114
6 Conformal Field Theory on the Plane
• rotation (or angular translation): i(( 0 − ¯
0 );
• special conformal transformation: 1 and ¯
1 .
The inversion defined by
inversion:
I
+ (z) := I (z) :=
1
z
(6.42a)
is not an element of SL(2, C). However, the inversion with a minus sign
I
− (z) := −I (z) = I (−z) = −
1
z
(6.42b)
is a SL(2, C) transformation.
A useful transformation is the circular permutation of (0, 1, ∞):
g ∞,0,1 (z) =
1
1 − z
.
(6.43)
6.2.3 Definition of a CFT
A CFT is characterized by its set of (composite) fields (also called operators) O(z, ¯
z)
which correspond to any local expression constructed from the fields appearing
in the Lagrangian and of their derivatives. 5 For example, in a scalar field theory, the
simplest operators are of the form ∂ m φ n .
Among the operators, two particular categories are distinguished according to
their transformation laws:
• primary operator:
∀f meromorphic :
O(z, ¯
z) =
df
dz
h
d ¯
f
d¯ z
¯
h
O
f (z), ¯
f (¯ z)
,
(6.44)
• quasi-primary (or SL(2, C) primary) operator:
∀f ∈ PSL(2, C) :
O(z, ¯
z) =
df
dz
h
d ¯
f
d¯ z
¯
h
O
f (z), ¯
f (¯ z)
.
(6.45)
The parameters (h, ¯
h) are the conformal weights of the operator O (both are
independent from each other), and combinations of them give the conformal
5 Not all CFTs admit a Lagrangian description. But, since we are mostly interested in string theories
defined from Polyakov’s path integral, it is sufficient to study CFTs with a Lagrangian.