Lectures on String Theory

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3.3.1 Solutions of the equations of motion
Here we are going to discuss the solutions of the string equations off motion
□X µ = 0
which arises upon fixing the conformal gauge.
• Closed strings. We have
X µ (σ, τ) = X µ L (τ + σ) + Xµ R
(τ − σ)
where
X µ R (τ − σ) = 1 2 xµ + pµ
(τ − σ) +
i
4πT
∑
√
4πT
n≠0
n≠0
1
n αµ ne −in(τ−σ) (3.41)
X µ L (τ + σ) = 1 2 xµ + pµ
(τ + σ) +
i ∑
√ nᾱµ 1
4πT
ne −in(τ+σ) (3.42)
4πT
Since X µ (σ, τ) are real then (x µ , p µ ) are real as well and
Let us define the zero modes as
Oscillators obey the Poisson relations
α µ −n = (α µ n) † , ᾱ µ −n = (ᾱ µ n) †
α µ 0 = ᾱ µ 0 = 1 √
4πT
p µ .
{α µ m, α ν n} = {ᾱ µ m, ᾱ ν n} = −imδ m+n η µν ,
{α µ m, ᾱ ν n} = 0 (3.43)
{x µ , p ν } = η µν .
The Virasoro constraints become
L m = 1 2
∞∑
n=−∞
α µ m−nα nµ , ¯Lm = 1 2
∞∑
n=−∞
ᾱ µ m−nᾱ nµ . (3.44)
• Open strings Solution of the wave equation with the open string boundary
conditions is
X µ (σ, τ) = x µ + pµ
πT τ +
√ i ∑
πT
n≠0
1
n αµ ne −inτ cos nσ . (3.45)