3.1 Scattering Amplitudes on Moduli Space
81
where the second equality follows by setting z = e τ . After analytic continuation
k 0 = −ik 0
M , X 0 = iX 0
M , x 0 = ix 0
M , and τ = it, we find [26, p. 186]
X
0
M = x
0
M + α
k
0
M t.
(3.41)
This indicates that the measure of the worldsheet time in (3.39) must be rescaled by
1/α k 0
M such that
Vol M K 0,2 −→
8π 2 i δ(0)
α k 0
M
=
C S 2 2π i δ(0)
2k 0
M
.
(3.42)
This is equivalent to rescale E by α k 0 and to use δ(ax) = a −1 δ(x).
Ultimately, the 2-point amplitude becomes (removing the subscript on k 0 )
A 0,2 (k, k
) = 2k
0 (2π)
D−1 δ
(D−1) (k + k
)
(3.43)
and matches the QFT formula (3.16). We see that taking into account the scale of
the coordinates is important to reproduce this result.
The computation displayed here presents some ambiguities because of the
regularization. However, this ambiguity can be fixed from unitarity of the scattering
amplitudes. A more general version of the Faddeev–Popov gauge fixing has been
introduced in [8] to avoid dealing altogether with infinities. It is an interesting
question whether these techniques can be extended to compute the tree-level 1- and
0-point amplitudes on the sphere. In most cases, the 1-point amplitude is expected
to vanish since 1-point correlation functions of primary operators other than the
identity vanish in unitary CFTs. 4 The 0-point function corresponds to the sphere
partition function: the saddle point approximation to leading order allows to relate
it to the spacetime action evaluated on the classical solution φ 0 , Z 0 ∼ e −S[φ 0 ]/¯ h .
Since the normalization is not known and because S[φ 0 ] is expected to be infinite,
only comparison between two spacetimes should be meaningful (à la Gibbons–
Hawking–York [15, sec. 4.1]). In particular, for Minkowski spacetime we find
naively
Z 0 ∼
δ (D) (0)
Vol K 0
,
(3.44)
which is not well-defined. This question has not yet been investigated.
4 The integral over the zero-mode gives a factor δ (D) (k) that implies k = 0. At zero momentum,
the time scalar X 0 is effectively described by unitary CFT. However, there can be some subtleties
when considering marginal operator.
81
where the second equality follows by setting z = e τ . After analytic continuation
k 0 = −ik 0
M , X 0 = iX 0
M , x 0 = ix 0
M , and τ = it, we find [26, p. 186]
X
0
M = x
0
M + α
k
0
M t.
(3.41)
This indicates that the measure of the worldsheet time in (3.39) must be rescaled by
1/α k 0
M such that
Vol M K 0,2 −→
8π 2 i δ(0)
α k 0
M
=
C S 2 2π i δ(0)
2k 0
M
.
(3.42)
This is equivalent to rescale E by α k 0 and to use δ(ax) = a −1 δ(x).
Ultimately, the 2-point amplitude becomes (removing the subscript on k 0 )
A 0,2 (k, k
) = 2k
0 (2π)
D−1 δ
(D−1) (k + k
)
(3.43)
and matches the QFT formula (3.16). We see that taking into account the scale of
the coordinates is important to reproduce this result.
The computation displayed here presents some ambiguities because of the
regularization. However, this ambiguity can be fixed from unitarity of the scattering
amplitudes. A more general version of the Faddeev–Popov gauge fixing has been
introduced in [8] to avoid dealing altogether with infinities. It is an interesting
question whether these techniques can be extended to compute the tree-level 1- and
0-point amplitudes on the sphere. In most cases, the 1-point amplitude is expected
to vanish since 1-point correlation functions of primary operators other than the
identity vanish in unitary CFTs. 4 The 0-point function corresponds to the sphere
partition function: the saddle point approximation to leading order allows to relate
it to the spacetime action evaluated on the classical solution φ 0 , Z 0 ∼ e −S[φ 0 ]/¯ h .
Since the normalization is not known and because S[φ 0 ] is expected to be infinite,
only comparison between two spacetimes should be meaningful (à la Gibbons–
Hawking–York [15, sec. 4.1]). In particular, for Minkowski spacetime we find
naively
Z 0 ∼
δ (D) (0)
Vol K 0
,
(3.44)
which is not well-defined. This question has not yet been investigated.
4 The integral over the zero-mode gives a factor δ (D) (k) that implies k = 0. At zero momentum,
the time scalar X 0 is effectively described by unitary CFT. However, there can be some subtleties
when considering marginal operator.