The Cosmic Landscape (43 page)

The term
Landscape
did not originate with string theorists or cosmologists. When I first used it in 2003 to describe the large number of String Theory vacuums, I was borrowing it from a much older field of science: the physics and chemistry of large molecules. The possible configurations of a large molecule, made of hundreds or thousands of atoms, had long been described as landscapes or, sometimes, energy landscapes. The Landscape of String Theory has much less in common with the impoverished landscapes of quantum field theory than with the “configuration space” of large molecules. Let’s pursue this point before returning to the exploration of String Theory.

Begin with a single atom. Three numbers are required in order to specify the location of the atom: the coordinates of the atom along the x-, y-, and z-axes. If you don’t like x, y, and z, you may use longitude, latitude, and altitude instead. Thus, the possible configurations of a single atom are the points of ordinary three-dimensional space.

The next-simplest system made of atoms is a diatomic molecule—a molecule composed of two atoms. Specifying the position of two atoms requires six coordinates: three for each atom. It would be natural to call the six coordinates x
₁
, y
₁
, z
₁
and x
₂
, y
₂
, z
₂
, the subscripts 1 and 2 referring to the two atoms. These six numbers describe two points of three-dimensional space, but we can also combine the six coordinates to form an abstract, six-dimensional space. That six-dimensional space is the landscape describing a diatomic molecule.

Now let’s jump to a molecule composed of one thousand atoms. For inorganic chemistry this would be a very large molecule, but for an organic biomolecule, it is fairly ordinary. How do we describe all the ways that the one thousand atoms can arrange themselves? This question is not entirely academic: biochemists and biophysicists who want to understand how protein molecules fold and unfold themselves think in terms of a molecular landscape.

Evidently, to specify the configuration of all one thousand atoms, we need to give three thousand numbers, which we can think of as the coordinates of a three-thousand-dimensional landscape: a landscape of possible molecular “designs.”

The collection of atoms has potential energy that varies as the atoms are moved around. For example, in the case of the diatomic molecule, if the two atoms are squeezed together, the potential energy becomes large. If the atoms move apart, they will eventually reach a point of minimum energy. Of course it is much more difficult to visualize the energy of one thousand atoms, but the principle is the same: the potential energy of the molecule varies as we move across the landscape. As in chapter 3, if we think of potential energy as altitude, the landscape will have a rich topography with mountains, valleys, ridges, and plains. It shouldn’t come as a surprise that the stable configurations of the molecule correspond to the bottoms of valleys.

The striking thing is that the number of these valleys is enormous: it grows exponentially with the number of atoms. For a large molecule the number of isolated valleys is way beyond millions or billions. The landscape of a molecule with one thousand atoms can easily have 10
¹⁰⁰
valleys. What does all of this have to do with the Landscape of vacuums and String Theory? The answer is that, like a molecule, a compactification of String Theory has a great many “moving parts.” Some of those parts we have already met. The compactification moduli were the quantities that determine the sizes and shapes of the various geometric features of Calabi Yau manifolds. In this chapter we are going to explore some additional moving parts and see why the Landscape is so complex and extraordinarily rich.

In chapter 8 I described how Ed Witten’s 1995 idea combined the multitude of String Theories into one big M(aster)-theory. But that theory had one serious problem: it needed new objects, objects that String Theory had not previously predicted. The theory would have to work something like this: each one of the String Theories must contain previously unsuspected objects deeply hidden in its mathematics. The fundamental strings of one version were not the same objects as the fundamental strings of another version. But as the moduli varied—as one moved through the Landscape—the new objects of version A would morph into the old objects of version B. One example that we have already seen is how the membranes of M-theory morph into the strings of Type IIa theory. Witten’s ideas were attractive—even compelling—but the nature of the new objects and their mathematical place in the theory was a complete mystery. That is, until Joe Polchinski discovered his branes.

Joe Polchinski has the good looks and sunny disposition of “the boy next door.” Commenting about food, Joe once said, “There are only two kinds of food—the kind you put chocolate sauce on and the kind you put ketchup on.” But the boyish exterior hides one of the deepest and most powerful minds to attack the problems of physics in the last half century. Even before Witten introduced his M-theory, Joe had been experimenting with a new idea in String Theory. More or less as a mathematical game, he postulated that there could be special places in space where strings could terminate. Picture a child holding the end of a jump rope and shaking it to make waves. The waves travel down to the far end of the rope, but what happens next depends on whether the far end is free to flop around or is attached to some anchor. Before Polchinski, open strings always had free ends—the floppy option—but Joe’s new idea was that there could be anchors in space that held string ends from flopping. The anchor could be a simple point in space: that would be more or less like a hand rigidly constraining the end from moving. But there are other possibilities. Suppose the end of the rope were attached to a ring that could slide up and down a pole. The end would be partly fixed but partly free to move. Although attached to the pole, the end would be free to slide along a line—the pole itself. What ropes with poles can do, so can strings, or so Polchinski reasoned. Why not have special lines in space to which string ends can attach? Like the rope and pole, the string end would be free to slide along the length of the line. The line might even be bent into a curve. But points and lines don’t exhaust the possibilities. The string end could be attached to a surface, a kind of membrane. Free to slide in any direction along the surface, the string end could not escape from the membrane.

