Kiefer C. Quantum gravity

292 STRING THEORY
√
X µ R (σ− )= xµ α
2 + ′
2 αµ 0 (τ − σ)+··· , (9.45)
√
X µ L (σ− )= ˜xµ α
2 + ′
2 ˜αµ 0 (τ + σ)+··· , (9.46)
where ··· stands for ‘oscillators’ (this part does not play a role in the following
discussion). The sum of both thus yields
X µ = 1 √ √
α
2 (xµ +˜x µ ′
α
)+
2 (αµ 0 +˜αµ 0 ) τ + ′
2 (˜αµ 0 − αµ 0 ) σ + ··· . (9.47)
The oscillators are invariant under σ → σ +2π, but the X µ transform as
√
α
X µ → X µ ′
+2π
2 (˜αµ 0 − αµ 0 ) . (9.48)
We now distinguish between a non-compact direction of space and a compact
direction; see in particular Polchinski (1998a) for the following discussion. In the
non-compact directions, the X µ must be unique. One then obtains for them from
(9.48)
√
˜α µ α
0 = αµ 0 = ′
2 pµ . (9.49)
This is the situation encountered before and one is back at Eqns (9.7) and (9.8).
For a compact direction the situation is different. Assume that there is one compact
direction with radius R in the direction µ = 25. The coordinate X 25 ≡ X
thus has period 2πR. Under σ → σ +2π, X can now change by 2πwR, w ∈ Z,
where w is called the ‘winding number’. These modes are called ‘winding modes’
because they can wind around the compact dimension. Since exp(2πiRp 25 )generates
a translation around the compact dimension which must lead to the same
state, the momentum p 25 ≡ p must be discretized,
p = n R , n ∈ Z . (9.50)
From
p = 1 √
2α
′ (˜α 0 + α 0 )
(α 0 is a shorthand for α 25
0 , etc.) one gets for these ‘momentum modes’ the relation
˜α 0 + α 0 = 2n √
α
′
R 2 . (9.51)
For the winding modes, one has from (9.48)
√
α
′
2π
2 (˜α 0 − α 0 )=2πwR