The Elegant Universe - Brian Greene [160]
The BPS properties exhaust only a small part of the full physics of a chosen string theory when its coupling constant is large, but they nonetheless give us a tangible grip on some of its strong coupling characteristics. As the coupling constant in a chosen string theory is increased beyond the realm accessible to perturbation theory, we anchor our limited understanding in the BPS states. Like a few choice words in a foreign tongue, we will find that they will take us quite far.
Duality in String Theory
Following Witten, let's start with one of the five string theories, say the Type I string, and imagine that all of its nine space dimensions are flat and unfurled. This, of course, is not at all realistic, but it makes the discussion simpler; we will return to curled-up dimensions shortly. We begin by assuming that the string coupling constant is much less than 1. In this case, perturbative tools are valid, and hence many of the detailed properties of the theory can and have been worked out with accuracy. If we increase the value of the coupling constant but still keep it a good deal less than 1, perturbative methods can still be used. The detailed properties of the theory will change somewhat—for instance, the numerical values associated with the scattering of one string off another will be a bit different because the multiple loop processes of Figure 12.6 make greater contributions when the coupling constant increases. But beyond these changes in detailed numerical properties, the overall physical content of the theory remains the same, so long as the coupling constant stays in the perturbative realm.
As we increase the Type I string coupling constant beyond the value 1, perturbative methods become invalid and so we focus only on the limited set of nonperturbative masses and charges—the BPS states—that are still within our ability to understand. Here is what Witten argued, and later confirmed through joint work with Joe Polchinski of the University of California at Santa Barbara: These strong coupling characteristics of Type I string theory exactly agree with known properties of Heterotic-O string theory, when the latter has a small value for its string coupling constant. That is, when the coupling constant of the Type I string is large, the particular masses and charges that we know how to extract are precisely equal to those of the Heterotic-O string when its coupling constant is small. This gives us a strong indication that these two string theories, which at first sight, like water and ice, seem totally different, are actually dual. It persuasively suggests that the physics of the Type I theory for large values of its coupling constant is identical to the physics of the Heterotic-O theory for small values of its coupling constant. Related arguments gave equally persuasive evidence that the reverse is also true: The physics of the Type I theory for small values of its coupling constant is identical to that of the Heterotic-O theory for large values of its coupling constant.9 Although the two string theories appear to be unrelated when analyzed using the perturbative approximation scheme, we now see that each transforms into the other—somewhat like the transmutation between water and ice—as their coupling constants are varied in value.
This central new kind of result, in which the strong coupling physics of one theory is described by the weak coupling physics of another theory, is known as strong-weak duality. As with the other dualities discussed previously, it tells us that the two theories involved are not actually distinct. Rather, they give two dissimilar descriptions of the same underlying theory. Unlike the English-Chinese trivial duality, strong-weak coupling duality is powerful. When the coupling constant of one