Warped Passages - Lisa Randall [159]
Everything about the eleven-dimensional supergravity theory with a curled-up dimension—even short distances and high-energy processes and objects—has a counterpart in the ten-dimensional superstring theory. Furthermore, the duality holds true for a dimension curled up in a circle of any size, no matter how large. Earlier, when we looked at a rolled-up dimension, we argued only that a small curled-up dimension would not be noticed.
But how can theories with different numbers of dimensions possibly be the same? After all, the number of dimensions of space is the number of coordinates that we need to specify the position of a point. The duality could be true only if superstring theory always uses an additional number to describe pointlike objects.
The key to the duality is that in superstring theory, there are special new particles that you can uniquely identify only by specifying the value of the momentum in nine spatial dimensions and also the value of a charge, whereas in eleven-dimensional supergravity you need to know momentum in ten spatial dimensions. Notice that even though you have nine dimensions in one case and ten in the other, in both cases you would need to specify ten numbers: nine values of momentum and a charge in one case, and ten values of momentum in the other.
Ordinary uncharged strings don’t pair with objects in the eleven-dimensional theory. Because you need to know eleven numbers to locate an object in the spacetime of the eleven-dimensional theory, only particles that carry charge have eleven-dimensional mates. And the partners of the objects in the eleven-dimensional theory particles turn out to be branes—namely, charged, pointlike branes called D0-branes. String theory and eleven-dimensional supergravity are dual because for every D0-brane of a given charge in ten-dimensional superstring theory,* there is a corresponding particle with a particular eleven-dimensional momentum. And vice versa. The objects of the ten-and eleven-dimensional theories (and their interactions as well) are exactly matched.
Although charge might seem very different from momentum in a particular direction, if every object with a particular momentum in the eleven-dimensional theory matches onto an object with a particular charge in the ten-dimensional one (and vice versa), it is up to you whether you want to call that number momentum or charge. The number of dimensions is the number of independent directions of momentum—that is, the number of different directions in which an object can travel. But if momentum along one of the dimensions can be replaced by a charge, the number of dimensions isn’t really well defined. The best choice is determined by the value of the string coupling.
This astonishing duality was one of the first analyses in which branes proved instrumental. Branes were the additional ingredients that were needed for different string theories to match onto each other. But the critical feature of string theory branes that is important for their application in physical theories is that they can house particles and forces. The next chapter explains why.
What to Remember
String theory is a misnomer: string theory also contains higher-dimensional branes. D-branes are a type of brane in string theory on which open strings (strings that don’t loop back on themselves) must end.
Branes played a role in many of the important string theory developments of the last decade.
Branes were critical in demonstrating duality, which showed that superficially different versions of string theory are in fact equivalent.
At low energies, ten-dimensional superstring theory is dual to eleven-dimensional supergravity—an eleven-dimensional theory that contains supersymmetry and gravity. Particles in one theory match onto