Lectures on String Theory

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Exercise 45. Consider conformal transformations z → z ′ = f(z). By definition,
primary fields are the fields which transform as tensors under conformal transformations
( ∂z
φ(z, ¯z) → φ ′ ′ ) h ( ∂¯z
′ )¯hφ(z
(z, ¯z) =
′ (z), ¯z ′ (¯z))
∂z ∂¯z
Find how a primary field transforms under infinitezimal conformal transformations
φ → φ + δ ξ,¯ξφ, where z ′ = z + ξ(z).
Exercise 46. Show that conformal fields of weight h have the following mode
expansion
φ(z) =
∑
z −n−h φ n .
n∈integer
Exercise 47. In the radial quantization products of fields is defined by putting
them in the radial order. Using the radial order prescription show that the conformal
transformation
δ ξ φ(w) = [T ξ , φ(w)]
can be written in the form
∮
δ ξ φ(w) =
C w
dz
ξ(z)T (z)φ(w)
2πi
Here C w is a small contour in a complex plane around point z.
Exercise 48.
formula
Using the previous exercise together with the Cauchy-Riemann
∮
C w
dz f(z)
2πi (z − w) = f (n−1) (w)
n (n − 1)!
show that any conformal field must have the following R-ordered operator product
with T (z):
T (z)φ(w) =
hφ(w)
(z − w) + ∂φ(w) + regular terms
2 (z − w)
Exercise 49. Show that the following operator product of the stress tensor
T (z)T (w) = c/2 2T (w) ∂T (w)
+ +
(z − w)
4
(z − w)
2
(z − w)
+ regular terms
corresponds to the following transformation law
δ ξ T (z) = c
12 ∂3 ξ(z) + 2∂ξ(z)T (z) + ξ(z)∂T (z)