Progress in Warped String Compactifications Gary Shiu University - - PowerPoint PPT Presentation

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Progress in Warped String Compactifications Gary Shiu University - - PowerPoint PPT Presentation

Jan 12, 2024 233 likes •767 views

Progress in Warped String Compactifications Gary Shiu University of Wisconsin Collaborators D-brane Inflation & Non-Gaussianities in CMB: Bret Underwood, Steven Kecskemeti, John Maiden, Diego Chialva, Xingang Chen, Min-xin Huang,

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Gary Shiu

University of Wisconsin

Progress in Warped String Compactifications

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Collaborators

Bret Underwood, Steven Kecskemeti, John Maiden, Diego Chialva, Xingang Chen, Min-xin Huang, Shamit Kachru Bret Underwood, Devin Walker, Kathryn Zurek

  • D-brane Inflation & Non-Gaussianities in CMB:
  • Warped Throats at the LHC:
  • D3-brane vacua in Stabilized Compactifications:
  • Dynamics of Warped Flux Compactifications:

Michael Douglas, Gonzalo Torroba, Bret Underwood Oliver DeWolfe, Liam McAllister, Bret Underwood

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Collaborators

Bret Underwood, Steven Kecskemeti, John Maiden, Diego Chialva, Xingang Chen, Min-xin Huang, Shamit Kachru Bret Underwood, Devin Walker, Kathryn Zurek

  • D-brane Inflation & Non-Gaussianities in CMB:
  • Warped Throats at the LHC:
  • D3-brane vacua in Stabilized Compactifications:
  • Dynamics of Warped Flux Compactifications:

Michael Douglas, Gonzalo Torroba, Bret Underwood Oliver DeWolfe, Liam McAllister, Bret Underwood

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Warped String Vacua: Open String Sector

A Gentle Landscape

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A Warped Landscape

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Motivation

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Motivation

Stabilizing moduli

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Motivation

Stabilizing moduli String Inflation

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Warped String Inflation

  • Warping invoked in slow-roll & DBI inflation models.
  • CMB Signatures depend strongly on geometries.

Ne η Ntot

GS, Underwood Baumann, Dymarsky, Klebanov, McAllister, Steinhardt KS AdS Mass Gap

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Non-Gaussianities

Chen, Huang, Kachru, GS [See also: Cheung, Cremenini, Fitzpatrick, Kaplan, Senatore]

fNL ∼ O(γ2)

k3 k2

ζk1ζk2ζk3 = (2π)3δ3(k1 + k2 + k3)F(k1, k2, k3)

  • Models with large non-Gaussianities & distinctive shape.
  • Exact numerical factors and shape for general single

field inflation computed.

Alishahiha, Silverstein, Tong

F(1, k2, k3)k2

2k2 3

fNL ∼ O()

Slow-roll DBI

[See talks of Shandera, Leblond, ...]

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Motivation

RS-like hierarchy Stabilizing moduli Inflation

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Hierarchy from Warping

KS AdS

Giddings, Kachru, Polchinski; KKLT; Dasgupta, Rajesh, Sethi; ...

Scale of inflation, electroweak scale, ...

GS, Underwood, Walker, Zurek

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Motivation

RS-like hierarchy Stabilizing moduli String Inflation

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Motivation

RS-like hierarchy Local model: explicit metric Gauge/gravity correspondence Stabilizing moduli String Inflation Cosmic strings, reheating, ... Scale of SUSY mediation scenario, sequestering, ...

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Warped Effective Theory

Douglas, Shelton,Torroba

  • Finding vacua & fluctuations around
  • Flatness of inflaton potential
  • Computation of soft SUSY terms
  • ...

Require warping corrections to N=1, D=4 SUGRA

0.5 1 1.5 2 S 0.05 0.1 0.15 0.2 V

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  • 1. Open String Sector
  • 2. Closed String Sector

Warped String Vacua

DeWolfe, McAllister, GS, Underwood GS, Torroba, Underwood, Douglas

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  • 1. Open String Sector
  • 2. Closed String Sector

Warped String Vacua

DeWolfe, McAllister, GS, Underwood GS, Torroba, Underwood, Douglas

(STUD)

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Warped String Vacua: Open String Sector

DeWolfe, McAllister, GS, Underwood

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D-BRANE MODULI

D-brane models of particle physics F-theory/open string landscape End of brane-inflation, reheating Multi-field effects in brane inflation ....

Talks of Vafa, Verlinde, Wijnholt, ...

Barnaby, Burgess, Cline; Kofman, Yi; Chialva, GS, Underwood; Frey, Mazumdar, Myers; Chen,Tye; ...

Huang, GS, Underwood; Easson et al; Shandera, Leblond; ... Gomis, Marchesano, Mateos; Collinucci, Denef, Esole; ...

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Multifield Effects of D-brane Inflation

Huang, GS, Underwood; [See also Easson et al]

D3-brane has angular coordinates: multi-field inflation Multi-field DBI inflation: interesting and significant effects on non-Gaussianities Transfer function:

TRS

depends on sharpness of turn, (weakly broken) isometry directions, ...

Entropy perturbations Curvature perturbations Langlois, Renaux-Petel, Steer, Tanaka

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Type IIB Flux Compactifications

Metric is warped product of 4d with (conformally) CY Fluxes Instantons

D7 or Euclidean D3 Generate warping & stabilize complex structure moduli. Stabilize Kahler moduli.

GKP; Dasgupta, Rajesh, Sethi, ... KKLT, ...

Stabilize D7 positions. Stabilize D3 positions.

