B Summary of Important Formulas
385
The OPE of T with itself defines the central charge c
T (z)T (w) ∼
c/2
(z − w) 4 +
2T (w)
(z − w) 2 +
∂T (w)
z − w
.
(B.23)
B.3.3 Hermitian and BPZ Conjugations
Both conjugations do not change the ghost number of a state.
Hermitian
The Hermitian conjugate of a general state built from n operators A i and a complex
number λ is
(λ A 1 · · · A n |0)
†
= λ
∗
0|A
†
n · · · A
†
1 .
(B.24)
BPZ
The BPZ conjugate of modes is
φ
t
n = (I
±
◦ φ) n = (−1)
h (±1)
n φ −n ,
(B.25)
where I ± (z) = ±1/z. The plus sign is usually used for the closed string, and the
minus sign for the open string. Given a general state built from n operators and a
complex number λ, the conjugation does not change the order of the operators and
does not conjugate complex numbers:
(λ A 1 · · · A n |0)
t
= λ 0|(A 1 )
t
· · · (A n )
t .
(B.26)
However, it reverses radial ordering such that operators must be (anti-)commuted in
radial ordered expressions.
The BPZ product satisfies
A, B = (−1)
|A||B|
B, A.
(B.27)
Moreover, the inner product is non-degenerate, so
∀A : :A|B = 0 |B = 0.
(B.28)
Denoting by {|φ r } a complete basis of states, then the conjugate basis {{φ c
r |} is
defined by the BPZ product as
φ
c
r |φ s = δ rs .
(B.29)