At The Monastery

By cjohnson | July 6, 2006 12:21 pm

CIRMI’ve decided to live in a monastery for two weeks, along with several other physicists and mathematicians. It is quite pleasant. We have breakfast at 7:30am, lunch at 12:30pm, and dinner at 7:30pm. The latter two are proper sit-down affairs with several courses. We are served by several stern French women of an appropriate level of uncompromising sternness and impatience, who are -frankly- slightly scary. I’m looking forward to tonight’s dinner especially, since the Thursday night special is one of my all-time favourite French dishes (and word)… Bouillabaisse. It is a local specialty. By now you’ve probably guessed, I’m in Provence (which I like to think of as a sort of prototype of Southern California, but that’s another story). In particular, just outside Marseille, on the edge of the Mediterranean. The monastery is in fact the Centre International de Rencontres Mathematiques: C.I.R.M., as you can read on the doorway. Each day I wake up, eat, go to lectures, work in my room, work in the library, talk to people, and work some more. Apparently it is quite beautiful in the surroundings, but I’ve not gone to look yet. Maybe I will on the weekend.

So what have I learned? Well, the setup is excellent. There are both mathematicians and physicists here, and a wide range of talks. In fact, the title of the workshop has “Affine Hecke Algebras” in it. If you stopped me in the street and asked me what an Affine Hecke algebra is, I’d have to tell you that I don’t know what the Helle they are in any precise sense, but that does not matter. I’m learning a lot about what’s been going on in some aspects of the physics and mathematics world in some areas, with a pleasingly European flavour which reminds me of my youth, since a lot of the European string theory and statistical mechanics stars from my past (gosh, almost 20 years ago!) seem to be here. At the same time, to my surprise, several relatively freshly minted young researchers that I know from over the water are here. It is nice.

There have been several excellent talks. Particularly so since the physicists have been careful to spend the first half of their talk giving a general introduction for the benefit of the mathematicians before diving into technical details. As a result, while I am not sure that the mathematicians really benefit from this as much as is hoped, I think it just makes for a great 90 minute (with 5 minute break) physics talk for the physicists. As a result, you get a good idea of what the issues (at least in the mind of the presenter) are, and a bit of the background and history, before they go on to tell you what they have actually done recently.

For example, Martin Schnabl just reported on what can only be described as spectacularly important nice results (see the paper, hep-th/0511286, here). Here he is, in action at the lovely multi-panelled blackboard which he used to great effect:

Martin Schnabl

D-brane Among other things, he has managed to prove -exactly- Ashoke Sen’s conjecture (see a review, hep-th/0410103, here) that in open string field theory, the famous open string tachyon instability indeed moves your vacuum from the one containing an unstable D-brane to a stable one corresponding to no D-brane, and that the energy difference between these two vacua corresponds to the mass-energy of the missing D-brane. (I’ve explained what D-branes are in a previous post, here. Diagram on left.) The picture to have in mind is that the system describes physics moving in a potential energy curve, V(T) , that looks like this:

tachyon condensation

The usual open string theory contains a “tachyon” field T which describes a negative mass-squared excitation (or particle, or vibration mode). Such modes simply mean that the vacuum is unstable, and the theory wants to go somewhere else. It corresponds to living at the top of the local hill on the curve. The theory is unstable to rolling off the top and settling into the stable vacuum, at the bottom of the local valley. Being at the top of the hill, the theory describes open string excitations corresponding to a D-brane (a D25-brane to be precise…the strings’ ends can be anywhere in spacetime), and the idea is that the bottom of the hill contains no D-brane. It has gone away completely, leaving only the 26 dimensional closed string background. The mass-energy of the D-brane is soaked up in the potential energy difference in moving down the hill. (Cute. That Sen guy makes nice, sharp, conjectures, doesn’t he? That’s just the start of the story -there are other components to the conjecture- but I’ll just focus on this aspect for now. There is a question mark at the bottom of the hill since it was not established exactly how this is to be described in its entirety, but the starting point is the String Field Theory.)

