2.3 Faddeev–Popov Gauge Fixing
57
2.3.5 Gauge Fixed Path Integral
As a conclusion of this section, we found that the partition function (2.28) can be
written as
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],
(2.136a)
=
M g
d
M g t
det(φ i , ˆ
μ j ) 2
ˆ
g
det(φ i , φ j ) ˆ
g
det
ˆ
P
†
1
ˆ
P 1
det(ψ i , ψ j ) ˆ
g
Z m [ ˆ
g]
ckv [ ˆ
g]
.
(2.136b)
after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings. It is
implicit that the factors for the CKV and moduli are respectively absent for g > 1
and g < 1. For g = 0 the CKV group is non-compact and its volume is infinite. It
looks like the partition vanishes, but there are subtleties which will be discussed in
Sect. 3.1.3.
Remark 2.14 (Weil–Petersson Metric) When the metric is chosen to be of constant
curvature ˆ
R = −1, the moduli measure together with the determinants form the
Weil–Petersson measure
d(WP) =
M g
d
M g t
det(φ i , ˆ
μ j ) ˆ
g
det(φ i , φ j ) ˆ
g
.
(2.137)
In (2.136), the background metric ˆ
g ab is fixed. However, the derivation holds for
any choice of ˆ
g ab : as a consequence, it makes sense to relax the gauge fixing and
allow it to vary while adding gauge symmetries. The first symmetry is background
diffeomorphisms:
σ
a
= ˆ
f
a (σ
b ), ˆ
g
(σ
) = f
∗
ˆ
g(σ ), φ
(σ
) = f
∗ φ(σ ), ,
(σ
) = f
∗ (σ ).
(2.138)
This symmetry is automatic for S m [ ˆ
g, ,] since S m [g, ,] was invariant under (2.5).
Similarly, the integration measures are also invariant. A second symmetry is found
by inspecting the decomposition (2.56)
g ab = f
∗
e
2φ
ˆ
g ab (t)
,
(2.139)
which is left invariant under a background Weyl symmetry (also called emergent):
g
ab (σ ) = e
2ω(σ ) g ab (σ ),
φ
(σ ) = φ(σ ) − ω(σ ),
,
(σ ) = (σ ).
(2.140)
Let us stress that it is not related to the Weyl rescaling (2.10) of the metric g ab .
The background Weyl rescaling (2.140) is a symmetry even when the physical
57
2.3.5 Gauge Fixed Path Integral
As a conclusion of this section, we found that the partition function (2.28) can be
written as
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],
(2.136a)
=
M g
d
M g t
det(φ i , ˆ
μ j ) 2
ˆ
g
det(φ i , φ j ) ˆ
g
det
ˆ
P
†
1
ˆ
P 1
det(ψ i , ψ j ) ˆ
g
Z m [ ˆ
g]
ckv [ ˆ
g]
.
(2.136b)
after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings. It is
implicit that the factors for the CKV and moduli are respectively absent for g > 1
and g < 1. For g = 0 the CKV group is non-compact and its volume is infinite. It
looks like the partition vanishes, but there are subtleties which will be discussed in
Sect. 3.1.3.
Remark 2.14 (Weil–Petersson Metric) When the metric is chosen to be of constant
curvature ˆ
R = −1, the moduli measure together with the determinants form the
Weil–Petersson measure
d(WP) =
M g
d
M g t
det(φ i , ˆ
μ j ) ˆ
g
det(φ i , φ j ) ˆ
g
.
(2.137)
In (2.136), the background metric ˆ
g ab is fixed. However, the derivation holds for
any choice of ˆ
g ab : as a consequence, it makes sense to relax the gauge fixing and
allow it to vary while adding gauge symmetries. The first symmetry is background
diffeomorphisms:
σ
a
= ˆ
f
a (σ
b ), ˆ
g
(σ
) = f
∗
ˆ
g(σ ), φ
(σ
) = f
∗ φ(σ ), ,
(σ
) = f
∗ (σ ).
(2.138)
This symmetry is automatic for S m [ ˆ
g, ,] since S m [g, ,] was invariant under (2.5).
Similarly, the integration measures are also invariant. A second symmetry is found
by inspecting the decomposition (2.56)
g ab = f
∗
e
2φ
ˆ
g ab (t)
,
(2.139)
which is left invariant under a background Weyl symmetry (also called emergent):
g
ab (σ ) = e
2ω(σ ) g ab (σ ),
φ
(σ ) = φ(σ ) − ω(σ ),
,
(σ ) = (σ ).
(2.140)
Let us stress that it is not related to the Weyl rescaling (2.10) of the metric g ab .
The background Weyl rescaling (2.140) is a symmetry even when the physical