DFT on Calabi-Yau threefolds and Axion Inﬂation

DFT on Calabi-Yau threefoldsand Axion Inflation

Ralph Blumenhagen

Max-Planck-Institut fur Physik, Munchen

(RB, Font, Fuchs, Herschmann, Plauschinn, arXiv:1503.01607)

(RB, Font, Fuchs, Herschmann, Plauschinn, Sekiguchi, Wolf, arXiv:1503.07634)

(RB, Font, Plauschinn, arXiv:1507.08059)

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Introduction

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Introduction

Moduli stabilization in string theory:

• Race-track scenario

• KKLT

• LARGE volume scenario

Based on instanton effects → exponential hierarchies → cangenerate Msusy ≪ MPl

Experimentally:

• Supersymmetry not found at LHC with M < 1TeV.

• Not excluded large field inflation: Minf ∼ MGUT

Contemplate scenario of moduli stabilization with onlypolynomial hierarchies → string tree-level with fluxes

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Introduction

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Introduction

PLANCK 2015, BICEP2 results:

• upper bound: r < 0.07

• spectral index: ns = 0.9667± 0.004 and its runningαs = −0.002± 0.013.

• amplitude of the scalar power spectrumP = (2.142± 0.049) · 10−9

Lyth bound

∆φ

Mpl∼ O(1)

√

r

0.01and

Minf = (Vinf)1

4 ∼( r

0.1

)1

4 × 1.8 · 1016GeV

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Introduction

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Introduction

Inflationary mass scales:

• Hubble constant during inflation: H ∼ 1014 GeV.

• mass scale of inflation: Vinf = M4inf = 3M2

PlH2inf ⇒

Minf ∼ 1016 GeV

• mass of inflaton during inflation: M2Θ = 3ηH2 ⇒

MΘ ∼ 1013 GeV

Large field inflation with ∆Φ > Mpl:

• Makes it important to control Planck suppressedoperators (eta-problem)

• Invoking a symmetry like the shift symmetry of axionshelps

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Axion inflation

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Axion inflation

Axions are ubiquitous in string theory so that many scenarioshave been proposed

• Natural inflation with a potentialV (θ) = Ae−SE(1− cos(θ/f)). Hard to realize in stringtheory, as f > 1 lies outside perturbative control.(Freese,Frieman,Olinto)

• Aligned inflation with two axions, feff > 1. (Kim,Nilles,Peloso)

• N-flation with many axions and feff > 1.(Dimopoulos,Kachru,McGreevy,Wacker)

Comment: These models have come under pressure by theweak gravity conjecture, which for instantons was proposed tobe f · SE < 1. (Montero,Uranga,Valenzuela),(Brown,Cottrell,Shiu,Soler)

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Axion monodromy inflation

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• Monodromy inflation: Shift symmetry is broken bybranes or fluxes unwrapping the compact axion →polynomial potential for θ. (Kaloper, Sorbo), (Silverstein,Westphal)

non-pert. fluxes

Discrete shift symmetry acts also on the fluxes, i.e. one getsdifferent branches → tunneling a la Coleman-de Lucia

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Recent proposal: Realize axion monodromy inflation via theF-term scalar potential induced by background fluxes.(Marchesano.Shiu,Uranga),(Hebecker, Kraus, Wittkowski),(Bhg, Plauschinn)

Advantages

• Generating the inflaton potential, supersymmetry isbroken spontaneously by the very same effect by whichusually moduli are stabilized

• Generic, as the field strengths Fp+1 = dCp +H ∧ Cp−2

involves the gauge potentials Cp−2.

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Objective

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Objective

For a controllable single field inflationary scenario, all modulineed to be stabilized such that

MPl > Ms > MKK > Minf > Mmod > Hinf > |MΘ|

Aim: Systematic study of realizing single-field fluxed F-termaxion monodromy inflation, taking into account the interplaywith moduli stabilization.

Continues the studies from (Bhg,Herschmann,Plauschinn), (Hebecker,

Mangat, Rombineve, Wittkowsky) by including the Kahler moduli.