Previous work in this area to check this so-called “tachyon condensation” process (starting with Sen and Zwiebach, but continued by many; see Schnabl’s paper for references, and the review of Taylor and Zwiebach, hep-th/0311017, here) over the last eight years or so was able to show that this was probably true, by working in various approximation schemes in the String Field Theory. The approximation schemes -coming from cutting off the frequency (called the “level”) at which the strings can vibrate in order to render their computer-aided computations tractable- were not satisfying (although surprisingly close to the correct answer even for crude truncations), but were the best that could be done.

Hmmm. For you to appreciate the enormity of the task, I should step back a touch and tell you what string field theory is. Remember, just like on a musical instrument, you can get strings to vibrate at higher and higher frequencies. Each of those separate modes of vibration correspond to a new particle in the theory (higher the vibration frequency, the “level”, the higher the mass of the particle), and specifically, there is a quantum field associated to such a particle that we have to incorporate into the physics. In ordinary physics, you have some number of particle species (say an electron and a photon, for studying the properties of electrons under the influence of electromagnetism….useful for getting the physics of your electronic devices right) and there is a quantum field for each species. Don’t worry about what a quantum field is, just know that the basic toolbox “Quantum field theory” is arguably the most successful computational tool -as applied to nature- ever devised, since it allows you to compute the properties of electrons (say) to an incredible accuracy which has been checked experimentally to something like fifteen significant figures. See the nice book of Richard Feynman called “QED” for an accessible account of this. Quantum field theory is what we use to build the Standard Model of particle physics which is tested at collider experiments, like the LHC to come. Search this blog for “Standard Model” and “LHC” for more.

Well, with the string, since it has an infinite number of modes of vibration (like your guitar, piano strings or violin, if they were perfect, in an approriate sense), there are an infinite number of particles, and hence quantum fields, to deal with in one go. So String Field Theory is an incredibly complicated piece of technology since it has an infinite number of fields. Nevertheless, you can write down (due to the work of early pioneers like Graeme Warren Siegel, and crucial understanding of the interaction terms (in terms of non-commutative geometry, star products, and other goodies) given by Edward Witten, all back in the 80s… and several other workers who will hate me for not mentioning them, but since they do not read blogs I can safely refer you to the references in Schnabl’s paper) a remarkably elegant and simple equation (the “action”) for the theory:

open string field theory

([Cheekily snipped from Martin's paper.] That’s just to show the non-expert that we write down real equations, and are not making stuff up -at least not here. I’m not going to explain this equation, but just thought you’d like to see what the basic object looks like.) Simple as it is (yes, in the scheme of things, really it is) for many questions (such as checking whether the structure of the vacuum is as Sen said it is) the physics of this equation is hard to study exactly, because of this business of having to deal with an infinite number of fields (hidden in the object denoted by the Greek letter psi) in the computation. This is why people have truncated the theory, hoping that the neglected high mass modes don’t affect the theory as much as the lighter modes. This turned out to be correct – a simple truncation to keep only the first two levels gets the answer (for the mass of the brane) 97% correct, and a few more levels gets you to 99% and beyond pretty quickly. (I’m quoting the content of the last sentence from memory, so I might be a little off. The exact number of levels does not matter to the point, really.)

But Schnabl has shown how to get the whole thing exactly, analytically. No computers involved, no level truncation. He works with all the infinite fields elegantly at once, if you like. He does this essentially by a clever choice of basis for his space of states in his conformal field theory computations, so that when he comes to evaluate the action, he can write rather elegant closed forms for the exact solutions for the equations of motion. (As an aside (and amusingly, the thing that caught the eyes of some of the mathematicians more than the actual physical result), if you’re a fan of the Bernoulli numbers, he’s found new identities for them that are quite novel and probably significant mathematically. The solution involves them explicitly.)

I’m not going to describe the technical details since that would be pointless here. Read the paper, if interested. Feel free to discuss it, and conclusions you draw from it, at any level you wish to in the comments.

