Small W0 neartheConifold
Transcript of Small W0 neartheConifold
Small W0 near the Conifold
Max Brinkmann
String Phenomenology Seminar Series
06.10.2020
The Big Picture
[adapted from Eran Palti’s review, 1903.06239]
Where does dS lie in this picture?
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The Little Picture
[adapted from Eran Palti’s review, 1903.06239]
Where does KKLT lie in this picture?
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KKLT – a recipe for dS
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
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KKLT – a recipe for dS [Kachru,Kallosh,Linde,Trivedi ’03]
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
[statistics: Douglas et al ’04/’05][Demirtas,Kim,McAllister,Moritz ’19]
[Sethi ’17][Moritz,Retolaza,Westphal ’17][Gautason,Van Hemelryck,Van Riet ’19][ Kallosh,Linde,McDonough,Scalisi ’18][Hamada,Hebecker,Shiu,Soler ’18][Carta,Moritz,Westphal ’19][Blumenhagen,M.B.,Makridou ’20]
[Klebanov,Strassler ’00][Giddings,Kachru,Polchinski ’02][Douglas,Shelton,Torroba ’07][Blumenhagen,Herschmann,Wolf ’16]
[Bena,Graña,Halmagy ’10][Michel,Mintun,Polchinski,Puhm,Saad ’14][Bena,Dudas,Graña,Lüst ’18][Blumenhagen,Kläwer,Schlechter ’19]
[...and many, many more!]
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KKLT – a recipe for dS
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
[Álvarez-García,Blumenhagen,M.B.,Schlechter ’20]
[Demirtas,Kim,McAllister,Moritz ’20]
[Demirtas,Kim,McAllister,Moritz ’19]
Max Brinkmann Small W0 near the Conifold 2/252/25
KKLT – a recipe for dS
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
[Álvarez-García,Blumenhagen,M.B.,Schlechter ’20]
[Demirtas,Kim,McAllister,Moritz ’20]
[Demirtas,Kim,McAllister,Moritz ’19]
KKLT and the Swampland Conjectures[Blumenhagen,M.B.,Makridou ’20]
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A recipe for |W0| ≪ 1
Idea: natural hierarchiesSmall values can naturally be generated by perturbative mechanisms, ifthe leading term vanishes analytically.
The prepotential of the complex structure moduli splits into a classicaland a nonperturbative part, if they are• at weak string coupling,• at large complex structure,• expressed in terms of the mirror variables.
To-Do List1. Find fluxes that solve pert. F-term eq. for vanishing superpotential.
2. Generate a small nonpert. potential for the remaining flat direction.
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A recipe for |W0| ≪ 1
Idea: natural hierarchiesSmall values can naturally be generated by perturbative mechanisms, ifthe leading term vanishes analytically.
The prepotential of the complex structure moduli splits into a classicaland a nonperturbative part, if they are• at weak string coupling,• at large complex structure,• expressed in terms of the mirror variables.
To-Do List1. Find fluxes that solve pert. F-term eq. for vanishing superpotential.
2. Generate a small nonpert. potential for the remaining flat direction.
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A recipe for |W0| ≪ 1
Setup• Orientifold X of a CY 3-fold.• Wrapped by D7-branes carrying −QD3.
• Period vector Π =(
Ua
Fa
)w/ sympl. pairing Σ =
(0 −11 0
).
• Prepotential F s.th. Fa = ∂UaF .• Inhomogeneous coordinates: U0 = 1, F0 = 2F − U iFi .
The prepotential splits into F(U) = Fpert(U) + Finst(U).
Including the axio-dilaton:
K = − log(−i Π†ΣΠ) − log(S + S) ,W = (F + i SH)T · Σ · Π .
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A recipe for |W0| ≪ 1
Perturbative Prepotential
Fpert(U) = − 13!KabcUaUbUc + 1
2aabUaUb + baUa + ξ
with• Kabc the triple intersection numbers of the mirror CY,
• ξ = − ζ(3)χ2(2πi)3 , and χ the Euler number of the mirror CY,
• aab, ba rational numbers related to the mirror CY.
