In part 1, I gave the standard background about why one wants to consider “compactifying” string theory. The idea is that string theory is consistent in ten dimensions, but we see four. The way to get from ten to four is to roll up the extra dimensions. However, when you do this, you introduce new particles into your theory that correspond to the sizes and shapes of the extra rolled up dimensions. To fix these sizes and shapes you must give the particles masses, and you do this by turning on certain generalized electric and magnetic fields inside the extra six dimensions.
There’s more background left to cover. Let’s get on with it.
When you have a magnetic or electric field passing through some surface—think of the fields as arrows of force—the amount of field passing through the surface is called flux. The generalized fields in string theory don’t pass through surfaces (only). They can also pass through hyper-surfaces (three-dimensional volumes). There are others that can pass through objects of even higher dimension, but these are not important for our purposes.
(ASIDE: What does it mean for a field to pass through a hyper-surface? The best way to think of this is to lower the dimension: imagine a sphere with arrows poking out of it—the number of arrows poking through it is the flux through the sphere. Now a hyper-surface is a three-dimensional analog of a surface. You can for instance imagine a hyper-sphere—this is a three dimensional object such that if you were inside this object, if you went in the same direction for a while, you’d come back to the same point no matter which way you set off in—up, down, left, right, forward, or backwards, or some combination—you’d always eventually come back to where you started.)
The amount of energy that corresponds to different configurations of the rolled up dimensions given a set of fluxes through them can be computed. In fact, the energy can be thought of as a function of the sizes and shapes of the extra dimensions along with the fluxes. This function is called the potential energy.
A simple example of potential energy is measuring the amount of energy it takes to take a ball from the floor and lift it to some height. Each height has its own potential energy, and in general, as you lift the ball, the potential energy is greater. So now, imagine having a terrain of hills, valleys. A ball rolling around in a valley will generally be mostly stuck there—it will roll against the sloping hills on the sides of the valley and, since it only has a finite amount of energy, it may climb up a ways, but eventually it rolls down again. So it is stuck in the valley. A ball rolling at the top of a hill will very soon find itself rolling down the slopes, away from the peak, and eventually, into some valley, where it will be trapped. The key here is that the potential energy on peaks is greater than that in valleys, and if you don’t have anything stabilizing the ball at the peaks, it will roll down to a place where it is stabilized—in the valleys.
The configuration of the extra dimensions behaves in an analogous way; if the potential energy of a configuration is a “peak” then the configuration will adjust until it finds itself in a well—a region of potential energy where there are hills on all sides. If such a region exists, then the extra rolled up dimensions of ten dimensional space stabilize. That is, a stable balance is achieved between the forces due to the fluxes that are pushing on the extra rolled up dimensions. These wells in the potential energy are called “vacua” or “minima”. If the extra dimensions end up rolled up small enough, then an observer like us will see the world as four dimensional. Furthermore, now that the extra dimensions have been stabilized, the additional particles that I mentioned before gain very large masses—so they are very hard to push around. Basically, they won’t be observable unless we access them through extremely high energy collisions. So by stabilizing the extra dimensions we have solved a key problem in connecting ten dimensional string theory to the four dimensional world: we have gotten an *effectively* four dimensional world and we got rid of the extra gunk from the hidden dimensions by making it very heavy.
I have to go now, so I will continue this discussion in Part 3, but before I depart, I’d like to insert some caveats into the discussion above. The procedure I am outlining above is called “flux compactification". The word “compactification” refers to rolling up the extra dimensions. The word “flux” refers to these electric and magnetic-like fields that we turn on to stabilize the sizes and shapes of the rolled up extra dimensions. There are actually a handful of different avenues for doing this. In our paper, we focus on one of the best studied avenues which arises from so-called Type IIB superstring theory. The caveat is this: I said that when we roll up the extra dimensions we get a bunch of different particles that correspond to random fluctuations in the sizes and shapes of the rolled up dimensions. I then said that fluxes stabilize these dimensions and give heavy masses to these particles. In the context of our paper, the fluxes actually only give masses to *some* of the particles—a class of them called “complex structure moduli”. There is another type called “Kahler moduli” that do not get fixed by our fluxes, but instead need other methods to stabilize them. In our paper we simply assert that these are stabilized in some way or other, and that we will not focus on the details of how to do so. This is okay as long as you’re honest about it—but it also means that one natural extension of our work is to treat these other particles more seriously and try to deal with stabilizing them in detail.
I must run!