Let me close by noting that, to me, this is a very significant result in the scheme of things. We’re ultimately looking to describing Nature with strings one day, and we need to understand the space of solutions of the theory. What are they? How many are there? Etc. This is very hard to explore with certainty, even in a simple theory such as the 26 dimensional vanilla open string theory which was the context above. We’ll need to do it for much more complicated string theories, and maybe even away from the point where the strings are weakly coupled and easy to identify as the appropriate degrees of freedom.

The above example is a rare case -especially in higher dimensional or “critical” string theory- where an exact computation allows one to track the dynamics of the vacuum of string theory in an off-shell formulation. (It is possible in several cute toy models of non-critical strings in low dimensions, such as those I’ve described in other posts here and here). This is the sort of thing of which we need a lot more before we have believable control over statements about what non-supersymmetric vacua of string theory are really up to. Until we have control over these issues, we’ve got to be very careful about what we’re doing in this context, taking explorations of such vacua -and phenomenological conclusions which may be drawn from them- as interesting, but highly tentative.


CATEGORIZED UNDER: Personal, Science
  • Sean

    I didn’t know about Schnabl’s paper, although I was aware of the issue with the decaying tachyon. Am I allowed to use my field-theory intuition to guess that, since the potential energy has changed, the background geometry has as well? (From Minkowski to anti-de Sitter, maybe?) Also, if we considered a real dynamical process, the potential energy wouldn’t just disappear, it would undergo a phase transition to a gas of closed strings (as in reheating after inflation). Is this an explicit part of the story, or are we artificially changing the tachyon by hand rather than conserving energy? (Nothing wrong with not solving the equations of motion explicitly, if you’re mostly interested in what the end state looks like.)

  • Moshe

    Sean, the energy from the decaying brane indeed finds itself in closed string radiation. In principle that radiation would back-react on the geometry but working in perturbation theory (in the string coupling, equivalently in h-bar) this back-reaction is pushed to the next order (namely *quantum* open string field theory, which is much less explored).

    Clifford: nice to have you back! Looks like really impressive result, I am wondering if it is better understood why the level truncation worked in the first place, but maybe I should go read the paper…

    (Also, side comment – the Siegel gauge is Warren’s, I believe…)

  • Cynthia


    Sad to hear that you are trapped in confinement. :-( … Nonetheless, glad to see that you are waxing more and waning less on D-branes. :-)

    Best wishes,

  • Uncle Enzo

    Forza Italia! Italia beats Germany! Italia goes to World Cup Final! Forza Italia!

    Forget physics. Forget string theory and loop quantum gravity. Let’s talk about Italia! Forza Italia!

    Italia all the way, baby! Forza! Forza!




  • Clifford


    (1) Thanks for the correction. I always get that wrong.

    (2) Contrary to popular misinterpretation, I had not actually gone anywhere, but it is always nice to be welcomed back.

    (3) Your comment to Sean about the closed stirng issue is spot on. Further, I should add (for Sean’s benefit since I guess you know this well) that the issue of properly identifying the closed string degrees of freedom that now should be there as quanta of the system, and verifying that the open ones have really gone, are commented on (I have bene told) in the paper, although I believe that there is still a lot to be understood there.

    The appearance and disappearance of the closed string stuff after condensation is something which I think is still way better understood in the non-critical string models. There, the closed string degrees of freedom are just “ripples on the sea” of double-scaled matrix eigenvalues sloshing around at the bottom of the potential. The creation or destruction of a D-brane is simply the raising or lowering of one of these eigenvalues into or out of the sea and placing it at another extremal point of the potential.

    Cynthia: I’ve still no idea what the waxing and waning to which you refer to is about, but thanks anyway.



  • Hmmmm

    Now, when I was at that same conference, last week, there wasnt a blogger in sight. Guess thats the difference between a math & physics conference…

    PS The beach is only a few minutes away… so though too many talks are scheduled, you can usually get a swim in between the last talk and dinner.

  • Clifford

    Well, maybe I’ll go and take my sketchpad or notebook and dip my feet in. I heard about the beach. Pleasant to go on a walk to it while having a think. Will pass on the swimming, I expect. Don’t have the equipment, for a start.



  • Hmmmm

    Didnt think there were that many mathematicians left this week. Kazhdan, vocally, I’d expect. Any others?

