The 9/11 Flip

Dijkgraaf Verlinde Verlinde 9 -11 flip figureOk, so here goes. A bit of physics linked to this all so significant date. There’s this term that people in string theory were using a lot in the middle to late 90’s, called the “9/11 flip”. I think maybe the Verlinde brothers, Erik and Hermann, possibly in conjunction with Robbert Dijkgraaf, made the term popular but I am not sure about that and I welcome a correction. [Update, since there is some confusion: I’m talking about the term here, not the technique itself, which is older.]

(On right (click for larger) is a snapshot of one of the figures from their influential 1997 “Matrix String Theory” paper. You can see the use the term there, and it is in the paper’s text too, and soon everyone was using it in seminars and other papers to follow.)

The flip became particularly useful when people were discovering the wonders of “M-theory”, which is the catch-all term for whatever is the parent theory of string theory, something we are still trying to formulate. There are a number of narrower uses of the term, however – some more justified than others. For a while, everyone was thinking about the five ten-dimensional supersymmetric string theories (“type IIA”, “type IIB”, “type I”, and two types of “heterotic” theory) which, prior to the middle of 1995 (actually as early as late 1994 or early 1995) all seemed like totally distinct theories, and in the post middle-1995 era (after Witten’s remarkable talk at strings 1995 here at USC: paper here) were recognized to be all part of the same theory. The universe (at least the continually expanding string theory one) changed radically overnight.

m-theory puzzle
(One of my preferred ways of presenting the puzzle that is M-theory, and how the ten dimensional string theories fit in the puzzle. This is a slide from one of my general talks on the subject.)

The 9/11 flip is really simple, although when a setting is complicated enough, it can end up showing that two very distinct settings, with rather different physics, are related to each other. This was the gift of M-theory that made many things so clear. d-braneOne of the lessons that began in 1994 and early 1995 (from excellent work by Hull and Townsend – see here and here) and was cemented in Witten’s mid-1995 work was that higher dimensional extended objects are really important in string theory, and crucial if you want to understand what the theories really are (in fancier terms: if you want to understand the theories “non-perturbatively”). In other words, not just strings, but also membranes and higher dimensional “branes” (the latter now being the general term). You’ve got to understand and incorporate how those things can wiggle and move about – their dynamics – and incorporate them into the theory (as Polchinski showed how to do later that year for a large class of such branes, the D-branes. I’ve posted about them here a lot).

It is using these objects that one can see that the five theories are all connected. Once you know they are connected, it is easy to see that branes are not just along for the ride, but absolutely forced on us, everywhere one cares to look in the theory, dooming strings’ prominent position in string theory, and essentially telling us that string theory is not a theory of strings. (I was pleased to co-author -with Kaloper, Khuri and Myers – an early paper that year on the subject, helping make the point with such a computation.)

The other key thing that helps us put the picture all together besides the other extended objects, is an additional dimension. These five string theories are all ten dimensional theories, in their simplest form. It turns out that they secretly all know about an eleventh dimension. When you allow for that extra dimension and think of them as eleven dimensional theories, you don’t get five new eleven dimensional theories – you get a single eleven dimensional theory. This is where I differ with many on what the parent theory ought to be called. Many call that eleven dimensional theory -a simple highly (super)symmetric eleven dimensional theory of gravity called “eleven dimensional supergravity”- many call that M-theory. But that is not really right. It is at best a low energy approximation to M-theory, since we know that theories of gravity on their own are incomplete as quantum theories, and we were only led to it in a low energy limit. But anyway, people often call 11D supergravity M-theory, and I guess we are stuck with that until we learn what M-theory (the fully formulated parent theory) actually is.

I digress. Technicalities aside, what do we have? Strings, membranes (2-branes), 3-branes, and so forth in various ten dimensional string theories, which if you look at them the right way, are actually all part of the same theory, a fact that is at least partly explained by adding a dimension. From 11D supergravity, we could understand the five different string theories – they were just the result of doing five different things to the 11D theory that result in hiding one of the dimensions.

So what is the 9/11 flip? Well, it is just the business of exchanging the hidden (eleventh) dimension with one of the dimensions you were aware of. In this way, you end up seeing that two different theories are connected, or that two very different objects in a given theory are actually closely related. Let me give an example of the latter. The “type IIA” theory contains strings that are historically native to the theory (the type IIA string), and the simplest branes it contains are even dimensional (0-branes (a class of point particles), 2-branes (a class of membranes), 4-branes, 6-branes, 8-branes), along with another sort of brane called a 5-brane.

