by Lee Smolin
But now, following Witten, string theorists proposed to revive the membrane theory in eleven dimensions. They did so because they noticed several amazing facts. First, if you take one of the eleven dimensions to be a circle, then you can wrap one dimension of a membrane around that circle (see Fig. 10). This leaves the other dimension of the membrane free to move in the remaining nine dimensions of space. This is a one-dimensional object moving in a nine-dimensional space. It looks just like a string!
Witten found that you could get all five of the consistent superstring theories by wrapping one dimension of a membrane in different ways around the circle; moreover, you got those five theories and no others.
This is not all. Recall that when a string is wrapped around a circle, there are transformations called T-dualities. As opposed to other kinds of dualities, these are known to be exact. We also find such dual transformations when one dimension of a membrane is wrapped around a circle. If we interpret these transformations in terms of the string theories we get from wrapping the membrane, they turn out to be exactly the strong-weak dualities that connect those string theories. Those particular dualities, you’ll recall, had been conjectured but not proved, outside of special cases. They were now understood to come from transformations of the eleven-dimensional theory. This is so pretty that it’s hard not to believe in the existence of the eleven-dimensional unifying theory. The only problem left open was to discover it.
Fig. 10. On the left, we have a two-dimensional membrane, which we can imagine is wrapped around a hidden dimension, which is a small circle. Seen from far enough away (right), it looks like a string wrapped around the large dimension.
Later that year, Witten gave the so-far-undefined theory a name. The act of naming it was brilliant: He called it simply M-theory. He didn’t want to say what “M” stood for, because the theory did not yet exist. We were invited to fill in the rest of the name by inventing the theory itself.
Witten’s talk raised many questions. If he was right, there was a lot to discover. One person listening was Joseph Polchinski, a string theorist working in Santa Barbara. As he tells it, “After Ed’s talk, I made a list of twenty homework problems for myself, to understand it better.”3 The homework led him to a discovery that would be key in the second superstring revolution—that string theory is not just a theory of strings. Other objects live in the ten-dimensional space-time.
People who don’t know much about aquariums think they are only about fish. But aquarium enthusiasts know that the fish are only what first attract your eye. A healthy aquarium is all about the plant life. If you try to stock an aquarium with just fish, it won’t go well. You will soon have a piscine morgue. It turns out that during the first superstring revolution, from 1984 to 1995, we were like amateurs trying to make aquariums with just fish. We missed most of what was necessary to make the system work until Polchinski discovered the missing essentials.
In the fall of 1995, Polchinski showed that a string theory, to be consistent, must include not only strings but surfaces of higher dimensions moving in the background space.3 These surfaces are also dynamical objects. Just like strings, they are free to move in space. If a string, which is a one-dimensional object, can be fundamental, why can’t a two-dimensional surface be fundamental? In higher dimensions, where there is a lot of room, why not a three-, four-, or even five-dimensional surface? Polchinski found that the dualities between string theories would not work out consistently unless there were higher-dimensional objects in the theory. He called them D-branes. (The term “brane” comes from “membrane,” which is a two-dimensional surface; the “D” refers to a technicality I won’t try to explain here.) Branes play a special role in the life of the strings: They are places where open strings can end. Normally, the ends of open strings travel freely through space, but sometimes the ends of a string can be constrained to live on the surface of a brane (see Fig. 11). This is because branes can carry electric and magnetic charges.
From the point of view of the strings, the branes are additional features of the background geometry. Their existence enriches string theory by greatly increasing the number of possible background geometries where a string could live. Besides wrapping the extra dimensions in some complicated geometry, you can wrap the branes around loops and surfaces in that geometry. You can have as many branes as you like, and they can wrap around the compactified dimensions an arbitrary number of times. In this way, you can make an infinite number of possible backgrounds for string theories. This scheme of Polchinski’s was to have enormous consequences.
The branes also deepen our understanding of the relationship between gauge theories and string theories. They do this by allowing a new way for symmetries to arise in string theory, a result of piling several branes one on top of another. As I just mentioned, open strings can end on the branes. But if several branes are in the same place, it doesn’t matter which of them a string ends on. This means that there is a kind of symmetry at work, and symmetries, as described in chapter 4, give rise to gauge theories. Consequently, we find a new connection between string theory and gauge theories.
Fig. 11. A two-dimensional brane, on which an open string ends.
Branes also opened up a whole new way of thinking about how our three-dimensional world might relate to the extra spatial dimensions of string theory. Some of the branes that Polchinski discovered are three-dimensional. By piling up three-dimensional branes, you get a three-dimensional world with whatever symmetries you like, floating in a higher-dimensional world. Could our three-dimensional universe be such a surface in a higher-dimensional world? This is a big idea, and it makes a possible connection to a field of research called brane worlds, in which our universe is seen as a surface floating in a higher-dimensional universe.
