186
8 BRST Quantization
which confirms that the BRST charge increases the ghost number by 1
[N gh , Q B ] = Q B .
(8.20)
One finds that the BRST charge is nilpotent
{Q B , Q B } = 0
(8.21)
and commutes with the energy–momentum tensor
[Q B , T (z)] = 0
(8.22)
only if the matter central charge corresponds to the critical dimension
c m = 26.
(8.23)
The most important commutator for the modes is
L n = {Q B , b n }.
(8.24)
Nilpotency of Q B then implies that Q B commutes with L n
[Q B , L n ] = 0.
(8.25)
8.3
BRST Cohomology: Two Flat Directions
The simplest case for studying the BRST cohomology is when the target spacetime
has at least two non-compact flat directions represented by two free scalar fields
(X 0 , X 1 ) (Sect. 7.1). The remaining matter fields are arbitrary as long as the critical
dimension c m = 26 is reached. The reason for introducing two flat directions is
that the cohomology is easily worked out by introducing light-cone (or complex)
coordinates in target spacetime.
The field X 0 can be spacelike or timelike 0 = ±1, while we consider X 1 to
be always spacelike, 1 = 1. The oscillators are denoted by α 0
m and α 1
m , and the
momenta of the Fock vacua by k = (k 0 , k 1 ) such that
k
2
= 0
k
0
2 +
k
1
2 .
(8.26)
The rest of the matter sector, called the transverse sector ⊥, is an arbitrary CFT
with energy–momentum tensor T ⊥ , central charge c ⊥ = 24 and Hilbert space
H ⊥ . The ghost together with the two scalar fields forms the longitudinal sector .
The motivation for the names longitudinal and transverse will become clear later:
8 BRST Quantization
which confirms that the BRST charge increases the ghost number by 1
[N gh , Q B ] = Q B .
(8.20)
One finds that the BRST charge is nilpotent
{Q B , Q B } = 0
(8.21)
and commutes with the energy–momentum tensor
[Q B , T (z)] = 0
(8.22)
only if the matter central charge corresponds to the critical dimension
c m = 26.
(8.23)
The most important commutator for the modes is
L n = {Q B , b n }.
(8.24)
Nilpotency of Q B then implies that Q B commutes with L n
[Q B , L n ] = 0.
(8.25)
8.3
BRST Cohomology: Two Flat Directions
The simplest case for studying the BRST cohomology is when the target spacetime
has at least two non-compact flat directions represented by two free scalar fields
(X 0 , X 1 ) (Sect. 7.1). The remaining matter fields are arbitrary as long as the critical
dimension c m = 26 is reached. The reason for introducing two flat directions is
that the cohomology is easily worked out by introducing light-cone (or complex)
coordinates in target spacetime.
The field X 0 can be spacelike or timelike 0 = ±1, while we consider X 1 to
be always spacelike, 1 = 1. The oscillators are denoted by α 0
m and α 1
m , and the
momenta of the Fock vacua by k = (k 0 , k 1 ) such that
k
2
= 0
k
0
2 +
k
1
2 .
(8.26)
The rest of the matter sector, called the transverse sector ⊥, is an arbitrary CFT
with energy–momentum tensor T ⊥ , central charge c ⊥ = 24 and Hilbert space
H ⊥ . The ghost together with the two scalar fields forms the longitudinal sector .
The motivation for the names longitudinal and transverse will become clear later: