52
2 Worldsheet Path Integral: Vacuum Amplitudes
The Jacobian is computed from
1 =
dξ e
−|ξ | 2
g = J
dξ
(0) dξ
e
−|ξ |
2
g −|ξ (0) |
2
g
= J
i
dα i e
−α i α j (ψi,ψj ) g
dξ
e
−|ξ |
2
g
= J
det(ψ i , ψ j ) g
−1/2 .
Note that the integration over the α i is a standard finite-dimensional integral.
This gives
dξ =
det(ψ i , ψ j ) g dξ
i
dα i .
(2.110)
Since nothing depends on the α i , they can be integrated over as in (2.53), giving
the volume of the CKV group
ckv [g] =
i
dα i .
(2.111)
Replacing the integration over ξ thanks to (2.106), the path integral becomes
Z g =
T g
d
M g t t gauge [ ˆ
g]
−1
×
d g φ d g ξ
det(φ i , μ j ) g
det(φ i , φ j ) g
ckv [g] −1
det(ψ i , ψ j ) g
FP [g] Z m [g]. (2.112)
Since the matter action and measure, and the Liouville measure are invariant
under reparametrizations, one can perform a change of variables
(f
∗
ˆ
g, f
∗ φ, f
∗ ) −→ ( ˆ
g, φ, ,)
(2.113)
such that everything becomes independent of f (or equivalently ξ ). Since the
measure for ξ is Weyl invariant, it is possible to separate it from the rest of the
expression, which yields an overall factor of Diff 0 [g]. This brings the partition
function to the form
Z g =
T g
d
M g t
Diff 0 [ ˆ
g]
gauge [ ˆ
g]
d g φ
det(φ i , μ j ) g
det(φ i , φ j ) g
ckv [g] −1
det(ψ i , ψ j ) g
FP [g] Z m [g],
(2.114)