17.2 Off-Shell Superstring Amplitudes
347
where S g,m,n is a M g,m,n -dimensional section of
P g,m,n parametrized by coordinates
t λ . The 1-form B corresponds to a generalization of the bosonic 1-form. It has ghost
number 1 and includes a correction to compensate the variation of the PCO locations
in terms of the moduli parameters:
B λ =
α
C α
dσ α
2π i
b(σ α )
∂F α
∂t λ
F
−1
α (σ α )
+
α
C α
d ¯
σ α
2π i
¯
b( ¯
σ α )
∂ ¯
F α
∂t λ
¯
F
−1
α ( ¯
σ α )
−
A
1
X (y A )
∂y A
∂t λ
∂ξ(y A ).
(17.39)
The last factor amounts to consider the combination
X (y A ) − ∂ξ(y A ) dy A
(17.40)
for each PCO insertion: 4 the correction is necessary to ensure that the BRST
identity (13.46) holds. This can be understood as follows: the derivative acting on
the PCO gives a term dX (z) = ∂X (z)dz that must be cancelled. This is achieved by
the second term since {Q B , ∂ξ(z)} = ∂X (z).
Remark 17.3 While it is sufficient to work with M g,n for on-shell bosonic amplitudes, on-shell superstring amplitudes are naturally expressed in
P g,m,n (with the
local coordinate removed) since the positions of the PCO must be specified even
on-shell.
Remark 17.4 (Amplitudes on the Supermoduli Space) Following Polyakov’s approach from Chaps. 2 and 3 to the superstring would lead to replace the moduli space
by the supermoduli space. The latter includes Grassmann odd moduli parameters in
addition to the moduli parameters from M g,m+n (in the same way the superspace
includes odd coordinates θ along with spacetime coordinates x). The natural
question is whether it is possible to split the integration over the even and odd moduli
and to integrate over the latter such that only an integral over M g,m+n remains.
In view of (17.38a), the answer seems positive. However, this is incorrect: it was
proven in [3] that there is no global holomorphic projection of the supermoduli
space to the moduli space. This is related to the problem of spurious poles described
below. But, this does not prevent to do it locally: in that case, implementing the
procedure carefully should give the rules of vertical integration [8, 28, 31].
4 The sum is formal since it is composed of 0- and 1-forms.
347
where S g,m,n is a M g,m,n -dimensional section of
P g,m,n parametrized by coordinates
t λ . The 1-form B corresponds to a generalization of the bosonic 1-form. It has ghost
number 1 and includes a correction to compensate the variation of the PCO locations
in terms of the moduli parameters:
B λ =
α
C α
dσ α
2π i
b(σ α )
∂F α
∂t λ
F
−1
α (σ α )
+
α
C α
d ¯
σ α
2π i
¯
b( ¯
σ α )
∂ ¯
F α
∂t λ
¯
F
−1
α ( ¯
σ α )
−
A
1
X (y A )
∂y A
∂t λ
∂ξ(y A ).
(17.39)
The last factor amounts to consider the combination
X (y A ) − ∂ξ(y A ) dy A
(17.40)
for each PCO insertion: 4 the correction is necessary to ensure that the BRST
identity (13.46) holds. This can be understood as follows: the derivative acting on
the PCO gives a term dX (z) = ∂X (z)dz that must be cancelled. This is achieved by
the second term since {Q B , ∂ξ(z)} = ∂X (z).
Remark 17.3 While it is sufficient to work with M g,n for on-shell bosonic amplitudes, on-shell superstring amplitudes are naturally expressed in
P g,m,n (with the
local coordinate removed) since the positions of the PCO must be specified even
on-shell.
Remark 17.4 (Amplitudes on the Supermoduli Space) Following Polyakov’s approach from Chaps. 2 and 3 to the superstring would lead to replace the moduli space
by the supermoduli space. The latter includes Grassmann odd moduli parameters in
addition to the moduli parameters from M g,m+n (in the same way the superspace
includes odd coordinates θ along with spacetime coordinates x). The natural
question is whether it is possible to split the integration over the even and odd moduli
and to integrate over the latter such that only an integral over M g,m+n remains.
In view of (17.38a), the answer seems positive. However, this is incorrect: it was
proven in [3] that there is no global holomorphic projection of the supermoduli
space to the moduli space. This is related to the problem of spurious poles described
below. But, this does not prevent to do it locally: in that case, implementing the
procedure carefully should give the rules of vertical integration [8, 28, 31].
4 The sum is formal since it is composed of 0- and 1-forms.