234
A. Cole and G. Shiu
axiodilaton and complex structure moduli are stabilized by turning on 3-form fluxes
that thread the internal manifold. We will study the resulting distribution of stabilized
axiodilaton and complex structure moduli VEVs using persistent homology. There
are of course other interesting quantities to study, including the VEVs of other moduli (which must be stabilized via other means), masses of various moduli, and the
magnitude of the flux superpotential (to be defined shortly). As we will see, restricting to the axiodilaton and complex structure moduli VEVs already gives plenty of
structure for us to explore.
We now review the relevant formulas for generating distributions of vacua. Readers who are not familiar with the terminologies of supergravity may find the following
discussion somewhat technical but the basic idea is simple, in that we are looking
for local minima of the potential energy. The most important equations for our purposes are (9.19), (9.20), and (9.22). We follow the conventions of [36]. Consider a
Calabi-Yau threefold M with h 2,1 complex structure moduli. Take a symplectic basis
{A
a
, B b } for the b 3 = 2h 2,1 + 2 three-cycles, with a, b = 1, . . . , h 2,1 + 1. We have
dual cohomology elements α a , β
b satisfying
A a
α b = δ
a
b ,
B b
β
a
= −δ
a
b ,
M
α a ∧ β
b
= δ
b
a
(9.12)
From the unique holomorphic three-form , we have the periods z
a
≡
A a
, G b ≡
B b
, which form the b 3 -vector Π(z) ≡ (G b , z
a
). Additionally
M
∧ = z
a
G a − z
a
G a = −Π
†
· Σ · Π
(9.13)
where we have the symplectic matrix
Σ =
0 1
−1 0
(9.14)
whose entries are (h 2,1 + 1) × (h 2,1 + 1) matrices. The NSNS and RR 3-form fluxes
are quantized and may be written in the α, β basis
F 3 = −(2π)
2
α
( f a α a + f a+h 2,1 +1 β
a
), H 3 = −(2π)
2
α
(h a α a + h a+h 2,1 +1 β
a
)
(9.15)
where we have defined the integer-valued b 3 -vectors f and h. From now on we set
(2π)
2
α
= 1. The Kähler potential truncated to the axiodilaton and complex structure
moduli is
K = − log
i
M
∧
− log
−i(φ − φ)
= − log(−iΠ † · Σ · Π) − log(−i(φ − φ))
(9.16)
A. Cole and G. Shiu
axiodilaton and complex structure moduli are stabilized by turning on 3-form fluxes
that thread the internal manifold. We will study the resulting distribution of stabilized
axiodilaton and complex structure moduli VEVs using persistent homology. There
are of course other interesting quantities to study, including the VEVs of other moduli (which must be stabilized via other means), masses of various moduli, and the
magnitude of the flux superpotential (to be defined shortly). As we will see, restricting to the axiodilaton and complex structure moduli VEVs already gives plenty of
structure for us to explore.
We now review the relevant formulas for generating distributions of vacua. Readers who are not familiar with the terminologies of supergravity may find the following
discussion somewhat technical but the basic idea is simple, in that we are looking
for local minima of the potential energy. The most important equations for our purposes are (9.19), (9.20), and (9.22). We follow the conventions of [36]. Consider a
Calabi-Yau threefold M with h 2,1 complex structure moduli. Take a symplectic basis
{A
a
, B b } for the b 3 = 2h 2,1 + 2 three-cycles, with a, b = 1, . . . , h 2,1 + 1. We have
dual cohomology elements α a , β
b satisfying
A a
α b = δ
a
b ,
B b
β
a
= −δ
a
b ,
M
α a ∧ β
b
= δ
b
a
(9.12)
From the unique holomorphic three-form , we have the periods z
a
≡
A a
, G b ≡
B b
, which form the b 3 -vector Π(z) ≡ (G b , z
a
). Additionally
M
∧ = z
a
G a − z
a
G a = −Π
†
· Σ · Π
(9.13)
where we have the symplectic matrix
Σ =
0 1
−1 0
(9.14)
whose entries are (h 2,1 + 1) × (h 2,1 + 1) matrices. The NSNS and RR 3-form fluxes
are quantized and may be written in the α, β basis
F 3 = −(2π)
2
α
( f a α a + f a+h 2,1 +1 β
a
), H 3 = −(2π)
2
α
(h a α a + h a+h 2,1 +1 β
a
)
(9.15)
where we have defined the integer-valued b 3 -vectors f and h. From now on we set
(2π)
2
α
= 1. The Kähler potential truncated to the axiodilaton and complex structure
moduli is
K = − log
i
M
∧
− log
−i(φ − φ)
= − log(−iΠ † · Σ · Π) − log(−i(φ − φ))
(9.16)