## Thursday, February 03, 2005

### Tachyons can change topology

In the last 2 hours, Allan Adams just told us about his supernew paper

and because I think that it is definitely an interesting paper, let me say a couple of words.

Imagine that you take a type II string theory and compactify it down to 8 dimensions, on a two-dimensional genus "g" Riemann surface.

Well, unless "g=1", it is a non-conformal theory, so you will have to deal with a time-dependent background. Let's not worry. Let's assume the string coupling to be weak throughout the story.

Imagine that you start with a genus 2 Riemann surface. It can degenerate into two genus 1 Riemann surfaces connected by a thin tube. The circle wrapped around this tube is homologically trivial, and you can show that the fermions will be antiperiodic around it: it will be a Scherk-Schwarz/Rohm compactification on a thermal circle. The reason for the antiperiodicity is the same like the reason that the closed strings in the NS-NS sector must have antiperiodic boundary conditions for the fermions assuming that the corresponding operators in the "z" plane don't introduce any branch cuts.

OK, imagine that the tube is very long. Because of the antiperiodic boundary conditions, the sign of the GSO condition in the sectors with odd windings is reverted, and one can find some tachyons there assuming that the radius is small enough so that the winding is not enough to make the squared mass positive. Equivalently, one can T-dualize along the circumference of the tube to obtain some sort of type 0 theory which has a bulk tachyon if the radius in the type 0 picture is large enough. Go exactly near the point where the first tachyon in the "w=1" sector starts to evolve. It's the first perturbative instability you encounter.

These guys then argue that the most obvious time evolution will take place. The tachyons start to get condensed, and the handle will pinch off. It can be seen as a perturbative instability although it is probably continuously connected to the non-perturbative stability called the Witten bubble, and they use various CFT techniques, Ricci flows, RG flows, N=1 and N=2 worldsheet supersymmetry to study the process quantitatively. They argue that the two ends of the tube don't talk to each other - the strings can't propagate through the critical region where the topology change takes place. I am not gonna write the math here because you can open the paper.

Such a process can reduce the genus of a Riemann surface. Recall the picture with Brian Greene's breakfast on PBS/NOVA: the topology of the coffee cup and the doughnut are identical, but once Brian bites doughnut, it is going to become a sphere. In this case, the TV program is exact, not just a lower-dimensional analogy of the conifold transition. ;-)

The same process, however, can divide a higher genus Riemann surface into pieces that don't interact at all. The world decays into pieces - baby universes and similar stuff. A lot of interesting stuff happens from the low-energy effective theory viewpoint - doubling of gravity, decoupling of various modes, gaps emerging and disappearing, and many other things. Note that spacetime supersymmetry is broken, but one can arrange the parameters of the geometry in such a way that the evolution is more or less controllable.

I still believe that similar kinds of topology changing transitions may eventually destabilize or eliminate most of the "landscape". If you start with a too convoluted Calabi-Yau space with fluxes, there will be many modes how it can decay - it is potentially able to split into two (or more) Calabi-Yau pieces. Instead of the 2-dimensional handles of the cylindrical type, there may be many higher-dimensional analogues of the cylinder although most of us so far seem to have trouble to find working higher-dimensional examples. Such processes do not have to be too likely, but there are just many channels in which such a complicated Calabi-Yau space can decay - the number of channels is large because the number of "simpler" minima in the landscape is claimed to be large as well. This largeness is, I believe, self-destructive for the landscape.

My intuition is that such a decay tends to simplify the homology of both final products - i.e. reduce their Hodge numbers. This is a reason to believe that the Calabi-Yaus with very small Hodge numbers will be preferred. Braun, He, Ovrut, Pantev have found a quasi-unique heterotic standard model based on a Calabi-Yau with h^{1,1}=h^{2,1}=3. Three is the smallest positive integer after one and two - a pretty good choice. Assuming that there is something right about this and previous paragraph, Braun et al. have a pretty good chance that they have found the theory of everything. ;-)

Meanwhile, Adams et al. have made useful steps to understand tachyons in string theory. Note that these new understood tachyons start to look like bulk tachyons. The first understood tachyons were open tachyons (Sen and others); then people (APS; but also Headrick) continued with the closed string twisted tachyons; now they're getting into the bulk.

1. The pinching process could have an analogue in TGD framework, where the genus of boundary component of 3-surface is identified as being responsible for the family replication. e,nu_e would correspond to spherical topology, etc. and the ordering of families would have a nice explanation.

