9 Towards the “Shape” of Cosmological Observables and the String …
235
The fluxes induce a superpotential for the complex structure moduli and axiodilaton
φ ≡ C 0 + ie
−ϕ [51]
W =
M
G 3 ∧ (z) = ( f − φh) · Π(z)
(9.17)
where G 3 ≡ F 3 − φ H 3 . The superpotential and Kähler potential are related to the
scalar potential via
V = e
K
g
ab D a W D b W − 3 W |
2
(9.18)
where D a W ≡ ∂ a W + W ∂ a K, and a runs over all complex structure moduli. (At
tree-level, the Kähler moduli do not appear in the scalar potential.) We are interested
in vacua with vanishing F-terms
D φ W =
1
φ − φ
( f − φh) · Π(z) = 0
(9.19)
D a W = ( f − φh) · (∂ a Π(z) + Π(z)∂ a K) = 0
(9.20)
These so-called F-flatness conditions (9.19) and (9.20) imply that the (3,0) and (1,2)
parts of the fluxes vanish, so that G 3 is imaginary self-dual, 6 G 3 = i G 3 .
The fluxes also contribute to the D3-brane charge
N flux =
M
F 3 ∧ H 3 = f · Σ · h
(9.21)
Since the corresponding flux threads a compact space, the total D3-brane charge must
vanish, a consistency condition called tadpole cancellation. For imaginary self-dual
fluxes, we have that N flux > 0. Therefore tadpole cancellation requires the presence
of negative D3-brane charges. A fixed amount of negative charge is induced by
orientifolding. Orientifolding is a particular discrete identification of the internal
manifold giving rise to fixed points with negative D-brane charge. For cancellation,
any difference L − N flux can be made up by mobile D3-branes spanning the fourdimensional spacetime. We will not consider an explicit orientifold, but will instead
take the orientifold charge as a free parameter L max , so that
0 < N flux ≤ L max
(9.22)
As we will see, L max sets the scale at which we observe interesting structure in the
moduli space distribution. To summarize, for a given Calabi-Yau, we choose fluxes
satisfying (9.22) and use the F-flatness conditions (9.19), (9.20) to solve for the VEVs
of corresponding stabilized moduli. We then study the distribution of these moduli
using persistent homology.
235
The fluxes induce a superpotential for the complex structure moduli and axiodilaton
φ ≡ C 0 + ie
−ϕ [51]
W =
M
G 3 ∧ (z) = ( f − φh) · Π(z)
(9.17)
where G 3 ≡ F 3 − φ H 3 . The superpotential and Kähler potential are related to the
scalar potential via
V = e
K
g
ab D a W D b W − 3 W |
2
(9.18)
where D a W ≡ ∂ a W + W ∂ a K, and a runs over all complex structure moduli. (At
tree-level, the Kähler moduli do not appear in the scalar potential.) We are interested
in vacua with vanishing F-terms
D φ W =
1
φ − φ
( f − φh) · Π(z) = 0
(9.19)
D a W = ( f − φh) · (∂ a Π(z) + Π(z)∂ a K) = 0
(9.20)
These so-called F-flatness conditions (9.19) and (9.20) imply that the (3,0) and (1,2)
parts of the fluxes vanish, so that G 3 is imaginary self-dual, 6 G 3 = i G 3 .
The fluxes also contribute to the D3-brane charge
N flux =
M
F 3 ∧ H 3 = f · Σ · h
(9.21)
Since the corresponding flux threads a compact space, the total D3-brane charge must
vanish, a consistency condition called tadpole cancellation. For imaginary self-dual
fluxes, we have that N flux > 0. Therefore tadpole cancellation requires the presence
of negative D3-brane charges. A fixed amount of negative charge is induced by
orientifolding. Orientifolding is a particular discrete identification of the internal
manifold giving rise to fixed points with negative D-brane charge. For cancellation,
any difference L − N flux can be made up by mobile D3-branes spanning the fourdimensional spacetime. We will not consider an explicit orientifold, but will instead
take the orientifold charge as a free parameter L max , so that
0 < N flux ≤ L max
(9.22)
As we will see, L max sets the scale at which we observe interesting structure in the
moduli space distribution. To summarize, for a given Calabi-Yau, we choose fluxes
satisfying (9.22) and use the F-flatness conditions (9.19), (9.20) to solve for the VEVs
of corresponding stabilized moduli. We then study the distribution of these moduli
using persistent homology.