4.2 Complex Representation of Path Integral
97
The last equation can inspire to search for a similar rewriting of the moduli
parameters. In fact, the moduli space itself is a complex manifold and can be
endowed with complex coordinates [7, 8]:
m I = t 2I −1 + it 2I ,
¯
m I = t 2I −1 − it 2I ,
I = 1, . . . , M
c
g
(4.32)
with the integration measure
d
M g t = d
2M c
g m.
(4.33)
The last ingredient to rewrite the vacuum amplitudes (2.136) is to obtain the
determinants. The inner products of vector and traceless symmetric fields also
factorize
(T 1 , T 2 ) = 2
d
2 σ
ˆ
g ˆ
g
ac g
bd T 1,ab T 2,cd = 4
d
2 z
T 1,zz T 2,¯ z¯ z + T 1,¯ z¯ z T 2,zz
,
(4.34a)
(ξ 1 , ξ 2 ) =
d
2 σ
ˆ
g ˆ
g ab ξ
a ξ
b
=
1
4
d
2 z
ξ
z
1 ξ
¯
z
2 + ξ
¯
z
1 ξ
z
2
.
(4.34b)
All inner products are evaluated in the flat background metric. For (anti)holomorphic fields, only one term survives in each integral: since each field
appears twice in the determinants (φ i , φ j ) and (φ i , φ j ), the final expression is a
square, which cancels against the squareroot in (2.136). The remaining determinant
involves the Beltrami differential (2.65b):
μ izz = ∂ i ¯
g zz ,
μ i ¯
z¯ z = ∂ i ¯
g ¯
z¯ z
(4.35)
( ¯
g zz = 0 in our coordinates system, but its variation under a shift of moduli is not
zero). The basis can be changed to a complex basis such that the determinant of
inner products between Beltrami and quadratic differentials is a modulus squared.
All together, the different formulas lead to the following rewriting of the vacuum
amplitude :
Z g =
M g
d
2M c
g m
| det(φ I , μ J )|
2
| det(φ I , ¯
φ J )|
det
P
†
1 P 1
| det(ψ I , ¯
ψ J )|
Z m [δ]
ckv [δ]
,
(4.36)
where the absolute values are to be understood with respect to the basis of P 1 and
P
†
1 , for example |f (m I )|
2
:= f (m I )f ( ¯
m I ).
The same reasoning can be applied to the ghosts. The c and b ghosts are
respectively a vector and a symmetric traceless tensor, both with two independent
components: it is customary to define
c := c
z ,
¯
c := c
¯
z ,
b:= b zz ,
¯
b := b ¯
z¯ z .
(4.37)
97
The last equation can inspire to search for a similar rewriting of the moduli
parameters. In fact, the moduli space itself is a complex manifold and can be
endowed with complex coordinates [7, 8]:
m I = t 2I −1 + it 2I ,
¯
m I = t 2I −1 − it 2I ,
I = 1, . . . , M
c
g
(4.32)
with the integration measure
d
M g t = d
2M c
g m.
(4.33)
The last ingredient to rewrite the vacuum amplitudes (2.136) is to obtain the
determinants. The inner products of vector and traceless symmetric fields also
factorize
(T 1 , T 2 ) = 2
d
2 σ
ˆ
g ˆ
g
ac g
bd T 1,ab T 2,cd = 4
d
2 z
T 1,zz T 2,¯ z¯ z + T 1,¯ z¯ z T 2,zz
,
(4.34a)
(ξ 1 , ξ 2 ) =
d
2 σ
ˆ
g ˆ
g ab ξ
a ξ
b
=
1
4
d
2 z
ξ
z
1 ξ
¯
z
2 + ξ
¯
z
1 ξ
z
2
.
(4.34b)
All inner products are evaluated in the flat background metric. For (anti)holomorphic fields, only one term survives in each integral: since each field
appears twice in the determinants (φ i , φ j ) and (φ i , φ j ), the final expression is a
square, which cancels against the squareroot in (2.136). The remaining determinant
involves the Beltrami differential (2.65b):
μ izz = ∂ i ¯
g zz ,
μ i ¯
z¯ z = ∂ i ¯
g ¯
z¯ z
(4.35)
( ¯
g zz = 0 in our coordinates system, but its variation under a shift of moduli is not
zero). The basis can be changed to a complex basis such that the determinant of
inner products between Beltrami and quadratic differentials is a modulus squared.
All together, the different formulas lead to the following rewriting of the vacuum
amplitude :
Z g =
M g
d
2M c
g m
| det(φ I , μ J )|
2
| det(φ I , ¯
φ J )|
det
P
†
1 P 1
| det(ψ I , ¯
ψ J )|
Z m [δ]
ckv [δ]
,
(4.36)
where the absolute values are to be understood with respect to the basis of P 1 and
P
†
1 , for example |f (m I )|
2
:= f (m I )f ( ¯
m I ).
The same reasoning can be applied to the ghosts. The c and b ghosts are
respectively a vector and a symmetric traceless tensor, both with two independent
components: it is customary to define
c := c
z ,
¯
c := c
¯
z ,
b:= b zz ,
¯
b := b ¯
z¯ z .
(4.37)