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A recipe for |W0| ≪ 1
Lemma: existence of a perturbatively flat vacuum
If a pair (M⃗, K⃗ ) ∈ Zn × Zn satisfies: Then:
−12M⃗ · K⃗ ≤ QD3, The tadpole bound is satisfied,
Nab = KabcMc is invertible, ∂W = 0 has solution U⃗ = p⃗S ,
K⃗T N−1K⃗ = 0, W = 0 in that solution,
p⃗ = N−1K⃗ lies in the Kähler cone, the minimum is trustable
a · M⃗ and b⃗ · M⃗ are integer-valued. and the fluxes are integer valued.[Demirtas,Kim,McAllister,Moritz ’19]
The appropriate choice of fluxes is given by
F = (b⃗ · M⃗, a · M⃗, 0, M⃗)T ,H = (0, K⃗ , 0, 0)T ,
s.th. the superpotential is hom. of deg. 2 in the U i .
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A recipe for |W0| ≪ 1
Instanton Contributions
Finst(U) ∼∑
q⃗e2πi q⃗·U⃗
with the sum running over effective curves.
Stabilizing the axio-dilaton: Racetrack
In the perturbative minimum: U⃗ = p⃗S .
Weff(S) ∼ Ma∂aFinst ∼∑
q⃗e2πiSp⃗ ·⃗q
This stabilizes S to exponentially small values if the two leadinginstanton contributions satisfy p⃗ · q⃗1 ≈ p⃗ · q⃗2.
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What’s next?
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
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Periods at the Conifold
The hardest part is finding the periods/prepotential at the conifold!
Method 1: Resumming with GV-nilpotent curves+ Elegantly avoid complete computation.
- Only special cases: conifold cycle mirror to shrinking curve.
- No full solution, only good approximation.
[Demirtas,Kim,McAllister,Moritz ’20]
Method 2: analytic computation of transition matrices+ Fairly general method
+ Complete solution, can check approximations numerically
- Fairy involved computations
[Álvarez-García,Blumenhagen,M.B.,Schlechter ’20; Lorenz’ talk]
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Periods at the Conifold
Transition MatrixPeriods in a symplectic basis hard to compute. Solving the Picard-Fuchsequations Di ω = 0 in any local basis is easy.
Π = m · ω
At LCS, monodromies fix the transition matrix m uniquely. At theconifold monodromies are not enough.
We can rewrite the fundamental period at the LCS as a hypergeometricfunction, which allowed us to analytically determine the transitionmatrix.
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Periods at the Conifold
Transition MatrixPeriods in a symplectic basis hard to compute. Solving the Picard-Fuchsequations Di ω = 0 in any local basis is easy.
Π = m · ω
At LCS, monodromies fix the transition matrix m uniquely. At theconifold monodromies are not enough.
With the analytic transition matrix we can efficiently compute theperiods at the conifold to very high order (30+), as well as theprepotential and mirror maps. (Lorenz’ talk)
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|W0| ≪ 1 at conifold
Setup• n-parameter CY orientifold• One modulus close to the conifold, others at LCS
Notation: X 0 = 1, X i = (Uα, Z )T , (i = 1, ... , n), (α = 1, ... , n − 1)
Prepotential
Fpert = − 13!KijkX iX jX k + 1
2AijX iX j + BiX i + C − Z 2 log Z2πi ,
Finst = 1(2πi)3
∑c⃗
ac⃗
n∏i=1
qini ,
Note that while the qi are still simply exponentials in the LCS moduli,the conifold modulus enters linearly!
