110
6 Conformal Field Theory on the Plane
− −−−−−−− →
Fig. 6.3 Map from the cylinder with two caps (half-spheres) to the Riemann sphere 0
since local properties of the CFT (e.g. the spectrum of operators) are determined by
the conformal algebra, they will be common to all surfaces.
Mathematically, a difference between 0 and 0,2 had to be expected since the
sphere has a positive curvature (and χ = −2) but the cylinder is flat (with χ = 0).
Punctures contribute negatively to the curvature (and thus positively to the Euler
characteristics).
Remark 6.1 The coordinate z is always used as a coordinate on the complex plane,
but the corresponding metric may be different—compare (6.5) and (6.18). As
explained previously, this does not matter since the theory is insensitive to the
conformal factor.
6.2
Classical CFTs
In this section, we consider an action S[] which is conformally invariant. We
first identify and discuss the properties of the conformal algebra and group, before
explaining how a CFT is defined.
6.2.1 Witt Conformal Algebra
Since the Riemann sphere is identified with the complex plane, they share the same
conformal group and algebra. Consider the metric (6.5)
ds
2
= dzd¯ z,
(6.19)
then, any meromorphic change of coordinates
z −→ z
= f (z),
¯
z −→ ¯
z
= ¯
f (¯ z)
(6.20)
6 Conformal Field Theory on the Plane
− −−−−−−− →
Fig. 6.3 Map from the cylinder with two caps (half-spheres) to the Riemann sphere 0
since local properties of the CFT (e.g. the spectrum of operators) are determined by
the conformal algebra, they will be common to all surfaces.
Mathematically, a difference between 0 and 0,2 had to be expected since the
sphere has a positive curvature (and χ = −2) but the cylinder is flat (with χ = 0).
Punctures contribute negatively to the curvature (and thus positively to the Euler
characteristics).
Remark 6.1 The coordinate z is always used as a coordinate on the complex plane,
but the corresponding metric may be different—compare (6.5) and (6.18). As
explained previously, this does not matter since the theory is insensitive to the
conformal factor.
6.2
Classical CFTs
In this section, we consider an action S[] which is conformally invariant. We
first identify and discuss the properties of the conformal algebra and group, before
explaining how a CFT is defined.
6.2.1 Witt Conformal Algebra
Since the Riemann sphere is identified with the complex plane, they share the same
conformal group and algebra. Consider the metric (6.5)
ds
2
= dzd¯ z,
(6.19)
then, any meromorphic change of coordinates
z −→ z
= f (z),
¯
z −→ ¯
z
= ¯
f (¯ z)
(6.20)