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?

Max Brinkmann Small W0 near the Conifold 1/251/25

The Little Picture

[adapted from Eran Palti’s review, 1903.06239]

Where does KKLT lie in this picture?


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


KKLT – a recipe for dS [Kachru,Kallosh,Linde,Trivedi ’03]


nonpert. effects110 120 130 140 150 160

-2.×10-15

-1.×10-15

1.×10-15

2.×10-15AdS vacuum


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!]




nonpert. effects110 120 130 140 150 160

-2.×10-15

-1.×10-15

1.×10-15

2.×10-15AdS vacuum


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]





nonpert. effects110 120 130 140 150 160

-2.×10-15

-1.×10-15

1.×10-15

2.×10-15AdS vacuum


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]



KKLT and the Swampland Conjectures[Blumenhagen,M.B.,Makridou ’20]


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.



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 .



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.



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 .



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.


What’s next?


nonpert. effects110 120 130 140 150 160

-2.×10-15

-1.×10-15

1.×10-15

2.×10-15AdS vacuum


small D3 uplift

110 120 130 140 150 160

1.5×10-15

2.×10-15

2.5×10-15

3.×10-15dS vacuum


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.


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]



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.



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)


|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!



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



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



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 .



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



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 )



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)



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.


|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 )



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.



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



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]


KKLT and the Swampland Conjectures


nonpert. effects110 120 130 140 150 160

-2.×10-15

-1.×10-15

1.×10-15

2.×10-15AdS vacuum


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]


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?


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.


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 .


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.


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.


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?


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!


IIA flux vacua - Freund-Rubin

• Backreaction of fluxes on the geometry important• Encoded by geometric fluxes• No dilute flux limit


• 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!


IIA flux vacua - Freund-Rubin

• Backreaction of fluxes on the geometry important• Encoded by geometric fluxes• No dilute flux limit


• 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?


small w0 neartheconifold

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