String Theory and M-Theory

6.1 The bosonic string and Dp-branes 201
coordinate ˜x 25 , of the circle, which is fixed at the position of the D-brane.
So these states describe a gauge field on the D24-brane.
When A25 is allowed to depend on the 25 noncompact space-time coordinates,
the transverse displacement of the D24-brane in the ˜x 25 direction can
vary along its world volume. Therefore, an A25 background configuration
can describe a curved D-brane world volume. More generally, starting with
a flat rigid Dp-brane, transverse deformations are described by the values
of the 25 − p world-volume fields that correspond to massless scalar openstring
states. These scalar fields are the 25 − p transverse components of
the higher-dimensional gauge field, and their values describe the transverse
position of the D-brane. These scalar fields on the D-brane world volume
can be interpreted as the Goldstone bosons associated with spontaneously
broken translation symmetry in the transverse directions. The translation
symmetry is broken by the presence of the D-branes.
This discussion illustrates the fact that condensates (or vacuum expectation
values) of massless string modes can have a geometrical interpretation.
There is a similar situation for gravity itself. String theory defined on a flat
space-time background gives a massless graviton in the closed-string spectrum,
and the corresponding field is the space-time metric. The metric can
take values that differ from the Lorentz metric, thereby describing a curved
space-time geometry. The significant difference in the case of D-branes is
that their geometry is controlled by open-string scalar fields.
EXERCISES
EXERCISE 6.1
Compute the mass squared of the ground state of an open string attached
to a flat Dp-brane in ¡ 25,1 .
SOLUTION
Let us label the coordinates that satisfy Neumann boundary conditions by
an index i = 0, . . . , p and the coordinates that satisfy Dirichlet boundary
conditions at both ends by an index I = p+1, . . . , 25. The mode expansions
for left- and right-movers are, as usual,
X µ
L = xµ + ˜x µ
+
2
1
2 l2 s p µ (τ + σ) + i
2 ls
m=0
1
m αµ me −im(τ+σ) ,