The problem is why g>2 families are absent/super heavy/very unstable. The key observation is that g<=2 geometry is always hyper-elliptic that is allows Z_2 as global conformal symmetry. Furthermore, the elementary particle vacuum functionals in the modular degrees of freedom constructible using theta functions and satisfying some very general constraints like modular
invariance and natural stability conditions, vanish for hyper-elliptic surfaces for g>2.

The conjecture is that for the maxima of the exponent of Kahler function playing the role of vacuum functional around which matrix elements are perturbatively calculable, boundary components are hyper-elliptic so that elementary particle vacuum functionals vanish in the lowest order for g>2. This might be enough to explain why g>2 families are so different.

The natural guess is that g>2 topologies are unstable and super-heaviness is more or less equivalent with the instability against pinching of the handles since lifetime is inversely proportional to mass. Z_2 symmetry would stabilize the lower genera against a rapid decay but higher genera would be unstable since Z_2 symmetry is not quantally possible.

A stringy view about the situation results when one considers extrema of Kaehler action of form X^2xY^2. X^2 subset M^4 and Y^2 subset CP_2 are minimal surfaces. Assume X^2 to be an open string with particles at ends labelled be genus of Y^2. Y^2 is any complex sub-manifold of CP_2 by general results about the connection between calibrations and minimal surfaces. There is a hierarchy of surfaces Y^2 defined as zeros of polynomials of two complex coordinates of CP_2, and degree and genus order naturally these surfaces with S^2 at the bottom, somewhat like states of harmonic oscillator. I expect that string tension is an integer multiple of the tension for Y^2= S^2(homologically non-trivial geodesic sphere). The decay process would gradually reduce degree and genus and the outcome would be X^2xS^2 strings with lowest genus particles at their ends.

Best,

Matti Pitkanen

2. That's an interesting paper and I am going to take the time to read it somewhat carefully soon. For now I'd like to comment on your thoughts on it, which I find most inspiring. It is my first impression that much of the motivation for the paper's subject you mention, Lubos, is based on your own thoughts.

Why should one study string theory with eight macroscopic dimensions on a genus g surface to begin with? Clearly, for phenomenological reasons. ;))

Just kidding, of course. One serious answer could be that anything that's part of the theory of everything is interesting. We have to collect as many technical results as we can because, obviously, our current understanding does not suffice to make quantitative contact with nature.

Along the way it may be important to respond to different ideas, like the landscape, which are somewhat incompatible with our idea of what string theory is really capable of.

This is why I enjoyed reading your comments. Destabilization, I find, is quite an interesting idea to kill off the landscape business. Another may be that the web of dualities becomes so dense on complicated CYs that in the end most of the gazillions of vacua are just dual to one another, even if we don't see it yet. A third is that there is a subtle dynamical mechanism that will eventually tell us that most of the vacua we thought define consistent theories are really anomalous and hence don't quite exist.

I personally like the 1st and 3rd idea, because I feel there is quite a bit of room for them and they keep alive the hope that string theory has *one* physical vacuum, which is our universe. Let me briefly say what I have in mind.

A lot of really wonderful research has been carried out in 2 decades of string theory and the best stuff was the physically inspired, not overly rigorous and technical. I believe this will continue to be true. However, the subtleties left aside may end up mattering. Who tells us that in the mathematical niceties, that nobody has understood yet, there isn't a subtle mechanism hidden, poised to kill most of today's vacua?

To be concrete look for example at the continuation from Lorentzian signature space to Euclidean space that we have to perform to define the Polyakov path integral. This construction is a crutch; it allows us to walk but we can't run or even stand straight. What exactly do we loose in the analytical continuation? Well, calculate for example the background field equation for an electric field using a beta function calculation. These field equation are nice, too nice, in fact, namely free of tadpoles. Too bad -- Schwinger explained that they shouldn't be: The electric background induces vacuum instability. One can track back where this loss of a major physical effect occurs. A Lorentzian metric (on the world sheet) typically defines a sense of world sheet-time orientation. Often it is true that a Lorentzian metric is equivalent to a Euclidean counter part plus a global time-like Killing vector field. The latter is completely lost in the naive continuation and it is the part that encodes information about particle fluxes due to vacuum instability. Remedying this, I believe, is a very difficult task, part of the difficulty stemming from the words "typically" and "often" I used above. ;))

But isn't it quite reasonable to think that Schwinger's effect may be important in addressing the question of whether the landscape is stable? Isn't it also conceivable that doing the Polyakov path integral right may give rise to new and subtle effects restricting the number of vacua?

I intentionally mentioned an example from good old world-sheet theory. After all, despite the impressive progress due to space time ideas, this remains our most reliable definition of string theory to date.

That's why string theory has never been more exciting and promising than today. It is tremendously successful and yet there is so much more to be discovered about it.

