Lectures on String Theory

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Exercise 38. Compute the three-point correlation function of the tachyon vertex
operators
〈0|V (k 4 , τ 4 )V (k 3 , τ 3 )V (k 2 , τ 2 )V (k 1 , τ 1 )|0〉
Exercise 40. Verify that ∂ n −X µ is a conformal operator. Find the corresponding
conformal dimension. Find the singular terms of the OPE
T −− (τ, σ) ∂ n −X µ (τ ′ , σ ′ ) .
Exercise 41. Find the singular terms of the OPE
T −− (τ, σ) V (k, τ ′ , σ ′ ) ,
where V (k, τ ′ , σ ′ ) is the vertex operator of tachyon.
Exercise 42. Find the singular terms of the OPE
T −− (τ, σ) T −− (τ ′ , σ ′ ) .
Exercise 43. Suppose that there are i = 1, . . . , n grassman (anticommuting)
variables η i and ¯η i . Let us define the integration rules as
∫
∫
dη = 0 , dηη = 1
for any η i and ¯η i . Show that for any n × n matrix M the following formula is valid
∫
detM = dηd¯η e¯ηMη .
Here ¯ηMη ≡ ¯η i M ij η j .
Exercise 44. Show that the stress-tensor for the ghost fields implies the following
expression for the ghost Virasoro generators of the closed string
L gh
m =
¯L gh
m =
∞∑
n=−∞
∞∑
n=−∞
(m − n) : b m+n c −n :
(m − n) : ¯b m+n¯c −n :