes of N=4 super Yang Mills.

I am editing in Linux Netscape that does not behave too well, so accept my apologies for the mess in this post.

Posted by: Lubos Motl on December 16, 2003 8:32 PM | Permalink | Reply to this

### Re: Witten and twistors

If del Pezzos should be viewed as target spaces of (2,1) strings, how would that explain the mysterious duality?

Looking at

D. Kutasov & E. Martinec New Principles for String/Membrane Unification

I find it extremely intriguing that the worldsheet supergravity can be interpreted as the target space theory of (2,1) strings. The natural question seems to be: Can the worldsheet theory of (2,1) strings itself be again interpreted as the target space theory of yet another type of strings?

In the introduction of the Kutasov&Martinec paper it says that the target space theory of $N=2$ strings admits a twistorial formulation. So is this the point where Witten’s recent paper makes contact with the mysterious duality?

Finally, I see speculation about explaining S-duality in terms of T-duality in various scenarios. I would be very grateful if anyone could provide me with some sort of overview of the current status of our understanding of the relation between S-duality and T-duality.

Posted by: Urs Schreiber on January 4, 2004 6:58 PM | Permalink | Reply to this

### Re: Witten and twistors

Hi Urs,

the idealistic purpose to define the N=(2,1) string theory - or perhaps another type of string theory - on the del Pezzo target space is nothing else than to provide a full nonperturbative formulation of string/M-theory on the appropriate torus. Such a formulation would imply the U-duality in a very manifest way. The del Pezzo surface would be, along the lines of Kutasov and Martinec (and analogously to Witten’s projective space) a generalized worldsheet or worldvolume, and the path integral over it would give you the S-matrix of theory. Well, the mysterious duality would be proved because a theory on the del Pezzo surface would be explicitly equivalent to M-theory.

By the way, yes, for an appropriate choice of the GSO projections of the (2,1) strings, Kutasov and Martinec have showed that the target space can be another copy of the (2,1) string theory itself - and in principle, you can imagine to repeat this process infinitely many times. (This idea was in fact mentioned in Green’s Worldsheets for Worldsheets in the 1980s.) All these features of the (2,1) strings are very attractive, and the main question is how to make it work, including the interactions, and a nontrivial del-Pezzo-like target space was a very natural choice to consider even though the details still don’t work.

All the best
Lubos

Posted by: Luboš Motl on January 4, 2004 8:40 PM | Permalink | Reply to this

### Re: Witten and twistors

Hi Luboš,

After further reading, I am surprised to see that also the bosonic string is part of the big picture of (2,1)-string target space theories. Weird! Would this imply that there might be a duality or something which relates the bosonic string to superstrings?

Another thing: In their 1996 paper Kutasov and Martinec say that (2,1) strings with $2+1$ dimensional target spaces are

likely [to] describe a membrane in a $10+1$ dimensional sopace

and leave a detailed study to future work. Has this been clarified since then?

Posted by: Urs Schreiber on January 5, 2004 3:57 PM | Permalink | Reply to this

### Re: Witten and twistors

Hi Urs,

sorry for the delay. Yes, you can get the bosonic string’s worldsheet as the target space of the (2,1) strings with some appropriate GSO projections - those that treat the 24 chiral fermions of the (2,1) string on equal footing.

This appearance of the bosonic string should not be terribly surprising. The bosonic string might be inconsistent - and unrelated to superstring theories - in spacetime, but the worldsheet of the bosonic string is essentially equally consistent as the worldsheet theory of the superstring, so why should not it appear as a target space of another string theory?

The membranes can indeed appear as the target space of the (2,1) strings, too, but as far as I know not too many new arguments that go beyond Kutasov and Martinec have appeared after their paper.

All the best
Lubos

Posted by: Lubos Motl on January 21, 2004 5:01 PM | Permalink | Reply to this

### Re: Witten and twistors

Hi Lubos -

Since the worldsheet theory of the bosonic string as a theory of quantum gravity in 1+1 dimensionsions is consistent it is maybe no surprise that one can realize the bosonic string as the target space of another string theory. What I found surprising, though, is that this other “worldsheets for worldsheets” theory is the same $\left(2,1\right)$ string that also has target spaces that correspond to the superstring and supermembrane.

The reason is that, maybe naively, I would expect that the different target space incarnations of the single $\left(2,1\right)$ theory are closely related and maybe even just different aspects of the same thing. This is indeed true for the various superstrings and the supermembrane, which are all related by dualities and/or (double) dimensional reduction.

I realize that probably this expectation is too naive, but I am still wondering about the following:

The bosonic string is considered unacceptable maninly because of its tachyon (also because of the absence of fermions in its spectrum). But today we know that at least the open string tachyon is not a sign of inconsistency but simply due to the fact that the uncharged D25 brane is unstable and that the open bosonic string theory sort of collapses to the closed bosonic string theory while the D25 brane decays.

Hence one is left wondering about the meaning of the closed string tachyon. I have never heard of a possible interpretation of it. But couldn’t it be that its presence just signals that, as before with the open bosonic string, the closed bosonic string background is sort of unstable and hence decays into something else - something which then finally might be a consistent string theory?