  • Clifford

    Several. But I don’t know most of them. Just had lots of fish with Ron Donagi and Katrin Wendland…. and there are others.


  • wolfgang


    two questions somewhat related to the question of Sean.
    Can this process perhaps be used to explain why the extra dimensions are compact (at least why the background cannot be flat)?
    And what about the closed string tachyon?

  • Clifford


    Some people have speculated about this for years. Needs to be worked on. Also, some earlier speculations about finding the heterotic string as a vacuum reachable from rolling from the bosonic string theory ought to be ket in mind. See papers by Englert, Houart, and Taormina in 2001 and 2002…. maybe their ideas can be given dynamical accompaniment?

    The clsoed string tachyon is less well understood in general, but there are some cases where a pure closed string instability of a vacuum can be seen to be radiated away….. See for example Adams, Polchinski and Silverstein, in 2001, for example. I don’t know if anyone has said anything about the flat space closed string tachyon of the bosonic string, but it is just begging to have a clean statement about it, if not. If only we had tricks as nice as Schnabl’s for the closes string sector, we could simply compute the analytic form of the energy given up in going to a neighbouring potential well and stare at it and see if it says anything to us…..


  • wolfgang

    > the heterotic string as a vacuum reachable from rolling from the bosonic string theory

    Thank you for the link(s), looks very interesting.
    M-theory = bosonic string 8-?

  • Stephan

    Clifford, it really is nice to have you back posting again.

  • Plato

    My sentiments as per #13

  • Plato

    Oh, and I had an slide from a ole lecture from 2001 that I thought I would link for additional information, if it is appropriate?

    A false vacuum is a classically stable excited state which is quantum mechanically unstable. In the quantum theory, matter which is in a false vacuum may `tunnel’ to its true vacuum state.

    I find it hard to reconcile “such thinking” so “mathematically endowed” with actual processes. I guess that is why we can learn so much from the blogopshere? :)

  • Nick Halmagyi

    people should feel free to convince Clifford that we should hire a couple of Vespas for the weekend. He doesnt seem to appreciate just quite how swank and cliched we would look in the south of france on vespas…

    attacking his sense of adventure I think would be an excellent starting point…

  • Clifford

    Stephan, Plato, thanks.

    wolfgang: -as as to your last sentence… it has been said before. It is an intriguing thought.


  • Clifford

    Bikes.. how about hiring some nice bikes? ;-)


  • Nick Halmagyi

    dont get me wrong, Im a big bike fan. I dont think I need to defend my credentials there…but…Im just not sure that when France wins that funny little soccer game on sunday, doing laps of the town fountain on a Brompton will really cut it.


  • Andrei Sobolevskii


    There is a nice English-language website on where to bike and what to see in Provence:

    Check the Itineraries section.


  • Cynthia


    Just when I have become fairly comfortable with the “high-altitudes” of the 10+1 dimensions, you are ascending to the “nose-bleed-altitudes” of 26 dimensions!

    What’s up?

  • Ambitwistor

    So what do affine Hecke algebras have to do with statistical mechanics?

  • Amara

    Is it possible to rent a bike there, and do you have time? If the answer to both is yes, let me recommend a some interesting places that I think are fantastic to visit by bicycle. You can travel to St. Remy de Provence, make that your base to take a couple of day rides to see Les Baux, Glanum: the Greek/Roman archeological ruins and the Ancien Moastere de St-Paul-de-Mausole, which is the monastery turned convelescent home and surroundings where van Gogh spent the last year of his life.

    Les Baux is great to visit by bike because it a vision that sneaks up on you. On my own bicycle trip there, my small road twisted and turned through the cliffs. I remember climbing on the twisty roads not being able to see very far around the mountains, and climbing, and try to see, and then I rounded a turn, and …. looked closer at the mountain in the distance …. and looked again. It was there! The Les Baux village is so well-blended into the rocks that one is not really quite sure that it is sitting there, until one spends some long minutes staring andadjusting one’s eyes. When I was sure that I saw it, I immediately pulled over my bicycle, and I found a good viewplace to sit and I watched Les Baux for about a 1/2 hour. Some cars raced around the curve and the occupants didn’t even notice the glorious sight in the near distance, because they were travelling too fast.