Note:

• There exist a no-go theorem for having an unconstrainedaxion in supersymmetric minima of N = 1 supergravitymodels (Conlon)

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Introduction

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Introduction

Realistic string model building =⇒ break N=2 → N=1 viaorientifold projection

• Scalar potential splits VDFT = VF + VD + VNS−tad

• Fits into 4d N = 1 SUGRA, i.e. VF can be computed viaa Kahler and superpotential

• Apply this to SPHENO, i.e. a high moduli stabilizationscheme and inflationary cosmology

Lit: Type IIB orientifolds on CY threefolds with geometricand non-geometric fluxes. (Shelton,Taylor,Wecht),(Aldazabal,Camara,Ibanez,Font), (Grana, Louis, Waldram), (Benmachiche, Grimm),(Micu, Palti, Tasinato)

Recently: (RB, Font, Fuchs, Herschmann, Plauschinn, Sekiguchi, Wolf,

arXiv:1503.07634) based on ideas from (Marchesano.Shiu,Uranga),(Hebecker,Kraus, Wittkowski),(Bhg, Plauschinn)

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Framework

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Framework

Framework: Type IIB orientifolds on CY threefolds withgeometric and non-geometric fluxes. (Shelton,Taylor,Wecht),(Aldazabal,Camara,Ibanez,Font), (Grana, Louis, Waldram), (Benmachiche, Grimm),(Micu, Palti, Tasinato)

Dimensional reduction of DFT on Calabi-Yau manifold with(small) fluxes, treated as perturbations.(RB, Font, Plauschinn,arXiv:1507.08059)

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Framework

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Framework

Generalization of the dimensional reduction a la Taylor/VafaSUGRA with H3, F3-form on CY (Taylor,Vafa, hep-th/9912152)

• Technical problem: DFT action contains the unknownmetric gii′ on the CY

• Solution: Rewrite the action (on CY) such that only Ω,J and ⋆-op. appear, like

⋆L = −e−2φ

2d10x

√−GHijk Hi′j′k′ gii

′

gjj′

gkk′

= −e−2φ

2H ∧ ⋆H ,

DFT contains many more terms that are not of this simpletype.

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Flux formulation of DFT

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(Siegel; Hohm, Kwak; Aldazabal, Geissbuhler, Marques, Nunez, Penas; ...)

In a generalized frame formalism with N-bein

EAI =

(

eai −ea

kBki

0 eai

)

,

the action in the NS-NS sector is given by

SNSNS =1

2κ210

∫

dDX e−2d

[

FABCFA′B′C ′

(

1

4SAA′

ηBB′

ηCC ′ − 1

12SAA′

SBB′

SCC ′ − 1

6ηAA′

ηBB′

ηCC ′

)

+ FAFA′

(

ηAA′ − SAA′

)

]

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Dimensional Reduction

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Dimensional Reduction

We perform the dimensional reduction such that

• The (unknown) metric of the Calabi-Yau three-foldgij(x) only depends on the usual coordinates.

• On a CY there are no non-trivial homological one-cycles,and hence FA = 0. The term

FABCFA′B′C ′ηAA′

ηBB′

ηCC ′

vanishes via Bianchiidentities.

• We give constant expectation values to the fluxes FIJK

and assume that they are only subject to quadraticconstraints (Bianchi identities).

• We are only interested in terms contributing to thescalar potential.

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Thus, we consider the part of the DFT action in the fluxformulation

SNSNS =1

2κ210

∫

dDX e−2dFIJKFI ′J ′K′

(1

4HII ′ηJJ

′

ηKK′ − 1

12HII ′HJJ ′HKK′

+ . . .)

and perform a further dimensional reduction on a CY.

The generalized flux FIJK splits into components

Fijk = Hijk , F ijk = F i

jk , Fijk = Qi

jk , F ijk = Rijk .

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Fluxes

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Fluxes

Geometric fluxes H,F - and F -flux and non-geometric fluxesQ,R are operators acting on p-forms

H ∧ = 13! Hijk dxi ∧ dxj ∧ dxk ,

F = 12! F

kij dxi ∧ dxj ∧ ιk ,

Q • = 12! Qi

jk dxi ∧ ιj ∧ ιk ,

R x = 13! R

ijk ιi ∧ ιj ∧ ιk .