Branes did all this, but they did even more. They made it possible to describe some special black holes within string theory. This discovery, by Andrew Strominger and Cumrun Vafa in 1996, was perhaps the greatest accomplishment of the second superstring revolution.
The relationship of branes to black holes is indirect but powerful. Here is how it goes: You begin by turning off the gravitational force (you do this by setting the string coupling constant to zero). It may seem strange to describe black holes, which are nothing but gravity, in this way, but watch what happens. With gravity turned off, we can consider geometries in which many branes are wrapped around the extra dimensions. We now draw on the fact that the branes carry electric and magnetic charges. It turns out that there’s a limit to how much charge a brane can have, which is related to the brane’s mass. The configurations with the highest possible charge are very special and are called extremal. These comprise one of the special situations we talked about before, where there are extra symmetries that allow us to do more precise calculations. In particular, such situations are characterized by having several different supersymmetries that relate fermions and bosons.
There is also a maximal amount of electric or magnetic charge that a black hole can have and still be stable. These are called extremal black holes, and they had been studied for many years by specialists in general relativity. If you study particles moving on these backgrounds, you also find several different supersymmetries.
Surprisingly, despite the fact that the gravitational force has been turned off, the extremal brane systems turn out to share some properties with extremal black holes. In particular, the thermodynamic properties of the two systems are identical. Thus, by studying the thermodynamics of extremal branes wrapped around the extra dimensions, we can reproduce the thermodynamic properties of extremal black holes.
One of the challenges of black-hole physics has been to explain Jacob Bekenstein’s and Stephen Hawking’s discoveries that black holes have entropy and temperature (see chapter 6). The new idea from string theory is that—at least, in the case of extremal black holes—you can make progress by studying the analogous system of extremal branes wrapped around the extra dimensions. In fact, many properties of the two sys
tems match exactly. This almost miraculous coincidence occurs because in both cases there are several different supersymmetry transformations relating fermions and bosons. These turn out to allow construction of a powerful mathematical analogy that forces the thermodynamics of the two systems to be identical.
But this was not the whole story. You could also study black holes that were almost extremal, in that they had slightly less charge than the maximal amount possible. On the brane side, you could also study collections of branes that had slightly less than the maximal charge. Does the correspondence between branes and black holes still hold? The answer is yes, and precisely so. As long as you stay very close to the extremal cases, the properties of the two systems match closely. This is a much stricter test of the correspondence. On each side, there are complicated and precise relationships between temperature and other quantities such as energy, entropy, and the charges. The two cases agree very well.
In 1996, I heard a young Argentinian postdoc named Juan Maldacena lecture about these results at a conference in Trieste, where I used to spend time during the summer. I was floored. The precision with which the behavior of the branes matched the physics of black holes immediately convinced me to set aside time to work on string theory again. I took Maldacena out to dinner at a pizzeria overlooking the Adriatic, and I found him to be one of the smartest and most perceptive young string theorists I had ever encountered. One thing we discussed that night over wine and pizza was whether the systems of branes might be more than just models of black holes. Did they provide a genuine explanation of the entropy and temperature of black holes?
We could not answer that question, and it has remained open. The answer depends on how significant these results are. Here we encounter the situation I described in other cases where extra symmetry led to very powerful findings. There are, again, two points of view. The pessimistic point of view holds that the relationship between the two systems is probably an accidental result of the fact that both have a lot of extra symmetry. To a pessimist, the fact that the calculations are beautiful does not imply that they lead to general insights about black holes. On the contrary, the pessimist worries that the calculations are beautiful because they depend on very special conditions that do not extend to typical black holes.
The optimist, however, believes that all black holes can be understood using the same ideas, and that the extra symmetries present in special cases simply allow us to calculate more precisely. As with strong-weak duality, we still don’t know enough to decide whether the optimists or the pessimists are right. In this case, there is an added worry, which is that the piles of branes are not black holes, because the gravitational force has been turned off. It is conjectured that they would become black holes if the gravitational force were slowly turned on. In fact, this can be imagined to happen in string theory, because the strength of the gravitational force is proportional to a field that can vary in space and time. But the problem is that such a process, where the gravitational field changes in time, has always been hard for string theory to describe concretely.
As wonderful as his work in black holes was, Maldacena was only getting started. In the fall of 1997, he released an astounding paper in which he proposed a new kind of duality.4 The dualities we have mentioned so far are between theories of the same kind, living in a spacetime of the same number of dimensions. Maldacena’s revolutionary idea was that a string theory could have a dual description in terms of a gauge theory. This is astounding because a string theory is a theory of gravity, whereas a gauge theory lives in a world without gravity, on a fixed-background spacetime. Moreover, the world described by the string theory has more dimensions than the gauge theory that represents it.