Let’s pick the type IIA string, and also consider the 2-brane. The point is that these are really arising from the same object: A membrane in eleven dimensions. You get to the type IIA string theory by simply curling up one of the eleven dimensional theory’s dimensions on a circle. It is that simple. Why did we not notice this long before? Well, we could only formulate strings at weak coupling (when they interact with each other weakly), and it turns out that the size of that hidden eleventh dimension sets the strength of the coupling of the resulting ten dimensional string theory. So as long as we stayed at weak coupling (or “perturbation theory”), we stayed in ten dimensions. Much the same thing works for the other string theories, but I won’t go into that.

Eleven dimensional supergravity contains just 2-branes and 5-branes. So you get to construct all the objects you can make in the resulting ten dimensional theories from these objects alone, or instead (or as well as in more complex cases) exploit aspects of the geometry of curling up a circle. I won’t do the whole story here, and just focus on how you get the type IIA string and the 2-brane. Well, to get the 2-brane is easy. It is just the object you have in eleven dimensions, now constrained to move only in the ten dimensions because you went and curled the eleventh one up into a tiny circle. Simple. A string in ten dimensions comes from doing something slightly different. When curling up your eleventh dimension, let the 2-brane lie with one of its dimensions in that direction, so that it will curl up too. Now you’re in ten dimensions, what does the membrane look like? A string!

This is not so had to imagine. Take a drinking straw. It is actually a two-dimensional sheet (a membrane, or 2-brane) that has one of its dimensions curled up on a circle. If that circle is very small, you don’t see it, and so the drinking straw (from far away for example) looks like a line – a string. That’s not really the whole deal though. We need one more step: Imagine that circle of the straw/membrane was instead curled in a direction that is not part of the three spatial ones we live in, but in a hidden fourth spatial direction say. Then it would not actually need to be made small at all, by the way – you’d just see the part of the membrane that lies in our three dimensions – and it would be a string. What’s going on in these M-theory constructions is both processes – the shrinking of the eleventh direction on a circle to give ten dimensions, and the wrapping of the membrane on that circle to make it a string.

The “flip” part of the procedure is just the business of changing things from one arrangement to the other. On the one hand, the 2-brane is not tangled in the hidden eleventh dimension, so it looks like a 2-brane in ten dimensions. But if you exchange (“flip”) the hidden dimension with the one of the ten dimensions along which the 2-brane is lying and curl that up instead, then you get a string. So I can get to the ten dimensional type IIA theory yielding either a string or a 2-brane, by just starting with a 2-brane in eleven. I get from one construction to the other by flipping back and forth between which of the directions I choose to be the hidden one or not: It is either along, or not along, one of the directions of the 2-brane. That’s the “flip”.

The term “9/11” itself is an accident of nomenclature and sort of a joke. We typically label the coordinates for the dimensions in physics starting with “0” for the time dimension. So the four dimensions of our observable universe are [tex](x^0,x^1,x^2,x^3)[/tex]. When you are working with a ten dimensional theory, you will end up with this list of coordinates going up to [tex]x^9[/tex], of course. Then came 1995 and people working a lot with another dimension. There was an issue of conventions. Some people just wrote [tex]x^{10}[/tex] and some people did not like that you needed two digits, which was a bit confusing when written down for example, so some groups (I think this might have started earlier in papers by people like Duff and his collaborators – they’d been doing a lot of work that can now be seen as pre-M-theory work on M-theory, going back to the 80’s!) started using other symbols for ten. Things like [tex]x^{\sharp}[/tex]. Regardless of how you wrote it, people started saying “the eleventh direction” a lot, and sometimes you even saw (but not too often, thank goodness) the use of [tex]x^{11}[/tex]. But I think it was because of the term “eleventh direction”, and because we typically (for no good reason) first think of curling up (or “compactifying”) the directions that have the highest labels in any discussion of removing or hiding extra dimensions, the term “9/11 flip” arose naturally. You swap the ninth direction (which is really the tenth, but labelled [tex]x^9[/tex] you see) with the hidden eleventh direction in order to change perspective.

Well, I hope that serves as a nice alternative 9/11 post, appearing late enough in the day so as not to interfere with material you might have understandably preferred to read earlier on in the spirit of memorial.

-cvj

Bookmark the permalink.

12 Responses to The 9/11 Flip

  1. Javier says:

    Just to stay that teh reason non-critial strings theories are in the news is because their prediction of a dispersion relation for vacuum lightspeed and the MAGIC observatory which has, maybe, found partial experimental evidences of it.

  2. Clifford says:

    I’m not sure what natural or generic mean, really. Depends upon your goals, for a start. I think it is certainly very interesting to study and explore in its own right, whether or not it is perceived as better (by some measure I’m not sure how best to define) than the CY approach. Sorry to be vague on this, I just don’t have a strong opinion one way or another.

