In Part 2, I described how the extra dimensions can be rolled up and stabilized by adding fluxes—generalized versions of electromagnetic fields—through them. In Part 1, I listed the two main themes of the work. We are now ready to start exploring them in a bit more depth.
The first theme involves the search for stable configurations of the extra dimensions. We refer to such a configuration as a “vacuum” since to a first approximation, if the universe settles into such a setup, it looks like a relatively empty four dimensional space. Small fluctuations around this vacuum then describe the various particles that we should observe.
There are two sets of choices to make when curling up the extra dimensions. The first is what shape you want to roll them up into. Think about taking a piece of paper and rolling it up into a cylinder—clearly there really is only one shape although the size of the cylinder is a choice. When you go to higher dimensions, there are many—perhaps infinitely—more options for how to roll things up. Needless to say, the options in six-dimensions are essentially limitless. That said, string theorists typically restrict themselves to rolling the extra six-dimensions on something called a “Calabi-Yau manifold”. Remarkably, even though there are again, probably an infinite number of these shapes, it has been conjectured that they are all closely related. In fact, Brian Greene and others have shown that certain very simple transformations that slightly alter the topology (the way things are connected) in these shapes can be used to transition between thousands of them. Furthermore, string theory provides a physical mechanism for such changes to occur. Thus, it is highly likely that if you choose an initial Calabi-Yau shape as a starting point, there may be physical processes (in string theory) that let you get to any of the others, so your initial choice wasn’t really much of a choice at all.
But let’s for the moment forget the connected nature of the set of Calabi-Yau manifolds and just pick a specific one. Once you’ve done that, you have another set of choices: you must choose the fluxes that stabilize the size and shape of the Calabi-Yau manifold. A typical Calabi-Yau manifold has hundreds of parameters that set its size and shape which in turn implies the need for hundreds of fluxes passing through the Calabi-Yau in some intricate manner. In our work, we looked at a family of Calabi-Yaus that are simpler: you could simply focus on two parameters that control the geometry of the manifold. Stabilizing these involves setting eight fluxes—a far more tractable problem to study (note to experts: by two parameters, I mean two real parameters. These can be combined into a single complex valued parameter, which is usually what is done).
So, we’ve picked one of these special types of Calabi-Yaus and we pick eight fluxes. Then you check to see if the potential energy of the configuration actually has a minimum—that is, is there a minimum energy that the system will naturally settle into, stabilizing the shape of the rolled up dimensions and leading to an effectively four dimensional theory. We conduct this search numerically, so we can only look at a finite portion of the potential energy and we find that whether or not there is a minimum is a hit-or-miss affair.
In general, the trial-and-error process of choosing fluxes and looking for these minima in a numerically generated (i.e. generated by computer) situation is tedious. One of the technical things we do is to automate this by adapting methods previously used to investigate the statistical distribution of these sorts of minima. The key is that certain choices of the parameter that controls the shape of the rolled up dimensions are special. In our examples there are three special choices. Our paper focuses on one such choice—the so-called “conifold point”—although the automated methods for finding minima could be adapted to any of the other special choices as well. The conifold point is a choice of the shape of the Calabi-Yau where part of it has collapsed to zero size. If you look in the vicinity of this collapsed region it resembles a cone, where the tip of the cone is the endpoint of the collapse. It turns out that in the vicinity of this point, we know completely analytically (i.e. via pen-and-paper—no computers necessary) the way the shape of the Calabi-Yau is configured. This let’s us do non-computer calculations and allows us to pinpoint configurations that minimize energy that are very close to the conifold point.
I should probably back-up here to explain something that may be confusing if you haven’t considered such things before. The Calabi-Yau manifold is a six dimensional collection of points, just like the space in your room is a three dimensional collection of points. To locate a point in a Calabi-Yau manifold you must give it a label with six numbers, just as you must give (x,y,z) coordinates to a point in your living room if you want to be able to locate it. There is a parameter that controls certain aspects of the shape of the Calabi-Yau manifold. To make this idea concrete, imagine if the walls of your living room were adjustable, but not independently of one-another. You have a dial that can be turned to any value. Given a setting on the dial, your living room walls will twist and stretch stopping at some preset configuration for the setting on the dial. So, you can imagine that for every number that you can choose using the dial, there is a configuration of the living room walls. The three dimensional space that is your living room is changed by choosing different settings on the dial.
The Calabi-Yau manifolds have a similar setup: any given shape is associated to a choice of two numbers. If you start at (0.421, 2.125) then the six dimensional Calabi-Yau will have some configuration. If you then alter these numbers a bit, changing them to (0.428, 2.121) then the shape of your Calabi-Yau will change a little bit. If you change the numbers a lot then the shape of the Calabi-Yau will differ by a lot. A key point is that even though the shape is changing, the topology—the way the points in the Calabi-Yau are connected—does not change in any substantial way. You are not poking any holes in the manifold, nor are you closing up the ones that may already be there. It’s like stretching and squeezing a donut, but without either breaking it, closing up the hole in it, or poking a new hole into it.
The conifold point is a choice of these two parameters such that the shape of the Calabi-Yau takes on a rather extreme form. A donut is a useful shape to keep in the back of your mind. As you adjust these two numbers so that you get closer to the conifold configuration, part of the Calabi-Yau begins to shrink. If you think of the donut, imagine taking your fingers, wrapping them around part of the donut and squeezing. Assuming this isn’t a crumbly donut, then by squeezing it, you are shrinking the part your fingers are wrapped around while the parts farther away don’t change much. Eventually, you can squeeze the part you are wrapping to a really tiny size—your donut will now look more like a very curved croissant whose ends are touching. This is analogous to choosing the Calabi-Yau parameters so that they are at the conifold configuration. Very near the point on the donut where it is crushed down to tiny size, the shape looks like a cone—that’s why the configuration is called a conifold—it’s a portmanteau of “cone” and “manifold”.
The conifold configuration of the Calabi-Yau—the configuration where it has a pinched point somewhere—is a major part of our analysis of the ways that space in string theory may undergo dramatic changes of shape. But we have to understand how quantum mechanics allows you to “tunnel” through barriers in order to understand this phenomenon. I think that that will have to wait for Part 4!
Happy New Year!