Maybe we should stop arguing with "loop quantum gravitists" and invest every hour we have into the real deal.

Best wishes,
Michael

3. Again I appreciate the visualizations.

Interplay between geometry and topology? It is very interesting to see this developing view.

Are we utilizing a predictive feature(what is it string amplitiudes)? in relation to tachyon condensation?

4. Hi Lubos,

Do you actually have any evidence for your idea about destabilizing the landscape by decays into simple Calabi-Yaus, or is just your usual pure wishful thinking.

Good to see that your commenters are sometimes as off-the wall as mine:

"the web of dualities becomes so dense on complicated CYs that in the end most of the gazillions of vacua are just dual to one another, even if we don't see it yet"

"That's why string theory has never been more exciting and promising than today. It is tremendously successful..."

Pretty hilarious stuff....

5. Peter,

if/when I had a quantitative evidence with detailed geometry, I would probably publish it, and not just on a blog.

This proposal is just a sketch of my general feeling that "convoluted" vacua simply won't be generated by the early cosmology, as we will once understand properly.

I admit it is a belief to a large extent, but the idea that all convoluted vacua are comparably likely like the simple vacua - and therefore the anthropic unpredictability is the issue for the day - is also just a belief.

Best
Lubos

6. Dear Michael,

I enjoyed reading your comment. By the way, just to be serious: the motivation for these papers is purely a conceptual one - to understand tachyons in string theory including the non-realistic vacua. I don't think that the authors were driven by phenomenological motivation. But I agree that it is an important motivation for us.

The Schwinger production of D-branes is something that the landscape modelers do study. It's controlled by an instanton representing the instability with respect to a spherical nucleation of a domain wall, which changes the amount of flux inside. This process itself can be kind of controlled and suppressed because the instanton's action is large. On the other hand, spontaneous topology change of a general kind is something that, I believe, has not been studied satisfactorily for the garden variety landscape vacua.

So far I have not understood your analytical continuation stuff, but it's probably because of the limited time I had for it.

All the best
Lubos

7. Dear Matti,

what you write is very interesting but I don't have the necessary background. What is the TGD framework? References welcome.

Best
Lubos

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10. but they also realized the power of using visual imagery to represent mathematical symbols.Ciara MuldoonAs soon as you used the word Tachyon, the anti- would automatically jump on the band wagon. Nobody questions the intelligence of this anti position, just why string/Mtheory should not be so?

We already have the amazing Randi.:)

Dialogue to debate the essence of why this math should not be used would be more constructive, and save a laymen from having to venture into further illusions, if that is what is felt.

There is no doubt in my mind that we need the physics to verify. Sean asks a good question in his blog about metaphors, and such leading math minds can readily say, why this vision is not a good one?

If you are going after mirror symmetry the anti- position is not doing very well right now with no arguments to back it up.:)

Michael might have been right about Loop, but there are intelligent people working in "all areas," that might debate the essence of these views constructively?

11. Hi Lubos,

let me explain in a few words what I meant. When you write down the geometric action of a point particle, it involves a square root of the displacement squared along the world line, something like

S = \int DX \sqrt{\dot X^\mu \dot X_\mu } ,

where the dot is the derivative with respect to the world line parameter. If the metric that raises/lowers \mu is Lorentzian, then the square root is zero whenever the path becomes time-like at any point -- as it does an arbitrary number of times in a given path integral configuration -- and whenever that happens you can choose the sign of the square root as you please. Just fixing this set of signs from the out start is not an honest way of dealing with this issue.

One can see this more clearly. Introduce a world-line einbein so that the action becomes quadratic. A moment's thought will convince you that the sign ambiguity in the original action is reflected in the question of whether the particle is moving forward or backward in its own proper time according to the einbein. If you just fix the sign, the direction can never reverse, but summing over all configurations requires the number of orientation changes along any given path to be arbitrary.

Now if the metric that raises/lowers \mu is Euclidean, than the issue of this sign ambiguity never arises to begin with. You just loose the sense of time-orientation on the world-line.

One can now argue that in the Euclidean version vacuum instability issues such as Schwinger's effect are simply absent. (I won't get into details here.) Vacuum instability seems to be reflected in the path integral formulation by the fact that trajectories that do not admit a sense of time-orientation, in the sense explained above, make a non-zero net contribution.

I don't have to explain to you that all these issues carry over to string theory; some issues are even much exacerbated there. In fact, it is much harder to make sense of the sense of time orientation on a world-sheet, but presumably this is important. As you know, Schwinger's effect is absent in beta-function calculations using the standard analytical continuation. After these explanations, even Peter Woit can guess why that is.

