Lectures on String Theory

– 140 –
as
Here
Exercise 50. Under finite transformations z → f(z) the stress tensor transforms
T (z) → T ′ (z) = (∂f) 2 T (f(z)) + c
12 D(f) z
D(f) z = ∂f(z)∂3 f(z) − 3 2 (∂2 f(z)) 2
(∂f) 2
is the Schwarzian derivative. Show that if f(z) = az+b
cz+d then D(f) z = 0.
Exercise 51. Consider a parallelogram on the complex plane determined by the
following identification
z ≡ z + nλ 1 + mλ 2 ,
n, m ∈ Z
Show that a general transformation
λ 1 → dλ 1 + cλ 2 , λ 2 → bλ 1 + aλ 2
with the condition ad − bc = 1 preserves the area of the parallelogram.
Exercise 52. Show that the modular transformations
T : τ → τ + 1 , S : τ → − 1 τ
applied to the fundamental region of the modular group
{
M g=1 = − 1 2 ≤ Reτ ≤ 0, |τ|2 ≥ 1 ∪ 0 < Reτ < 1 }
2 , |τ|2 > 1
generate the whole upper-half plane.
Exercise 53. Show that τ = i and τ = e 2πi
3 are the fixed points of S and ST
transformations respectively.
Exercise 54. Verify that the action
S = − 1 ∫
d 2 σ
(∂ α X µ ∂ α X µ + 2i
8π
¯ψ
)
µ ρ α ∂ α ψ µ .
is invariant under supersymmetry transformations w
δ ɛ X µ = i¯ɛψ µ ,
δ ɛ ψ µ = 1 2 ρα ∂ α X µ ɛ
δ ɛ ¯ψµ = − 1 2¯ɛρα ∂ α X µ