17.2 Off-Shell Superstring Amplitudes
345
The completeness relations are
1 =
r
|φ r φ
c
r | ,
(17.31)
H T and
1 =
r
(−1)
|φ r |
|φ
c
r φ r |
(17.32)
on
H T .
Finally, the operator G is defined as
G =
1
NS sector,
X 0 R sector.
(17.33)
Note the following properties
[G, L
±
0 ] = [G, b
±
0 ] = [G, Q B ] = 0.
(17.34)
It will appear in the propagator and kinetic terms of the superstring field theory.
17.2 Off-Shell Superstring Amplitudes
In this section, we are going to build the scattering amplitudes. The procedure is
very similar to the bosonic case, except for the PCO insertions and of the Ramond
sector. For this reason, we will simply state the result and motivate the modifications
with respect to the bosonic case.
17.2.1 Amplitudes
External states can be either NS or R: the Riemann surface corresponding to the
g-loop scattering of m external NS states and n external R states is denoted by
g,m,n . R states must come in pairs because they correspond to fermions. As in the
bosonic case, the amplitude is written as the integration of an appropriate p-form
(g,m,n)
p
over the moduli space M g,m,n (or, more precisely, of a section of a fibre
bundle with this moduli space as a basis). From the geometric point of view, nothing
distinguishes the punctures, and thus,
M g,m,n := dim M g,m,n = 6g − 6 + 2m + 2n.
(17.35)
345
The completeness relations are
1 =
r
|φ r φ
c
r | ,
(17.31)
H T and
1 =
r
(−1)
|φ r |
|φ
c
r φ r |
(17.32)
on
H T .
Finally, the operator G is defined as
G =
1
NS sector,
X 0 R sector.
(17.33)
Note the following properties
[G, L
±
0 ] = [G, b
±
0 ] = [G, Q B ] = 0.
(17.34)
It will appear in the propagator and kinetic terms of the superstring field theory.
17.2 Off-Shell Superstring Amplitudes
In this section, we are going to build the scattering amplitudes. The procedure is
very similar to the bosonic case, except for the PCO insertions and of the Ramond
sector. For this reason, we will simply state the result and motivate the modifications
with respect to the bosonic case.
17.2.1 Amplitudes
External states can be either NS or R: the Riemann surface corresponding to the
g-loop scattering of m external NS states and n external R states is denoted by
g,m,n . R states must come in pairs because they correspond to fermions. As in the
bosonic case, the amplitude is written as the integration of an appropriate p-form
(g,m,n)
p
over the moduli space M g,m,n (or, more precisely, of a section of a fibre
bundle with this moduli space as a basis). From the geometric point of view, nothing
distinguishes the punctures, and thus,
M g,m,n := dim M g,m,n = 6g − 6 + 2m + 2n.
(17.35)