String Theory Demystified

CHAPTER 8 Compactifi cation and T-Duality 163
This means that for X 25 we have:
25 25
25 25 x x
0 + 0
25
XL ( τ + σ) = XL
( τ + σ) = + α′ p0
( τ + σ)
+ i
2
′
− ( + )
∑
≠ n
( − ) = − ( −
e
25
α αn
in τ σ
2 n 0
X µ µ
R τ σ XR
τ σ)
= ( )
− + x x
+ ′ p − −i
′
−
∑ n e
(8.38)
25 25 0 0
25
α 0 τ σ
2
25
α αn in( τ−σ) 2
Adding together to get the total mode expansion gives
25 25 25
X = x+ α′ p σ + i
Now using Euler’s famous formula:
0
0
n≠0
25
α ′ αn − in(
τ+ σ) − −
∑ ⎡e
− e
2 n
⎣
n≠0
− +
− in( τ+ σ) −in( τ−σ) ⎡e
− e
e − e = 2i⎢
⎣ 2i
=−2ie
in( τ σ) −in( τ−σ) −inτ
⎡e
− e
⎢
⎣ 2i
inσ −inσ
This means that the mode expansion can be written as
25 25 25
X = x+ α′ p σ + 2α′
0
= x
25
0
0
∑
n≠0
αn
n
K
αn
+ α′ σ + 2α′
∑
R n≠0
n
25
25
⎤
⎥
⎦
⎤
⎥
⎦
=−2ie
in( τ σ)
−in
−inτ
e sin nσ
−inτ
e sin nσ
⎤
⎦
τ sin nσ
Now we can analyze this expression to discover the properties of open strings in
the dual theory. The fi rst item to notice is:
• The expression for X 25
has no linear terms that contain the worldsheet time
coordinate τ. Physically, This means that the dual string has no momentum
in the 25th dimension.
• If the string carries no momentum for µ = 25, it must be fi xed. What does a
fi xed vibrating string do? The motion is oscillatory.
• Notice that the expansion contains a sin nσ term, which of course satisfi es
sin nσ = 0 at σ = 0, π .