Maybe one should supplement the Euclidean (world-sheet) metric with a Killing vector field that defines the time orientation. But how exactly do you incorporate it into the action? Worse: Such a Killing field need not exist.

Best wishes,
Michael

12. Oops, I just saw that I mangled the one formula I typed. I meant of course:

S= \int DX \exp ( -i m \int d \tau \sqrt{\dot X^\mu \dot X_\mu} )

DX is the path integral over all paths the particle can take, and \tau parameterizes the world line, and m is the particles mass.

--Michael

13. Dear Michael,

are you arguing that the Schwinger effect does not exist? It definitely looks so.

The Schwinger effect, or various vacuum decays etc., are easy to describe with an exact solution in the Euclidean spacetime. The Euclidean path integral is the better behaved one in which your various subtle issues about the sign of the action - and imaginary actions for spacelike paths etc. - do not arise. The Euclidean path integral is just a mathematical trick, and one must be very careful how this solution is interpreted in the Minkowski spacetime.

The right interpretation is that there is a nucleation of a bubble whose boundary moves nearly by the speed of light. Inside the bubble, you have the "new", energetically favored vacuum - it is the same vacuum that you find near the origin of the Euclidean solution. The solution is only relevant for t>0. For t<0, you assume that the spacetime is in the unstable phase.

The full calculation in the Euclidean language certainly does not require the spacetimes/diagrams to have a globally well-defined orientation. If it were so, a sphere could not contribute to string theory amplitudes because you can't comb a sphere.

Moreover, the non-gravitational instantons - signs of instabilities - including the Schwinger pair production are described by the usual, causally trivial Euclidean spacetime, so you can define the orientation very easily.

Best
Lubos

14. Hey Lubos!

Am I denying the existence of the Schwinger effect? I do not think so.

Sure, you can see Schwinger's effect in Euclidean space in certain descriptions. One can use instantons. And Schwinger reinterpreted an imaginary vacuum energy density as the pair production rate.

Can we do the same in string theory? Of course. hep-th/9209032, e.g., studies a theory of open strings, with Chan-Paton charges at their ends, in an electric background and in a limit where gravity is decoupled. They found that the effective string action also picks up an imaginary part, and interpreted that as the string production rate following Schwinger.

So everything is nice and dandy, right Lubos? Just turn gravity on! Now what? Are you going to couple gravity to your imaginary energy density? Gravity does not care about Schwinger's reinterpretation. Imaginary energy densities are just fatal in a theory of gravity.

The effective string action is not the place where you want to see the effects of vacuum instability. One should expect to find this as tadpoles in the background field equations. Anyone who has studied beta function calculations knows that there are no such tadpoles in the theory obtained from standard analytical continuation.

This problem has its origin in the issues I described in my previous posting.

You say one has to be careful interpreting Euclidean space solutions in Minkowski space. It is simply not true that every aspect of physics in Minkowski space finds it counterpart in the Euclidean continuation. There are things that require more than being careful.

To summarize, in field theory the situation is satisfactory. An imaginary energy may seem awkward, but there is nothing wrong with it if you have the correct interpretation in mind. In string theory we do not have an appropriate description, unless you turn gravity off. But what is string theory without gravity worth?

Best wishes,
Michael

15. Hi Michael,

what you write almost sounds like Tom Banks. I am very interested in these comments - you say that low energy effective action is not a good starting point to study vacuum decay and vacuum selection issues. Could you please be a little bit more specific what are "your"/the new rules (from string theory and quantum gravity) and especially how they differ from the effective rules?

Concerning your last paragraph. The goal of string theory is not to be always different, but to predict physics in the unification regime. So your slogan "what would string theory be worth without gravity" has nothing to do with the question whether gravity in string theory qualitatiely changes the predictions of the instabilities from the low energy effective action, or whether it preserves some of them and which etc. String theory, when calculated properly, has a unique answer to this question - at least unique in every particular context. And we can't find this answer by ideology, I think. ;-)

I don't know why you mention imaginary energy density (in Minkowski space?). In my viewpoint it is unphysical and I've never used a concept like that. Should I have?

All the best
Lubos

16. FYI, perhaps no more baryonic dark matter problem?
http://www.wired.com/news/space/0,2697,66487,00.html

17. Hey Lubos,

I agree that Tom Banks' arguments overlap with what I am trying to say. It is my opinion that the background field equations should not just relate classical fields to each other, but should be equations relating fields and fluxes. That's what I would expect to be the natural description of quantized matter plus gravity and it seems reasonable to me that a more careful definition of the Polyakov path integral would reveal just that.

