2.4 Ghost Action
59
use of this point of view is to rewrite a massive vector field as a massless gauge field
together with an axion [30].
Remark 2.16 (Two-Dimensional Gravity) In 2d gravity, one does not work in the
critical dimension (2.125) and c L = 0. Thus, the Liouville mode does not decouple:
the conformal anomaly breaks the Weyl symmetry at the quantum level which
gives dynamics to gravity, even if it has no degree of freedom classically. As a
consequence, one chooses gauge = Diff .
Since the role of the classical Weyl symmetry is not as important as for string
theory, it is even not necessary to impose it classically. This leads to consider
non-conformal matter [1, 2, 4, 14, 15]. Following the arguments from Sect. 2.1,
the existence of the emergent Weyl symmetry (2.140) implies that the total action
S grav [ ˆ
g, φ] + S m [ ˆ
g, ,] must be a CFT for a flat background ˆ
g = δ, even if the two
actions are not independently CFTs.
2.4
Ghost Action
2.4.1 Actions and Equations of Motion
It is well-known that a determinant can be represented with two anti-commuting
fields, called ghosts. The fields carry indices dictated by the map induced by the
operator of the Faddeev–Popov determinant: one needs a symmetric and traceless
anti-ghost b ab and a vector ghost c a fields:
FP [g] =
d
g b d
g c e
−S gh [g,b,c] ,
(2.144)
where the prime indicates that the ghost zero-modes are omitted. The ghost action
is
S gh [g, b, c] :=
1
4π
d
2 σ
√
g g
ab g
cd b ac (P 1 c) bd
(2.145a)
=
1
4π
d
2 σ
√
g g
ab
b ac ∇ b c
c
+ b bc ∇ a c
c
− b ab ∇ c c
c
.
(2.145b)
The ghosts c a and anti-ghosts b ab are associated respectively to the variations due
to the diffeomorphisms ξ a and to the variations perpendicular to the gauge slice.
The normalization of 1/4π is conventional. In Minkowski signature, the action is
multiplied by a factor i.
Since b ab is traceless, the last term of the action vanishes and could be removed.
However, this implies to consider traceless variations of the b ab when varying the
action (to compute the equations of motion, the energy–momentum tensor, etc.).
On the other hand, one can keep the term and consider unconstrained variation of
b ab (since the structure of the action will force the variation to have the correct
59
use of this point of view is to rewrite a massive vector field as a massless gauge field
together with an axion [30].
Remark 2.16 (Two-Dimensional Gravity) In 2d gravity, one does not work in the
critical dimension (2.125) and c L = 0. Thus, the Liouville mode does not decouple:
the conformal anomaly breaks the Weyl symmetry at the quantum level which
gives dynamics to gravity, even if it has no degree of freedom classically. As a
consequence, one chooses gauge = Diff .
Since the role of the classical Weyl symmetry is not as important as for string
theory, it is even not necessary to impose it classically. This leads to consider
non-conformal matter [1, 2, 4, 14, 15]. Following the arguments from Sect. 2.1,
the existence of the emergent Weyl symmetry (2.140) implies that the total action
S grav [ ˆ
g, φ] + S m [ ˆ
g, ,] must be a CFT for a flat background ˆ
g = δ, even if the two
actions are not independently CFTs.
2.4
Ghost Action
2.4.1 Actions and Equations of Motion
It is well-known that a determinant can be represented with two anti-commuting
fields, called ghosts. The fields carry indices dictated by the map induced by the
operator of the Faddeev–Popov determinant: one needs a symmetric and traceless
anti-ghost b ab and a vector ghost c a fields:
FP [g] =
d
g b d
g c e
−S gh [g,b,c] ,
(2.144)
where the prime indicates that the ghost zero-modes are omitted. The ghost action
is
S gh [g, b, c] :=
1
4π
d
2 σ
√
g g
ab g
cd b ac (P 1 c) bd
(2.145a)
=
1
4π
d
2 σ
√
g g
ab
b ac ∇ b c
c
+ b bc ∇ a c
c
− b ab ∇ c c
c
.
(2.145b)
The ghosts c a and anti-ghosts b ab are associated respectively to the variations due
to the diffeomorphisms ξ a and to the variations perpendicular to the gauge slice.
The normalization of 1/4π is conventional. In Minkowski signature, the action is
multiplied by a factor i.
Since b ab is traceless, the last term of the action vanishes and could be removed.
However, this implies to consider traceless variations of the b ab when varying the
action (to compute the equations of motion, the energy–momentum tensor, etc.).
On the other hand, one can keep the term and consider unconstrained variation of
b ab (since the structure of the action will force the variation to have the correct