The Shape of Curled-Up Dimensions

The equations of string theory dictate a certain class of six-dimensional geometrical forms as the possibilities for the shape of curled-up dimensions. These forms are known as Calabi-Yau spaces, named after Eugenio Calabi and Shing-Tung Yau for their pre-string work with these geometrical shapes. (Recent work has shown that string theory actually requires seven curled-up dimensions; the shapes associated with this are called Joyce manifolds. However, the discussion will focus on Calabi-Yau shapes since they have been researched more fully.) The mathematics associated with Calabi-Yaus is complex, but the diagram to the right conveys the basic idea. (See ToolTip for source information.) However, keep in mind that it is a two-dimensional representation of a higher-dimensional figure and therefore has fundamental limitations. This image depicts one of the vast number of Calabi-Yau spaces meeting the requirements emerging from string theory.

The shape in the diagram is one possible shape that could be taken by the curled-up dimensions. This means that this shape is essentially "affixed" to every point in our familiar world of three extended spatial dimensions. In other words, by traversing a distance in the familiar three dimensions, you also traverse distances in the other dimensions curled up into this form; however, they are so tiny that any motion you make, no matter how small, would invariably circumnavigate them many times over. Their tiny spatial extent means there is not enough space for a large object - like a human - to move freely. After a motion of any kind, you have in fact traversed the Calabi-Yau dimensions, but you are thoroughly oblivious to this fact.

Experimental Hallmarks of String Theory and Curled-Up Dimensions

Previously, experiment and theory in physics have always gone hand in hand. However, with the advent of string theory, physicists are theoretically exploring objects of such tiny spatial extent that our particle accelerators cannot possibly search for them. In fact, based on scaling, a particle accelerator the size of the galaxy is needed to probe strings, and more detailed study has revealed that perhaps scaling is inaccurate and we would need an accelerator the size of the currently observed universe. However, physicists instead want to focus on physical properties predicted by string theory that are within our current range of experiment.

The first step toward such a goal has been taken in the realization that Calabi-Yau spaces contain holes of various numbers of dimensions which can affect a string's vibrational pattern. This goes a long way toward answering one of particle physics' most intriguing questions: why are there three families of elementary particles? why not one, or four, or any other number? The answer proposed by string theory is as follows.

Connected with each hole in the Calabi-Yau space is a group of low-energy string vibrational patterns. Since it has been previously determined that our familiar elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups, or families. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi-Yau has three holes, then three families of vibrational patterns and thus three families of particles will be observed experimentally.

There is, however, a problem with this reasoning: no one knows which of the tens of thousands of eligible Calabi-Yau spaces is the actual form of the curled-up dimensions. If the actual Calabi-Yau has three holes, we will have received impressive evidence supporting string theory. But some eligible Calabi-Yaus have as many as 480 holes. If the appropriate criteria for selecting a specific shape can be found, then this argument will provide powerful validation of string theory.

Other Possible Experimental Consequences

Logically, since strings vibrate through all the dimensions, the shape of the curled-up ones will affect their vibrations and thus the properties of the elementary particles observed. For example, Andrew Strominger and Edward Witten have shown that the masses of particles depend on the manner of the intersection of the various holes in a Calabi-Yau. In other words, the positions of the holes relative to one another and to the substance of the Calabi-Yau space was found by Strominger and Witten to affect the masses of particles in a certain way. This, of course, is true of all particle properties.

Choosing Calabi-Yau Spaces

The approximate equations of string theory were written using a method called perturbation theory, which "glosses over" the criteria required for selecting a Calabi-Yau from the vast numbers of possibilities. Current research, as will be discussed later, focuses on new nonperturbative methods that hopefully will yield more exact equations and thus the criteria for selecting a shape.

As yet, although we do not have any defined criteria for selecting a Calabi-Yau, we can make some headway toward narrowing the choices a bit. It is logical to start with those Calabi-Yaus that have three holes, and thus yield three families. Problematically, a Calabi-Yau can be smoothly deformed to yield an infinite number of other topologically equivalent shapes, as illustrated in the diagram. However, since the holes in Calabi-Yaus will change in shape, size, and position through these transitions, and the string vibrations that determine particles' properties are dependent on the properties of the holes, many of these Calabi-Yaus can be eliminated.

Most available Calabi-Yaus that have survived the elimination process to this point do not yield physical properties consistent with those observed in our universe. Nevertheless, these shapes cannot be completely eliminated because the equations used to determine the physics arising from a given Calabi-Yau are only approximations, which could cause significant errors in calculation. Physicists have performed calculations with the limited equations available and found that a few shapes correspond roughly to the physical properties observed in our universe, but they cannot refine those calculations until nonperturbative techniques become more refined.

General Experimental Indicators of String Theory

Since direct experimental evidence of string theory is impossible to obtain, indirect general observations may also go a long way towards supporting string theory. One such observation would be the discovery of supersymmetry, or rather, its predicted particles, but this would not "clinch it" for strings; supersymmetry has been incorporated into the point-particle model as well. Another, better experimental indicator of string theory is partially charged particles, which have no explanation in point-particle models but are possible with string theory. Currently, all observed electric charges come either in thirds or in wholes: quarks and antiquarks have charges of 2/3 or 1/3 (and their negatives), and the other particles have charges of 0, 1, or -1. However, within the framework of string theory, electric charges can be a multitude of values. This is due to the fact that certain eligible Calabi-Yaus have holes with an unusual property: they require that strings "wrapped" around them can disentangle themselves only by unwrapping a certain number of times, which is manifested in the denominator of the possible fractional charges.