Twisted differential D

D = d−H ∧ −F −Q • −R x .

Nilpotency, D2 = 0, leads to a set of quadratic constraints onthe fluxes.

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Bianchi identities

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Bianchi identitiesThese constraints correspond to Bianchi identities

0 = Hm[abFmcd] ,

0 = Fm[bc F

da]m +Hm[abQc]

md ,

0 = Fm[ab]Qm

[cd] − 4F [cm[aQb]

d]m +HmabRmcd ,

0 = Qm[bcQd

a]m +Rm[ab F c]md ,

0 = Rm[abQmcd] ,

0 = Rmn[aF b]mn ,

0 = RamnHbmn − F amnQb

mn ,

0 = Q[amnHb]mn ,

0 = RmnlHmnl .ETH Zurich, 21.01.2016 – p.16/42

DFT on CY

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DFT on CY

Withχ := e−BD

(

eB+iJ)

and Ψ := e−B D(

eB Ω)

, the DFT actionin the flux formulation

SNSNS =1

2κ210

∫

dDX e−2dFIJKFI ′J ′K′

(1

4HII ′ηJJ

′

ηKK′ − 1

12HII ′HJJ ′HKK′

+ . . .)

can be rewritten as (very tedious computation)

SNSNS = − 1

2κ210

∫

e−2φ

[

1

2χ ∧ ⋆χ +

1

2Ψ ∧ ⋆Ψ

−1

4

(

Ω ∧ χ)

∧ ⋆(

Ω ∧ χ)

− 1

4

(

Ω ∧ χ)

∧ ⋆(

Ω ∧ χ)

]

.

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R-R sector

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R-R sector

The DFT action in the R-R sector can be expressed as

⋆LRR = −1

2G ∧ ⋆G .

with

G = F (3) + e−B D(

eBC)

.

We have managed to reexpress the DFT action compactified on

a CY just in terms of quantities that do not explicitly contain

the CY metric. → now can further dimensionally reduce the

Lagrangian along Taylor/Vafa

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Further Reduction on CY

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On CY X introduce a symplectic basis as

αΛ, βΛ ∈ H3(X ) , Λ = 0, . . . , h2,1 .

For the even cohomology of X we introduce bases of theform

ωA ∈ H1,1(X ) , σA ∈ H2,2(X )

with A = 1, . . . , h1,1.It is convenient to extend this with the zero- and six-form

ωA =

√g

V dx6, ωA

,

σA =

1, σA

,A = 0, . . . , h1,1 .

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One defines the constant fluxes via the relations

DαΛ = qΛAωA + fΛAσ

A , DβΛ = qΛAωA+ fΛAσA ,

DωA = −fΛAαΛ+ fΛAβΛ , DσA = qΛAαΛ− qΛ

AβΛ .

with

fΛ0 = rΛ , fΛ0 = rΛ ,

qΛ0 = hΛ , qΛ0 = hΛ .

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Using then the combined basis ωA, σA, we can define a

complex (2h1,1 + 2)-dimensional vector V1 in the followingway

V1 =

16 κABCJ AJ BJC

J A

112 κABCJ BJ C

,

We introduce a (2h2,1 + 2)-dimensional vector as

V2 =

(

XΛ

−FΛ

)

.

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V =1

2

(

FT + CT · OT)

· M1 ·(

F+O · C)

+e−2φ

2V T1 · OT · M1 · O · V1

+e−2φ

2V T2 · O ·M2 · OT · V 2

− e−2φ

4V V T2 · C · O ·

(

V1 × VT1 + V 1 × V T

1

)

· OT · CT · V 2 .

with

M1,2 =

(

1 ReN0 1

)(

−ImN 0

0 −ImN−1

)(

1 0

ReN 1

)

related to the period matrices of the CY.ETH Zurich, 21.01.2016 – p.22/42


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This scalar potential was shown to agree with the one of N=2gauged supergravity. (D’Auria, Ferrara, Trigiante, hep-th/0701247)

Scalar potential of DFT on fluxed CY⇐⇒

Scalar potential of N=2 GSUGRA

Orientifold projection: N=2 → N=1 susy

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Orientifold projection

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Perform an orientifold projection

σ∗ : J → J , σ∗ : Ω → −Ω ,

with O7- and O3-planes.Splits the cohomology into even and odd parts

Hp,q(X ) = Hp,q+ (X )⊕Hp,q

− (X ) .

