String Theory Demystified

224 String Theory Demystifi ed
a a
Given x = x , the Dirichlet boundary condition can be written as
a a a
X ( 0, τ) = X ( π, τ)
= x a= p+ 1,
… , d
(13.6)
Notice that we can also specify the Dirichlet boundary conditions by defi ning:
a a a
δX = X ( π, τ) − X ( 0, τ)
a= p+ 1, … , d (13.7)
Then we could write the Dirichlet boundary condition as
δ X a = 0 (13.8)
The coordinates are divided into two groups and given labels depending on boundary
conditions that are applied:
• The coordinates with indices µ =± , i= 2,
..., p are called NN coordinates
since they satisfy Neumann boundary conditions at both ends.
• The coordinates with indices a= p+1, ..., d are called DD coordinates
since they satisfy Dirichlet boundary conditions at both ends.
A simplifi ed illustration of the boundary conditions is shown in Fig. 13.2. To
summarize, a Dp-brane is located at x a and has extension along the x i directions.
D-brane
Directions normal to D-brane-Dirichlet
boundary conditions
Directions along D-brane-Neumann boundary conditions
Figure 13.2 A visualization of the boundary conditions and an open string.