String Theory Demystified

164 String Theory Demystifi ed
The last point is particularly important. Recall that Dirichlet boundary conditions
on the string are
µ
µ
X = X = 0
σ = 0 σ= π
Looking at the expression for the dual fi eld, notice that
At σ = π we have
Hence,
D-Branes
25
X ( σ = 0,
τ)
= x
25
0
25
25 K 25
X ( σ = π, τ) = x
x 0 + α′ π = 0 + 2KπR′
R
25
X X 25
( σ = π, τ) − ( σ = 0, τ) = 2KπR′
This tells us that the dual string winds around the dual dimension of radius R ′ with
winding number K.
Summarizing
• T-duality transforms Neumann boundary conditions into Dirichlet boundary
conditions.
• T-duality transforms Dirichlet boundary conditions into Neumann boundary
conditions.
• T-duality transforms a bosonic string with momentum but no winding into a
string with winding but no momentum.
• For the dual string, the string endpoints are restricted to lie on a 25-dimensional
hyperplane in space-time.
• The endpoints of the dual string can wind the circular dimension an integer
number of times given by K.
The hyperplane that the open string is attached to carries special signifi cance. A
D-brane is a hypersurface in space-time. In the examples worked out in this chapter,
it is a hyperplane with 24 spatial dimensions. The dimension which has been
excluded in this example is the dimension which has been compactifi ed. The D is
short for Dirichlet which refers to the fact that the open strings in the theory have