13.3 Properties of Amplitudes
279
The energy–momentum tensor generates changes of coordinates. Hence, T s = T ∂ s
is precisely the generator associated to an infinitesimal change of the coordinate x s
on P g,n . The latter is given by the vector ∂ s . For this reason, one can write
dx s {Q B , B s } = dx s T s = dx s ∂ s = d,
(13.51)
where d is the exterior derivative on P g,n . The minus signs arise if the states V i are
Grassmann odd.
13.3 Properties of Amplitudes
In order for the p-form (13.12) to be non-vanishing, its total ghost number should
match the ghost number anomaly
N gh
ω p (V 1 , . . . , V n )
=
n
i=1
N gh (V i ) − p = 6 − 6g,
(13.52)
using N gh (B) = −1. For an amplitude, one has p = M g,n = 6g − 6 + 2n, and thus,
N gh (ω M g,n ) = 6 − 6g ⇒
n
i=1
N gh (V i ) = 2n.
(13.53)
This condition holds automatically for on-shell states since N gh (c ¯
cV i ) = 2.
13.3.1 Restriction to ˆ
P g,n
The goal of this section is to explain why amplitudes must be described in terms of
a section of ˆ
P g,n (12.10) instead of P g,n . This means that one should identify local
coordinates differing by a global phase rotation.
The off-shell amplitudes (13.16) are multi-valued on P g,n . Indeed, the amplitude
depends on the local coordinates 1 and changes by a factor under a global phase
rotation of any local coordinate w i → e iα w i . However, such a global rotation leaves
the surface unchanged, since the flat metric |dw i |
2 is invariant. This means that
the same surface leads to different values for the amplitude. To prevent this multivaluedness of the amplitudes, it is necessary to identify local coordinates differing
by a constant phase.
A second way to obtain this condition is to require that the section S g,n is globally
defined: every point of the section should correspond to a single point of the moduli
space M g,n . However, there is a topological obstruction that prevents finding a
1 The current argument does not apply for on-shell amplitudes.
279
The energy–momentum tensor generates changes of coordinates. Hence, T s = T ∂ s
is precisely the generator associated to an infinitesimal change of the coordinate x s
on P g,n . The latter is given by the vector ∂ s . For this reason, one can write
dx s {Q B , B s } = dx s T s = dx s ∂ s = d,
(13.51)
where d is the exterior derivative on P g,n . The minus signs arise if the states V i are
Grassmann odd.
13.3 Properties of Amplitudes
In order for the p-form (13.12) to be non-vanishing, its total ghost number should
match the ghost number anomaly
N gh
ω p (V 1 , . . . , V n )
=
n
i=1
N gh (V i ) − p = 6 − 6g,
(13.52)
using N gh (B) = −1. For an amplitude, one has p = M g,n = 6g − 6 + 2n, and thus,
N gh (ω M g,n ) = 6 − 6g ⇒
n
i=1
N gh (V i ) = 2n.
(13.53)
This condition holds automatically for on-shell states since N gh (c ¯
cV i ) = 2.
13.3.1 Restriction to ˆ
P g,n
The goal of this section is to explain why amplitudes must be described in terms of
a section of ˆ
P g,n (12.10) instead of P g,n . This means that one should identify local
coordinates differing by a global phase rotation.
The off-shell amplitudes (13.16) are multi-valued on P g,n . Indeed, the amplitude
depends on the local coordinates 1 and changes by a factor under a global phase
rotation of any local coordinate w i → e iα w i . However, such a global rotation leaves
the surface unchanged, since the flat metric |dw i |
2 is invariant. This means that
the same surface leads to different values for the amplitude. To prevent this multivaluedness of the amplitudes, it is necessary to identify local coordinates differing
by a constant phase.
A second way to obtain this condition is to require that the section S g,n is globally
defined: every point of the section should correspond to a single point of the moduli
space M g,n . However, there is a topological obstruction that prevents finding a
1 The current argument does not apply for on-shell amplitudes.