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.
Branes of Every Dimension
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.
The D-brane is a model of a world with elementary particles behaving much like the real elementary particles. The only thing missing on the D-brane is gravity. That’s because the graviton is a closed string—a string with no ends. A string with no ends would not be stuck to the brane at all.
Could the real world (with the exception of the graviton) of electrons, photons, and all the other elementary particles—as well as atoms, molecules, people, stars, and galaxies—all take place on a brane? To the majority of theorists working on these problems, it seems the most likely possibility.
Branes and Compactification
All kinds of things can be done with branes. Take a D2-brane—a membrane—and curve it into a 2-sphere. You’ve made a balloon. The trouble is that the tension of the membrane makes it quickly collapse like a punctured balloon. You could shape the D2-brane to form the surface of a torus, but this, too, would collapse.
But now imagine a brane that is stretched from one end of the universe to the other. The simplest example to visualize is an infinite D1-brane stretched right across the universe like an infinite cable. An infinite D-brane has no way to shrink and collapse. You can imagine that two cosmic giants hold its ends in place, but since the D-brane is infinite, the giants are infinitely far away.
There is no need to stop at D1-branes: an infinite sheet stretched across the universe is also stable. This time we would need many giants to hold the edges in place, but again, they would be infinitely far off. The infinite membrane would be a world with elementary particles that might resemble a “flatland” version of our own universe. You might think the creatures on the membrane would have no way of telling that more dimensions exist, but that would not be quite right. The giveaway would be the properties of the gravitational force. Remember that gravity is caused by gravitons jumping between objects. But gravitons are closed strings without ends. They have no reason to stick to the brane. Instead, they freely travel through all of space. They can still be exchanged between objects on the brane but only by traveling out into the extra dimensions, then back to the brane. Gravity would be like a science-fiction “message” telling the flatland creatures that there are more dimensions out there and that they are imprisoned on a two-dimensional surface.
The “unobserved” dimensions of gravity would in fact be easy to detect. When objects collide they can radiate gravitons, just as when electrons collide, they radiate photons. But typically the radiated gravitons will fly off into space and never return to the brane. Energy would be lost from the brane in this way. The flatland creatures would discover that energy doesn’t get converted to heat, potential energy, or chemical energy: it just disappears.
Now imagine that space has more dimensions than the usual three. Infinite D-3 branes could be stretched through space in the same way, and on a 3-brane all the usual things of our world could exist—except that gravity would be all wrong. The gravitational force law would reflect the fact that the graviton moves through more dimensions. Gravity would be “diluted” by spreading out in the extra dimensions. The result would be calamitous. Gravity would be much weaker, and galaxies, stars, and planets would be poorly held together. In fact gravity would be too weak to hold us to the earth even if the earth were somehow kept together.
Let’s take the extra dimensions—the ones that we can’t explore but the graviton can—and roll them up into a microscopically small compact space. The three dimensions of ordinary experience form an infinite room, but
the other directions have walls, ceilings, and floors. The points on opposite walls or on ceiling and floor are matched just as I described in chapter 8.
To help visualize, let’s return to the example in which we compactified three-dimensional space by rolling up one direction. Beginning with an infinite room, each point of the ceiling was identified with the point on the floor directly beneath it. But now the floor has a carpet that stretches to infinity in infinite directions. The carpet is a D-brane. Imagine the carpet-brane slowly moving through the vertical dimension. It slowly rises from the floor like a magic carpet in the Arabian Nights. It continues to levitate and rise until it just touches the ceiling. And abracadabra—zap! The carpet instantly reappears at the floor.
The graviton is still not attached to the carpet-brane, but now it can’t get very far away. There is very little room for it to move in the extra dimension. And if the extra dimension is microscopically small, then it is hard to tell if the graviton is off the brane. The result: gravity is almost exactly as it would be if, like everything else, the graviton moved on the brane. And of course there is nothing new if we replace the membrane with a D3-brane in a higher dimensional space. A D3-brane in the nine-dimensional space of String Theory would be very similar to our world if the extra six dimensions were tightly rolled up.
Most string theorists think we really do live on a brane-world, floating in a space with six extra dimensions. And perhaps there are other branes floating nearby, microscopically separated from us but invisible (to us) because our photons stick to our own brane, and theirs stick to their brane. Though invisible, these other branes would not be impossible to detect: gravity, formed of closed strings, would bridge the gap. But isn’t that exactly what dark matter is: invisible matter whose gravitational pull is felt by our own stars and galaxies? Polchinski’s D-branes open up all sorts of new directions. From our point of view, a universe with many brane-worlds living peacefully side by side is just one more possibility that can be found in the Landscape. Calabi Yau spaces of incredible complexity, hundreds of moduli, brane-worlds, fluxes (yet to come): the universe is starting to look like a world that only Rube Goldberg’s mother could love. To paraphrase the famous experimental physicist I. I. Rabi, “Who ordered all that stuff?”1
But by no means have we exhausted all the gimmicks and gadgets with which Rube can play. Here’s another: in addition to floating in the compact space, branes can also be wrapped around the compact directions. The simplest example is to go back to the infinite cylinder and wind a D1-brane around it. This would look the same as winding an ordinary string around the cylinder, except the string is replaced by a D1-brane. This object, from a distance, would look like a point particle on a one-dimensional line. On the other hand, suppose the compact space were an ordinary 2-sphere. You could try to wrap a string or D1-brane around the equator of the sphere like a belt around the middle of a fat man. But the belt could slip off the spherical fat man. A string or D1-brane wrapped on a sphere is not stable—it would not stay there for long. In the words of the physicist Sidney Coleman, “You can’t lasso a basketball.”
