Orbifolds : The story before the Big Bang

Calabi-Yau Manifold (Six-Dimensional Parts of ourUniverse)

by : Aria Ratmandanu 

        The COBE (Cosmic background explorer Satellite) results have given physicists confidence that we understand the origin of the universe to within a fraction of a second after the Big Bang. However, we are still left with the embarrassing questions of what preceded the Big Bang and why it occurred. General relativity, if taken to its limits, ultimately “yields nonsensical answers. Einstein, realizing that general relatively simply breaks down at those enormously small distances, tried to extend general relativity into a more comprehensive theory that could explain these phenomena.

       At the instant of the Big Bang, we expect quantum effects to be the dominant force, overwhelming gravity. The key to the origin of the Big Bang, therefore, is a quantum theory of gravity. So far, the only theory that can claim to solve the mystery of what happened before the Big Bang is the ten-dimensional superstring theory. Scientists are just now conjecturing how the ten-dimensional universe split into a four- and a six-dimensional universe. What does our twin universe look like ?

      One physicist who is struggling with these cosmic questions is Cumrum Vafa, a Harvard professor who has spent several years studying how our ten-dimensional universe may have been torn into two smaller universes. He is, ironically, also a physicist torn between two worlds. Living in Cambridge, Massachusetts, Vafa is originally from Iran, which has been racked by political convulsions for the past decade. On the one hand, he wishes eventually to return to his native Iran, perhaps when the social tumult “has calmed down. On the other hand, his research takes him far from that troubled region of the world, all the way to the far reaches of six-dimensional space, long before the tumult in the early universe had a chance to stabilize.

        Imagine a simple video game, he says. A rocket ship can travel in the video screen, he points out, until it veers too far to the right. Any video-game player knows that the rocket ship then suddenly appears from the left side of the screen, at exactly the same height. Similarly, if the rocket ship wanders too far and falls off the bottom of the screen, it rematerializes at the top of the screen. Thus, Vafa explains, there is an entirely self-contained universe in that video screen. You can never leave the universe defined by that screen. Even so, most teenagers have never asked themselves what that universe is actually shaped like. Vafa points out, surprisingly enough, that the topology of the video screen is that of an inner tube!

Figure 1.  If a rocket disappears off the right side of a video-game screen, it re-emerges on the left. If it disappears at the top, it re-emerges at the bottom. Let us now wrap the screen so that identical points match. We first match the top and bottom points by wrapping up the screen. Then we match the points on the left-and right-hand sides by rolling up the screen like a tube. In this way, we can show that a video-game screen has the topology of a doughnut.

          Think of the video screen as a sheet of paper. Since points at the top of the screen are identical to the points at the top of the screen are identical to the points at the bottom, we can seal the top and bottom sides together with glue. We now have rolled the sheet of paper into a tube. But the points on the left side of the tube are identical to the points on the right side of the tube. One way to glue these two ends is to bend the tube carefully into a circle, and seal the two open ends together with glue.

      What we have done is to turn a sheet of paper into a doughnut. A rocket ship wandering on the video screen can be described as moving on the surface of an inner tube. Every time the rocket vanishes off the video screen and reappears on the other side of the screen, this corresponds to the rocket ship moving across the glued joint of the inner tube.

       Vafa conjectures that our sister universe has the shape of some sort of twisted six-dimensional torus. Vafa and his colleagues have pioneered the concept that our sister universe can be described by what mathematicians call an orbifold. In fact, his proposal that our sister universe has the topology of an orbifold seems to fit the observed data rather well.

        To visualize an orbifold, think of moving 360 degrees in a circle. Everyone knows that we come back to the same point. In other words, if I dance 360 degrees around a May pole, I know that I will come back to the same spot. In an orbifold, however, if we move less than 360 degrees around the May pole, we will still come back to the same point. Although this may sound preposterous, it is easy to construct orbifolds. Think of Flatlanders living on a cone. If they move less than 360 degrees around the apex of the cone, they arrive at the same spot. Thus an orbifold is a higher-dimensional generalization of a cone.

      To get a feel for orbifolds, imagine that some Flatlanders live on what is called a Z-orbifold, which is equivalent to the surface of a square bean bag (like those found at carnivals and country fairs). At first, nothing seems different from living in Flatland itself. As they explore the surface, however, they begin to find strange happenings. For example, if a Flatlander walks in any direction long enough, he returns to his original position as though he walked in a circle. However, Flatlanders also notice that there is something strange about certain points in their universe (the four points of the bean bag). When walking around any of these four points by 180 degrees (not 360 degrees), they return to the same place from which they started.

      The remarkable thing about Vafa’s orbifolds is that, with just a few assumptions, we can derive many of the features of quarks and other subatomic particles. (This is because, as we saw earlier, the geometry of space in Kaluza-Klein theory forces the quarks to assume the symmetry of that space.) This gives us confidence that we are on the right track. If these orbifolds gave us totally meaningless results, then our intuition would tell us that there is something fundamentally wrong with this construction.

      If none of the solutions of string theory contains the Standard Model, then we must throw away superstring theory as another promising but ultimately incorrect theory. However, physicists are excited by the fact that it is possible to obtain solutions that are tantalizingly close to the Standard Model.”

Figure 2. If we join points A and B, then we form a cone, which is the simplest example of an orbifold. In string theory, our four-dimensional universe may have a six-dimensional twin, which has the topology of an orbifold. However, the six-dimensional universe is so small that it is unobservable.

       Mathematicians for the past 80 years have been working out the properties of these weird surfaces in higher dimensions, ever since the French mathematician Henri Poincaré pioneered the subject of topology in the early twentieth century. Thus the ten-dimensional theory is able to incorporate a large body of modern mathematics that previously seemed quite useless.