6.1 The Riemann Sphere
107
the metric reads
ds
2
= dx
2
+ dy
2
= dzd¯ z.
(6.5)
The relations between the derivatives in the two coordinate systems are easily
found:
∂ := ∂ z =
1
2
(∂ x − i∂ y ),
¯
∂ := ∂ ¯
z =
1
2
(∂ x + i∂ y ).
(6.6)
The indexed form will be used when there is a risk of confusion. If the index is
omitted, then the derivative acts directly to the field next to it, for example,
∂φ(z 1 )∂φ(z 2 ) := ∂ z 1 ∂ z 2 φ(z 1 )φ(z 2 ).
(6.7)
Generically, the meromorphic and anti-meromorphic parts of an object will be
denoted without and with a bar, see (6.55) for an example.
The extended complex plane ¯
C can be covered by two coordinate patches z ∈ C
and w ∈ C. In the first, the point at infinity (north pole) is removed, in the second,
the origin (south pole) is removed. On the overlap, the transition function is
w =
1
z
.
(6.8)
This description avoids to work with the infinity: studying the behaviour of f (z) at
z = ∞ is equivalent to study f (1/w) at w = 0.
Since any two-dimensional metric is locally conformally equivalent to the flat
metric, it is sufficient to work with this metric in each patch. This is particularly
convenient for the Riemann sphere since one patch covers it completely except for
one point.
6.1.2 Relation to the Cylinder: String Theory
The worldsheet of a closed string propagating in spacetime is locally topologically
a cylinder R × S 1 of circumference L. In this section, we show that the cylinder
can also be mapped to the complex plane—and thus to the Riemann sphere—after
removing two points. Since the cylinder has a clear physical interpretation in string
theory, it is useful to know how to translate the results from the plane to the cylinder.
It makes also sense to define two-dimensional models on the cylinder independently of a string theory interpretation since the compactification of the spatial
direction from R to S 1 regulates the infrared divergences. Moreover, it leads to a
natural definition of a “time” and of a Hamiltonian on the Euclidean plane.
107
the metric reads
ds
2
= dx
2
+ dy
2
= dzd¯ z.
(6.5)
The relations between the derivatives in the two coordinate systems are easily
found:
∂ := ∂ z =
1
2
(∂ x − i∂ y ),
¯
∂ := ∂ ¯
z =
1
2
(∂ x + i∂ y ).
(6.6)
The indexed form will be used when there is a risk of confusion. If the index is
omitted, then the derivative acts directly to the field next to it, for example,
∂φ(z 1 )∂φ(z 2 ) := ∂ z 1 ∂ z 2 φ(z 1 )φ(z 2 ).
(6.7)
Generically, the meromorphic and anti-meromorphic parts of an object will be
denoted without and with a bar, see (6.55) for an example.
The extended complex plane ¯
C can be covered by two coordinate patches z ∈ C
and w ∈ C. In the first, the point at infinity (north pole) is removed, in the second,
the origin (south pole) is removed. On the overlap, the transition function is
w =
1
z
.
(6.8)
This description avoids to work with the infinity: studying the behaviour of f (z) at
z = ∞ is equivalent to study f (1/w) at w = 0.
Since any two-dimensional metric is locally conformally equivalent to the flat
metric, it is sufficient to work with this metric in each patch. This is particularly
convenient for the Riemann sphere since one patch covers it completely except for
one point.
6.1.2 Relation to the Cylinder: String Theory
The worldsheet of a closed string propagating in spacetime is locally topologically
a cylinder R × S 1 of circumference L. In this section, we show that the cylinder
can also be mapped to the complex plane—and thus to the Riemann sphere—after
removing two points. Since the cylinder has a clear physical interpretation in string
theory, it is useful to know how to translate the results from the plane to the cylinder.
It makes also sense to define two-dimensional models on the cylinder independently of a string theory interpretation since the compactification of the spatial
direction from R to S 1 regulates the infrared divergences. Moreover, it leads to a
natural definition of a “time” and of a Hamiltonian on the Euclidean plane.