182
8 BRST Quantization
with the mode index. The total central charge, energy–momentum tensor and Hilbert
space are denoted by
c = c m + c gh = c m − 26,
T(z)= T
m (z) + T
gh (z),
H = H m ⊗ H gh .
(8.1)
The goal is to find the physical states in the cohomology, that is, which are BRST
closed
Q B |ψ = 0
(8.2)
but non-exact (Sect. 3.2): the latter statement can be understood as an equivalence
between closed states under shift by exact states
|ψ ∼ |ψ + Q B | .
(8.3)
We introduce the BRST current and study its CFT properties. Then, we give a
computation of the BRST cohomology when the matter CFT contains at least two
scalar fields.
8.2
BRST in the CFT Formalism
The BRST current can be found from (3.50) to be [12]
j B (z) = :c(z)
T
m (z) +
1
2
T
gh (z)
: + κ ∂
2 c(z)
(8.4a)
= c(z)T
m (z) + :b(z)c(z)∂c(z) : + κ ∂
2 c(z),
(8.4b)
and similarly for the anti-holomorphic sector. This can be derived from (3.53): the
generator of infinitesimal changes of coordinates (given by the Lie derivative) is
the energy–momentum tensor. The factor of 1/2 comes from the expression (7.99)
of the ghost energy–momentum tensor: the second term does not contribute, while
the first has a factor of 2. Since the transformation of c in (3.53) has no factor, the
1/2 is necessary to recover the correct normalization. Finally, one finds that the
transformation of b is reproduced. The different computations can be checked using
the OPEs given below. The last piece is a total derivative and does not contribute to
the charge: for this reason, it cannot be derived from (3.53), and its coefficient will
be determined below. Note that it is the only total derivative of dimension 1 and of
ghost number 1.
The BRST charge is then obtained by the contour integral
Q B = Q B,L + Q B,R ,
Q B,L =
dz
2π i
j B (z),
Q B,R =
d¯ z
2π i
¯
j B (¯ z).
(8.5)
8 BRST Quantization
with the mode index. The total central charge, energy–momentum tensor and Hilbert
space are denoted by
c = c m + c gh = c m − 26,
T(z)= T
m (z) + T
gh (z),
H = H m ⊗ H gh .
(8.1)
The goal is to find the physical states in the cohomology, that is, which are BRST
closed
Q B |ψ = 0
(8.2)
but non-exact (Sect. 3.2): the latter statement can be understood as an equivalence
between closed states under shift by exact states
|ψ ∼ |ψ + Q B | .
(8.3)
We introduce the BRST current and study its CFT properties. Then, we give a
computation of the BRST cohomology when the matter CFT contains at least two
scalar fields.
8.2
BRST in the CFT Formalism
The BRST current can be found from (3.50) to be [12]
j B (z) = :c(z)
T
m (z) +
1
2
T
gh (z)
: + κ ∂
2 c(z)
(8.4a)
= c(z)T
m (z) + :b(z)c(z)∂c(z) : + κ ∂
2 c(z),
(8.4b)
and similarly for the anti-holomorphic sector. This can be derived from (3.53): the
generator of infinitesimal changes of coordinates (given by the Lie derivative) is
the energy–momentum tensor. The factor of 1/2 comes from the expression (7.99)
of the ghost energy–momentum tensor: the second term does not contribute, while
the first has a factor of 2. Since the transformation of c in (3.53) has no factor, the
1/2 is necessary to recover the correct normalization. Finally, one finds that the
transformation of b is reproduced. The different computations can be checked using
the OPEs given below. The last piece is a total derivative and does not contribute to
the charge: for this reason, it cannot be derived from (3.53), and its coefficient will
be determined below. Note that it is the only total derivative of dimension 1 and of
ghost number 1.
The BRST charge is then obtained by the contour integral
Q B = Q B,L + Q B,R ,
Q B,L =
dz
2π i
j B (z),
Q B,R =
d¯ z
2π i
¯
j B (¯ z).
(8.5)