2.3 Faddeev–Popov Gauge Fixing
51
The volume of the group generated by the vectors orthogonal to the CKV is
denoted as
Diff 0
[g] :=
Diff 0
[ ˆ
g] =
d g ξ
.
(2.103)
As explained in the beginning of this section, one should extract the volume
of the full Diff 0 group, not only the volume
Diff 0
[g]. Since the two sets of
vectors are orthogonal, we can expect the measures, and thus the volumes, to
factorize. However, a Jacobian can and does arise: its role is to take into account
the normalization of the zero-modes. Denoting by ψ i a basis (not necessarily
orthonormal) for the zero-modes
ker P 1 = Span{ψ i },
i = 1, . . . , K g ,
(2.104)
the change of variables
ξ
−→ ξ
(2.105)
reads
d g ξ
=
1
det(ψ i , ψ j ) g
d g ξ
ckv [g]
,
(2.106)
where ckv [g] is the volume of the CKV group. The determinant is necessary when
the basis is not orthonormal. The relation between the gauge volumes is then
Diff 0 [g] =
det(ψ i , ψ j ) g ckv [g]
Diff 0
[g].
(2.107)
Note that the CKV volume is given in (2.111) and depends only on the topology but
not on the metric. By using arguments similar to the ones which lead to (2.61a), one
can expect that each term is independently invariant under Weyl rescaling: this is
indeed true (Sect. 2.3.3).
Computation: Equation (2.106)
Let us expand ξ (0) on the zero-mode basis
ξ
(0)
= α i ψ i ,
(2.108)
where the α i are real numbers, such that one can write the changes of variables
ξ −→ (ξ
, α i ).
(2.109)