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|W0| ≪ 1 at conifold
Setup• n-parameter CY orientifold• One modulus close to the conifold, others at LCS
Notation: X 0 = 1, X i = (Uα, Z )T , (i = 1, ... , n), (α = 1, ... , n − 1)
Prepotential
Fpert = − 13!KijkX iX jX k + 1
2AijX iX j + BiX i + C − Z 2 log Z2πi ,
Finst = 1(2πi)3
∑c⃗
ac⃗
n∏i=1
qini ,
Note that while the qi are still simply exponentials in the LCS moduli,the conifold modulus enters linearly!
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|W0| ≪ 1 at conifold
To-Do List1. Find fluxes that solve pert. eq. for vanishing superpotential at Z = 0.2. Stabilize conifold modulus at leading order to WZ = O(Z ).3. Include instanton corrections for the remaining flat direction.
Choosing FluxesWant the superpotential to be degree 2 in the LCS moduli:
F =
BiM i
(AiαM i , Mn)T
0M⃗
, H =
0K⃗00
, M⃗, K⃗ ∈ Zn
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|W0| ≪ 1 at conifold
To-Do List1. Find fluxes that solve pert. eq. for vanishing superpotential at Z = 0.2. Stabilize conifold modulus at leading order to WZ = O(Z ).3. Include instanton corrections for the remaining flat direction.
Choosing FluxesWant the superpotential to be degree 2 in the LCS moduli:
F =
BiM i
(AiαM i , Mn)T
0M⃗
, H =
0K⃗00
, M⃗, K⃗ ∈ Zn
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|W0| ≪ 1 at conifold
To-Do List1. Find fluxes that solve pert. eq. for vanishing superpotential at Z = 0.2. Stabilize conifold modulus at leading order to WZ = O(Z ).3. Include instanton corrections for the remaining flat direction.
Choosing FluxesWant the superpotential to be degree 2 in the LCS moduli:
F =
BiM i
(AiαM i , Mn)T
0M⃗
, H =
0K⃗00
, M⃗, K⃗ ∈ Zn
⇒ BiM i , AiαM i ∈ Z⇒ Bi , Aiα must be rational
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|W0| ≪ 1 at conifold
Step 1: first order, Z=0Stabilizing the LCS moduli at the conifold locus:
W = 12NαβUαUβ + iSKαUα = 0 (N−1)αβKαKβ = 0
∂αW = 0 Uα = pαS
with Nαβ = KiαβM i and pα = −i(N−1)αβKβ .
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|W0| ≪ 1 at conifold
Step 1: first order, Z=0Stabilizing the LCS moduli at the conifold locus:
W = 12NαβUαUβ + iSKαUα = 0 (N−1)αβKαKβ = 0
∂αW = 0 Uα = pαS
with Nαβ = KiαβM i and pα = −i(N−1)αβKβ .
⇒ Nαβ = KiαβM i invertible
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|W0| ≪ 1 at conifold
Step 2: Stabilize ZIntegrating out the LCS moduli Uα:
Wpert(S, Z ) = −mZ log(Z )2πi + m1Z + n1SZ + O(Z 2) ,
DZ W = ∂Z W + ∂Z K · O(Z )
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|W0| ≪ 1 at conifold
Step 2: Stabilize ZIntegrating out the LCS moduli Uα:
Wpert(S, Z ) = −mZ log(Z )2πi + m1Z + n1SZ + O(Z 2) ,
DZ W = ∂Z W + ∂Z K · O(Z )
⇒ Z0 = ζ0 e−2πpZ S , WZ = mZ2πi + O(Z 2)
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|W0| ≪ 1 at conifold
Step 3: Instanton Contributions and RacetrackThe CS moduli are stabilized to log(Z ) ∼ Uα ∼ S .
Weff(S) = −M i∂iFinst + mZ2πi ∼
∑an ecnS .
That this can be stabilized to small W0 has to be checked case by case.
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|W0| ≪ 1 at conifold - Example
Example: 3-parameter CY P1,1,2,8,12[24]
K111 = 8, K112 = 2, K113 = 4, K123 = 1, K133 = 2 ,
A33 =(1
2 + 3 − 2 log(2π)2πi
), B =
(236 , 1, 23
12
)T.
