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Witten argued, and others since have filled in important details, that all five string theories are linked through a network of such dualities.3 Their overarching union, called M-theory (we’ll see why in a moment), combines insights from all five formulations, stitched together through the various duality relationships, to gain a far more refined understanding of each. One such insight, central to the theme we’re pursuing, showed that there’s much more to string theory than strings.

Branes

When I started studying string theory, I asked the very question that many in the years since have asked me: Why are strings considered so special? Why focus solely on fundamental ingredients that have only length? After all, the theory itself requires that the arena within which its ingredients exist—the spatial universe—has nine dimensions, so why not consider entities shaped like two-dimensional sheets or three-dimensional blobs or their higher-dimensional cousins? The answer I learned as a graduate student in the 1980s, and explained frequently when I lectured on the subject through the mid-1990s, was that the mathematics describing fundamental constituents with more than one spatial dimension suffered from fatal inconsistencies (such as quantum processes that would have negative probabilities, a meaningless mathematical result). But when the same mathematics was applied to strings, the inconsistencies canceled themselves out, leaving a cogent description.†4 Strings were definitely in a class of their own.

Or so it seemed.

Armed with the newfound calculational methods, physicists started analyzing their equations much more precisely and produced a range of unexpected results. One of the most surprising established that the rationale for excluding anything but strings was rickety. Theorists realized that the mathematical problems encountered when studying higher-dimensional ingredients, such as discs and blobs, were artifacts of the approximations being used. Using the more precise methods, a small army of theorists established that ingredients with various numbers of spatial dimensions do lurk in string theory’s mathematical shadows.5 The perturbative techniques were too coarse to expose these ingredients but the new methods finally could. By the late 1990s, it was abundantly clear that string theory was not just a theory that contained strings.

The analyses revealed objects, shaped like Frisbees or flying carpets, with two spatial dimensions: membranes (one meaning of the “M” of M-theory), also called two-branes. But there was more. The analyses revealed objects with three spatial dimensions, so-called three-branes; objects with four spatial dimensions, four-branes, and so on, all the way up to nine-branes. The mathematics made clear that all of these entities could vibrate and wiggle, much like strings; indeed, in this context, strings are best thought of as one-branes—a single entry on an unexpectedly long list of the theory’s basic building blocks.

An allied revelation, just as flabbergasting to those who’d spent the better part of their professional lives working on the subject, was that the number of spatial dimensions the theory requires is not actually nine. It’s ten. And if we fold in the dimension of time, the total number of spacetime dimensions is eleven. How could this be? Remember the “(D–10) times Trouble” consideration, recounted in Chapter 4, underlying the conclusion that string theory needs ten spacetime dimensions. The mathematical analysis that produced that equation was, once again, based on a perturbative approximation scheme that assumed the string coupling was small. Surprise, surprise, that approximation missed one of the theory’s spatial dimensions. The reason, Witten showed, is that the size of the string coupling directly controls the size of the hitherto unknown tenth spatial dimension. By taking the coupling small, researchers had unwittingly made this spatial dimension small, too—so small as to be invisible to the mathematics itself. The more precise methods rectified this failing,