128
6 Conformal Field Theory on the Plane
The mode expansions have no branch cut (fractional power of z or ¯
z) for periodic
fields (bosonic untwisted or fermionic twisted). We will see explicit examples of
such operators later in this book.
Under Euclidean conjugation (6.95), the modes are related by
(O
‡ ) −m,−n = (O m,n )
† .
(6.106)
In particular, if the operator is Hermitian (under the Euclidean adjoint), the reality
condition on the modes relates the negative modes with the conjugated positive
modes
O
‡
= O ⇒ (O m,n )
†
= O −m,−n .
(6.107)
When no confusion is possible (for Hermitian operators), we will write O
†
m,n instead
of (O m,n ) † .
For a holomorphic field φ(z), the above expansion becomes
φ(z) =
n∈Z+h+ν
φ n
z n+h .
(6.108)
Conversely, the modes are recovered from the field through
φ n =
C 0
dz
2π i
z
n+h−1 φ(z),
(6.109)
where the integration is counter-clockwise around the origin.
If the field is Hermitian, then
φ
‡
= φ ⇒ (φ n )
†
= φ −n .
(6.110)
The operators φ n have a conformal weight of −n (since the weight of z is −1). The
BPZ conjugate of the modes is
φ
t
n = (I
±
◦ φ) n = (−1)
h (±1)
n φ −n .
(6.111)
Computation: Equation (6.111)
φ
t
n = (I
±
◦ φ) n =
dz
2π i
z
n+h−1 I
±
◦ φ(z)
=
dz
2π i
z
n+h−1
∓
1
z 2
h
φ
±
1
z