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#### From p-branes to D-branes

A special class of p-branes in string theory are called D branes. Roughly speaking, a D brane is a p-brane where the ends of open strings are localized on the brane.
D-branes were discovered by investigating T-duality for open strings. Open strings don't have winding modes around compact dimensions, so one might think that open strings behave like particles in the presence of circular dimensions. However, the stringiness of open strings in the presence of compact dimensions exhibits itself in a more subtle manner, and the T-dual of an open string theory is anything but uninteresting.
The normal open string boundary conditions in the string oscillator expansion comes from the requirement that there be no momentum exiting or entering through the ends of an open string. This translates into what are called Neumann boundary conditions at the ends of the string at (s=0) and (s=p):

Suppose d-1-p of the space dimensions are compactified on a torus with radius R, and p of the space dimensions are left noncompact as before. In the T-dual of this string theory, the boundary conditions in those d-1-p directions are changed from Neumann to Dirichlet boundary conditions

This T-dual theory has strings with ends localized in d-1-p directions. So the T-dual of open strings compactified on a torus of radius R is open strings with their ends fixed to static p-branes, which we then call D-branes.
D branes have been very important in understanding string theory in general (see below) but also of crucial importance in understanding black holes in string theory, especially in counting the quantum states that lead to black hole entropy.

#### How many dimensions?

Before string theory won the full attention of the theoretical physics community, the most popular unified theory was an eleven dimensional theory of supergravity, which is supersymmetry combined with gravity. The eleven-dimensional spacetime was to be compactified on a small 7-dimensional sphere, leaving four spacetime dimensions visible to observers at large distances.
This theory didn't work as a unified theory of particle physics, because an eleven-dimensional quantum field theory theory based on point particles is not renormalizable. Also, chiral fermions cannot be defined in spacetimes with an odd number of dimensions. But this eleven dimensional theory would not die. It eventually came back to life in the strong coupling limit of superstring theory in ten dimensions.

#### The theory currently known as M

Technically speaking, M theory is the unknown eleven-dimensional theory whose low energy limit is the supergravity theory in eleven dimensions discussed above. However, many people have taken to also using M theory to label the unknown theory believed to be the fundamental theory from which the known superstring theories emerge as special limits.
We still don't know the fundamental M theory, but a lot has been learned about the eleven-dimensional M theory and how it relates to superstrings in ten spacetime dimensions.
Recall that one of the p-brane spacetimes that are stabilized by supersymmetry is a two-brane in eleven spacetime dimensions. This object is called the M2 brane for short.
Type IIA superstring theory has a stable one-brane solution called the fundamental string. If we take M theory with the tenth space dimension compactified into a circle of radius R, and wrap one of the dimensions of the M2 brane around that circle, then the result is the fundamental string of the type IIA theory. When the M2 brane is not around that circle, then the result is the two-dimensional D-brane, the D2 brane, of the type IIA theory.
If the two theories are identified, the type IIA coupling constant turns out to be proportional to the radius R of the compactified tenth dimension in the M theory. So the weakly coupled limit of type IIA superstring theory, which is the usual ten-dimensional theory, is also an expansion around small R. The strong coupling limit of type IIA theory is where R becomes very large, and the extra dimension of spacetime is revealed. So type IIA superstring theory lives in ten spacetime dimensions in the weak coupling limit, but eleven spacetime dimensions in the strongly coupled limit.
We still don't know what the fundamental theory behind string theory is, but judging from all of these relationships, it must be a very interesting and rich theory, one where distance scales, coupling strengths and even the number of dimensions in spacetime are not fixed concepts but fluid entities that shift with our point of view.

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