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

2.6 Operator Algebra of Chiral Quasi-Primary Fields 33
quasi-primary fields and their derivatives 1 . To this end we make the ansatz
φi(z) φ j(w) = �
k,n≥0
C k ij
a n ijk
n!
1
(z − w) hi +h j −hk−n �n φk(w) , (2.51)
where the (z−w) part is fixed by the scaling behaviour under dilations z ↦→ λz.Note
that we have chosen our ansatz such that an ijk only depends on the conformal weights
hi, h j, hk of the fields i, j, k (and on n), while Ck ij contains further information about
the fields.
Let us now take w = 1 in Eq. (2.51) and consider it as part of the following
three-point function:
��
� �
φi(z) φ j(1) φk(0) = �
l,n≥0
C l ij
a n ijl
n!
1
(z − 1) hi +h j −hl−n
� � n φl(1) φk(0) � .
Using then the general formula for the two-point function (2.49), we find for the
correlator on the right-hand side that
� � n z φl(z)φk(0) �� � �z=1 = � n z
l,n≥0
� dlk δhl,hk
z 2hk
�� ���z=1
= (−1) n � �
2hk + n − 1
n!
n
dlk δhl,hk .
We therefore obtain
�
φi(z) φ j(1) φk(0) � = �
C l ij dlk a n � �
2hk + n − 1 (−1)
ijk
n
n
(z − 1) hi +h j −hk−n . (2.52)
However, we can also use the general expression for the three-point function (2.50)
with values z1 = z, z2 = 1 and z3 = 0. Combining then Eq. (2.50) with Eq. (2.52),
we find
�
C
l,n≥0
l ij dlk a n � �
2hk + n − 1 (−1)
ijk
n
n
(z − 1) hi
! Cijk
=
+h j −hk−n (z − 1) hi +h j −hk zhi +hk−h j ,
�
C l ij dlk a n � �
2hk + n − 1
ijk
(−1)
n
n (z − 1) n ! Cijk
= � �hi +hk−h j
1 + (z − 1) .
l,n≥0
Finally, we use the following relation with x = z − 1 for the term on the right-hand
side of the last formula:
1 The proof that the OPE of two quasi-primary fields involves indeed just other quasi-primary
fields and their derivatives is non-trivial and will not be presented.