270
13 Off-Shell Amplitudes
where ∂ s =
∂
∂x s
and x s are the coordinates (12.13). It is antisymmetric in any pair
of two indices
ω i 1 i 2 ···i p = −ω i 2 i 1 ···i p ,
(13.2)
and multilinearity implies that
ω p
V
(1) , . . . , V
(p)
= ω p
V
(1)
s 1
∂ s 1 , . . . , V
(p)
s p ∂ s p
= ω i 1 ···i p V
(1)
s 1
· · · V
(p)
s p ,
(13.3)
given vectors V (α) = V
(α)
s ∂ s .
The p-forms that are needed to define off-shell amplitudes depend on the
external states V i (i = 1, . . . , n) inserted at the punctures z i . They are maps from
p T P g,n × H n to a function on P g,n . The dependence on the states is denoted
equivalently as
ω p (V 1 , . . . , V n ) := ω p (⊗ i V i ).
(13.4)
The simplest way to get a function on P g,n from the states V i is to compute a CFT
correlation function of the operators inserted at the points z i = f i (0) on the surface
g,n described by the point in M g,n .
The 0-form is just a function and is defined by
ω 0 = (2π i)
−M c
g,n
n
i=1
f i ◦ V i (0)
g,n
.
(13.5)
For simplicity, the dependence in the local coordinates f i is kept implicit in the rest
of the chapter.
A natural approach for constructing p-forms is to build them from elementary
1-forms and to use ghosts to enforce the antisymmetry. Remembering the Beltrami
differentials found in Chap. 2, the contour integral of ghosts b(z) weighted by some
vector field is a good starting point. In the current language, it is defined by its
contraction with a vector V = (v, C) ∈ T P g,n defined in (12.23)
B(V ) :=
C
dz
2π i
b(z)v(z) +
C
d¯ z
2π i
¯
b(¯ z) ¯
v(¯ z),
(13.6)
where b(z) and ¯
b(¯ z) are the b ghost components, and v is the vector field on g,n
defining V . The contours run anti-clockwise. If the contour C includes several
circles (C = ∪ α C α ), B(V ) is defined as the sum of the contour integrals on each
circle
B(V ) :=
α
C α
dz
2π i
b(z)v(z) + c.c..
(13.7)
13 Off-Shell Amplitudes
where ∂ s =
∂
∂x s
and x s are the coordinates (12.13). It is antisymmetric in any pair
of two indices
ω i 1 i 2 ···i p = −ω i 2 i 1 ···i p ,
(13.2)
and multilinearity implies that
ω p
V
(1) , . . . , V
(p)
= ω p
V
(1)
s 1
∂ s 1 , . . . , V
(p)
s p ∂ s p
= ω i 1 ···i p V
(1)
s 1
· · · V
(p)
s p ,
(13.3)
given vectors V (α) = V
(α)
s ∂ s .
The p-forms that are needed to define off-shell amplitudes depend on the
external states V i (i = 1, . . . , n) inserted at the punctures z i . They are maps from
p T P g,n × H n to a function on P g,n . The dependence on the states is denoted
equivalently as
ω p (V 1 , . . . , V n ) := ω p (⊗ i V i ).
(13.4)
The simplest way to get a function on P g,n from the states V i is to compute a CFT
correlation function of the operators inserted at the points z i = f i (0) on the surface
g,n described by the point in M g,n .
The 0-form is just a function and is defined by
ω 0 = (2π i)
−M c
g,n
n
i=1
f i ◦ V i (0)
g,n
.
(13.5)
For simplicity, the dependence in the local coordinates f i is kept implicit in the rest
of the chapter.
A natural approach for constructing p-forms is to build them from elementary
1-forms and to use ghosts to enforce the antisymmetry. Remembering the Beltrami
differentials found in Chap. 2, the contour integral of ghosts b(z) weighted by some
vector field is a good starting point. In the current language, it is defined by its
contraction with a vector V = (v, C) ∈ T P g,n defined in (12.23)
B(V ) :=
C
dz
2π i
b(z)v(z) +
C
d¯ z
2π i
¯
b(¯ z) ¯
v(¯ z),
(13.6)
where b(z) and ¯
b(¯ z) are the b ghost components, and v is the vector field on g,n
defining V . The contours run anti-clockwise. If the contour C includes several
circles (C = ∪ α C α ), B(V ) is defined as the sum of the contour integrals on each
circle
B(V ) :=
α
C α
dz
2π i
b(z)v(z) + c.c..
(13.7)