Lectures on String Theory

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Exercise 31. Compute in the light-cone gauge the commutator of the orbital
momenta
[l i− , l j− ] =?
Exercise 32. Show that in the quantum theory the eigenvalues of the covariant
number operator
∞∑
N =
are always nonnegative.
n=1
α µ −nα n,µ
Exercise 33. Using the previous exercise show that for any fixed state all but
a finite number of positively moded Virasoro operators automatically annihilate the
state without imposing any conditions. More precisely, show that any state |Φ〉 with
the number eigenvalue N ≥ 0 automatically satisfies
L n |Φ〉 = 0 for n > N.
Exercise 34. Compute the open string propagator
〈X(τ, σ)X(τ ′ , σ ′ )〉 = T ( X(τ, σ)X(τ ′ , σ ′ ) ) − : X(τ, σ)X(τ ′ , σ ′ ) : .
Exercise 35. Show that the vertex operator of the open string
V (k, τ) = e 1 √
πT
P ∞
n=1
kµα µ −n
e n
inτ
e ikµ(xµ + pµ
πT
} {{ τ)
} e − 1 P
√ ∞
πT n=1
V − V 0 V +
} {{ }
kµα µ n
n
e−inτ
} {{ }
is the conformal operator with the conformal dimension ∆ = α ′ k 2 .
Exercise 36. Compute the two-point correlation function of the tachyon vertex
operators
〈0|V (k 2 , τ 2 )V (k 1 , τ 1 )|0〉
Exercise 37. Compute the three-point correlation function of the tachyon vertex
operators
〈0|V (k 3 , τ 3 )V (k 2 , τ 2 )V (k 1 , τ 1 )|0〉