    Les Baux de Provence is an old village and castle ruins located on a detached (cliffs on all sides) portion of the Alpilles mountains. This village and its attached castle was built in the 12th (10th?) century by the warrior lords of Baux. The history of the place contains small and large battle skirmishes between the powerful ruling powers but Les Baux was also famous as a court of love in the 13th century. To become a member, the women had to be of noble birth, well-read and beautiful. Troubadours, often high lords, came from all of the southern provinces and composed passionate verses in praise of these ladies. The prize awarded to the best poet was a crown of peacock feathers and a kiss from the lady in question.

    The “plateau des antiques” where the Roman Monuments: Mausoleum, Arc Municpal, and the archeological site: Glanum is located is pretty cool but what I think I enjoyed more was 15 minutes away by foot: the convalescet home seeing the landscape/environment that inspired van Gogh and where he spent the last year year of his life.

    The psychiatric convalescent home is still in use, but the owners have set up rooms for visitors interested in learning more about the place, and about Vincent van Gogh. Also, from the views from the windows and from walking around the gardens and the rest of the home, you can gain same feelings for the Provence area that interested van Gogh so much. I remembering seeing the same mountains that he saw, similar twisted olive trees, the same beautiful yellows and blues and purples. Provence is colorful, and perhaps the area around St Remy, even more so. This last year of van Gogh’s life was his most prolific painting time- He painted about a hundred paintings: many colorful natural seens, self-portraits, and he painted the “Starry Night over St. Remy”, which is a favorite of every astronomer I know.

    “Everything is extraordinarity beautiful here! Everything and everywhere the color of the sky is an admirable blue, the sun shines like pale sulfur, it is sweet and charming like the combinations of blues and yellows in the skies of the Vermeer of Delft.”

    My favorite van Gogh quote is this one:

    “Don’t forget that little emotions are the great captains of our lives.” (Vincent van Gogh, Arles, 1888).

  • Clifford

    Hi Amara,

    The answers are “yes” and “not much”, so I will have to do this all some other time. Thanks to you and Andrei Sobolevskii for your suggestions.



  • Amara

    Oops, I left out the author of the previous quote, it was van Gogh too (writing to his brother Theo in 1888).

  • Clifford

    Cynthia, I don’t understand your question. The 26 dimensional bosonic string is a very common model to study. The first one most people cut their teeth on, in fact.

    Ambitwistor, I am not an expert -or even close- in this, and so I’ll leave it to someone else to give a better answer than I can.



  • Cynthia


    My bad… I wrongfully assumed that 25+1 dimensions were declared obsolete during the rise of the second superstring revolution. More specifically, I erroneously thought that this revolution announced that all string models merge together to form a unified 10+1 dimensional model called M-theory. However, I gather from your message that bosonic strings are not part of this unification and still uniquely remain in the 26 dimensions.

    Regardless of my misunderstanding, enjoy teething on 26 dimensions… :-)

    Have a nice weekend!

  • Clifford

    Cynthia… dimension is an elusive thing in string theory. See my comments in replies to others above about how the 26 dimensional bosonic string may not be unconencted to ten dimensional string theory. We don’t have a good handle on the possible vacua and dynamics of string theory yet…. there may be a lot to learn about how dimensions may change….


  • i

    I really like the last VVG quote: it is so true!

  • John G

    Horowitz and Susskind even wrote about the 27-dim bosonic M-theory:

    If lots of things get figured out using the bosonic string, people may not notice if they still think it’s a toy. The reasons for going away from the bosonic string may not be valid concerns any more. There’s even a 28-dim bosonic F-theory.

    The 26-27-28-dims relate to E6-E7-E8.

  • Plato

    IN light of “current information” with regards to the QGP, does one not think the “singularites” have to be rewritten?

  • Plato

    Can orbitals be held “consistently” in relation to the idea of “fermions” held to the brane?

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