# How many dimensions are there?

When the fall quarter started, there were six papers that I absolutely had to finish by the end of the term. Three have been completed, two are very close, and the last one — sadly, I think the deadline has irrevocably passed, and it’s not going to make it. So here’s the upshot.

About a year ago I gave a talk at the Philosophy of Science Association annual meeting in Austin. The topic of the session was “The Dimensions of Space,” and my talk was on “Why Three Spatial Dimensions Just Aren’t Enough” (pdf slides). I gave an overview of the idea of extra dimensions, how they arose historically and the role they currently play in string theory.

But in retrospect, I didn’t do a very good job with one of the most basic questions: how many dimensions does spacetime really have, according to string theory? The answer used to be easy: ten, with six of them curled up into a tiny manifold that we couldn’t see. But in the 1990’s we saw the “Second Superstring Revolution,” featuring ideas about D-branes, duality, and the unification of what used to be thought of as five distinct versions of string theory.

One of the most important ideas in the second revolution came from Ed Witten. Ordinarily, we like to examine field theories and string theories at weak coupling, where perturbation theory works well (QED, for example, is well-described by perturbation theory because the fine-structure constant α = 1/137 is a small number). Witten figured out that when you take the *strong*-coupling limit of certain ten-dimensional string theories, new degrees of freedom begin to show up (or more accurately, begin to become light, in the sense of having a low mass). Some of these degrees of freedom form a series of states with increasing masses. This is precisely what happens when you have an extra dimension: modes of ordinary fields that wrap around the extra dimension will have a tower of increasing masses, known as Kaluza-Klein modes.

In other words: the strong-coupling limit of certain ten-dimensional string theories is an eleven-dimensional theory! In fact, at low energies, it’s eleven-dimensional supergravity, which had been studied for years, but whose connection to string theory had been kind of murky. Now we know that 11-d supergravity and the five ten-dimensional string theories are just six different low-energy weakly-coupled limits of some single big theory, which we call M-theory even though we don’t know what it really is. (Even though the 11-d theory can arise as the strong-coupling limit of a 10-d string theory, it is itself weakly coupled in its own right; this is an example of strong-weak coupling duality.)

So … how many dimensions are there really? If one limit of the theory is 11-dimensional, and others are 10-dimensional, which is right?

I’ve heard respected string theorists come down on different sides of the question: it’s really ten-dimensional, it’s really eleven. (Some have plumped for twelve, but that’s obviously crazy.) But it’s more accurate just to say that there is no unique answer to this question. “The dimensionality of spacetime” is not something that has a well-defined value in string theory; it’s an approximate notion that is more or less useful in different circumstances. If you look at spacetime a certain way, it can look ten-dimensional, and another way it can look like eleven. In yet other configurations, thank goodness, it looks like four!

And it only gets worse. According to Juan Maldacena’s famous gravity-gauge theory correspondence (AdS/CFT), we can have a theory that is equally well described as a ten-dimensional theory of gravity, or a four-dimensional gauge theory without any gravity at all. It might sound like the degrees of freedom don’t match up, but ultimately infinity=infinity, so a lot of surprising things can happen.

This story is one of the reasons for both optimism and pessimism about the prospects for connecting string theory to the real world. On the one hand, string theory keeps leading us to discover amazing new things: it wasn’t as if anyone guessed ahead of time that there should be dualities between theories in different dimensions, it was forced on us by pushing the equations as far as they would go. On the other, it’s hard to tell how many more counterintuitive breakthroughs will be required before we can figure out how our four-dimensional observed universe fits into the picture (if ever). But it’s nice to know that the best answer to a seemingly-profound question is sometimes to unask it.