2.4 Ghost Action
61
2.4.2 Weyl Ghost
Ghosts have been introduced for the reparametrizations (generated by ξ a ) and the
traceless part of the metric (the gauge field associated to the transformation): one
may wonder why there is not a ghost c w associated to the Weyl symmetry along
with an anti-ghost for the trace of the metric (i.e. the conformal factor). This can
be understood from several viewpoints. First, the relation between a metric and its
transformation—and the corresponding gauge fixing condition—does not involve
any derivative: as such, the Jacobian is trivial. Second, one could choose
F
⊥
ab =
√
gg ab −
ˆ
g ˆ
g ab = 0
(2.152)
as a gauge fixing condition instead of (2.63), and the trace component does not
appear anywhere. Finally, a local Weyl symmetry is not independent from the
diffeomorphisms.
Remark 2.17 (Local Weyl Symmetry) The topic of obtaining a local Weyl symmetry
by gauging a global Weyl symmetry (dilatation) is very interesting [16, chap. 15,
19]. Under general conditions, one can express the new action in terms of the Ricci
tensor (or of the curvature): this means that the Weyl gauge field and its curvature
are composite fields.
Moreover, one finds that local Weyl invariance leads to an off-shell condition
while diffeomorphisms give on-shell conditions. This explains why one imposes
only Virasoro constraints (associated to reparametrizations) and no constraints for
the Weyl symmetry in the covariant quantization.
However, it can be useful to introduce a ghost field c w for the Weyl symmetry
nonetheless. In view of the previous discussion, this field should appear as a
Lagrange multiplier which ensures that b ab is traceless. Starting from the action
(2.145), one finds
S
gh [g, b, c, c w ] =
1
4π
d
2 σ
√ g g
ab
b ac ∇ b c
c
+ b bc ∇ a c
c
+ 2b ab c w
,
(2.153)
where b ab is not traceless anymore. The ghost c w is not dynamical since the action
does not contain derivatives of it, and it can be integrated out of the path integral to
recover (2.145).
The equations of motion for this modified action are
∇ a c b + ∇ b c a + 2g ab c w = 0,
∇
a b ab = 0,
g
ab b ab = 0.
(2.154)
Contracting the first equation with the metric gives
c w = −
1
2
∇ a c
a ,
(2.155)
61
2.4.2 Weyl Ghost
Ghosts have been introduced for the reparametrizations (generated by ξ a ) and the
traceless part of the metric (the gauge field associated to the transformation): one
may wonder why there is not a ghost c w associated to the Weyl symmetry along
with an anti-ghost for the trace of the metric (i.e. the conformal factor). This can
be understood from several viewpoints. First, the relation between a metric and its
transformation—and the corresponding gauge fixing condition—does not involve
any derivative: as such, the Jacobian is trivial. Second, one could choose
F
⊥
ab =
√
gg ab −
ˆ
g ˆ
g ab = 0
(2.152)
as a gauge fixing condition instead of (2.63), and the trace component does not
appear anywhere. Finally, a local Weyl symmetry is not independent from the
diffeomorphisms.
Remark 2.17 (Local Weyl Symmetry) The topic of obtaining a local Weyl symmetry
by gauging a global Weyl symmetry (dilatation) is very interesting [16, chap. 15,
19]. Under general conditions, one can express the new action in terms of the Ricci
tensor (or of the curvature): this means that the Weyl gauge field and its curvature
are composite fields.
Moreover, one finds that local Weyl invariance leads to an off-shell condition
while diffeomorphisms give on-shell conditions. This explains why one imposes
only Virasoro constraints (associated to reparametrizations) and no constraints for
the Weyl symmetry in the covariant quantization.
However, it can be useful to introduce a ghost field c w for the Weyl symmetry
nonetheless. In view of the previous discussion, this field should appear as a
Lagrange multiplier which ensures that b ab is traceless. Starting from the action
(2.145), one finds
S
gh [g, b, c, c w ] =
1
4π
d
2 σ
√ g g
ab
b ac ∇ b c
c
+ b bc ∇ a c
c
+ 2b ab c w
,
(2.153)
where b ab is not traceless anymore. The ghost c w is not dynamical since the action
does not contain derivatives of it, and it can be integrated out of the path integral to
recover (2.145).
The equations of motion for this modified action are
∇ a c b + ∇ b c a + 2g ab c w = 0,
∇
a b ab = 0,
g
ab b ab = 0.
(2.154)
Contracting the first equation with the metric gives
c w = −
1
2
∇ a c
a ,
(2.155)