The leading instanton contributions are
Finst = −5i qU136π3 −
493 q2U1
10368π3 + 5i qU1qZ36π3 + ... ,
qU1 = 864 e2πiU1 , qU2 = 4(π
i Z )4 e2πiU2 , qZ = (πi Z )
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|W0| ≪ 1 at conifold - Example
First ChecksBiM i , AiαM i ∈ Z ⇒ Bi , Aiα must be rational ✓
After steps 1 and 2: Z0 ≪ 1 for ℜ(S) ≫ 1 ✓
Choosing FluxesThe fluxes must be chosen to satisfy• BiM i , AiαM i ∈ Z,• Nαβ = KiαβM i invertible,• (N−1)αβKαKβ = 0,• Instanton series: 1 > |qi | ∼ |ef (M⃗,K⃗)S | ↔ f (M⃗,K⃗) < 0.
A simple search finds many reasonably small fluxes satisfying these.
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|W0| ≪ 1 at conifold - Example
A concrete solution
Choosing M⃗ = (−24, 120, 24)T , K⃗ = (−9, 3, −4)T :
⟨U1⟩ = 2.79 i , ⟨U2⟩ = 8.36 i , ⟨Z ⟩ = 1.36 · 10−6i , ⟨S⟩ = 22.3 ,
|qi | = (2 · 10−5, 0.2, 4 · 10−6) ,
W0 = −3.10 · 10−6 .
22.0 22.5 23.0 23.5 24.0
5.×10-15
1.×10-14
1.5×10-14
2.×10-14
2.5×10-14
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|W0| ≪ 1 at conifold - Example
MassesFull access to the Periods allows us to compute the masses.
{m2} = {6 · 1014︸ ︷︷ ︸Z
, 1 · 103, 3 · 102︸ ︷︷ ︸U i
, 2 · 10−11︸ ︷︷ ︸S
}M2pl .
Note that m2S ≈ |W0|2 ≈ m2
KKLT :The axio-dilaton cannot be integrated out before KKLT starts!
Search Results• Inexhaustive search over fluxes• Semianalytic computation, no numerical checks• O(104) vacua with |W0| ≤ 10−6
• More detailed search described by [Demirtas,Kim,McAllister,Moritz ’20]
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|W0| ≪ 1 at conifold - Example
MassesFull access to the Periods allows us to compute the masses.
{m2} = {6 · 1014︸ ︷︷ ︸Z
, 1 · 103, 3 · 102︸ ︷︷ ︸U i
, 2 · 10−11︸ ︷︷ ︸S
}M2pl .
Note that m2S ≈ |W0|2 ≈ m2
KKLT :The axio-dilaton cannot be integrated out before KKLT starts!
Search Results• Inexhaustive search over fluxes• Semianalytic computation, no numerical checks• O(104) vacua with |W0| ≤ 10−6
• More detailed search described by [Demirtas,Kim,McAllister,Moritz ’20]
Max Brinkmann Small W0 near the Conifold 19/2519/25
KKLT and the Swampland Conjectures
CS moduli stabilizedat |W0| ≪ 1
nonpert. effects110 120 130 140 150 160
-2.×10-15
-1.×10-15
1.×10-15
2.×10-15AdS vacuum
strong warping(conifold)
small D3 uplift
110 120 130 140 150 160
1.5×10-15
2.×10-15
2.5×10-15
3.×10-15dS vacuum
KKLT and the Swampland:The AdS Distance Conjecture
[Blumenhagen,M.B.,Makridou ’20]
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AdS distance conjecture [Lüst, Palti, Vafa ’19]
AdS distance conjecture (ADC)For an AdS vacuum the limit Λ → 0 is at infinite distance in field spaceand there is a tower of light states with
mtower = cAdS |Λ|α
for α > 0. Supersymmetric vacua have α = 1/2.
