8.2 BRST in the CFT Formalism
185
where
Q B =
m =0
c −m L
m
m −
1
2
m,n =0
m+n =0
(m − n)
c −m c −n b m+n
,
(8.14b)
M =
m =0
m c −m c m .
(8.14c)
The interest of this decomposition is that L 0 , M and
Q do not contain b 0 or c 0 ,
which make it very useful to act on states decomposed according to the zeromodes (7.169). The nilpotency of the BRST operator implies the relations
[L 0 , M] = [
Q B , M] = [
Q B , L 0 ] = 0,
Q
2
B = L 0 M.
(8.15)
Moreover, one has N gh (
Q B ) = 1 and N gh (M) = 2.
8.2.3 Commutators
From the various OPEs, one can compute the (anti-)commutators of the BRST
charge with the other operators. For the ghosts and a weight h primary field φ,
one finds
{Q B , b(z)} = T (z),
(8.16a)
{Q B , c(z)} = c(z)∂c(z),
(8.16b)
[Q B , φ(z)] = h ∂c(z)φ(z) + c(z)∂φ(z).
(8.16c)
This reproduces correctly (3.53).
Two facts will be useful in string theory. First, (8.16c) is a total derivative for
h = 1
[Q B , φ(z)] = ∂
c(z)φ(z)
.
(8.17)
Second, c(z)φ(z) is closed if h = 1
{Q B , c(z)φ(z)} = (1 − h)c(z)∂c(z)φ(z).
(8.18)
The commutator with the ghost current is
[Q B , j (z)] = −j B (z),
(8.19)