13.3 Properties of Amplitudes
281
This can be interpreted as a gauge fixing condition (Sect. 10.5), which could in
principle be relaxed. However, the decoupling of physical states (equivalent to
gauge invariance in SFT) happens only after integrating over the moduli space. This
requires having a globally defined section.
As a consequence, off-shell states are elements of the semi-relative Hilbert space
V i ∈ H
−
∩ ker L
−
0 ,
(13.56)
and the amplitudes are defined by integrating the form ω M g,n over a section S g,n ⊂
ˆ
P g,n .
Computation: Equation (13.54)
The operator associated to the state through |A i = A i (0) |0 transforms as
V i (0) −→ (e
iα i )
h (e
−iα i )
¯
h
V i (0),
(13.57)
which translates into
|V i −→ e
iα i (L 0 − ¯
L 0 )
|V i
(13.58)
for the state, using the fact that the vacuum is invariant under L 0 and ¯
L 0 . Then,
requiring the invariance of the state leads to (13.54).
13.3.2 Consequences of the BRST Identity
Two important properties of the on-shell amplitudes can be deduced from the
BRST identity (13.46): the independence of physical results on the choice of local
coordinates and the decoupling of pure gauge states.
Given BRST closed states, the LHS of (13.46) vanishes identically
∀i : Q B |V i = 0 ⇒ dω p−1 (V 1 , . . . , V n ) = 0.
(13.59)
Using this result, one can compare the on-shell amplitudes computed for two
different sections S and S
S
ω M g,n −
S
ω M g,n =
∂T
ω M g,n −1 =
T
dω M g,n −1 = 0,
(13.60)
using Stokes’ theorem and where T is the surface delimited by the two sections
(Fig. 13.3). This implies that on-shell amplitudes do not depend on the section,
and thus on the local coordinates. In obtaining the result, one needs to assume
that the vertical segments do not contribute. The latter corresponds to boundary
281
This can be interpreted as a gauge fixing condition (Sect. 10.5), which could in
principle be relaxed. However, the decoupling of physical states (equivalent to
gauge invariance in SFT) happens only after integrating over the moduli space. This
requires having a globally defined section.
As a consequence, off-shell states are elements of the semi-relative Hilbert space
V i ∈ H
−
∩ ker L
−
0 ,
(13.56)
and the amplitudes are defined by integrating the form ω M g,n over a section S g,n ⊂
ˆ
P g,n .
Computation: Equation (13.54)
The operator associated to the state through |A i = A i (0) |0 transforms as
V i (0) −→ (e
iα i )
h (e
−iα i )
¯
h
V i (0),
(13.57)
which translates into
|V i −→ e
iα i (L 0 − ¯
L 0 )
|V i
(13.58)
for the state, using the fact that the vacuum is invariant under L 0 and ¯
L 0 . Then,
requiring the invariance of the state leads to (13.54).
13.3.2 Consequences of the BRST Identity
Two important properties of the on-shell amplitudes can be deduced from the
BRST identity (13.46): the independence of physical results on the choice of local
coordinates and the decoupling of pure gauge states.
Given BRST closed states, the LHS of (13.46) vanishes identically
∀i : Q B |V i = 0 ⇒ dω p−1 (V 1 , . . . , V n ) = 0.
(13.59)
Using this result, one can compare the on-shell amplitudes computed for two
different sections S and S
S
ω M g,n −
S
ω M g,n =
∂T
ω M g,n −1 =
T
dω M g,n −1 = 0,
(13.60)
using Stokes’ theorem and where T is the surface delimited by the two sections
(Fig. 13.3). This implies that on-shell amplitudes do not depend on the section,
and thus on the local coordinates. In obtaining the result, one needs to assume
that the vertical segments do not contribute. The latter corresponds to boundary