2.3 Faddeev–Popov Gauge Fixing
49
Zero-modes ξ (0) of P 1 are called conformal Killing vectors (CKV)
ξ
(0)
∈ K g := ker P 1
(2.91)
and satisfy the conformal Killing equation (see also Sect. 5.1):
(P 1 ξ
(0) ) ab = ∇ a ξ
(0)
b + ∇ b ξ
(0)
a − g ab ∇ c ξ
(0)c
= 0.
(2.92)
CKVs correspond to reparametrizations which can be absorbed by a change of the
conformal factor. They should be removed from the ξ integration in order to not
double-count the corresponding metrics. The dimension of the zero-modes CKV
space depends on the genus [26]:
K g := dim R K g = dim R ker P 1 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
6 g = 0,
2 g = 1,
0 g > 1.
(2.93)
The associated transformations will be interpreted later (Chap. 5). The groups
generated by the CKVs are
g = 0 : K 0 = SL(2, C),
g = 1 : K 1 = U(1) × U(1).
(2.94)
Note that the first group is non-compact while the second is compact.
A general vector ξ can be separated into a zero-mode part and its orthogonal
complement ξ :
ξ = ξ
(0)
+ ξ
,
(2.95)
such that
(ξ
(0) , ξ
) g = 0
(2.96)
for the inner product (2.34b). Because zero-modes are annihilated by P 1 , the correct
change of variables in the partition function (2.66) maps to ξ only:
(P 1 ξ, ,) −→ (ξ
, φ).
(2.97)
Integrating over ξ at this stage would double-count the CKV (since they are already
described by the φ integration). The appropriate Jacobian reads
d g d g (P 1 ξ) = d g φ d g ξ
FP [g],
(2.98)
49
Zero-modes ξ (0) of P 1 are called conformal Killing vectors (CKV)
ξ
(0)
∈ K g := ker P 1
(2.91)
and satisfy the conformal Killing equation (see also Sect. 5.1):
(P 1 ξ
(0) ) ab = ∇ a ξ
(0)
b + ∇ b ξ
(0)
a − g ab ∇ c ξ
(0)c
= 0.
(2.92)
CKVs correspond to reparametrizations which can be absorbed by a change of the
conformal factor. They should be removed from the ξ integration in order to not
double-count the corresponding metrics. The dimension of the zero-modes CKV
space depends on the genus [26]:
K g := dim R K g = dim R ker P 1 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
6 g = 0,
2 g = 1,
0 g > 1.
(2.93)
The associated transformations will be interpreted later (Chap. 5). The groups
generated by the CKVs are
g = 0 : K 0 = SL(2, C),
g = 1 : K 1 = U(1) × U(1).
(2.94)
Note that the first group is non-compact while the second is compact.
A general vector ξ can be separated into a zero-mode part and its orthogonal
complement ξ :
ξ = ξ
(0)
+ ξ
,
(2.95)
such that
(ξ
(0) , ξ
) g = 0
(2.96)
for the inner product (2.34b). Because zero-modes are annihilated by P 1 , the correct
change of variables in the partition function (2.66) maps to ξ only:
(P 1 ξ, ,) −→ (ξ
, φ).
(2.97)
Integrating over ξ at this stage would double-count the CKV (since they are already
described by the φ integration). The appropriate Jacobian reads
d g d g (P 1 ξ) = d g φ d g ξ
FP [g],
(2.98)