386
B Summary of Important Formulas
We have
φ r |φ
c
s = (−1)
|φ r | δ rs .
(B.30)
B.3.4 Scalar Field
The simplest matter CFT is a set of D scalar fields X μ (z, ¯
z) such that i∂X μ and
i ¯
∂X μ are of weights h = (1, 0) and h = (0, 1)
i∂X
μ
=
n
α
μ
n
z n+1 ,
i ¯
∂X
μ
=
n
¯
α
μ
n
¯
z n+1 .
(B.31)
The commutation relations between the modes are
[α
μ
m , α
ν
n ] = mδ m+n,0 η
μν ,
[ ¯
α
μ
m , ¯
α
ν
n ] = mδ m+n,0 η
μν ,
[α
μ
m , ¯
α
ν
n ] = 0.
(B.32)
The zero-modes of both operators are equal and correspond to the (centre-of-mass)
momentum
α
μ
0 = ¯
α
μ
0 =
α
2
p
μ .
(B.33)
The conjugate of p μ is the centre-of-mass position x μ :
[x
μ , p
ν
] = η
μν .
(B.34)
Vertex operators are defined by
V k (z, ¯
z) = :e
ik·X(z,¯ z)
:,
h= ¯
h =
α 2 k 2
4
.
(B.35)
The scalar vacuum |k is annihilated by all positive-frequency oscillators, and it
is characterized by its eigenvalue for the zero-mode operator
p
μ
|k = k
μ
|k ,
∀n > 0 : α
μ
n |k = 0, ¯
α
μ
n |k = 0.
(B.36)
The vacuum is associated to the vertex operator V k :
|k = V k (0, 0) |0 = e
ik·x
|0 .
(B.37)