2.2 Path Integral
35
The variation of the action under the transformations (2.7) vanishes on-shell if the
energy–momentum tensor is conserved
∇
a T m,ab = 0
(on-shell).
(2.24)
On the other hand, the variation under (2.11) vanishes off-shell (i.e. without using
the equations of motion) if the energy–momentum tensor is traceless:
g
ab T m,ab = 0
(off-shell).
(2.25)
The conserved charges associated to the energy–momentum tensor generate worldsheet translations
P
a
:=
dσ T
0a
m .
(2.26)
The first component is identified with the worldsheet Hamiltonian P 0 = H and
generates time translations; the second component generates spatial translations.
Remark 2.4 (Tracelessness of the Energy–Momentum Tensor) In fact, the trace can
also be proportional to the curvature
g
ab T m,ab ∝ R.
(2.27)
Then, the equations of motion are invariant since the integral of R is topological.
The theory is invariant even if the action is not. Importantly, this happens for fields
at the quantum level (Weyl anomaly), for the Weyl ghost field (Sect. 2.4) and for the
Liouville theory (two-dimensional gravity coupled to conformal matter).
2.2
Path Integral
The quantization of the system is achieved by considering the path integral, which
yields the genus-g vacuum amplitude (or partition function):
Z g :=
d g g ab
gauge [g]
Z m [g],
Z m [g] :=
d g e
−S m [g,,]
(2.28)
at fixed genus g (not to be confused with the metric). The integration over g ab is
performed over all metrics of the genus-g Riemann surface g : g ab ∈ Met(( g ).
The factor gauge [g] is a normalization inserted in order to make the integral finite:
it depends on the metric (but only through the moduli parameters, as we will show
later) [11, p. 931], which explains why it is included after the integral sign. Its
value will be determined in the next section by requiring the cancellation of the
infinities due to the integration over the gauge parameters. This partition function
35
The variation of the action under the transformations (2.7) vanishes on-shell if the
energy–momentum tensor is conserved
∇
a T m,ab = 0
(on-shell).
(2.24)
On the other hand, the variation under (2.11) vanishes off-shell (i.e. without using
the equations of motion) if the energy–momentum tensor is traceless:
g
ab T m,ab = 0
(off-shell).
(2.25)
The conserved charges associated to the energy–momentum tensor generate worldsheet translations
P
a
:=
dσ T
0a
m .
(2.26)
The first component is identified with the worldsheet Hamiltonian P 0 = H and
generates time translations; the second component generates spatial translations.
Remark 2.4 (Tracelessness of the Energy–Momentum Tensor) In fact, the trace can
also be proportional to the curvature
g
ab T m,ab ∝ R.
(2.27)
Then, the equations of motion are invariant since the integral of R is topological.
The theory is invariant even if the action is not. Importantly, this happens for fields
at the quantum level (Weyl anomaly), for the Weyl ghost field (Sect. 2.4) and for the
Liouville theory (two-dimensional gravity coupled to conformal matter).
2.2
Path Integral
The quantization of the system is achieved by considering the path integral, which
yields the genus-g vacuum amplitude (or partition function):
Z g :=
d g g ab
gauge [g]
Z m [g],
Z m [g] :=
d g e
−S m [g,,]
(2.28)
at fixed genus g (not to be confused with the metric). The integration over g ab is
performed over all metrics of the genus-g Riemann surface g : g ab ∈ Met(( g ).
The factor gauge [g] is a normalization inserted in order to make the integral finite:
it depends on the metric (but only through the moduli parameters, as we will show
later) [11, p. 931], which explains why it is included after the integral sign. Its
value will be determined in the next section by requiring the cancellation of the
infinities due to the integration over the gauge parameters. This partition function