**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 12 - Beyond Strings: In Search of M-Theory**

Glimmers of M-Theory

The view now is very different. At Strings '95, Witten argued that if we start with the Type IIA string and increase its coupling constant from a value much less than 1 to a value much greater than 1, the physics we are still able to analyze (essentially that of the BPS saturated configurations) has a low-energy approximation that

When Witten announced this discovery, it stunned the audience and it has since rocked the string theory community. For almost everyone in the field, it was a completely unexpected development. Your first reaction to this result may echo that of most experts in the field:

The answer is of deep significance. To understand it, we must describe Witten's result more precisely. Actually, it's easier first to illustrate a closely related result discovered later by Witten and a postdoctoral fellow at Princeton University, Petr Hofava, that focuses on the Heterotic-E string. They found that the strongly coupled Heterotic-E string also has an eleven-dimensional description, and Figure 12.7 shows why. In the leftmost part of the figure, we take the Heterotic-E string coupling constant to be much smaller than 1. This is the realm that we have been describing in previous chapters and that string theorists have studied for well over a decade. As we move to the right in Figure 12.7, we sequentially increase the size of the coupling constant. Prior to 1995, string theorists knew that this would make the loop processes (see Figure 12.6) increasingly important and, as the coupling constant got larger, would ultimately invalidate the whole perturbative framework. But what no one suspected is that as the coupling constant is made larger, a new dimension becomes visible! This is the "vertical" dimension shown in Figure 12.7. Bear in mind that in this figure the two-dimensional grid with which we begin represents all nine spatial dimensions of the Heterotic-E string. Thus, the new, vertical dimension represents a

Moreover, Figure 12.7 illustrates a profound consequence of this new dimension. The

This realization does not invalidate any of the conclusions we have drawn in previous chapters, but it does force us to see them within a new framework. For instance, how does this all mesh with the one time and nine space dimensions required by string theory? Well, recall from Chapter 8 that this constraint arises from counting the number of independent directions in which a string can vibrate, and requiring that this number be just right to ensure that quantum-mechanical probabilities have sensible values. The new dimension we have just uncovered is

For technical reasons, Witten first came upon the eleventh dimension in his studies of the strong coupling properties of the Type IIA string, and there the story is quite similar. As in the Heterotic-E example, there is an eleventh dimension whose size is controlled by the Type IIA coupling constant. When its value is increased, the new dimension grows. As it does, Witten argued, the Type IIA string, rather than stretching into a ribbon as in the Heterotic-E case, expands into an "inner tube," as illustrated in Figure 12.8. Once again, Witten argued that although theorists have always viewed Type IIA strings as one-dimensional objects, having only length but no thickness, this view is a reflection of the perturbative approximation scheme in which the string coupling constant is assumed to be small. If nature

But what is this eleven-dimensional theory? At low energies (low compared to the Planck energy), Witten and others argued, it is approximated by the long-neglected eleven-dimensional supergravity quantum field theory. But for higher energies, how can we describe this theory? This topic is currently under intense scrutiny. We know from Figures 12.7 and 12.8 that the eleven-dimensional theory contains two-dimensional extended objects—two-dimensional membranes. And as we shall soon discuss, extended objects of other dimensions play an important role as well. But beyond a hodgepodge of properties,

Whatever the eleven-dimensional theory is, Witten has provisionally named it

*is*eleven-dimensional supergravity.When Witten announced this discovery, it stunned the audience and it has since rocked the string theory community. For almost everyone in the field, it was a completely unexpected development. Your first reaction to this result may echo that of most experts in the field:

*How can a theory specific to eleven dimensions be relevant to a different theory in ten?*The answer is of deep significance. To understand it, we must describe Witten's result more precisely. Actually, it's easier first to illustrate a closely related result discovered later by Witten and a postdoctoral fellow at Princeton University, Petr Hofava, that focuses on the Heterotic-E string. They found that the strongly coupled Heterotic-E string also has an eleven-dimensional description, and Figure 12.7 shows why. In the leftmost part of the figure, we take the Heterotic-E string coupling constant to be much smaller than 1. This is the realm that we have been describing in previous chapters and that string theorists have studied for well over a decade. As we move to the right in Figure 12.7, we sequentially increase the size of the coupling constant. Prior to 1995, string theorists knew that this would make the loop processes (see Figure 12.6) increasingly important and, as the coupling constant got larger, would ultimately invalidate the whole perturbative framework. But what no one suspected is that as the coupling constant is made larger, a new dimension becomes visible! This is the "vertical" dimension shown in Figure 12.7. Bear in mind that in this figure the two-dimensional grid with which we begin represents all nine spatial dimensions of the Heterotic-E string. Thus, the new, vertical dimension represents a

*tenth*spatial dimension, which, together with time, takes us to a total of eleven spacetime dimensions.**Figure 12.7**As the Heterotic-E string coupling constant is increased, a new space dimension appears and the string itself gets stretched into a cylindrical membrane shape.

