Introduction to Conformal Field Theory: With Applications to String ...

2.6 Operator Algebra of Chiral Quasi-Primary Fields 31
where d12 is called a structure constant. Finally, the invariance of the two-point
function under L1 essentially implies the invariance under transformations f (z) =
for which we find
− 1
z
� φ1(z) φ2(w) � → � 1
=
z
!
=
z 2h1
1
2h1w 2h2
d12
1
w2h2 φ1
� − 1
z
(z − w) h1+h2
� 1�
� 1 ��
− φ2 −
z w
d12
�
1 h1+h2
+ w
which can only be satisfied if h1 = h2. We therefore arrive at the result that
The SL(2, C)/Z2 conformal symmetry fixes the two-point function of
two chiral quasi-primary fields to be of the form
� φi(z) φ j(w) � = dij δhi ,h j
(z − w) 2hi
. (2.49)
As an example, let us consider the energy–momentum tensor T (z). From the
OPE shown in Eq. (2.41) (and using the fact that one-point functions of conformal
fields on the sphere vanish), we find that
� � c/2
T (z) T (w) = .
(z − w) 4
The Three-Point Function
After having determined the two-point function of two chiral quasi-primary fields
up to a constant, let us now consider the three-point function. From the invariance
under translations, we can infer that
�
φ1(z1) φ2(z2) φ3(z3) � = f (z12, z23, z13) ,
where we introduced zij
can be expressed as
= zi − z j. The requirement of invariance under dilation
�
φ1(z1) φ2(z2) φ3(z3) � → � λ h1φ1(λz1) λ h2φ2(λz2) λ h3φ3(λz3) �
from which it follows that
= λ h1+h2+h3 f ( λz12,λz23,λz13 )
!
= f ( z12, z23, z13 )