One way to understand Maldacena’s proposal is to recall the idea we discussed in chapter 7, in which a string theory can arise from studying the lines of flux of the electric field. Here, the electric field’s lines of flux become the basic objects of the theory. Being one-dimensional, they look like strings. You can say that the lines become emergent strings. In most cases, emergent strings that arise from gauge theories do not behave like the kinds of strings that string theorists talk about. In particular, they do not appear to have anything to do with gravity, and they do not provide a unification of forces.
However, Alexander Polyakov had suggested that in certain cases the emergent strings associated with a gauge theory might behave like fundamental strings. Yet the gauge-theory strings would not exist in our world; instead, in one of the most remarkable feats of imagination in the history of the subject, Polyakov conjectured that they would move in a space that had one additional dimension.5
How did Polyakov succeed in conjuring up an extra dimension for his strings to move in? He found that when treated quantum-mechanically, the strings that arise from the gauge theory have an emergent property, which, it turns out, can be described by a number attached to each point on the string. A number can also be interpreted as a distance. In this case, Polyakov proposed that the number attached to each point of the string be interpreted as giving the position of that point in an additional dimension.
Taking this new emergent property into account, it was most natural to see the lines of electric flux of the field as living in a space with one more dimension. Thus, Polyakov was led to propose a duality between a gauge field in a world with three spatial dimensions and a string theory in a world with four spatial dimensions.
While Polyakov had made a general suggestion of this kind, it was Maldacena who refined the idea. In the world he studied, our three dimensions of space host the maximally super theory—the gauge theory with the maximal amount of supersymmetry. He studied the emergent strings that would arise as a dual description of that gauge theory. Extending Polyakov’s argument, he found evidence that the string theory describing those emergent strings is actually a ten-dimensional supersymmetric string theory. Of the nine dimensions of space in which these strings live, four of them are like the ones in Polyakov’s conjecture. There are, then, five dimensions left over, which are extra dimensions as described by Kaluza and Klein (see chapter 3). The extra five dimensions are arranged as a sphere. The four dimensions of Polyakov are curved, too, but in the opposite way from a sphere; such spaces are sometimes called saddle-shaped (see Fig. 12). These correspond to universes with dark energy, but where the dark energy is negative.
Maldacena’s conjecture was much bolder than Polyakov’s original proposal. It sparked an enormous response, and it has been the subject of thousands of papers written since. It has so far not been proved, but a great deal of evidence has accumulated that there is at least an approximate correspondence between string theory and gauge theory.
There was—and is—a great deal at stake here. If the Maldacena duality conjecture is right and the two theories are equivalent, then we have an exact quantum description of a quantum string theory. Any question we want to ask about the supersymmetric string theory can be translated into a question about the maximally super theory, which is a gauge theory. This is in principle much more than we had in other cases, where the string theory was defined at the background-dependent level only by a series of approximations.
Fig. 12. A saddle-shaped surface, which is the geometry of space in universes with negative energy density.
There are, however, several caveats. Even if it is true, the duality conjecture can be useful only if one side of the duality can be defined precisely. So far, it has been possible to define the relevant version of string theory only in certain special cases. Thus, the hope was to go the other way and use the conjecture to define string theory in terms of the maximally super theory. However, while we knew much more about the maximally super theory, that theory also had not yet been precisely defined. There were hopes that we could do better, but they rested on difficult technical issues.
If Maldacena’s conjecture is false, then the maximally super theory and string theory are not equivalent. However, even in this case, there is considerable evidence that a
t some levels of approximation there are useful relationships between the two. These approximations may not be strong enough to define one theory in terms of the other, but they do make it possible to calculate some properties of one relative to the other. A great deal of fruitful work along these lines has been done.
For example, at the lowest level of approximation, the ten-dimensional theory is just a version of general relativity extended to ten dimensions and enhanced by supersymmetry. This has no quantum mechanics and is well defined. It is easy to do some calculations in this theory, such as studying the propagation of different kinds of waves in the ten-dimensional spacetime geometry. Remarkably, even if Maldacena’s conjecture proves true only at the lowest order of approximation, this has allowed us to calculate some properties of the corresponding gauge theory in our three-dimensional world.
This in turn led to insights into the physics of the other gauge theories. As a result, there is good evidence that, at least at the lowest level of approximation, string theories and gauge theories are related in the way Maldacena imagined. Whether the strong form of the Maldacena conjecture is true or false—indeed, even if string theory itself is false—we have gained powerful tools for understanding supersymmetric gauge theories.
After several years of intensive work, these matters remain confused. At issue is what exactly the relationship is between string theory and the maximally super theory. Most of the evidence is explained by a weak form of the Maldacena conjecture, which requires only that certain quantities in one theory be calculable using methods in the other and then only in a certain approximation. This, as I have noted, is already a result with important applications. But most string theorists believe in the strongest form of the conjecture, according to which the two theories are equivalent.
