3.2 BRST Quantization
83
a delta function
Z g =
M g
d M g t
ckv [g]
d g g ab d g d g b d
g c δ
√
gg ab −
ˆ
g ˆ
g ab
×
M g
i=1
(φ i , b) g e
−S m [g,,]−S gh [g,b,c] .
(3.47)
Note that it is necessary to use the traceless gauge fixing condition (2.152) as it will
become clear. The delta function is Fourier transformed in an exponential, thanks to
an auxiliary bosonic field
Z g =
M g
d M g t
ckv [g]
d g g ab d g B
ab d g d g b d
g c
×
M g
i=1
(φ i , b) g e
−S m [g,,]−S gh [g, ˆ
g,B]−S gh [g,b,c] ,
(3.48)
where the gauge fixing action reads
S gf [g, ˆ
g, B] = −
i
4π
d
2 σ B
ab
√
gg ab −
ˆ
g ˆ
g ab
.
(3.49)
Varying the action with respect to the auxiliary field B ab , called the Nakanishi–
Lautrup field, produces the gauge fixing condition.
The BRST transformations are
δ g ab = i L c g ab ,
δ = i L c ,
δ c
a
= i L c c
a ,
δ b ab = B ab ,
δ B ab = 0,
(3.50)
where is a Grassmann parameter (anti-commuting number) independent of the
position. If the traceless gauge fixing (2.152) is not used, then B ab is not traceless:
in that case, the variation δ b ab will generate a trace, which is not consistent. Since
the transformations act on the matter action S m as a diffeomorphism with vector c a ,
it is obvious that it is invariant by itself. It is easy to show that the transformations
(3.50) leave the total action invariant in (3.48). The invariance of the measure is
given in [16].
Remark 3.3 (BRST Transformations with Weyl Ghost) One can also consider the
action (2.153) with the Weyl ghost. In this case, the transformation law of the metric
is modified and the Weyl ghost transforms as a scalar
δ g ab = i L c g ab + i g ab c w ,
δ c w = i L c c w .
(3.51)
83
a delta function
Z g =
M g
d M g t
ckv [g]
d g g ab d g d g b d
g c δ
√
gg ab −
ˆ
g ˆ
g ab
×
M g
i=1
(φ i , b) g e
−S m [g,,]−S gh [g,b,c] .
(3.47)
Note that it is necessary to use the traceless gauge fixing condition (2.152) as it will
become clear. The delta function is Fourier transformed in an exponential, thanks to
an auxiliary bosonic field
Z g =
M g
d M g t
ckv [g]
d g g ab d g B
ab d g d g b d
g c
×
M g
i=1
(φ i , b) g e
−S m [g,,]−S gh [g, ˆ
g,B]−S gh [g,b,c] ,
(3.48)
where the gauge fixing action reads
S gf [g, ˆ
g, B] = −
i
4π
d
2 σ B
ab
√
gg ab −
ˆ
g ˆ
g ab
.
(3.49)
Varying the action with respect to the auxiliary field B ab , called the Nakanishi–
Lautrup field, produces the gauge fixing condition.
The BRST transformations are
δ g ab = i L c g ab ,
δ = i L c ,
δ c
a
= i L c c
a ,
δ b ab = B ab ,
δ B ab = 0,
(3.50)
where is a Grassmann parameter (anti-commuting number) independent of the
position. If the traceless gauge fixing (2.152) is not used, then B ab is not traceless:
in that case, the variation δ b ab will generate a trace, which is not consistent. Since
the transformations act on the matter action S m as a diffeomorphism with vector c a ,
it is obvious that it is invariant by itself. It is easy to show that the transformations
(3.50) leave the total action invariant in (3.48). The invariance of the measure is
given in [16].
Remark 3.3 (BRST Transformations with Weyl Ghost) One can also consider the
action (2.153) with the Weyl ghost. In this case, the transformation law of the metric
is modified and the Weyl ghost transforms as a scalar
δ g ab = i L c g ab + i g ab c w ,
δ c w = i L c c w .
(3.51)