Warped Passages (49 page)

Andy told me how he and Joe would discuss their research progress every day over lunch. Andy would talk about
p
-branes, and Joe would discuss D-branes. Although they were both studying branes, like all other physicists they initially thought that their two types of brane were two different things. Joe eventually realized that they were not.

Andy’s work demonstrated that the
p
-branes he was studying are critically important in string theory because in some spacetime geometries they give rise to new types of particle. Even if string theory’s non-intuitive and remarkable premise is true, and particles arise as oscillation modes of strings, string oscillations don’t necessarily account for all particles. Andy showed that there could still be additional particles that arise independently of strings.

Branes come in different shapes, forms, and sizes. Although we’ve focused on branes as the places where strings end, branes themselves are independent objects that can interact with their environment. Andy considered
p
-branes that wrap around a very tiny curled-up region of space, and he found that these tightly wrapped branes can act like particles. A wrapped
p
-brane that acts like a particle can be compared to a tightly cinched lasso. Just as a loop of rope becomes tiny once you pull it tightly around a pole or a bull’s horn, a brane can wrap around a compact region of space. And if that region of space is tiny, then the brane that is wrapped around it will be tiny as well.

These small branes, like more familiar macroscopic objects, have a mass that grows with their size. More of something (like lead pipe or dirt or cherries) is heavier, and less of it is lighter. Because a brane wrapped around a tiny region of space is so small, it will also be extremely light. And Andy’s calculations showed that in the extreme
case when the brane is as minuscule as you can imagine, this tiny brane looks like a new massless particle. Andy’s result was crucial in that it showed that even the most basic hypothesis of string theory—that everything arises from strings—is not always correct. Branes, too, contribute to the particle spectrum.

Joe’s remarkable observation of 1995 was that these new particles that arise from tiny
p
-branes could also be explained with D-branes. In fact, in his paper establishing the relevance of D-branes, Joe showed that D-branes and
p
-branes were actually the same thing. At energies where string theory makes the same predictions as general relativity, D-branes morph into
p
-branes. Joe and Andy, though they didn’t realize it at first, had actually been studying the same objects. This result meant that the significance of D-branes could no longer be questioned: they were no less important than the
p
-branes that had preceded them, and those
p
-branes were essential to the string theory spectrum of particles. Furthermore, there is a beautiful way to understand why
p
-branes are equivalent to D-branes. It is based on the subtle and important notion of
duality
.

Mature Branes and Duality

Duality is one of the most exciting concepts of the last ten years in particle physics and string theory. It has played a major role in recent advances in both quantum field theory and string theory, and, as we will soon see, it has especially important implications for theories with branes.

Two theories are dual when they are the same theory with different descriptions. In 1992 the Indian physicist Ashoke Sen was one of the first to recognize duality in string theory. In his work, which followed up the idea of duality that the physicists Claus Montonen and David Olive had originally introduced in 1977, he showed that a particular theory remained exactly the same if the particles and strings of the theory were interchanged. In the 1990s, the Israeli-born physicist Nati Seiberg, who was then at Rutgers University, also demonstrated remarkable dualities between different supersymmetric field theories with superficially different forces.

To understand duality’s significance, it helps to know a little about how string theorists generally do calculations. String theory’s predictions depend on the string’s tension. But they also depend on the value of a number called the
string coupling
, which determines the strength with which strings interact. Do they brush past each other, corresponding to a weak coupling, or do they collude about their mutual fates, corresponding to a strong coupling? If we knew the value of the string coupling, we could study string theory for only that particular value. But because we don’t yet know the value of the string coupling, we can hope to understand the theory only when we can make predictions for any string interaction strength. Then we can find out which one works.

The problem was that since the inception of string theory, the strongly coupled theory had appeared to be intractable. In the 1980s only string theory with weakly interacting strings was understood. (I’m using the adjective “weak” to describe the strength of string interactions, but don’t be misled by the word—this has nothing to do with the weak force.) When strings interact very strongly, it’s enormously difficult to calculate anything. Just as it’s simpler to untie a loose knot than a tight one, a theory with only weak interactions is much more manageable than a theory with strong ones. When strings interact with each other very strongly, they can get into a tangled mess that is too difficult to unravel. Physicists have tried various ingenious approaches for calculations involving strongly interacting strings, but have found no methods that they could usefully apply to the real world.

In fact, not only string theory, but all fields of physics are easier to understand when interactions are weak. That’s because if the weak interaction is only a small
perturbation
, or alteration, to a solvable theory—usually a theory with no interactions—then you can use a technique called
perturbation theory
. Perturbation theory lets you creep up on the answer to a question in the weakly interacting theory by starting from the theory with no interactions and calculating small improvements in incremental stages. Perturbation theory is a systematic procedure that tells you how to refine a calculation in successive steps until you reach any desired level of precision (or until you get tired, whichever comes first).

Using perturbation theory to approximate a quantity in an unsolvable theory might be compared to mixing paint to approximate a desired color. Suppose you’re striving for a subtle blue with hints of green that resembles the Mediterranean at its most beautiful. You might start with blue, and then mix in smaller and smaller amounts of green, alternating at times with a bit more blue, until you’ve achieved (almost) the precise color you’re after. Perturbing your paint mixture in this fashion is a way of proceeding in stages to obtain as close an approximation as you want to the desired color. Similarly, perturbation theory is a method for closely approximating the correct answer to whatever problem you’re studying by making incremental progress, starting from a problem you already know how to solve.

