String Theory Demystified

162 String Theory Demystifi ed
Open Strings and T-Duality
The situation is a little different when considering compactifi cation in the open
string case. This is because open strings cannot wind around the compact dimension.
We summarize this by saying:
• Open strings do not have winding modes when a dimension is compactifi ed
into a circle of radius R. Hence they have winding number n = 0.
Let’s quickly review a couple of facts about open strings in bosonic string theory.
In order to satisfy Poincaré invariance, we choose Neumann boundary conditions
for the open string:
X µ
∂
= 0 for σ = 0,
π
(8.35)
∂σ
The modal expansion for the open string with Neumann boundary conditions is
given by
µ
µ µ µ αn
−inτ
X ( στ , ) = x0 + 2α′ p0 + i 2 α′
∑ e cosnσ(8.36)
n≠
n
We can write left-moving and right-moving modes for the open string as
X
X
µ
L
µ
R
x x
p i
n e
µ µ
µ
0 + 0 µ α ′ αn
( τ + σ) = + α′ 0 ( τ + σ)
+ ∑
2 2
0
− in(
τ+ σ)
µ µ
x0 − x0
µ α ′
( τ − σ) = + α′ p0( τ − σ)
+ i
2 2
n≠0
αα µ
n
∑
n≠0
n e
−in( τ−σ) (8.37)
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Note that x will be the position coordinate along the compactifi ed dimension.
0
µ
Here we have added and subtracted x , which is the coordinate of the compactifi ed
0
dimension in the dual space.
We can go through the compactifi cation procedure by simply applying the
T-duality transformation (which is why we have written the open string modes
in terms of left movers and right movers). We let
X →X and
X →−X
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L L R R