String Theory Demystified

CHAPTER 13 D-Branes 223
The Space-Time Arena
The easiest way to describe a Dp-brane mathematically is to use the light-cone
gauge. To specify the D-brane, we need to choose which coordinates will
satisfy Neumann boundary conditions and which coordinates satisfy Dirichlet
boundary conditions. To use the light-cone gauge, we also need to define lightcone
coordinates that will satisfy Neumann boundary conditions, these will
include:
• Time
• One spatial coordinate, which we choose to be X 1 ( σ, τ)
For a Dp-brane, we let i= 2, … , p in the light-cone gauge. Then as usual we
defi ne:
X
±
( στ , )
X ± X
=
2
0 1
Neumann boundary conditions can be written as
µ
∂ X 0 = 0
σ
(13.1)
(13.2)
σ= , π
So, the coordinates chosen to satisfy Neumann boundary conditions are
+ −
i
X ( σ, τ) X ( σ, τ) X ( σ, τ) i= 2, … , p (13.3)
Let us suppose that the D-brane is located at x a . That is, letting a= p+1, … , d :
a a
x = x
(13.4)
The remaining spatial coordinates will satisfy Dirichlet boundary conditions. We
use a= p+1, … , d to denote these coordinates. In bosonic string theory we take
d = 25 while in superstring theory d = 9. So the Dirichlet boundary conditions will
be applied to
a
X ( σ, τ ) a= p+1, … , d
(13.5)