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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)
ETH Zurich, 21.01.2016 – p.1/42
Introduction
ETH Zurich, 21.01.2016 – p.2/42
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
ETH Zurich, 21.01.2016 – p.2/42
Introduction
ETH Zurich, 21.01.2016 – p.3/42
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
ETH Zurich, 21.01.2016 – p.3/42
Introduction
ETH Zurich, 21.01.2016 – p.4/42
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
ETH Zurich, 21.01.2016 – p.4/42
Axion inflation
ETH Zurich, 21.01.2016 – p.5/42
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)
ETH Zurich, 21.01.2016 – p.5/42
Axion monodromy inflation
ETH Zurich, 21.01.2016 – p.6/42
Axion monodromy inflation
• 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
ETH Zurich, 21.01.2016 – p.6/42
Axion monodromy inflation
ETH Zurich, 21.01.2016 – p.7/42
Axion monodromy inflation
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.
ETH Zurich, 21.01.2016 – p.7/42
Objective
ETH Zurich, 21.01.2016 – p.8/42
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)
ETH Zurich, 21.01.2016 – p.8/42
Introduction
ETH Zurich, 21.01.2016 – p.9/42
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)
ETH Zurich, 21.01.2016 – p.9/42
Framework
ETH Zurich, 21.01.2016 – p.10/42
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)
ETH Zurich, 21.01.2016 – p.10/42
Framework
ETH Zurich, 21.01.2016 – p.11/42
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.
ETH Zurich, 21.01.2016 – p.11/42
Flux formulation of DFT
ETH Zurich, 21.01.2016 – p.12/42
Flux formulation of DFT
(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′
)
]
ETH Zurich, 21.01.2016 – p.12/42
Dimensional Reduction
ETH Zurich, 21.01.2016 – p.13/42
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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Flux formulation of DFT
ETH Zurich, 21.01.2016 – p.14/42
Flux formulation of DFT
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
ETH Zurich, 21.01.2016 – p.15/42
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
ETH Zurich, 21.01.2016 – p.16/42
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
ETH Zurich, 21.01.2016 – p.17/42
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)
]
.
ETH Zurich, 21.01.2016 – p.17/42
R-R sector
ETH Zurich, 21.01.2016 – p.18/42
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
ETH Zurich, 21.01.2016 – p.19/42
Further Reduction on CY
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 .
ETH Zurich, 21.01.2016 – p.19/42
Further Reduction on CY
ETH Zurich, 21.01.2016 – p.20/42
Further Reduction on CY
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 .
ETH Zurich, 21.01.2016 – p.20/42
Further Reduction on CY
ETH Zurich, 21.01.2016 – p.21/42
Further Reduction on CY
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
)
.
ETH Zurich, 21.01.2016 – p.21/42
Further Reduction on CY
ETH Zurich, 21.01.2016 – p.22/42
Further Reduction on CY
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
Further Reduction on CY
ETH Zurich, 21.01.2016 – p.23/42
Further Reduction on CY
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
ETH Zurich, 21.01.2016 – p.24/42
Orientifold projection
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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Orientifold projection
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Orientifold projection
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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Orientifold projection
ETH Zurich, 21.01.2016 – p.26/42
Orientifold projection
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
ETH Zurich, 21.01.2016 – p.27/42
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
ETH Zurich, 21.01.2016 – p.28/42
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
ETH Zurich, 21.01.2016 – p.29/42
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
ETH Zurich, 21.01.2016 – p.30/42
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.
ETH Zurich, 21.01.2016 – p.30/42
Objective
ETH Zurich, 21.01.2016 – p.31/42
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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ETH Zurich, 21.01.2016 – p.32/42
results could be presented in the discussion session
Thank You!
ETH Zurich, 21.01.2016 – p.32/42
A representative model
ETH Zurich, 21.01.2016 – p.33/42
A representative model
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
ETH Zurich, 21.01.2016 – p.33/42
A representative model
ETH Zurich, 21.01.2016 – p.34/42
A representative model
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
ETH Zurich, 21.01.2016 – p.34/42
Mass scales
ETH Zurich, 21.01.2016 – p.35/42
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
ETH Zurich, 21.01.2016 – p.35/42
Axion inflaton
ETH Zurich, 21.01.2016 – p.36/42
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 .
ETH Zurich, 21.01.2016 – p.36/42
Toy model
ETH Zurich, 21.01.2016 – p.37/42
Toy model
Backreaction of the other moduli adiabatically adjustingduring the slow-roll of θ flattens the potential(Dong,Horn,Silverstein,Westphal)
θ
Vback
ETH Zurich, 21.01.2016 – p.37/42
Effective potential
ETH Zurich, 21.01.2016 – p.38/42
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
ETH Zurich, 21.01.2016 – p.38/42
Tensor-to-scalar ratio
ETH Zurich, 21.01.2016 – p.39/42
Tensor-to-scalar ratio
λ
r
With decreasing λ the model changes from chaotic toStarobinsky-like inflation.
ETH Zurich, 21.01.2016 – p.39/42
Parametric control
ETH Zurich, 21.01.2016 – p.40/42
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.
ETH Zurich, 21.01.2016 – p.40/42
Conclusions
ETH Zurich, 21.01.2016 – p.41/42
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?
ETH Zurich, 21.01.2016 – p.41/42
ETH Zurich, 21.01.2016 – p.42/42
Thank You!
ETH Zurich, 21.01.2016 – p.42/42
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