Lectures on String Theory

– 87 –
moduli space is equal to the number of lineally independent quadratic differentials
on a given Riemann surface of genus g. The theory of quadratic differentials was
developed by Kurt Strebel (I will add more on Strebel theory in due course).
The kernel of the operator ∇ (1)
z is spanned by vectors from V (1) :
∇ (1)
z V z = h z¯z ∂V ¯z = 0 =⇒ ∂V ¯z = 0 .
The globally defined vector fields which span a kernel of ∇ (1)
z are called conformal
Killing vectors. They generate conformal Killing group (or the group of conformal
isometries, i.e. globally defined diffeomorphisms which can be completely absorbed
by the Weyl rescalings).
Riemann-Roch theorem
An important question of how many moduli for a Riemann surface of genus g exists
is answered by the Riemann-Roch theorem. Define the index of ∇ (n)
z as the number of
its zero modes minus the number of zero modes of its adjoint ∇ z (n+1). The Riemann-
Roch theorem states that
ind∇ (n)
z
= dim ker∇ (n)
z − dim ker∇ z (n+1) = −(2n + 1)(g − 1) = 1 2 (2n + 1)χ g .
For n = 1 we therefore have
#complex moduli − #conformal Killing vectors = 3g − 3
One can find the number of conformal Killing vectors for a compact Riemann
surface in an independent way. These are globally defined analytic vector fields whose
norm is finite
∫
√
||V || 2 = hhz¯z V z V ¯z < ∞ .
M g
Here V z = V n z n . For the case of sphere with metric ds 2 =
there exists three independent conformal Killing vectors
Indeed, for the norm we have
||V || 2 = 2π
∂ z , z∂ z , z 2 ∂ z .
∫ ∞
0
⎧
8π
8
⎨ k = 0
3
rdr
(1 + r 2 ) 4 r2k 4π
= k = 1
⎩
3
8π
k = 2
3
4dzd¯z
(1+|z| 2 ) 2
one finds that
where k = 0, 1, 2 for the three vector fields in question. We see that if k ≥ 3 the
integral becomes divergent, therefore, for instance, the field z 3 ∂ z has an infinite norm,