246
11 Introduction to Off-Shell String Theory
Example 11.3: 2-Punctured Torus 1,2
The positions of the punctures are denoted by z i with i = 1, 2. One puncture can
be fixed using the single CKV of the surface, which leaves one position. Together
with the moduli parameter τ of the torus, this gives M 1,2 = 4, and the coordinates
of M 1,2 are {z 2 , τ }.
The g-loop n-point scattering amplitude with external states {V i } can be written
as an integral over M g,n of some M g,n -form ω
(g,n)
M g,n
:
A g,n (V 1 , . . . , V n ) =
M g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(11.54)
The integration over M g,n has the correct dimension to reproduce the formulas from
Sect. 3.1.
While it is possible to derive this amplitude from the path integral (see the
comments at the end of Sect. 3.1.2), we will make only use of the properties of CFT
on Riemann surfaces in the next chapter. This provides an alternative point of view
on the computation of scattering amplitudes and how to derive the formulas, which
can be helpful when the manipulation of the path integral is more complicated (for
example, with the superstring).
The expression of the form ω
g,n
M g,n
must (1) provide a measure on the moduli space
and (2) extract a function of the moduli from the states V i . It is natural to achieve
the second point by computing a correlation function on the Riemann surfaces g,n .
Moreover, Chaps. 2 and 3 indicate that the ghosts are part of the definition of the
measure. Hence, one can expect the ω
g,n
M g,n
to have the form:
ω
g,n
M g,n
(V 1 , . . . , V n ) =
ghosts ×
n
i=1
V i
g,n
×
M g,n
λ=1
dt λ .
(11.55)
We will motivate an expression in Chap. 14 before checking that it has the correct
properties. The ghost insertions are necessary to saturate the number of zero-modes
to obtain a non-vanishing result. By convention, the ghosts are inserted on the left:
while this does not make difference for on-shell closed states, this will for offshell states and for the other types of strings (open and supersymmetric) since the
operators can be Grassmann odd.
11.3.2 Local Coordinates
The next step is to consider off-shell states V i ∈ H. As motivated previously, one
needs to introduce local coordinates defined by the maps:
z = f i (w i ),
z i = f i (0).
(11.56)
11 Introduction to Off-Shell String Theory
Example 11.3: 2-Punctured Torus 1,2
The positions of the punctures are denoted by z i with i = 1, 2. One puncture can
be fixed using the single CKV of the surface, which leaves one position. Together
with the moduli parameter τ of the torus, this gives M 1,2 = 4, and the coordinates
of M 1,2 are {z 2 , τ }.
The g-loop n-point scattering amplitude with external states {V i } can be written
as an integral over M g,n of some M g,n -form ω
(g,n)
M g,n
:
A g,n (V 1 , . . . , V n ) =
M g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(11.54)
The integration over M g,n has the correct dimension to reproduce the formulas from
Sect. 3.1.
While it is possible to derive this amplitude from the path integral (see the
comments at the end of Sect. 3.1.2), we will make only use of the properties of CFT
on Riemann surfaces in the next chapter. This provides an alternative point of view
on the computation of scattering amplitudes and how to derive the formulas, which
can be helpful when the manipulation of the path integral is more complicated (for
example, with the superstring).
The expression of the form ω
g,n
M g,n
must (1) provide a measure on the moduli space
and (2) extract a function of the moduli from the states V i . It is natural to achieve
the second point by computing a correlation function on the Riemann surfaces g,n .
Moreover, Chaps. 2 and 3 indicate that the ghosts are part of the definition of the
measure. Hence, one can expect the ω
g,n
M g,n
to have the form:
ω
g,n
M g,n
(V 1 , . . . , V n ) =
ghosts ×
n
i=1
V i
g,n
×
M g,n
λ=1
dt λ .
(11.55)
We will motivate an expression in Chap. 14 before checking that it has the correct
properties. The ghost insertions are necessary to saturate the number of zero-modes
to obtain a non-vanishing result. By convention, the ghosts are inserted on the left:
while this does not make difference for on-shell closed states, this will for offshell states and for the other types of strings (open and supersymmetric) since the
operators can be Grassmann odd.
11.3.2 Local Coordinates
The next step is to consider off-shell states V i ∈ H. As motivated previously, one
needs to introduce local coordinates defined by the maps:
z = f i (w i ),
z i = f i (0).
(11.56)