Lectures on String Theory

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i.e. the torus is characterized by one complex modulus τ.
For g ≥ 2 there is theorem that states that the corresponding manifold always admits
a metric with constant negative curvature. For n ≠ 0 this means that − 1 nR > 0 and,
2
therefore, dim ker∇ z (n)
= 0. The Riemann surfaces with g ≥ 2 have no conformal
isometries and by the Riemann-Roch theorem that means that the number of complex
moduli n = 1 is 3g − 3.
For n = 0 we have dim ker∇ (0)
z = 1, because the corresponding kernel is spanned by
constants. The Riemann-Roch theorem implies then that
dim ker∇ z (1) − dim ker∇ (0)
z = (2n + 1) (g − 1) = g − 1 ,
} {{ } } {{ }
=1
n=0
i.e. dim ker∇ z (1) = g. The kernel of ∇z (1)
is spanned by one forms
∇ z (1)ω z = h z¯z ¯∂ωz = 0 =⇒ ¯∂ω z = 0 .
Thus we arrive at another important consequence of the Riemann-Roch theorem:
on a Riemann surface of the genus g there exists precisely g linearly independent
(globally defined) analytic one-forms. These analytic differential forms are called
abelian differentials of the first kind.
The information we obtained by using the Riemann-Roch theorem is summarized
in the Table below.
g dim ker∇ z
(n) dim ker∇ z (n+1)
0 2n + 1 0
1 1 1
> 1 1 for n = 0 g
0 for n > 0 (2n + 1)(g − 1)
Moduli space of tori
Here we would like to look more closely at the moduli space which describes conformally
non-equivalent tori – the Riemann surfaces of the genus g = 1. The torus can
be obtained by performing the following identification on the complex plane
z ≡ z + nλ 1 + mλ 2 , n, m ∈ Z , λ 1 , λ 2 ∈ C .
The parameters λ 1,2 are subject to conformal transformations z → λz and, therefore,
only their ratio τ = λ 2
λ 1
is scale-invariant. By using this freedom (the U(1)-rotation
+ real rescaling) one can always bring the parallelogram defining the torus upon