    Cheers,

    -cvj

  3. mark says:

    Hi Clifford,
    Do you think that compactifying M-theory on a manifold with G2 holonomy where there are no U(1) isometries is more “natural”/”generic” than compactifying it on a circle times a Calabi-Yau?

    Thanks you!

  4. Clifford says:

    Ok…. what I meant to say was that as ten dimensional theories the distinction needn’t be made for this discussion.

    Cheers,

    -cvj

  5. Clifford says:

    That’s just type I (as is type IB) The distinction is not so crucial at this level.. I could say more, but have to run.

    -cvj

  6. Alejandro Rivero says:

    No role in the puzzle for “type IA”?

  7. Clifford says:

    Hi,

    It is very tempting to give a humourous answer based on the term “critical” versus “non-critical” string theory, but I will resist.

    They are all part of the same thing. I’ve mentioned non-critical strings before in some blog posts, but it is probably easier to write something again than try to find it, so here goes:

    String theory (like many theories) comes with a large set of symmetries. These symmetries allow you to sometimes find nice computational schemes to make progress some places. Critical string theory arises from using the symmetries to pick a particularly nice framework that allowed a lot of progress to be made on many aspects. The price you pay for this is the remarkable fact that the string theory likes to choose to propagate in ten dimensions (if supersymmetric or 26 if bosonic), and all dimensions are on the same footing -ten dimensional local Lorentz invariance (the symmetry of Special Relativity) is built in. Those are the famous “critical” numbers of dimensions that people talk a lot about. Then there’s the whole wonderful and interesting endeavour to understand how to get four dimensional physics from this, by doing various things to the other six dimensions.

    Somehow, a most people forgot that there are other choices that can be made. These allow you to end up in other dimensions right at the outset. But that’s ok, since who (or what experiment) said that we need anything more than four dimensional Lorentz invariance? It was just a simplifying assumption. In fact, most of the activity in critical string theory goes about messing up that local Lorentz invariance in the higher dimensions anyway, so why fight hard to keep it? So you can give that up, finding that some directions are different than others… typically the string coupling and other fields can have preferred dependences on some directions, and so forth. The price is that it is very hard to understand those string theories in many settings (although there are some where we understand a lot), and for some dimensions (including four dimensions) we don’t even know how to formulate it. I suspect that this is an important clue, and we ought to return to that problem to learn interesting things – maybe about our world.

    I’m pleasantly surprised to hear that non-critical strings are in the news, since it is high time that we told more of the story of string theory, and stopped being so hung up on the idea that we have to start with ten dimensions, which is really quite an over-simplification of the issues. This is one of the many reasons why -again and again- I keep trying to remind people here and elsewhere that there is so much wonderful stuff to figure out in string theory. It is not only a work in progress, it might just have barely begun.

    Cheers,

    -cvj

  8. cecil kirksey says:

    Dr. Johnson:
    This maybe a little off subject but can you explain in lay (engineering background) terms the differences between critical and non-critical string theory? Do both relie on a common set of assumptions but just use different mathematical techniques to arrive at a final theory?

    Non-critical strings has been in the news lately but with no clear cut statement as to the viability of this aspect of string theory. Is non-critical string theory part of string theory or not? Are you a supporter of this approach?

  9. Clifford says:

    Hi Robert,

    Yes. There’s no question that the concept goes to much earlier. I’m talking about the terminology “9/11 flip”. I think that the term’s wide use came as a result of their matrix string paper and subsequent lectures.

    (I edited your comment slightly. Chief result: it now has the latex you desired. Look in the sidebar for [tex]\LaTeX[/tex] instructions.)

    Cheers,

    -cvj

  10. Robert says:

    I remember Hermann Verline using this term extensively at the 1998 Trieste spring school (where somebody else was giving lectures with musical lecture notes). But I would agree with David that the concept (the realization that the [tex]S\in SL(2,Z)[/tex] of the IIB S-duality is geometric in M-Theory) obviously predates matrix string theory.

  11. Clifford says:

    The technique is a lot older, yes, but the term itself is what I’m talking about (I should be sure to emphasize). It certainly took off in a big way after that paper of theirs and their use of the term in a lot of their talks on the subject, and lectures at various schools. But I can certainly believe that the term also started from earlier, but I do not know who. Yes, it would be nice if anyone who has some clues or guesses can comment…. maybe point to a paper where the term is used.

    Cheers,

    -cvj

  12. David says:

    Hi,

    Not sure the name 9/11 flip is due to the either Verlinde or Robbert. If I recall it was a supergravity person; part of the solution generating business in supergravity and almost predating m-theory though I might go for Hull and Townsend U-duality paper. Does anyone know?
    David