• Usually this tower is assumed to be the KK-tower.• Satisfied by most treelevel vacua.• DGKT are counterexamples for the strong version!• Strong version forbids scale separation.• Is SUSY the wrong distinguishing feature for strong version?
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KKLT: ADC
KKLT AdS minimum
A(2aτ + 3) = −3W0eaτ , Λ = −a2A2
6τe−2aτ .
ADC: what is the real KK scale?• Λ → 0 when τ → ∞, at infinite distance in field space. ✓• Naive KK scale: mKK ∼ 1/τ ≫ |Λ|α. ADC violated!?• Warped throat: KK modes near tip redshifted [Blumenhagen, Kläwer, Schlechter ’19]
m2KK ∼ 1
log2 |Λ||Λ|
13 .
• Up to log-corrections, ADC satisfied with α = 1/6. ✓• Since α < 1/2, scale separation is possible.
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Other quantum vacua: LVS
We find very similar behavior for the large volume scenario:
LVS AdS minimum
VLVS = λ
√τse−2aτs
V− µ
τse−aτs
V2 + ν
V3 , Λ ∼ −e−3aτs
τs
(1 + O( 1
τs))
.
ADC: satisfied with α = 2/9 up to log-corrections ✓• Λ → 0 as τs → ∞. ✓• For LVS, no constraint on W0 to be small, naive KK-scale justified:
m2KK ∼ 1
V43
∼ 1
τ23s
e− 43 aτs ∼ 1
log29 |Λ|
|Λ|49 .
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Log-Corrections to Swampland Conjectures
ADC - classicalThe limit Λ → 0 is at infinite distance in field space and there is a tower oflight states with
mtower = cAdS |Λ|α
for α > 0. Supersymmetric vacua have α = 1/2.
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Log-Corrections to Swampland Conjectures
ADC - quantumThe limit Λ → 0 is at infinite distance in field space and there is a tower oflight states with
mtower = cAdS |Λ|α 1log |Λ|β
for α, β > 0. Vacua without dilute flux limit have α = 1/2.
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Summary
• KKLT is a recipe for dS which has never actually been cooked.• Understanding the interplay between ingredients is important.• We have shown that small |W0| can be found near a conifold.• Progress has shown no big holes in the argument yet.• However, KKLT violates AdS and dS swampland conjectures.• Intrinsically quantum in nature – can this be the reason?
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Thank you for your attention!
IIA flux vacua - DGKT
• Closed string moduli stabilized via RR, H3 fluxes• Dilute flux limit exists
Example: isotropic 6-torus; moduli S , T =Tk , U =Uk
• W = if0T 3 − 3if4T + ih0S + 3ih1U• KK scales: m2
KK,i ∼ |Λ|7/9 (weak) ADC ✓
This minimum is supersymmetric, strong ADC is violated!
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IIA flux vacua - Freund-Rubin
• Backreaction of fluxes on the geometry important• Encoded by geometric fluxes• No dilute flux limit
Example: isotropic 6-torus; moduli S , T =Tk , U =Uk
• W = f6 + 3f2T 2 − ω0ST − 3ω1UT• (naive) KK scales: m2
KK,1 ∼ |Λ|ω2
1, m2
KK,2 ∼ |Λ|ω1ω2
• Looks like ωi allow for scale separation!
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IIA flux vacua - Freund-Rubin
• Backreaction of fluxes on the geometry important• Encoded by geometric fluxes• No dilute flux limit
Example: isotropic 6-torus; moduli S , T =Tk , U =Uk
• W = f6 + 3f2T 2 − ω0ST − 3ω1UT• Must take backreaction into account! [Font, Herráez, Ibáñez ’19]
• KK scales: m2KK,i ∼ |Λ| strong ADC ✓
Difference between these models are flux backreactions on thegeometry, i.e geometric fluxes. Is this the relevant feature fordistinguishing strong/weak ADC cases?
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