String Theory and M-Theory

2.3 String sigma-model action: the classical theory 33
If this choice is made for all µ, these boundary conditions respect Ddimensional
Poincaré invariance. Physically, they mean that no momentum
is flowing through the ends of the string.
• Open string with Dirichlet boundary conditions. In this case the positions
of the two string ends are fixed so that δX µ = 0, and
X µ |σ=0 = X µ
0 and X µ |σ=π = X µ π , (2.32)
where X µ
0 and Xµ π are constants and µ = 1, . . . , D − p − 1. Neumann
boundary conditions are imposed for the other p + 1 coordinates. Dirichlet
boundary conditions break Poincaré invariance, and for this reason
they were not considered for many years. But, as is discussed in Chapter
6, there are circumstances in which Dirichlet boundary conditions are
unavoidable. The modern interpretation is that X µ
0 and Xµ π represent the
positions of Dp-branes. A Dp-brane is a special type of p-brane on which a
fundamental string can end. The presence of a Dp-brane breaks Poincaré
invariance unless it is space-time filling (p = D − 1).
Solution to the equations of motion
To find the solution to the equations of motion and constraint equations it
is convenient to introduce world-sheet light-cone coordinates, defined as
σ ± = τ ± σ. (2.33)
In these coordinates the derivatives and the two-dimensional Lorentz metric
take the form
∂± = 1
2 (∂τ ± ∂σ) and
η++
η+−
= − 1
0
2 1
1
.
0
(2.34)
µ
X (σ,τ)
η−+ η−−
µ
X (σ,τ)
σ=0 σ=π σ=0 σ=π
Fig. 2.5. Illustration of Dirichlet (left) and Neumann (right) boundary conditions.
The solid and dashed lines represent string positions at two different times.