58
2 Worldsheet Path Integral: Vacuum Amplitudes
Weyl rescaling (2.10) is not. Together, the background diffeomorphisms and Weyl
symmetry have three gauge parameters, which is sufficient to completely fix the
background metric ˆ
g up to moduli.
In fact, the combination of both symmetries is equivalent to invariance under
the physical diffeomorphisms. To prove this statement, consider two metrics g
and g related by a diffeomorphism F and both gauge fixed to pairs (f, φ, ˆ
g) and
(f , φ , ˆ
g ):
g
ab = F
∗ g ab ,
g
ab = f
∗
e
2φ ˆ
g
ab
,
g ab = f
∗
e
2φ
ˆ
g ab
.
(2.141)
Then, the gauge fixing parametrizations are related by background symmetries
( ˆ
F , ω) as
ˆ
F = f
−1
◦ F ◦ f,
φ
= ˆ
F
∗ (φ − ω),
ˆ
g
ab = ˆ
F
∗ (e
2ω
ˆ
g ab ).
(2.142)
Moreover, this also implies that there is a diffeomorphism ˜
f = F ◦ f such that g
is gauge fixed in terms of (φ, ˆ
g):
g
ab = ˜
f
∗
e
2φ
ˆ
g ab
.
(2.143)
Computation: Equation (2.142)
The functions F , f , f , φ, φ and the metrics g ab , g
ab , ˆ
g ab and ˆ
g
ab are all fixed
and one must find ˆ
F and ω such that the relations (2.141) are compatible. First,
one rewrites g
ab in terms of ˆ
g ab and compare with the expression with ˆ
g
ab :
g
ab = F
∗ g ab = F
∗
f
∗
e
2φ
ˆ
g ab
= F
∗
f
∗
e
2(φ−ω) e
2ω
ˆ
g ab
= f
∗
e
2φ ˆ
g
ab
.
In the third equality, we have introduced ω because ˆ
g
ab = ˆ
F ∗ ˆ
g ab is not true in
general since there are 3 independent components but ˆ
F has only 2 parameters,
so we cannot just define f = F ◦ f and φ = φ. This explains the importance
of the emergent Weyl symmetry.
Remark 2.15 (Gauge Fixing and Field Redefinition) Although it looks like we are
undoing the gauge fixing, this is not exactly the case since the original metric
is not used anymore. One can understand the procedure of this section as a field
redefinition: the degrees of freedom in g ab are repackaged into two fields (φ, ˆ
g ab )
adapted to make some properties of the system more salient. A new gauge symmetry
is introduced to maintain the number of degrees of freedom. The latter helps to
understand the structure of the action on the background. Finally, in this context,
the Liouville action is understood as a Wess–Zumino action, which is defined as the
difference between the effective actions evaluated in each metric. Another typical
2 Worldsheet Path Integral: Vacuum Amplitudes
Weyl rescaling (2.10) is not. Together, the background diffeomorphisms and Weyl
symmetry have three gauge parameters, which is sufficient to completely fix the
background metric ˆ
g up to moduli.
In fact, the combination of both symmetries is equivalent to invariance under
the physical diffeomorphisms. To prove this statement, consider two metrics g
and g related by a diffeomorphism F and both gauge fixed to pairs (f, φ, ˆ
g) and
(f , φ , ˆ
g ):
g
ab = F
∗ g ab ,
g
ab = f
∗
e
2φ ˆ
g
ab
,
g ab = f
∗
e
2φ
ˆ
g ab
.
(2.141)
Then, the gauge fixing parametrizations are related by background symmetries
( ˆ
F , ω) as
ˆ
F = f
−1
◦ F ◦ f,
φ
= ˆ
F
∗ (φ − ω),
ˆ
g
ab = ˆ
F
∗ (e
2ω
ˆ
g ab ).
(2.142)
Moreover, this also implies that there is a diffeomorphism ˜
f = F ◦ f such that g
is gauge fixed in terms of (φ, ˆ
g):
g
ab = ˜
f
∗
e
2φ
ˆ
g ab
.
(2.143)
Computation: Equation (2.142)
The functions F , f , f , φ, φ and the metrics g ab , g
ab , ˆ
g ab and ˆ
g
ab are all fixed
and one must find ˆ
F and ω such that the relations (2.141) are compatible. First,
one rewrites g
ab in terms of ˆ
g ab and compare with the expression with ˆ
g
ab :
g
ab = F
∗ g ab = F
∗
f
∗
e
2φ
ˆ
g ab
= F
∗
f
∗
e
2(φ−ω) e
2ω
ˆ
g ab
= f
∗
e
2φ ˆ
g
ab
.
In the third equality, we have introduced ω because ˆ
g
ab = ˆ
F ∗ ˆ
g ab is not true in
general since there are 3 independent components but ˆ
F has only 2 parameters,
so we cannot just define f = F ◦ f and φ = φ. This explains the importance
of the emergent Weyl symmetry.
Remark 2.15 (Gauge Fixing and Field Redefinition) Although it looks like we are
undoing the gauge fixing, this is not exactly the case since the original metric
is not used anymore. One can understand the procedure of this section as a field
redefinition: the degrees of freedom in g ab are repackaged into two fields (φ, ˆ
g ab )
adapted to make some properties of the system more salient. A new gauge symmetry
is introduced to maintain the number of degrees of freedom. The latter helps to
understand the structure of the action on the background. Finally, in this context,
the Liouville action is understood as a Wess–Zumino action, which is defined as the
difference between the effective actions evaluated in each metric. Another typical