106
6 Conformal Field Theory on the Plane
− −−−−−−− →
Fig. 6.1 Map from the Riemann sphere to the complex plane. The south and north poles are
denoted by the letter S and N , and the equatorial circle by E
One speaks about “the point at infinity” because all the points at infinity (i.e. the
points z such that |z| → ∞)
lim
r→∞
r e
iθ
:= ∞
(6.2)
are identified (the limit is independent of θ ).
The identification can be understood by mapping (say) the south pole to the
origin of the plane and the north pole to infinity 2 (Fig. 6.1) through the stereographic
projection
z = e
iφ cot
θ
2
,
(6.3)
where (θ, φ) are angles on the sphere. Any circle on the sphere is mapped to a
circle in the complex plane. Conversely, the Riemann sphere can be viewed as a
compactification of the complex plane.
Introducing Cartesian coordinates (x, y) related to the complex coordinates by 3
z = x + iy,
¯
z = x − iy,
(6.4a)
x =
z + ¯
z
2
,
y =
z − ¯
z
2i
,
(6.4b)
2 Note that the points are distinguished in order to write the map, but they have nothing special by
themselves (i.e. they are not punctures).
3 General formulas can be found in Sect. 4.1 by replacing (τ, σ ) with (x, y). In most cases, the
conformal factor is set to zero (φ = 0) in this chapter.