String Theory and M-Theory

194 T-duality and D-branes
directions, as can be seen by comparing the original field (6.27) with the
T-dualized field (6.31). Explicitly, the boundary conditions are
X(τ, 0) = ˜x and X(τ, π) = ˜x + πK
R = ˜x + 2πK R, (6.32)
where we have used p = K/R and R = α ′ /R = 1/(2R) for the dual radius.
Observe that this string wraps the dual circle K times. This winding mode
is topologically stable, since the end points of the string are fixed by the
Dirichlet boundary conditions. Therefore, this string cannot unwind without
breaking.
Fig. 6.2. Dp-branes and open strings ending on them.
D-branes
T-duality has transformed a bosonic open string with Neumann boundary
conditions on a circle of radius R to a bosonic open string with Dirichlet
boundary conditions on a circle of radius R. We started with a string that
has momentum and no winding in the circular direction and ended up with
a string that has winding but no momentum in the dual circular direction.
The ends of the dual open string are attached to the hyperplane X = ˜x, and
they can wrap around the circle an integer number of times. The hyperplane
X = ˜x is an example of a Dirichlet-brane or a D-brane for short. In general,
a D-brane is defined as a hypersurface on which an open string can end, as
illustrated in Fig. 6.2. The important point to appreciate, though, is that
this is not just an arbitrary location in empty space. Rather, it is a physical
object. Usually one specifies the dimension of the brane and calls it a Dpbrane,
where p denotes the number of spatial dimensions. In the example
given here p = 24. By applying a T-duality transformation to open bosonic