These points, lines, and surfaces where strings could end needed a name. Joe called them Dirichlet-branes or just D-branes. Peter Dirichlet was a nineteenth-century German mathematician who had nothing whatever to do with String Theory. But 150 years earlier he had studied the mathematics of waves and how they reflected off fixed objects. By all rights the new objects should be called Polchinski-branes, but the term
P-branes
was already in use by string theorists for another kind of object.

Joe is a good friend of mine. Over a period of twenty-five years we had worked closely together on a number of physics projects. The first I heard of D-branes was over coffee in Quackenbush’s Intergalactic Café and Espresso bar in Austin, Texas. I think the year was 1994. The idea seemed amusing but not the stuff of revolutions. I wasn’t alone in underestimating their importance. D-branes were not high on the to-do list of anyone at that time—maybe not even Joe’s list. It wasn’t until shortly after Witten’s 1995 lecture that D-branes exploded into the consciousness of theoretical physicists.

What is the connection with Witten’s lecture? A few months later, in November, Joe wrote a paper that has had tremendous repercussions throughout all areas of theoretical physics. The new objects that Witten needed were exactly Joe’s D-branes. Armed with D-branes, physicists could now complete Witten’s project of replacing several apparently different theories by one single theory with many solutions.

What’s so special about strings? What is it about one-dimensional filaments of energy that makes string theorists so certain that they are the building blocks of all matter? The more we learn about the theory, the more certain we are becoming that nothing is very special about them. In the previous chapter we encountered the Magical Mystery aMazing eleven-dimensional M-theory. That theory doesn’t have strings at all. It has membranes and gravitons but no strings. As we saw, the strings appear only when we compactify M-theory, and even then the strings are just limits of ribbonlike membranes that become truly stringlike only when the compact dimension shrinks to a vanishing size. In other words, String Theory is a theory of strings only in certain limiting regions of the Landscape.

In a world with three space dimensions there are three types of objects that string theorists call branes. The simplest is a point particle. Since a point has no extension in any direction, it is common to think of the point as a
zero-dimensional space.
Life on a point would be very dull; there are no directions to explore. String theorists refer to point particles as 0-branes, the 0 representing dimensionality of the particle. According to the String Theory lingo, a 0-brane, on which strings can terminate, is called a D0-brane.

After the 0-branes come the 1-branes, or strings. A string has extension in only one direction. Living on a string is still very limiting, but at least you would have one dimension in which to move. There are two kinds of 1-branes in String Theory—the original strings and D1-strings: the one-dimensional objects where the ordinary strings can end.

Finally there are 2-branes, or membranes—flexible sheets of matter. Life is infinitely more varied on a 2-brane but still not as interesting as in three-dimensional space. In fact we could call our three-dimensional world a 3-brane, but unlike the 0-, 1-, and 2-branes, we cannot move the 3-brane around in space. It
is
space. But suppose we lived in a world with four space dimensions. The extra direction of space would allow a 3-brane freedom to move. In a world with four space dimensions, it is possible to have 0-, 1-, 2-, and 3-branes.

How about in the 9+1 dimensional world of String Theory? It is possible that branes might exist all the way from 0-branes to 8-branes. This in itself does not mean that a given theory actually has such objects. That depends on the basic constituents of matter and how they can be assembled. But it does mean that there are enough dimensions to contain such branes. The ten space directions of M-theory are enough to contain one more kind of brane: the 9-brane.

Just because ten different kinds of branes can fit into the ten dimensions of space, it doesn’t mean that M-theory actually has all of them as possible objects. In fact M-theory does not. It is a theory of gravitons, membranes, and 5-branes. No other branes exist. To explain why would take us far afield into the abstract mathematics of supersymmetric general relativity, but we don’t need to go there: it’s enough to know that eleven-dimensional supergravity (that’s 10+1-dimensional) is a theory of membranes and 5-branes interacting gravitationally by tossing gravitons back and forth.

The ten-dimensional String Theories each have a variety of D-branes. One version—Type IIa String Theory—has even-dimensional branes: D0, D2, D4, D6, and D8. Type IIb theory has the odd-dimensional branes: D1, D3, D5, D7, and D9.

Just as you could attach more than one rope to the same pole, any number of strings can terminate on a D-brane. In fact a single string can have both its ends attached to the same D-brane just like both ends of the jump rope could be attached to the same pole. These segments of string would be free to move along the brane, but they couldn’t get off it. They are creatures confined to live out their lives on the D-brane.

The thing that makes these small segments of string so interesting is that they behave just like elementary particles. Take for example D3-branes. The short strings, with both ends attached to the brane, are free to move throughout the three-dimensional volume of the D-3 brane. They can come together, attach to form a single segment, vibrate, and disconnect. They move and interact just like the particles that String Theory was originally cooked up to explain. But now they live on a brane.