The 9/11 Flip

Ok, 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.

(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. One 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