40
2 Worldsheet Path Integral: Vacuum Amplitudes
Fig. 2.1 The space of
metrics decomposed in gauge
orbits. Two metrics related by
a gauge transformation lie on
the same orbit. Choosing a
gauge slice amounts to pick
one metric in each orbit, and
the projection gives the space
of metric classes
One can perform the change of variables
r = x − y,
y = a
(2.46)
such that
Z =
R
da
∞
0
e
−r 2 =
√
π
2
Vol(R),
(2.47)
and Vol(R) is to be interpreted as the volume of the gauge group (translation by
a real number a).
Remark 2.7 Mathematically, the Faddeev–Popov procedure consists in identifying
the orbits (class of equivalent metrics) under the gauge group G and to write the
integral in terms of G-invariant objects (orbits instead of individual metrics). This
can be done by decomposing the tangent space into variations generated by G and
its complement. Then, one can define a foliation of the field space which equips it
with a fibre bundle structure: the base is the push-forward of the complement and
the fibre corresponds to the gauge orbits. The integral is then defined by selecting a
section of this bundle.
2.3.1 Metrics on Riemann Surfaces
According to the above procedure, each metric g ab ∈ Met(( g ) has to be expressed
in terms of gauge parameters (ξ and ω) and of a metric ˆ
g ab which contains the
2 Worldsheet Path Integral: Vacuum Amplitudes
Fig. 2.1 The space of
metrics decomposed in gauge
orbits. Two metrics related by
a gauge transformation lie on
the same orbit. Choosing a
gauge slice amounts to pick
one metric in each orbit, and
the projection gives the space
of metric classes
One can perform the change of variables
r = x − y,
y = a
(2.46)
such that
Z =
R
da
∞
0
e
−r 2 =
√
π
2
Vol(R),
(2.47)
and Vol(R) is to be interpreted as the volume of the gauge group (translation by
a real number a).
Remark 2.7 Mathematically, the Faddeev–Popov procedure consists in identifying
the orbits (class of equivalent metrics) under the gauge group G and to write the
integral in terms of G-invariant objects (orbits instead of individual metrics). This
can be done by decomposing the tangent space into variations generated by G and
its complement. Then, one can define a foliation of the field space which equips it
with a fibre bundle structure: the base is the push-forward of the complement and
the fibre corresponds to the gauge orbits. The integral is then defined by selecting a
section of this bundle.
2.3.1 Metrics on Riemann Surfaces
According to the above procedure, each metric g ab ∈ Met(( g ) has to be expressed
in terms of gauge parameters (ξ and ω) and of a metric ˆ
g ab which contains the