String Theory and M-Theory

202 T-duality and D-branes
X µ
R = xµ − ˜x µ
+
2
1
2 l2 s p µ (τ − σ) + i
2 ls
1
m
m=0
αµ me −im(τ−σ) .
The mode expansions for the fields with Neumann and Dirichlet boundary
conditions are
X i = X i L + X i R and X I = X I L − X I R ,
respectively. The two ends of the string have
X I (0, τ) = X I (π, τ) = ˜x I ,
which specifies the position of the D-brane. In uncompactified space-time
there can be no winding modes, so p I = 0.
The energy–momentum tensor
T++ = ∂+X i ∂+Xi + ∂+X I ∂+XI = ∂+X µ
L ∂+XLµ
has the same mode expansion as in Chapter 2, independent of p, and thus
the Virasoro generators, the zero-point energy, and the mass formula, are
the same as before
M 2 = 2(N − 1)/l 2 s .
The main difference is that this is now the mass of a state in the (p + 1)dimensional
world volume of the Dp-brane, whereas in Chapter 2 only the
space-time-filling p = 25 case was considered. The mass squared of the
open-string ground state therefore is
M 2 = −2/l 2 s = −1/α ′ .
EXERCISE 6.2
Consider a relativistic point particle with mass m and electric charge e
moving in an electromagnetic potential Aµ(x). The action describing this
particle is
S =
− m − ˙ X µ ˙ Xµ − e ˙ X µ
Aµ dτ.
Suppose that one direction is compactified on a circle of radius R. Show
that a constant vector potential along this direction, given by
A = − θ
2πR ,
✷