Just to make the connection to the origin of this discussion: If I am right, don't you agree that this might have far reaching consequences for the number of stable vacua and related issues?

The imaginary energy density is standard in field theories with unstable vacuum. Look at Schwinger's pioneering paper

Phys. Rev. D82, #5, 664 (1951).

This paper is so beautiful that it's a must read anyways.

Best wishes,
Michael

18. Michael, that classic Schwinger paper has well over 2000 citations. Do you know if it is available to view on the web somewhere? I simply don't have access to Physical Review that far back. Anyone with a pdf of it could email it to me at sdmedit@hotmail.com. Id like to see it but cant unless someone emails it or locates it online somewhere. The paper with the string version looks nice too. Lubos, maybe you could say a bit more about your idea of destabilising the Landscape via decays into C Yaus?
Regards, Steve M

19. Michael said:
"The imaginary energy density is standard in field theories with UNSTABLE VACUUM. Look at Schwinger's pioneering paper"

I find such wording amusing! Can you give an EXACT definition exactly what is vacuum and what is not vacuum? What do you mean "unstable" vacuum? Is "unstable vacuum" a vacuum or not a vacuum?

If "unstable vacuum" is not a vacuum, you have been abusing the English language. If it is a kind of vacuum, then we have at least two kinds of vacuum, the usual stable kind we know, and your unstable kind. Then there could be some sort phase transition between these two kind of vacuum and it can be observed. Then, the whole thing is no longer vacuum by the very definition of vacuum!!!!

There could only be just one kind of vacuum and that's the stable kind, by the very definition what is vacuum. Anything else is NOT called vacuum! It's not even physics. It's pure linguistics and logic!

The "imaginary energy" is nonsense. Put it into general relativity it means we could have spacetime curvature which is imaginary. What does imaginationary curvature mean? It doesn't even have a mathematical meaning!

Quantoken

20. Here is a criticism of the so called Tachyon:

Quantoken

21. Quantoken,
an unstable vacuum state is defined as a state that is the legitimate physical vacuum state of a given instant in time, but evolves non-trivially under the Hamiltonian action. It will therefore look like a state containing many particles at a later time with respect to the physical vacuum of that later instant. Vacuum states of different times are typically related to one another through Bogoliubov transformations. There is nothing "funny" about this.

There is indeed a conceptual difficulty in defining the physical vacuum in dynamic curved spaces. I recommend the papers by Leonard Parker on this issue for those who are interested. Anyway, this was not the subject of our present discussion. It also wasn't what you had in mind, Quantoken, because you didn't know it until I told you. I am explaining this to you although I know that you have no appreciation or understanding of these concepts. You are just here to heckle and I wish you weren't.

--Michael

22. Steve M, your hotmail account is apparently too small for the paper. I tried to send it and got an error message back.

Best, Michael

23. Hi Michael,
Sorry about that! I forgot that email account gets blasted with spam everyday and rapidly fills up! I have'nt cleaned it out in a few days! Can you email to
smllr@titan71.fsnet.co.uk

Many thanks! It is much appreciated.
Steve M

24. Quantoken...
"A fool may be thought wise and intelligent if he keeps his mouth shut"
Proverbs.17:27

25. Michael said:
"an unstable vacuum state is defined as a state that is the legitimate physical vacuum state of a given instant in time". Bla bla bla.

I am sorry. I am asking you to give an exactly definition of what is vacuum and what is not vacuum. You have NOT give me the definition of vacuum. Until you can define precisely what is vacuum, there is no point discussing vacuum state. I could well say that a space containing one photon is a vacuum state that is excited. You could also expand the concept and call everything an excited or unstable vacuum state. This is actually all theoretists have been saying. There are plenty of talk of various "vacuum states" but no one ever seem to mean anything about none-vacuum. Sure according to Penrose the whole universe is just one big bubble excited from a vacuum state, using "borrowed energy", as if you can borrow 10^100 Joules of energy for 1.4x10^10 years, without breaking the energy conservation law, and not even have to pay interest either :-)

Give me a precise definition of vacuum please. Or at least give a criterial to distinguish vacuum and none-vacuum.

Quantoken

26. Quantoken,
The physical vacuum state is the one annihilated by all lowering operators of the full (interacting) quantum fields.

--Michael

27. Hi Michael,
I don't wish to be a nuisance but if you could have a go at emailing the Schwinger paper again sometime it would be well appeciated.
Best regards
Steve M

28. Quantoken does'nt know about how to define vacuum states since he has never actually understood or even opened a quantum field theory book. His not having encountered a field theory book is not surprising really since in most bookstores the field theory books are generally not to be found on the same shelves as the coloring in books.