352
17 Superstring
where
η = η 0 ⊗ 1
⊗n−1
+ · · · + 1
⊗n−1
⊗ η 0 .
(17.55)
The BRST cohomology is trivial in the large Hilbert space: thus, if A (p) | is
closed, it must be exact in this space:
A
(p)
| ==α
(p)
| Q,
(17.56)
where the state α (p) (called gauge amplitude) must be in the large Hilbert space.
This is consistent withA (p) | η = 0 only if
α
(p)
| ηQ = 0
(17.57)
(Q and η anti-commute). It is then natural to interpret the state on which Q acts as
an amplitude with one less PCO
A
(p−1)
| ==α
(p)
| η
(17.58)
since N pic (η) = −1.
Continuing this procedure leads to an amplitude A (0) | without any PCO insertion, and thus without spurious singularities. Consistency with the picture number
anomaly requires the external state to have non-canonical picture numbers. But, this
should not be a puzzle since the amplitude states should be viewed as intermediate
object to obtain the final amplitude.
Hence, the amplitude A (p) | can be constructed by starting with A (0) |: inserting
ξ(z) in the amplitude leads to the gauge amplitude α (1) |, whose BRST variation
yields A (2) |. Continuing recursively helps to construct the desired amplitude.
Moreover, Q and ξ insertions automatically take care of the corrections at the
interfaces of the components.
Showing that the amplitude is independent of the non-physical data (i.e. gauge
invariance) is trivial since it is expressed as a BRST exact expression.
Vertical Integration: Small Hilbert Space
The section of
P g,m,n is given by a series of discontinuous components linked
by vertical segments. On the vertical segment, the PCO configuration interpolates
continuously between the components, and the integrand can encounter a spurious
pole. Since the integrand is not a total derivative in terms of the fibre coordinates,
its integration over a segment depends on the path followed and not only on the end
points. This implies that it diverges when it encounters the spurious pole. However,
there is a specific prescription that avoids these problems. When only one PCO
varies, the integrand is a total derivative of the PCO location and can thus be
integrated directly, giving a difference in the two end points. In this case, the result
is independent from the specific path and from the presence of the spurious pole.
17 Superstring
where
η = η 0 ⊗ 1
⊗n−1
+ · · · + 1
⊗n−1
⊗ η 0 .
(17.55)
The BRST cohomology is trivial in the large Hilbert space: thus, if A (p) | is
closed, it must be exact in this space:
A
(p)
| ==α
(p)
| Q,
(17.56)
where the state α (p) (called gauge amplitude) must be in the large Hilbert space.
This is consistent withA (p) | η = 0 only if
α
(p)
| ηQ = 0
(17.57)
(Q and η anti-commute). It is then natural to interpret the state on which Q acts as
an amplitude with one less PCO
A
(p−1)
| ==α
(p)
| η
(17.58)
since N pic (η) = −1.
Continuing this procedure leads to an amplitude A (0) | without any PCO insertion, and thus without spurious singularities. Consistency with the picture number
anomaly requires the external state to have non-canonical picture numbers. But, this
should not be a puzzle since the amplitude states should be viewed as intermediate
object to obtain the final amplitude.
Hence, the amplitude A (p) | can be constructed by starting with A (0) |: inserting
ξ(z) in the amplitude leads to the gauge amplitude α (1) |, whose BRST variation
yields A (2) |. Continuing recursively helps to construct the desired amplitude.
Moreover, Q and ξ insertions automatically take care of the corrections at the
interfaces of the components.
Showing that the amplitude is independent of the non-physical data (i.e. gauge
invariance) is trivial since it is expressed as a BRST exact expression.
Vertical Integration: Small Hilbert Space
The section of
P g,m,n is given by a series of discontinuous components linked
by vertical segments. On the vertical segment, the PCO configuration interpolates
continuously between the components, and the integrand can encounter a spurious
pole. Since the integrand is not a total derivative in terms of the fibre coordinates,
its integration over a segment depends on the path followed and not only on the end
points. This implies that it diverges when it encounters the spurious pole. However,
there is a specific prescription that avoids these problems. When only one PCO
varies, the integrand is a total derivative of the PCO location and can thus be
integrated directly, giving a difference in the two end points. In this case, the result
is independent from the specific path and from the presence of the spurious pole.