The fields of the ten-dimensional theory transform as

ΩP(−1)FL =

g, φ, C(0), C(4) even ,

B, C(2) odd .

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The holomorphic three-form Ω is expanded in H3−(X )

Ω = Xλαλ − Fλβλ .

The Kahler form J and the components of theten-dimensional form fields are combined into the complexfields

τ = C(0) + ie−φ ,

Ga = ca + τ ba ,

Tα = − i

2καβγ t

β tγ + ρα +1

2καab c

abb − i

4eφκαabG

a(G−G)b ,

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These moduli can be encoded in a complex, even multi-formΦevc as

Φevc = τ +Gaωa + Tασ

α .

with ωa ∈ H11− and σα ∈ H22

+ .One finds the orientifold even fluxes

F (3) : Fλ , Fλ ,

H : hλ , hλ ,

F : fλ α

, f λα , fλa , fλa ,

Q : qλa , qλ a , qλ

α , qλα ,

R : rλ, rλ .

αλ ∈ H21− and αλ ∈ H21

+ .ETH Zurich, 21.01.2016 – p.26/42

F-term potential

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F-term potential

Kahler potential:

K = − log[

−i(τ − τ)]

− 2 log V − log

[

i

∫

XΩ ∧ Ω

]

,

Implicite in the Tα and Ga moduli.

The superpotential in the presence of all fluxes is

W =

∫

X

(

F (3) +DΦevc

)

∧ Ω .

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F-term potential

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F-term potentialAfter a tedious computation, one can rewrite VF as

VF =M4

Pl eφ

2V2

∫

X

(

e−2φ

[

1

2(Re χ) ∧ ⋆(Re χ) +

1

2Ψ ∧ ⋆Ψ

− 1

4

(

Ω ∧ χ)

∧ ⋆(

Ω ∧ χ)

− 1

4

(

Ω ∧ χ)

∧ ⋆(

Ω ∧ χ)

]

+1

2G ∧ ⋆G

−[

(Im τ)− (ImGa)ωa + (ImTα)σα]

∧ DF (3)

)

.

• Prefactor= M4s so that the first two lines match with the

orientifold projected DFT action

• The third line corresponds to NS-NS tadpole terms,which will be cancelled by local sources.

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D-term potential

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D-term potentialOnly (Re χ) appeared, as (Im χ) ∈ H3

+(X ).Its contribution in the DFT Lagrangian is

⋆LH3+= −1

2e−2φ (Im χ) ∧ ⋆(Im χ) .

Defining

Dλ = rλ(

V − 12 καab t

αbabb)

+ qλa κaαb tαbb − f λα t

α ,

Dλ= −r

λ

(

V − 12 καab t

αbabb)

− qλa κaαb t

αbb + fλα

tα .

one can compute

⋆LH3+=− 1

2e−2φ

(

D

D

)T

· M1 ·(

D

D

)

(Robins,Wrase, arXiv:0709.2186)ETH Zurich, 21.01.2016 – p.29/42

D-term potential

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D-term potential

Transforming into a basis where rλ = qλa = f λα = 0, this canbe written as a D-term for the h21+ abelian gauge fields

VD =M4

Pl

2

[

(Ref)−1]ab

DaDb .

Summary: Performing an orientifold projection, the even partof the DFT scalar potential splits into three pieces

V = VF + VD + VNS-tad ,

with the NS-NS tadpole cancelled after R-R tadpolecancellation.

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Objective

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Objective

Scheme of moduli stabilization such that the followingaspects are realized:

• There exist non-supersymmetric minima stabilizing thesaxions in their perturbative regime.

• All mass eigenvalues are positive semi-definite, where themassless states are only axions.

