410
C Quantum Field Theory
At level 0, one introduces the pair
(B 0a 0 , ¯
c 0a 0 ) := (B
0
0a 0
, ¯
c
0
0a 0
)
(C.106)
and the associated antifields. The field ¯
c 0 := b is the Faddeev–Popov anti-ghost
associated to c 0 , and the trivial pair satisfies
|B 0 | = | 0 |,
| ¯
c 0 | = −| 0 |,
N gh (B 0 ) = 0,
N gh (c 0 ) = −1.
(C.107)
For level 1, two additional pairs
(B
0
1a 1
, ¯
c
0
1a 1
),
( ¯
B
1a 1
1 , c
1a 1
1 ),
(C.108)
and the corresponding antifields are introduced. The motivation for adding an
additional pair is that the level-0 pair only fixes m 0 − m 1 of the generators: the
additional m 1 extra ghosts c
1a 1
1 can be fixed by the residual level-0 symmetry. The
first level-1 pair fixes the level-1 symmetry.
Then, the gauge fixed action enjoys a BRST symmetry acting only on the fields
δ θ ψ
r
= θ sψ
r
= θ
∂ R S
∂ψ ∗
r
ψ ∗
r =∂ r
.
(C.109)
Note that this BRST operator is generically nilpotent only on-shell
s
2
∝ eom.
(C.110)
C.3.3 Quantum BV
At the quantum level, one considers the path integral
Z =
dψ
r dψ
∗
r e
−W [ψ r ,ψ ∗
r ]/¯ h ,
(C.111)
where W is called the quantum master action. The reason for distinguishing it from
the classical master action S is that the measure is not necessarily invariant by
itself under the generalized BRST transformation—this translates into a non-gauge
invariance of the measure of the physical fields, i.e. a gauge anomaly.
Quantum BRST transformation is generated by the quantum BRST operator σ
δ θ F = θ σ F = (W, F ) − ¯
h hF,
(C.112)
C Quantum Field Theory
At level 0, one introduces the pair
(B 0a 0 , ¯
c 0a 0 ) := (B
0
0a 0
, ¯
c
0
0a 0
)
(C.106)
and the associated antifields. The field ¯
c 0 := b is the Faddeev–Popov anti-ghost
associated to c 0 , and the trivial pair satisfies
|B 0 | = | 0 |,
| ¯
c 0 | = −| 0 |,
N gh (B 0 ) = 0,
N gh (c 0 ) = −1.
(C.107)
For level 1, two additional pairs
(B
0
1a 1
, ¯
c
0
1a 1
),
( ¯
B
1a 1
1 , c
1a 1
1 ),
(C.108)
and the corresponding antifields are introduced. The motivation for adding an
additional pair is that the level-0 pair only fixes m 0 − m 1 of the generators: the
additional m 1 extra ghosts c
1a 1
1 can be fixed by the residual level-0 symmetry. The
first level-1 pair fixes the level-1 symmetry.
Then, the gauge fixed action enjoys a BRST symmetry acting only on the fields
δ θ ψ
r
= θ sψ
r
= θ
∂ R S
∂ψ ∗
r
ψ ∗
r =∂ r
.
(C.109)
Note that this BRST operator is generically nilpotent only on-shell
s
2
∝ eom.
(C.110)
C.3.3 Quantum BV
At the quantum level, one considers the path integral
Z =
dψ
r dψ
∗
r e
−W [ψ r ,ψ ∗
r ]/¯ h ,
(C.111)
where W is called the quantum master action. The reason for distinguishing it from
the classical master action S is that the measure is not necessarily invariant by
itself under the generalized BRST transformation—this translates into a non-gauge
invariance of the measure of the physical fields, i.e. a gauge anomaly.
Quantum BRST transformation is generated by the quantum BRST operator σ
δ θ F = θ σ F = (W, F ) − ¯
h hF,
(C.112)