6.2 Classical CFTs
113
where the additional division by Z 2 is clearer when studying an explicit representation. It corresponds with ker P 1 defined in (2.91):
K 0 = PSL(2, C).
(6.35)
A matrix representation of SL(2, C) is
g =
a b
c d
,
a, b, c, d ∈ C,
det g = ad − bc = 1,
(6.36)
which shows that this group has six real parameters
K 0 := dim SL(2, C) = 6.
(6.37)
The associated transformation on the complex plane reads
f g (z) =
az + b
cz + d
.
(6.38)
The quotient by Z 2 is required since changing the sign of all parameters does not
change the transformation. These transformations have received different names:
Möbius, projective, homographic, linear fractional transformations. . .
Holomorphic vector fields are then of the form
v(z) = β + 2αz + γ z
2 ,
¯
v(¯ z) = ¯
β + 2 ¯
α ¯
z + ¯
γ ¯
z
2 ,
(6.39)
where
a = 1 + α,
b = β,
c= −γ,
d = 1 − α.
(6.40)
The finite transformations associated with (5.12) are
translation:
f g (z) = z + a,
a ∈ C,
(6.41a)
rotation:
f g (z) = ζ z,
|ζ | = 1,
(6.41b)
dilatation:
f g (z) = λ z,
λ ∈ R,
(6.41c)
SCT:
f g (z) =
z
cz + 1
,
c∈ C.
(6.41d)
Investigation leads to the following association between the generators and transformations:
• translation: −1 and ¯
−1 ;
• dilatation (or radial translation): (( 0 + ¯
0 );