Trying to find the answer to a problem about a theory with strong coupling, on the other hand, is more like trying to reproduce a Jackson Pollack painting by randomly pouring paint. Each time you poured some paint, the picture would change completely. Your painting would be no closer to the desired goal after twelve iterations than it would be after eight. In fact, each time you poured the paint you would as likely as not cover up much of your previous attempt, changing the picture so much that you would essentially be starting afresh each time.

Perturbation theory is similarly useless when a solvable theory is perturbed by a strong interaction. As with futile attempts to reproduce a modern splattered masterpiece, systematic attempts to approximate a quantity of interest in a strongly interacting theory will not succeed. Perturbation theory is useful and calculations are under control only when interactions are weak.

Sometimes, in certain exceptional situations, even when perturbation theory is useless, you can still understand the qualitative features of a strongly interacting theory. For example, the physical description of your system might resemble the weakly interacting theory in gross outline, even though the details are likely to be rather different. More often, however, it is impossible to say anything at all about a theory with strong interactions. Even the qualitative features of a strongly interacting system are often completely different from those of a superficially similar, weakly interacting system.

So, there are two things you might expect for strongly interacting
ten-dimensional string theory. You might believe that no one can solve it and you can’t say anything about it at all, or you might expect strongly interacting ten-dimensional string theory to look, at least in gross outline, like the weakly coupled string theory. Paradoxically, in some cases neither of these options turns out to be correct. In the case of a particular type of ten-dimensional string theory called IIA, the strongly interacting string looks nothing like the weakly interacting string. But we can nonetheless study its consequences because it is a tractable system in which calculations are possible.

At Strings ’95, a conference held at the University of Southern California in March of that year, Edward Witten flabbergasted the audience by demonstrating that at low energies, a version of ten-dimensional superstring theory with strong coupling was completely equivalent to a theory that most people would have thought was entirely different: eleven-dimensional supergravity, the eleven-dimensional supersymmetric theory that contains gravity. And the objects in this equivalent supergravity theory interacted weakly, so perturbation theory could be usefully applied.

This meant, paradoxically, that you could use perturbation theory to study the original strongly interacting, ten-dimensional superstring theory. You would not use perturbation theory in the strongly interacting string theory itself, but in a superficially entirely different theory: weakly interacting, eleven-dimensional supergravity. This remarkable result, which Paul Townsend of Cambridge University had previously also observed, meant that despite their different packaging, at low energies, ten-dimensional superstring theory and eleven-dimensional supergravity were in fact the same theory. Or, as physicists would say, they were dual.

We can illustrate the idea of duality with our paint analogy. Suppose that we started off with blue paint, but then “perturbed” it by adding green. A good description of our paint mixture would then be blue paint with a hint of green. But suppose instead that the green paint we added wasn’t a small perturbation: suppose that we added an enormous amount of green paint. If that amount far exceeded the amount of the original blue paint, a better, “dual” description of the mixture would be green paint with a hint of blue. The preferred description entirely depends on the quantities of each color involved.

Similarly, a theory might have one description when a coupling of an interaction is small. But when that coupling is sufficiently large, perturbation theory is no longer useful in the original description. Nonetheless, in certain remarkable situations, the original theory can be completely repackaged in such a way that perturbation theory applies. That would be the dual description.

It’s as if someone presented you with the ingredients for a five-course meal. Even with all the ingredients, you might not know where to start. To make the meal work, you’d have to figure out which ingredients are intended for which course, how the spices interact with the food and with one another, and what to cook and when. But if caterers delivered the same ingredients pre-organized and prepared into salad, soup, appetizer, main course, and dessert, I expect that anyone could manage to turn that into a meal. With the same ingredients organized the right way, making a dinner goes from a complicated to a trivial problem.

Duality in string theory works in this way. Although strongly interacting, ten-dimensional superstring theory looked completely intractable, the dual description automatically organizes everything into a theory in which perturbation theory can be applied. Calculations which are difficult in one theory become manageable in the other. Even when the coupling in one theory is too big to use perturbation theory, the coupling of the other is sufficiently small to allow you to carry out perturbation calculations. However, we have yet to understand duality fully. For example, no one knows how to compute anything when the string coupling is neither very small nor very large. But when one of the couplings is either very small or very large (and the other is, respectively, very large or very small), then we can do calculations.

The duality of strongly coupled superstring theory and weakly coupled, eleven-dimensional supergravity theory tells you that you can calculate everything you would want to know in a strongly interacting, ten-dimensional superstring theory by performing calculations in a theory that is superficially entirely different. Everything predicted by the strongly interacting, ten-dimensional superstring theory can be extracted from weakly interacting, eleven-dimensional supergravity theory. And vice versa.

The feature of this duality that makes it so incredible is that both descriptions involve only
local interactions
—interactions with nearby objects. Even if corresponding objects exist in both descriptions, duality is only a truly surprising and interesting phenomenon if both descriptions have local interactions. After all, a dimension is more than a collection of points: it is a way of organizing things according to whether they are nearby or far apart. A computer dump might contain everything I want to know and be equivalent to an organized set of files and documents, but it wouldn’t be a simple description unless the information were coherently organized with the relevant information contiguous. The local interactions in both the ten-dimensional superstring theory and the eleven-dimensional supergravity theory are what makes the dimensions in both theories—and therefore the theories themselves—meaningful and useful.