11.3 Off-Shell Amplitudes
247
There is one local coordinate for each operator, which is inserted at the origin. Local
coordinates on the surfaces can be seen in two different fashions (Fig. 11.1): either as
describing patches on the surface, in which case the maps f i correspond to transition
functions, or one can interpret them by cutting disks centred at the punctures and
whose interiors are mapped to complex planes, and the maps f i tell how to insert
the plane inside the disk.
When the amplitude A g,n is defined in terms of local coordinates, it will depend
on the maps f i , and one needs to ensure that this cancels when the A i s are on-shell.
But, the choice of the maps f i is arbitrary: selecting a specific set hides that all
choices are physically equivalent and should lead to the same results on-shell. For
this reason, the geometry can be enriched with the local coordinates, in the same
way that the puncture locations were added as a fibre to the moduli space M g to get
the marked moduli space M g,n . Hence, the fundamental geometrical object is the
fibre bundle P g,n with M g,n being the base and the local coordinates of the fibre.
Since there is an infinite number of functions, the fibre is infinite-dimensional, and
so is the space P g,n .
Every point of P g,n corresponds to a genus-g Riemann surface with n punctures
together with a choice of local coordinates around the punctures. The form ω
g,n
M g,n
is defined in this bigger space, and the integration giving the off-shell amplitude (11.54) is performed over a M g,n -dimensional section S g,n ⊂ P g,n (Fig. 11.2):
A g,n (V 1 , . . . , V n ) S g,n =
S g,n
ω
g,n
M g,n
(V 1 , . . . , V n )
S g,n
.
(11.57)
The subscript in the LHS indicates that the amplitudes depend on S g,n through
the choice of local coordinates. The on-shell independence of A g,n on the local
coordinates translates into the independence on the choice of the section:
∀S g,n : A g,n (V 1 , . . . , V n ) S g,n = A g,n (V 1 , . . . , V n )
(on-shell).
(11.58)
The section is taken to be continuous, which means that two neighbouring surfaces
of the moduli space must have close local coordinates.
In order to define the amplitude, one needs to find the expression of the M g,n -
form ω
g,n
M g,n
on P g,n . It is in fact simpler to define general p-forms ω
g,n
p on P g,n , in
particular, for proving general properties about the forms and the amplitudes. Given
a manifold, a p-form is an element of the cotangent space, and it can be defined
through its contraction with vectors (tangent space). Since vectors correspond to
small variation of the manifold coordinates, it is necessary to find a parametrization
of P g,n . The geometry of P g,n —and of its relevant subspaces and tangent space—
is studied in the next chapter. Then, we will come back on the construction of the
amplitudes in Chap. 13.
247
There is one local coordinate for each operator, which is inserted at the origin. Local
coordinates on the surfaces can be seen in two different fashions (Fig. 11.1): either as
describing patches on the surface, in which case the maps f i correspond to transition
functions, or one can interpret them by cutting disks centred at the punctures and
whose interiors are mapped to complex planes, and the maps f i tell how to insert
the plane inside the disk.
When the amplitude A g,n is defined in terms of local coordinates, it will depend
on the maps f i , and one needs to ensure that this cancels when the A i s are on-shell.
But, the choice of the maps f i is arbitrary: selecting a specific set hides that all
choices are physically equivalent and should lead to the same results on-shell. For
this reason, the geometry can be enriched with the local coordinates, in the same
way that the puncture locations were added as a fibre to the moduli space M g to get
the marked moduli space M g,n . Hence, the fundamental geometrical object is the
fibre bundle P g,n with M g,n being the base and the local coordinates of the fibre.
Since there is an infinite number of functions, the fibre is infinite-dimensional, and
so is the space P g,n .
Every point of P g,n corresponds to a genus-g Riemann surface with n punctures
together with a choice of local coordinates around the punctures. The form ω
g,n
M g,n
is defined in this bigger space, and the integration giving the off-shell amplitude (11.54) is performed over a M g,n -dimensional section S g,n ⊂ P g,n (Fig. 11.2):
A g,n (V 1 , . . . , V n ) S g,n =
S g,n
ω
g,n
M g,n
(V 1 , . . . , V n )
S g,n
.
(11.57)
The subscript in the LHS indicates that the amplitudes depend on S g,n through
the choice of local coordinates. The on-shell independence of A g,n on the local
coordinates translates into the independence on the choice of the section:
∀S g,n : A g,n (V 1 , . . . , V n ) S g,n = A g,n (V 1 , . . . , V n )
(on-shell).
(11.58)
The section is taken to be continuous, which means that two neighbouring surfaces
of the moduli space must have close local coordinates.
In order to define the amplitude, one needs to find the expression of the M g,n -
form ω
g,n
M g,n
on P g,n . It is in fact simpler to define general p-forms ω
g,n
p on P g,n , in
particular, for proving general properties about the forms and the amplitudes. Given
a manifold, a p-form is an element of the cotangent space, and it can be defined
through its contraction with vectors (tangent space). Since vectors correspond to
small variation of the manifold coordinates, it is necessary to find a parametrization
of P g,n . The geometry of P g,n —and of its relevant subspaces and tangent space—
is studied in the next chapter. Then, we will come back on the construction of the
amplitudes in Chap. 13.