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B Summary of Important Formulas
B.3.2 General Properties
A primary holomorphic field φ(z) of weight h transforms as
f ◦ φ(z) =
df
dz
h
φ
f (z)
(B.15)
for any local change of coordinates f . A quasi-primary operator transforms like this
only for f ∈ SL(2, C). Its mode expansion reads
φ(z) =
n
φ n
z n+h ,
φ n =
C 0
dz
2π i
z
n+h−1 φ(z),
(B.16)
where the integration is counter-clockwise around the origin.
The SL(2, C) vacuum |0 is defined by
∀n ≥ −h + 1 : φ n |0 = 0.
(B.17)
Its BPZ conjugate 0| satisfies
∀n ≤ h − 1 : :0|φ n = 0.
(B.18)
The state–operator correspondence associates a state |φ to each operator φ(z):
|φ := φ(0) |0 = φ −h |0 .
(B.19)
The operator corresponding to the vacuum is the identity 1. 1 The Hermitian and
BPZ conjugated states are
φ
‡
| := =0|I ◦ φ
† (0) = lim
z→∞
z
2h
0|φ
† (z),
φ| := =0|I
±
◦ φ(0) = (±1)
h lim
z→∞
z
2h
0|φ(z).
(B.20)
The energy–momentum tensor is a quasi-primary operator of weight h = 2
T (z) =
n
L n
z n+2 .
(B.21)
The OPE between T and a primary operator h of weight h is
T (z)φ(w) ∼
h φ(w)
(z − w) 2 +
∂φ(w)
z − w
.
(B.22)
1 Exceptionally, the state |0 and the operator 1 do not have the same symbol.