• For both the values of the moduli in the minima and themass of the heavy moduli one has parametric control interms of ratios of fluxes.

• One has either parametric or at least numerical controlover the mass of the lightest (massive) axion, i.e. theinflaton candidate.

• The moduli masses are smaller than the string and theKaluza-Klein scale.

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results could be presented in the discussion session

Thank You!

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A representative model

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Kahler potential is given by

K = −3 log(T + T )− log(S + S) .

Fluxes generate superpotential

W = −i f+ ihS + iqT ,

with f, h, q ∈ Z. Resulting scalar potential

V =(hs+ f)2

16sτ3− 6hqs− 2qf

16sτ2− 5q2

48sτ+

θ2

16sτ3

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Non-supersymmetric, tachyon-free minimum with

τ0 =6 f

5q, s0 =

f

h, θ0 = 0 .

Mass eigenvalues

M2mod,i = µi

hq3

16 f2

M2Pl

4π,

with µi > 0.

Gravitino-mass scale: M 3

2

≃pMmod

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Mass scales

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Mass scales

Cosmological constant in AdS minimum:

V0 = −µChq3

16 f2

M4Pl

4π

Uplift: It is possible to find (new) scaling type minima with

V0 ≥ 0 by including an uplifting D3-brane (D-term)

V = VF +A

V 4

3

+ (VD)

(Bhg, Damian, Font, Fuchs, Herschmann, Sun, arXiv:1510.01522)

Relation of mass scales:

Ms&pMKK≃

pMmod

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Axion inflaton

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Axion inflaton

Generate a non-trivial scalar potential for the massless axionΘ by turning on additional fluxes fax and deform

Winf = λW + fax∆W .

This quite generically leads to

Mmod&pMΘ =⇒ Mmod

&pMKK

Toy model with uplifted scalar potential

V = λ2(

(hs+ f)2

16sτ3− 6hqs− 2qf

16sτ2− 5q2

48sτ

)

+θ2

16sτ3+ Vup .

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Toy model

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Toy model

Backreaction of the other moduli adiabatically adjustingduring the slow-roll of θ flattens the potential(Dong,Horn,Silverstein,Westphal)

θ

Vback

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Effective potential

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Effective potential

Large field regime: θ/λ ≫ f. The potential in the large-fieldregime becomes

Vback(Θ) =25

216

hq3λ2

f2

(

1− e−γΘ)

.

with γ2 = 28/(14 + 5λ2) (similar to Starobinsky-model).

• For θ/λ ≪ f: 60 e-foldings from the quadratic potential

• Intermediate regime: linear inflation

• For θ/λ ≫ f: Starobinsky inflation

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Tensor-to-scalar ratio

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Tensor-to-scalar ratio

λ

r

With decreasing λ the model changes from chaotic toStarobinsky-like inflation.

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Parametric control

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Parametric control

From UV-complete theory point of view, large-field inflationmodels require a hierarchy of the form

MPl > Ms > MKK > Minf > Mmod > Hinf > MΘ ,

where neighboring scales can differ by (only) a factor ofO(10).

Main observation

• the larger λ, the more difficult it becomes to separatethe high scales on the left

• for small λ, the smaller (Hubble-related) scales on theright become difficult to separate.

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Conclusions

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Conclusions

• Systematically investigated the flux induced scalarpotential for non-supersymmetric minima, where we haveparametric control over moduli and the mass scales.

• All moduli are stabilized at tree-level → the frameworkfor studying F-term axion monodromy inflation.

• Since the inflaton gets its mass from a tree-level effect,one gets a high susy breaking scale.

• As all mass scales are close to the Planck-scale, it isdifficult to control all hierarchies. Does large fieldinflation necessarily must include stringy/KK effects?

• The (MS)SM could arise on a set of intersectingD7-branes → mutual constraints between fluxes andbranes (Freed/Witten anomalies). Is sequestering,Msoft ≪ M 3

2

, possible?

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Thank You!

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DFT on Calabi-Yau threefolds and Axion Inﬂation

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Transcript of DFT on Calabi-Yau threefolds and Axion Inﬂation