B Summary of Important Formulas
387
The conjugate vacuum is
k|p
μ
= =k|k
μ ,
k| = |k
† ,
−k| = |k
t .
(B.38)
B.3.5 Reparametrization Ghosts
The reparametrization ghosts are described by an anti-commuting first-order system
with the parameters (Chap. 7 and Table 7.1):
= 1,
λ= 2,
c gh = −26,
q gh = −3,
a gh = −1.
(B.39)
We focus on the holomorphic sector.
The b and c ghosts have weights:
h(b) = 2,
h(c)= −1,
(B.40)
such that the mode expansions are
b(z) =
n∈Z
b n
z n+2 ,
c(z)=
n∈Z
c n
z n−1 ,
(B.41a)
b n =
dz
2π i
z
n+1 b(z),
c n =
dz
2π i
z
n−2 c(z).
(B.41b)
The anti-commutators between the modes b n and c n read
{b m , c n } = δ m+n,0 ,
{b m , b n } = 0,
{c m , c n } = 0.
(B.42)
The energy–momentum tensor and the Virasoro modes are, respectively,
T = −2 : b∂c: − :∂b c:,
(B.43a)
L m =
n
n + m
: b m−n c n : =
n
(2m − n) : b n c m−n :.
(B.43b)
The expression of the zero-mode is
L 0 = −
n
n : b n c −n : =
n
n : b −n c n :.
(B.44)
The commutators between the L n and the ghost modes are
[L m , b n ] =
m − n
b m+n ,
[L m , c n ] = −(2m + n)c m+n .
(B.45)