*structure*of the Heterotic-E string changes as this dimension grows. It is stretched from a one-dimensional loop into a ribbon and then a deformed cylinder as we increase the size of the coupling constant! In other words, the Heterotic-E string is*actually a two-dimensional membrane*whose width (the vertical extent in Figure 12.7) is controlled by the size of the coupling constant. For over a decade, string theorists have always used perturbative methods that are firmly rooted in the assumption that the coupling constant is very small. As argued by Witten, this assumption has made the fundamental ingredients look and behave like one-dimensional strings even though they actually have a hidden, second spatial dimension. By relaxing the assumption that the coupling constant is very small and considering the physics of the Heterotic-E string when the coupling constant is large, the second dimension becomes manifest.This realization does not invalidate any of the conclusions we have drawn in previous chapters, but it does force us to see them within a new framework. For instance, how does this all mesh with the one time and nine space dimensions required by string theory? Well, recall from Chapter 8 that this constraint arises from counting the number of independent directions in which a string can vibrate, and requiring that this number be just right to ensure that quantum-mechanical probabilities have sensible values. The new dimension we have just uncovered is

*not*one in which a Heterotic-E string can vibrate, since it is a dimension that is locked within the structure of the "strings" themselves. Put another way, the perturbative framework that physicists used in deriving the requirement of a ten-dimensional spacetime assumed from the outset that the Heterotic-E coupling constant is small. Although it was not recognized until much later, this implicitly enforced two mutually consistent approximations: that the width of the membrane in Figure 12.7 is small, making it look like a string, and that the eleventh dimension is so small that it is beyond the sensitivity of the perturbative equations. Within this approximation scheme, we are led to envision a ten-dimensional universe filled with one-dimensional strings. Now we see that this is but an approximation to an eleven-dimensional universe containing two-dimensional membranes.For technical reasons, Witten first came upon the eleventh dimension in his studies of the strong coupling properties of the Type IIA string, and there the story is quite similar. As in the Heterotic-E example, there is an eleventh dimension whose size is controlled by the Type IIA coupling constant. When its value is increased, the new dimension grows. As it does, Witten argued, the Type IIA string, rather than stretching into a ribbon as in the Heterotic-E case, expands into an "inner tube," as illustrated in Figure 12.8. Once again, Witten argued that although theorists have always viewed Type IIA strings as one-dimensional objects, having only length but no thickness, this view is a reflection of the perturbative approximation scheme in which the string coupling constant is assumed to be small. If nature

*does*require a small value of this coupling constant then it is a trustworthy approximation. Nevertheless, Witten's arguments and those of other physicists during the second superstring revolution do give strong evidence that the Type IIA and Heterotic-E "strings" are, fundamentally, two-dimensional membranes living in an eleven-dimensional universe.**Figure 12.8**As the Type IIA string coupling constant is increased, strings expand from one-dimensional loops to two-dimensional objects that look like the surface of a bicycle-tire inner tube.

*no one knows what this eleven-dimensional theory is.*Are membranes its fundamental ingredients? What are its defining properties? How does it purport to make contact with physics as we know it? If the respective coupling constants are small, our best current answers to these questions are described in previous chapters, since at small coupling constants we are led back to the theory of strings. But if the coupling constants are not small, no one currently knows the answers.Whatever the eleven-dimensional theory is, Witten has provisionally named it

*M-theory*. The name stands for as many things as people you poll. Some samples: Mystery Theory, Mother Theory (as in "Mother of all Theories"), Membrane Theory (since, whatever it is, membranes seem to be part of the story), Matrix Theory (after some recent work by Tom Banks of Rutgers University, Willy Fischler of the University of Texas at Austin, Stephen Shenker of Rutgers University, and Susskind that offers a novel interpretation of the theory). But even without having a firm grasp on its name or its properties, it is already clear that M-theory provides a unifying substrate for pulling together all five string theories.