What about the torus—the surface of a bagel? Can a D1-brane be wrapped on the torus in a stable way? Yes, and in more ways than one. There are two ways to “belt the bagel.” One way is to run the belt through the hole. Try it. Take a bagel or donut and run a string through the hole. Wrap it around and tie it. The string can’t come off. Can you see the other way to belt the torus?
The deciding factor is the “topology” of the torus. Topology is the mathematical subject that distinguishes spheres from tori (plural of torus) and more complicated spaces. An interesting extension of the torus is a surface with two holes in it. Take a lump of clay and mold it into a ball. The surface is a sphere. Now poke a hole through it so that it resembles a donut: the surface is a torus. Next, poke a second hole. The surface is a two-holed generalization of a torus. There are more ways that you could wind a D1-brane on the two-hole torus than on the one-hole torus. A mathematician would call the sphere a zero genus surface, the torus a genus one surface, and the two-hole torus a genus two surface. Obviously you can poke any number of holes to make surfaces of any genus. The higher the genus, the more ways there are to wrap branes.
Having nine space dimensions, String Theory has six extra dimensions to hide by compactification. Six-dimensional spaces are vastly more complicated than two-dimensional spaces. Not only can you wrap D-1 branes but also there are higher dimensional versions of donut holes that allow you to wrap D2-, D3-, D4-, D5-, and D6-branes in hundreds of ways.
So far we have mainly thought about branes one at a time. But in fact you can have stacks of them. Think of the carpet in an infinite room. But why not have two carpets, one lying on top of the other? In fact it is possible to stack them up like stacks of carpets in a Persian bazaar. Just as the carpets could float freely of one another, a stack of D-branes can separate into several freely floating branes. But the D-branes are a bit like sticky carpets. If you bring them together, they will stick, forming a compound brane. This gives Rube Goldberg more options in designing his machine. He can place several carpet stacks at different heights in the room. He has new flexibility to make worlds with all sorts of properties. In fact with five carpets, stuck together in a stack of two and a stack of three, he can make a world with Laws of Physics that have many similarities to the Standard Model!
The locations of branes in the compact space are new variables to add to the moduli when counting the possibilities for creating a universe. From a distance, when the compact directions are microscopic—too small to see—the brane positions just appear to be additional scalar fields that define the Landscape.
Fluxes
Fluxes have emerged as one of the most important ingredients in the Landscape. They, more than anything else, make the Landscape prodigiously large. Fluxes are a bit more abstract, and harder to visualize, than branes. They are interesting new ingredients, but the bottom line is simple. From a distance they just look like even more scalar fields. The most familiar examples of fluxes are the electric and magnetic fields of Faraday and Maxwell. Faraday was not a mathematician, but he had a powerful ability to visualize. He must almost have been able to see the electromagnetic fields in his experimental apparatuses. His picture of the field of a magnet was lines of force emanating out of the North Pole and flowing back into the South Pole. At every point in space, the lines of force specify the direction of the magnetic field, while the density of the lines (how close together they are) specifies the field’s intensity.
Faraday pictured the electric field in the same way—lines flowing out of positive charges and into negative charges. Picture an imaginary sphere surrounding an isolated charged object with lines of electric force flowing out and receding off to infinity. The lines of force must pass through the sphere. These imaginary lines passing through the sphere are an example of the electric flux through a surface.
There is a measure of the total amount of flux passing through a surface. Faraday pictured it as the number of lines of force passing through the surface. Had he known calculus, he might have described it as a surface integral of the electric field. The idea of the number of lines was an even better one than Faraday knew. The flux through a surface happens to be one of those things that modern quantum mechanics tells us is quantized. Like photons, the unit of flux cannot be subdivided into fractions. Indeed, the flux cannot vary continuously but must be thought of in terms of discrete lines, so that the flux through any surface is an integer.
Ordinary electric and magnetic fields point along directions of three-dimensional space, but it is also possible to think of fluxes that point along the six compact directions of space. In a six-dimensional space the mathematics of fluxes is more complicated, but you can still think of lines or surfaces of force, winding their way over a Calabi Yau space, and passing through its donut holes.
To go more deeply into flux on a Calabi Yau space would require a good deal of modern geometry
and topology. But the important conclusions are not so hard. As in the case of magnetic fields, the flux through the various donut holes is quantized. It is always an integer multiple of some basic flux unit. This means that to specify the flux completely all you need to specify is a number of integers—how many units of flux there are through each hole in the space.
How many integers are needed to describe the flux on a Calabi Yau space? The answer depends on the number of holes the surface has. Calabi Yau surfaces are far more complicated than a simple torus and typically have several hundred holes. Thus, hundreds of flux integers are part of the description of a point on the Landscape!
Conifold Singularities
Thus far a typical setup can involve a few hundred moduli to fix the size and shape of the compact space, some branes located at various positions on the space, and now an additional few hundred flux integers. What more can we provide for Rube?
There are many more things to play with, but to keep this book of manageable size I will explain only one more—the conifold singularity. A soccer ball is a sphere. If you ignore the texture and seams on the surface, it is smooth. An American football, by contrast, is smooth everywhere except at the ends, where it comes to points. An infinitely sharp point somewhere on a smooth surface is an example of a singularity. In the case of the football, the singularities are called conical singularities. The pointy shape of the ends is like the tip of a cone.
Singularities in higher dimensional spaces—places where the space is not smooth—are more complicated. They have more complex topology. The conifold is one such singularity that can exist on a Calabi Yau space. Although complicated, as its name suggests, it is similar to the tip of a cone. For our purposes we can think of the conifold as a pointy conical place in the geometry.