2.3 Faddeev–Popov Gauge Fixing
55
If the number of non-compact dimensions is D, it means that the central charge
(2.21) of the transverse CFT satisfies
c ⊥ = 26 − D.
(2.126)
In this case, the integrand does not depend on the Liouville mode (because Diff is
invariant under Weyl transformations) and the integration over φ can be factored out
and yields the volume of the Weyl group (2.60b)
d g φ = Weyl [ ˆ
g].
(2.127)
Then, taking
gauge [ ˆ
g] = Diff [ ˆ
g] × Weyl [ ˆ
g]
(2.128)
removes the infinite gauge contributions and gives the partition function
Z g =
M g
d
M g t
det(φ i , ˆ
μ j ) ˆ
g
det(φ i , φ j ) ˆ
g
ckv [ ˆ
g] −1
det(ψ i , ψ j ) ˆ
g
FP [ ˆ
g] Z m [ ˆ
g].
(2.129)
2.3.4 Ambiguities, Ultralocality and Cosmological Constant
Different ambiguities remain in the previous computations, starting with the
definitions of the measures (2.32) and (2.34), then in obtaining the volume of
the diffeomorphism (2.60a) and Weyl (2.60b) groups, and finally in deriving the
conformal anomaly (2.118).
These different ambiguities can be removed by renormalizing the worldsheet
cosmological constant. This implies that the action
S μ [g] =
d
2 σ
√ g
(2.130)
must be added to the classical Lagrangian, where μ 0 is the bare cosmological
constant. This means that Weyl invariance is explicitly broken at the classical
level. After performing all the manipulations, μ 0 is determined by removing all
ambiguities and enforcing invariance under the Weyl symmetry at the quantum
level. This amounts to set the renormalized cosmological constant to zero (since
it breaks the Weyl symmetry). The possibility to introduce a counter-term violating
a classical symmetry arises because the symmetry itself is broken by a quantum
anomaly, so there is no reason to enforce it in the classical action.
We now review each issue separately. First, consider the inner product of a single
tensor (2.32): the determinant det γ g depends on the metric and one should be more
55
If the number of non-compact dimensions is D, it means that the central charge
(2.21) of the transverse CFT satisfies
c ⊥ = 26 − D.
(2.126)
In this case, the integrand does not depend on the Liouville mode (because Diff is
invariant under Weyl transformations) and the integration over φ can be factored out
and yields the volume of the Weyl group (2.60b)
d g φ = Weyl [ ˆ
g].
(2.127)
Then, taking
gauge [ ˆ
g] = Diff [ ˆ
g] × Weyl [ ˆ
g]
(2.128)
removes the infinite gauge contributions and gives the partition function
Z g =
M g
d
M g t
det(φ i , ˆ
μ j ) ˆ
g
det(φ i , φ j ) ˆ
g
ckv [ ˆ
g] −1
det(ψ i , ψ j ) ˆ
g
FP [ ˆ
g] Z m [ ˆ
g].
(2.129)
2.3.4 Ambiguities, Ultralocality and Cosmological Constant
Different ambiguities remain in the previous computations, starting with the
definitions of the measures (2.32) and (2.34), then in obtaining the volume of
the diffeomorphism (2.60a) and Weyl (2.60b) groups, and finally in deriving the
conformal anomaly (2.118).
These different ambiguities can be removed by renormalizing the worldsheet
cosmological constant. This implies that the action
S μ [g] =
d
2 σ
√ g
(2.130)
must be added to the classical Lagrangian, where μ 0 is the bare cosmological
constant. This means that Weyl invariance is explicitly broken at the classical
level. After performing all the manipulations, μ 0 is determined by removing all
ambiguities and enforcing invariance under the Weyl symmetry at the quantum
level. This amounts to set the renormalized cosmological constant to zero (since
it breaks the Weyl symmetry). The possibility to introduce a counter-term violating
a classical symmetry arises because the symmetry itself is broken by a quantum
anomaly, so there is no reason to enforce it in the classical action.
We now review each issue separately. First, consider the inner product of a single
tensor (2.32): the determinant det γ g depends on the metric and one should be more