11.3 Off-Shell Amplitudes
245
11.3.1 Amplitudes from the Marked Moduli Space
In Chap. 3, the scattering amplitudes were written as an integral over the moduli
space M g of the Riemann surface g . As a consequence, the moduli of M g and the
positions of the vertex operators are not treated on an equal footing. Moreover, the
insertions of operators are not symmetric since some are integrated, and the others
have factors of c. These problems can be solved by reinterpreting the scattering
amplitudes in a more geometrical way.
The key is to consider the punctures where vertex operators are inserted as part
of the geometry and not as external data added on top of the Riemann surface g .
Then, the worldsheet with the external states is described as a punctured (or
marked) Riemann surface g,n , which is a Riemann surface g with n punctures
(marked points) z i . The Euler number of such a surface was given in (3.4):
χ g,n := χ(( g,n ) = 2 − 2g − n.
(11.52)
This makes sense since punctures can be interpreted as disks (boundaries). Note that
the punctures are labelled and thus distinguishable.
Since the marked points are distinguished, marked Riemann surfaces with
identical g and n but with punctures located at different points are seen as different
(this statement requires some care for g = 0 and g = 1 due to the presence of
CKV). The corresponding moduli space is denoted by M g,n , and it can be viewed
as a fibre bundle with M g as the base and the puncture positions as the fibre. The
dimension of M g,n is
M g,n := dim R M g,n = 6g − 6 + 2n,
for
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
g ≥ 2,
g = 1, n ≥ 1,
g = 0, n ≥ 3.
(11.53)
These cases are equivalent to χ g,n < 0, that is, when the surfaces have a negative
curvature. The corresponding coordinates are denoted by t λ , λ = 1, . . . , M g,n .
Comparing (11.53) with (2.51) and (2.93), this corresponds to the situation where
g,n has no CKV left unfixed.
Example 11.2: 4-Punctured Sphere 0,4
The positions of the punctures are denoted by z i with i = 1, . . . , 4. Since there
are three CKVs, the positions of three punctures (say z 1 , z 2 and z 3 ) can be fixed,
leaving only one position that characterizes 0,4 . Hence, the moduli space has
dimension M 0,4 = 2, and M 0,4 is parametrized by {z 4 }.
245
11.3.1 Amplitudes from the Marked Moduli Space
In Chap. 3, the scattering amplitudes were written as an integral over the moduli
space M g of the Riemann surface g . As a consequence, the moduli of M g and the
positions of the vertex operators are not treated on an equal footing. Moreover, the
insertions of operators are not symmetric since some are integrated, and the others
have factors of c. These problems can be solved by reinterpreting the scattering
amplitudes in a more geometrical way.
The key is to consider the punctures where vertex operators are inserted as part
of the geometry and not as external data added on top of the Riemann surface g .
Then, the worldsheet with the external states is described as a punctured (or
marked) Riemann surface g,n , which is a Riemann surface g with n punctures
(marked points) z i . The Euler number of such a surface was given in (3.4):
χ g,n := χ(( g,n ) = 2 − 2g − n.
(11.52)
This makes sense since punctures can be interpreted as disks (boundaries). Note that
the punctures are labelled and thus distinguishable.
Since the marked points are distinguished, marked Riemann surfaces with
identical g and n but with punctures located at different points are seen as different
(this statement requires some care for g = 0 and g = 1 due to the presence of
CKV). The corresponding moduli space is denoted by M g,n , and it can be viewed
as a fibre bundle with M g as the base and the puncture positions as the fibre. The
dimension of M g,n is
M g,n := dim R M g,n = 6g − 6 + 2n,
for
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
g ≥ 2,
g = 1, n ≥ 1,
g = 0, n ≥ 3.
(11.53)
These cases are equivalent to χ g,n < 0, that is, when the surfaces have a negative
curvature. The corresponding coordinates are denoted by t λ , λ = 1, . . . , M g,n .
Comparing (11.53) with (2.51) and (2.93), this corresponds to the situation where
g,n has no CKV left unfixed.
Example 11.2: 4-Punctured Sphere 0,4
The positions of the punctures are denoted by z i with i = 1, . . . , 4. Since there
are three CKVs, the positions of three punctures (say z 1 , z 2 and z 3 ) can be fixed,
leaving only one position that characterizes 0,4 . Hence, the moduli space has
dimension M 0,4 = 2, and M 0,4 is parametrized by {z 4 }.