Branes

Branes are physicists' new names for strings since it was discovered that string theory does not limit itself to one-dimensional objects. A one-brane is the new name for a string, which is a one-dimensional object; a two-brane is a membrane or a string extended into two dimensions; a three-brane is the same extended into three dimensions, and so on. Branes are predicted by the increase of a value in the five string theories called the coupling constant, which will be discussed later in the series. They arise mainly when string theory is taken to have eleven, not ten, total dimensions, and offer many interesting possibilities in the field of indirect experimental proof.

Wound Strings

There is one essential feature of string theory that is impossible in particle physics: wound strings. These are strings wound, or "wrapped" around, a curled dimension. For a lower-dimensional illustration, see the diagram at right. The first string is moving across the surface of the cylinder, but the second is wrapped around the circular dimension of the cylinder and is moving laterally along it. When a string is wrapped, the technical term is that it is in a winding mode of motion. Clearly this represents a new area of physics since this type of configuration is an absurdity in point-particle physics.

A wound string has a certain minimum mass determined by the size of its enclosed dimension and the number of times it encircles the dimension (its winding number). By E=mc², we can also see than the string has a minimum energy, called its winding energy. The vibrations of the string determine its overall energy greater than this minimum.

A major detail of winding modes involves size. According to string theory, physical processes that take place while the radius of the encircled dimension is below the Planck length and decreasing are exactly identical to those that take place when the radius is longer than the Planck length and increasing. This means that, as the encircled dimension collapses, its radius will hit the Planck length and bounce back again, reexpanding with the radius grater than the Planck length again. In other words, attempts by the encircled dimension to shrink smaller than the Planck length will actually cause expansion.

The Logic Behind Contraction/Expansion Relationships

Wound strings' energy come from two different sources: the familiar vibrational motion and the new winding energy. Vibrational motion can be separated into two categories: ordinary and uniform vibrations. Ordinary vibrations are the usual oscillations discussed in preceding pages; for simplicity, they will temporarily be ignored. Uniform vibrations are the simple motion of a string's sliding from one place to another. There are two important observations related to uniform motion that will lead to the essence of the contraction/expansion relationship.

First, uniform vibrational energies are inversely proportional to the encircled dimension's radius. By the uncertainty principle, a smaller radius confines a string to a smaller area and thus increases the energy of its motion. Second, winding energies are directly proportional to the radius because the radius causes a string to have a minimum mass, which can be translated into energy. These two conclusions show that large radii imply large winding energies and small vibrational energies, and small radii imply small winding energies and large vibrational energies.

This conclusion yields the essential realization: for any large radius there is a corresponding small radius in which the winding energies of the former are the vibrational energies in that latter and vice versa. Since physical properties depend on the total energy of a string, not its individual winding or vibrational energy, there is no observable physical difference between the corresponding radii.

Now consider an example of the preceding principle. Imagine that the radius of the encircled dimension is 5 times the Planck length (R=5). A string can encircle this dimension any number of times; this number is called the winding number. The energy from winding is determined by the product of the radius and the winding number. The uniform vibrational patterns, which are inversely proportional to the radius, are in this case proportional to whole-number multiples (due to the fact that energy comes in discrete packets, or quanta) of the reciprocal of the radius (1/R). This calculation yields the vibration number. If the radius is decreased in size to R=1/10, the winding and vibration numbers simply switch, yielding the same total energy (see table below).

Definition of Distance

These results tell us that there are two competing ways to define distance in string theory - one related to unwound, freely moving strings, and one related to wound strings. Unwound strings are free to move about and probe the whole circumference of a circle - a length by definition proportional to R, the radius of the circle. On the other hand, wound strings have minimum energies proportional to R and therefore, by the uncertainty principle, can probe distances of 1/R. If each is used to measure the radius of the circle, the unwound string will return a value of R and the wound string will return its reciprocal, 1/R. However, the nature of these distances ensures that whenever they deviate much from 1 (the Planck length), as they do in our universe, one measurement is simple (the one returning R) and the other is difficult (the one returning 1/R). For this reason, we go on defining distance in the ordinary manner without being aware of another, vastly more difficult way to perform this operation. This discrepancy is due to the masses of the two configurations - the high-energy, and thus small-size, strings are extremely massive, but the low-energy, and thus large-size, strings are very light and therefore accessible to present technologies. When the current size of the universe is measured, it is measured with the light string modes, and a figure of about 10⁶¹ times the Planck length is obtained. However, were it technologically feasible, we should be able to use the heavy string modes to measure the reciprocal of this distance.

It is due to this fact that contraction beyond the Planck length cannot occur. If measurements are always carried out in the easiest manner available, the result will always be at least the Planck length. Imagine that a measurement carried out using unwound modes reveals a vast but shrinking universe. When R=1, the two modes will become equally easy and a value of the Planck length will always be obtained. When R shrinks even smaller, the winding modes become lighter than the unwound modes, and thus they are used for measurement. This will return the reciprocal of the result given by the unwound modes, so the result will imply that the universe is once again larger than the Planck length and expanding.