38
2 Worldsheet Path Integral: Vacuum Amplitudes
symmetries. Hence, one can expect difficulties for imposing it at the quantum level
and ensuring that the Liouville mode in (2.16) remains without dynamics.
The metric variation (symmetric tensor) is decomposed in its trace and traceless
parts
δg ab = g ab δδ + δg
⊥
ab ,
δδ=
1
2
g
ab δg ab ,
g
ab δg
⊥
ab = 0.
(2.37)
In this decomposition, both terms are decoupled in the inner product
|δg ab |
2
g = 4u|δδ|
2
g + |δg
⊥
μν |
2
g
,
(2.38)
where the norm of δδ is the one of a scalar field (2.34a). The norm for δg ⊥
ab is
equivalent to (2.34c) with u = 0 (since it is traceless). Requiring positivity of the
inner product for a non-traceless tensor imposes the following constraint on u:
u > 0.
(2.39)
One can absorb the coefficient with u in δδ, which will just contribute as an overall
factor: its precise value has no physical meaning. The simple choice u = 1/4 sets the
coefficient of |δδ| 2
g to 1 in (2.38) (another common choice is u = 1/2). Ultimately,
this implies that the measure factorizes as
d g g ab = d g d g g
⊥
ab .
(2.40)
Computation: Equation (2.38)
G
abcd δg ab δg cd =
G
abcd
⊥
+ u g
ab g
cd
g ab δδ + δg
⊥
ab
g cd δδ + δg
⊥
cd
=
2u g
cd δδ + G
abcd
⊥ δg
⊥
ab
g cd δδ + δg
⊥
cd
= 4u (δδ)
2
+ G
abcd
⊥ δg
⊥
ab δg
⊥
cd
= 4u δδ
2
+ 2g
ac g
bd δg
⊥
ab δg
⊥
cd .
Remark 2.5 Another common parametrization is
G
abcd
= g
ac g
bd
+ c g
ab g
cd .
(2.41)
It corresponds to (2.35) up to a factor 1/2 and setting u = 1 + 2c.
Remark 2.6 (Matter and Curved Background Measures) As explained previously,
matter fields carry a representation and the inner product must yield an invariant
combination. In particular, spacetime indices must be contracted with the spacetime
metric G μν (X) (which is the non-linear sigma model metric appearing in front of
2 Worldsheet Path Integral: Vacuum Amplitudes
symmetries. Hence, one can expect difficulties for imposing it at the quantum level
and ensuring that the Liouville mode in (2.16) remains without dynamics.
The metric variation (symmetric tensor) is decomposed in its trace and traceless
parts
δg ab = g ab δδ + δg
⊥
ab ,
δδ=
1
2
g
ab δg ab ,
g
ab δg
⊥
ab = 0.
(2.37)
In this decomposition, both terms are decoupled in the inner product
|δg ab |
2
g = 4u|δδ|
2
g + |δg
⊥
μν |
2
g
,
(2.38)
where the norm of δδ is the one of a scalar field (2.34a). The norm for δg ⊥
ab is
equivalent to (2.34c) with u = 0 (since it is traceless). Requiring positivity of the
inner product for a non-traceless tensor imposes the following constraint on u:
u > 0.
(2.39)
One can absorb the coefficient with u in δδ, which will just contribute as an overall
factor: its precise value has no physical meaning. The simple choice u = 1/4 sets the
coefficient of |δδ| 2
g to 1 in (2.38) (another common choice is u = 1/2). Ultimately,
this implies that the measure factorizes as
d g g ab = d g d g g
⊥
ab .
(2.40)
Computation: Equation (2.38)
G
abcd δg ab δg cd =
G
abcd
⊥
+ u g
ab g
cd
g ab δδ + δg
⊥
ab
g cd δδ + δg
⊥
cd
=
2u g
cd δδ + G
abcd
⊥ δg
⊥
ab
g cd δδ + δg
⊥
cd
= 4u (δδ)
2
+ G
abcd
⊥ δg
⊥
ab δg
⊥
cd
= 4u δδ
2
+ 2g
ac g
bd δg
⊥
ab δg
⊥
cd .
Remark 2.5 Another common parametrization is
G
abcd
= g
ac g
bd
+ c g
ab g
cd .
(2.41)
It corresponds to (2.35) up to a factor 1/2 and setting u = 1 + 2c.
Remark 2.6 (Matter and Curved Background Measures) As explained previously,
matter fields carry a representation and the inner product must yield an invariant
combination. In particular, spacetime indices must be contracted with the spacetime
metric G μν (X) (which is the non-linear sigma model metric appearing in front of