2.4 Ghost Action
63
It is simpler to first focus on the b ghost (to avoid the problems related to the
CKV). The path integral (2.151) can be rewritten as
Z g =
M g
d
M g t
ckv [ ˆ
g] −1
det(ψ i , ψ j ) ˆ
g
d ˆ
g d ˆ
g b d
ˆ
g c
M g
i=1
(b, ˆ
μ i ) ˆ
g e
−S m [ ˆ
g,,]−S gh [ ˆ
g,b,c] .
(2.159)
In this expression, c zero-modes are not integrated over, only the b zero-modes are.
This is the standard starting point on Riemann surfaces with genus g ≥ 1. The inner
product reads explicitly
(b, ˆ
μ i ) ˆ
g =
d
2 σ
ˆ
g G
abcd
⊥ b ab ˆ
μ i,cd =
d
2 σ
ˆ
g g
ac g
bd b ab ˆ
μ i,cd .
(2.160)
Computation: Equation (2.159)
Since the zero-modes of b are in the kernel of P
†
1 , it means that the quadratic
differentials (2.76) also provide a suitable basis:
b = b 0 + b
,
b 0 = b 0i φ i ,
where the b 0i are Grassmann-odd coefficients. The first step is to find the
Jacobian for the changes of variables b → (b , b 0i ):
1 =
d ˆ
g b e
−|b| 2
ˆ
g = J
d ˆ
g b
i
db 0i e
−|b |
2
ˆ
g −|b 0i φ i | 2 = J
det(φ i , φ j ).
Next, (2.151) has no zero-modes, so one must insert M g of them at arbitrary
positions σ 0
j to get a non-vanishing result when integrating over d M g b 0i . The
result of the integral is
d
M g b 0i
j
b 0 (σ
0
j ) =
d
M g b 0i
j
b 0i φ i (σ
0
j )
= det φ i (σ
0
j ).
The only combination of the φ i which does not vanish is the determinant due
to the antisymmetry of the Grassmann numbers. Combining both results leads
to
d ˆ
g b
det(φ i , φ j ) ˆ
g
=
d ˆ
g b
det φ i (σ 0
j )
M g
j =1
b(σ
0
j ).
(2.161)
63
It is simpler to first focus on the b ghost (to avoid the problems related to the
CKV). The path integral (2.151) can be rewritten as
Z g =
M g
d
M g t
ckv [ ˆ
g] −1
det(ψ i , ψ j ) ˆ
g
d ˆ
g d ˆ
g b d
ˆ
g c
M g
i=1
(b, ˆ
μ i ) ˆ
g e
−S m [ ˆ
g,,]−S gh [ ˆ
g,b,c] .
(2.159)
In this expression, c zero-modes are not integrated over, only the b zero-modes are.
This is the standard starting point on Riemann surfaces with genus g ≥ 1. The inner
product reads explicitly
(b, ˆ
μ i ) ˆ
g =
d
2 σ
ˆ
g G
abcd
⊥ b ab ˆ
μ i,cd =
d
2 σ
ˆ
g g
ac g
bd b ab ˆ
μ i,cd .
(2.160)
Computation: Equation (2.159)
Since the zero-modes of b are in the kernel of P
†
1 , it means that the quadratic
differentials (2.76) also provide a suitable basis:
b = b 0 + b
,
b 0 = b 0i φ i ,
where the b 0i are Grassmann-odd coefficients. The first step is to find the
Jacobian for the changes of variables b → (b , b 0i ):
1 =
d ˆ
g b e
−|b| 2
ˆ
g = J
d ˆ
g b
i
db 0i e
−|b |
2
ˆ
g −|b 0i φ i | 2 = J
det(φ i , φ j ).
Next, (2.151) has no zero-modes, so one must insert M g of them at arbitrary
positions σ 0
j to get a non-vanishing result when integrating over d M g b 0i . The
result of the integral is
d
M g b 0i
j
b 0 (σ
0
j ) =
d
M g b 0i
j
b 0i φ i (σ
0
j )
= det φ i (σ
0
j ).
The only combination of the φ i which does not vanish is the determinant due
to the antisymmetry of the Grassmann numbers. Combining both results leads
to
d ˆ
g b
det(φ i , φ j ) ˆ
g
=
d ˆ
g b
det φ i (σ 0
j )
M g
j =1
b(σ
0
j ).
(2.161)