If I am allowed to ask a wildly speculative question: Couldn’t it be that maybe somehow 16 excess dimensions “decay” in some sense and let the open bosonic string evolve into the, say heterotic string - or something like that.

I hope you see what I am getting at.

Posted by: Urs Schreiber on January 25, 2004 10:43 PM | Permalink | Reply to this

### Re: Witten and twistors

Dear Urs,

thanks for your response. Yes, the absence of fermions might be unpleasant for bosonic string phenomenology, but it is not a sign of the inconsistency. The tachyon is more serious.

You are certainly not the first person who asks about the fate of the closed string tachyons. There have been several papers - less convincing than Sen’s treatment of open string tachyons - and the most convincing insight is about *twisted* tachyons by APS - Adams, Polchinski, Silverstein. The bulk closed string tachyons are harder to stabilize, and no solution has been universally accepted.

Your comment that the existence of the closed string tachyon signals instability of the whole 26D background is tautology. Yes, of course, this is what every tachyon does. The problem is that none knows what it decays to. Various people suggested e.g. lower-dimensional theories with linear dilaton. The bosonic string might decay by losing the dimensions, until you get into 2 dimensions where the “tachyon” is really massless and does not imply any further instability. Others proposed that the bosonic string spacetime might spontaneously create 10-dimensional “brane” worldvolumes described by superstring theory, and so on.

Yes, I know what you’re getting at, and I have considered identical ideas in the past - including the precise topological gauge bundle that can reduce the dimension. Even though it is very far from the full convincing solution, let me tell you about it.

The group E8 is known to have pi_3 nonzero - isomorphic to Z - this is why there are instantons in E8 - and the next nonzero homotopy is pi_15, which means that you can have nonzero

Tr(F /\ F /\ F /\ F /\ F /\ F /\ F /\ F)

There are eight copies of the field strength. Of course, pi_15 is exactly the right number to generate codimension 16 solutions. If you start with a E8 gauge theory in 26 dimensions, you can get 10-dimensional worldvolumes localized at the core of the generalized instanton with the nontrivial trace of F to eighth power. E8 in bosonic string theory or bosonic M-theory is not terribly natural, but the number 16 seems attractive in this way.

If you get further with these ideas, let me know… ;-) Good luck.

All the best
Lubos

Posted by: Lubos Motl on January 26, 2004 12:49 AM | Permalink | Reply to this

### Re: Stability

You say “The simplest (but not the only) way out is to just think of holography as a semi-classical imprint (to steal Bousso’s term) of the underlying quantum theory, whatever that may be. Thus, if we can’t work semi-classically, we can’t apply holography.”

Why should that be? Why do people think that holography or more precisely entropy bounds are a property of a quantum gravity given that support for it comes only from classical or at best semi-classical arguments:

There is the classical argument (of Hawking and possibly others) that it’s a consequence of some positive energy condition that horizont area increases.

Then there is the realization that the second law of thermodynamics (that for closed system the entropy never decreases) doesn’t work if you have a black hole and the outside universe doesn’t behave like a closed system.

This suggests the idea that horizont area somehow behaves like entropy and using the expression for the Hawking temperature and the Schwarzschild mass you determine the coefficient 1/4 — a semi-classical argument.

And then come the arguments (to which the criticism of Marolf and Sorkin applies) that you can save the second law for the closed system consisting of the outside universe plus all horizonts only if the entropy content of space regions (with some gravitational mass) is bounded from above by some expression.

But who tells you you should think of the outside universe plus all horizonts as a closed system?

And then based on these arguments (it’s interesting to look at some old papers on that subject, he has to include fudge factors of order 10 exclude obvious contradictions) people come to extremely strong conclusions such as the non-existence of string compactifications to 1+1 dimension.

So my real question is: Why do people believe that this is a fundamental principle and not just an observation that these bounds are usually weak enough not to be violated?

Posted by: Robert on December 16, 2003 4:10 PM | Permalink | Reply to this

### Re: Stability

Dear Gentlemen!

A very interesting discussion has been started on news.groups. Please feel more than free to write anything about string theory at news.groups right now, because this newsgroup belongs to string theory for a couple of weeks. ;-)

You can access this newsgroup via the web, too.

A guy interested in string theory pointed out the difference between the narrow and the broad meaning of string theory, and he said a couple of things about Brian Greene’s book and a comparison of a newsgroup on cosmic research with sci.physics.strings.

Come to news.groups to discuss the proposal to found sci.physics.strings - the new and kewl newsgroup. Your decision whether you will participate or not will influence the opinion of other readers whether string theory is an irrelevant obscure theory studied by a couple of crazy people who are not able to use the internet, but who eat a lot of money from the state budgets worldwide! :-)

Posted by: Luboš Motl on December 17, 2003 12:55 AM | Permalink | Reply to this

### Re: Stability

Luboš,

You’ll get a wider audience if you post a new blog entry, rather than tacking this announcement onto the end of a long and convoluted comment thread.

You should also note that, since I’ve announced this blog to the world, and since, just today, it has been linked to by at least two other blogs (I, II), your audience is much bigger than the aforementioned “gentlemen.”

Usenet $\ne$ The Internet

Posted by: Jacques Distler on December 17, 2003 1:17 AM | Permalink | Reply to this

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