Lectures on String Theory

– 88 –
i.e. it is not normalizable. The three conformal Killing vectors are well behaved at
∞ as can be seen by making conformal transformation w = 1/z under which
−w 2 ∂ w , − w∂ w − ∂ w .
The limit z → ∞ corresponds now to w → 0 and we see that these fields are wellbehaved
at this point 17 . The three conformal Killing vectors we found correspond to
the Virasoro generators L 0 and L ±1 and they span (over the complex field) the Lie
algebra sl(2, C). Therefore, sl(2, C) is the Lie algebra of analytic globally defined
maps of the Riemann sphere onto itself.
Thus, the Riemann sphere has three conformal Killing vectors and, according to
the Riemann-Roch theorem, no moduli. That means that all metrics on the sphere
are conformally equivalent, or, in other words, there is a unique Riemann surface of
genus zero.
One can count the number of conformal Killing vectors for higher genus Riemann
surfaces as well. To this end one has to use the Ricci identity
∇ z (n+1)∇ (n)
z
Let V (n) ∈ ker∇ (n)
z . Then we have
0 = (∇ (n)
z
− ∇ (n−1)
z ∇ z (n) = 1 2 nR .
V (n) |∇ (n)
z V (n) ) = −(V (n) |∇ z (n+1) ∇(n) z V (n) ) =
= − 1 2 (V (n) |∇ z (n+1) ∇(n) z V (n) ) − 1 2 (V (n) |∇ z (n+1) ∇(n) z V (n) )
= − 1 2 (V (n) |∇ (n−1)
z ∇ z (n) + 1 2 nR) − 1 2 (V (n) |∇ z (n+1) ∇(n) z V (n) )
= 1 h
(∇ (n)
z
2
V (n) |∇ (n)
z V (n) ) + (∇ z (n) V (n) |∇ z (n) V (n) ) − 1 2 nR(V (n) |V (n) i
) .
Therefore, for any vector from the kernel of ∇ (n)
z
the following equality is valid
(∇ (n)
z V (n) |∇ z (n) V (n) ) + (∇ z
} {{ }
(n)V (n) |∇ z (n)V (n) ) − 1 } {{ } 2 nR(V (n) |V (n) ) = 0 .
non−negative
non−negative
Consider the case of a torus g = 1. On a torus there is a globally defined flat metric
ds 2 = dzd¯z which gives R = 0. Therefore, the equality above leads to two equations
∂V (n) = ¯∂V (n) = 0 ,
i.e. V (n) = const and, therefore, dim ker∇ z
(n) = 1. Thus, there is a unique generator
of conformal isometries, it corresponds to the rigid U(1)×U(1) rotations of the torus.
The Riemann-Roch theorem gives for g = 1
#complex moduli − #conformal Killing vectors = 3 − 3 = 0 ,
} {{ }
=1
17 According to our general discussion of Riemann surfaces the sphere requires at least two coordinate
patches to make an atlas. Transformation from one patch to another is analytic and is given
by w = 1/z.