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A. Cole and G. Shiu
One subtlely is that we must fix a gauge, since multiple flux choices can be
equivalent under modular transformations. We will fix our gauge by mapping moduli
VEVs to a fundamental domain. It is sensible to perform our gauge fixing in the
moduli space because this is where we will perform our persistent homology analysis.
If we were to perform gauge-fixing by imposing certain conditions on the fluxes, some
inequivalent vacua with the same moduli VEVs would be mapped to different parts
of the moduli space. Moreover, if we were to choose to fix our gauge by restricting
the moduli to a collection of regions other than a domain (e.g.. multiple disconnected
regions), the resulting structure of the distribution would depend on our particular
choice of regions. On the other hand, as was checked in [30], the choice of particular
domain does not affect the persistent homology results.
Finally, there is physical motivation for gauge-fixing in this way. Given the large
number of vacua in the landscape, one might find themselves with an attitude of
“anything goes,” i.e. any consistent 4-dimensional physics is realized. However,
as has recently been emphasized by the swampland program [99], this does not
seem to be true. In other words, requiring that a 4-dimensional theory arises from
UV-complete quantum gravity imposes nontrivial restrictions. In our context, these
restrictions arise as the presence of correlations in distributions of vacua and an
inability to fine-tune certain quantities beyond a particular resolution. This latter
point, the existence of a minimum resolution, is best captured for moduli VEVs
when one fixes gauge in the way we have proposed.
9.4.2 Rigid Calabi-Yau
Consider a toy Calabi-Yau with no complex structure moduli, studied in [8, 34, 36].
We have b 3 = 2. A symplectic basis for H 3 (M) can be written as {α, β}. We take the
periods of the holomorphic three-form to be
β
= 1,
α
= i
(9.23)
The superpotential is
W = Aφ + B
(9.24)
where A ≡ −h 1 − ih 2 , B ≡ f 1 + i f 2 . The D3-brane charge induced by the fluxes
is
N flux = f 1 h 2 − f 2 h 1
(9.25)
A. Cole and G. Shiu
One subtlely is that we must fix a gauge, since multiple flux choices can be
equivalent under modular transformations. We will fix our gauge by mapping moduli
VEVs to a fundamental domain. It is sensible to perform our gauge fixing in the
moduli space because this is where we will perform our persistent homology analysis.
If we were to perform gauge-fixing by imposing certain conditions on the fluxes, some
inequivalent vacua with the same moduli VEVs would be mapped to different parts
of the moduli space. Moreover, if we were to choose to fix our gauge by restricting
the moduli to a collection of regions other than a domain (e.g.. multiple disconnected
regions), the resulting structure of the distribution would depend on our particular
choice of regions. On the other hand, as was checked in [30], the choice of particular
domain does not affect the persistent homology results.
Finally, there is physical motivation for gauge-fixing in this way. Given the large
number of vacua in the landscape, one might find themselves with an attitude of
“anything goes,” i.e. any consistent 4-dimensional physics is realized. However,
as has recently been emphasized by the swampland program [99], this does not
seem to be true. In other words, requiring that a 4-dimensional theory arises from
UV-complete quantum gravity imposes nontrivial restrictions. In our context, these
restrictions arise as the presence of correlations in distributions of vacua and an
inability to fine-tune certain quantities beyond a particular resolution. This latter
point, the existence of a minimum resolution, is best captured for moduli VEVs
when one fixes gauge in the way we have proposed.
9.4.2 Rigid Calabi-Yau
Consider a toy Calabi-Yau with no complex structure moduli, studied in [8, 34, 36].
We have b 3 = 2. A symplectic basis for H 3 (M) can be written as {α, β}. We take the
periods of the holomorphic three-form to be
β
= 1,
α
= i
(9.23)
The superpotential is
W = Aφ + B
(9.24)
where A ≡ −h 1 − ih 2 , B ≡ f 1 + i f 2 . The D3-brane charge induced by the fluxes
is
N flux = f 1 h 2 − f 2 h 1
(9.25)