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Warped Deformed Conifold

Geometry of KS throat:

  • Approximately

AdS5xT1,1 far from tip

  • Topologically
  • Finite size S3 at tip
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Warped Deformed Conifold

Before KKLT effects, D3 moduli space = M6

Geometry of KS throat:

  • Approximately

AdS5xT1,1 far from tip

  • Topologically
  • Finite size S3 at tip
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D3 Moduli Stabilization

D7 branes wrapped on Σ4

Backreaction of D3 on V ol(Σ4)

D3 pushed to the tip by D7-brane

Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan; [See also Berg, Haack, Kors; Giddings, Maharana; Ganor]

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D3 Moduli Stabilization

D7 branes wrapped on Σ4

Backreaction of D3 on V ol(Σ4)

D3 pushed to the tip by D7-brane D3 vacua depend on 4-cycle embeddings

Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan; [See also Berg, Haack, Kors; Giddings, Maharana; Ganor]

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D3 Moduli Stabilization

D7 branes wrapped on Σ4

Backreaction of D3 on V ol(Σ4)

D3 pushed to the tip by D7-brane D3 vacua depend on 4-cycle embeddings

Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan; [See also Berg, Haack, Kors; Giddings, Maharana; Ganor]

break throat isometries

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Wrapped Branes in Throats

Deformed conifold coordinates: Embedding of D7 defined by a holomorphic function:

  • ACR or Ouyang-type

Arean, Crooks, Ramallo Karch, Katz

  • P. Ouyang
  • Kuperstein-type
  • S. Kuperstein
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SUSY Flavor Branes

H.-Y. Chen, Ouyang, GS (in progress)

ˆ F = ˆ B2 + 2πα′F ,

Flavor branes defined by holomorphic functions:

ˆ F2,0 = ˆ F0,2 = 0 .

e−A ˆ J ∧ ˆ F = tan θ e2A 2 ˆ J ∧ ˆ J − 1 2 ˆ F ∧ ˆ F

  • f(z) = 0 or f(w) = 0

κ symmetry requires further that:

Marino, Minasian, Moore, Strominger

  • nly been checked for the Kuperstein embedding.

Naively not satisfied for other embeddings, but field theory dual has SUSY moduli space in IR. Explicit construction of F

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D3 Vacua in KS Throat

Solve F-term equations:

D3 generically stabilized at pts

DeWolfe, McAllister, GS, Underwood

Using the DG Kahler potential (later):

ρ K = −3 log e4u = −3 log(ρ + ¯ ρ − γk(Y, ¯ Y )/3)

D3 vacua on S3: and above.

Wnp

pushes away from D7

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D3 vacua in KS Throat

General ACR-type:

Preserves

Ouyang-type:

No SUSY D3 vacua!

Kuperstein-type:

D3 stabilized at points

4

  • A=1
  • zA2 = −2 (w1w2 − w3w4) = 2

Deformed conifold coordinates:

DeWolfe, McAllister, GS, Underwood

Isometry further broken weakly by bulk effects.

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Warped String Vacua: Closed String Sector

GS, Torroba, Underwood, Douglas (STUD)

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Issues with Strong Warping

D=10 String Theory D=10 SUGRA with fluxes D=4 N=1 SUGRA EFT

Ex: GKP and KKLT

Type IIB String Theory in D=10 Low Energy Low Energy

KK Dimensional Reduction

String vacua, inflation, de-Sitter, MSSM…

IIB Supergravity in D=10

KK Dimensional Reduction

N=1 SUGRA in D=4

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Issues with Strong Warping

D=10 String Theory D=10 SUGRA with fluxes D=4 N=1 SUGRA EFT

Low Energy

KK Dimensional Reduction

String vacua, inflation, de-Sitter, MSSM… Many subtleties with warped KK reduction:

  • General KK ansatz (compensators)
  • Mixing/sourcing of KK modes with moduli
  • Backreaction of moduli on warp factor
  • 10D Gauge redundancies
  • 10D Constraint equations

In warped backgrounds these issues are all highly coupled to each other!

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KK Scale in Warped Background

KK modes Moduli Unwarped

m2

z ∼ 1

α

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KK Scale in Warped Background

KK modes Moduli Unwarped Strong warping

DeWolfe, Giddings; Giddings, Maharana; Frey, Maharana; Burgess, Camara, de Alwis, Giddings, Maharana, Quevedo, Suruliz; ...

m2

z ∼ 1

α

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KK Scale in Warped Background

KK modes Moduli Unwarped Strong warping

Fields localize to region of strong warping.

DeWolfe, Giddings; Giddings, Maharana; Frey, Maharana; Burgess, Camara, de Alwis, Giddings, Maharana, Quevedo, Suruliz; ...

m2

z ∼ 1

α

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KK Scale in Warped Background

KK modes Moduli Unwarped Strong warping

Fields localize to region of strong warping.

DeWolfe, Giddings; Giddings, Maharana; Frey, Maharana; Burgess, Camara, de Alwis, Giddings, Maharana, Quevedo, Suruliz; ...

Masses redshifted

m2

z ∼ 1

α

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KK Scale in Warped Background

KK modes Moduli Unwarped Strong warping

Fields localize to region of strong warping.

DeWolfe, Giddings; Giddings, Maharana; Frey, Maharana; Burgess, Camara, de Alwis, Giddings, Maharana, Quevedo, Suruliz; ...

Masses redshifted No mass hierarchy between moduli and KK modes for integrating out heavy fields.

m2

z ∼ 1

α

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Towards a Warped EFT

Previous proposals of 4D Warped EFT:

DeWolfe, Giddings

did not account for these issues. Ansatz for fluctuations: ... does not solve 10D EOM!

(DeWolfe, Giddings) Giddings, Maharana; STUD

More general ansatz does, but extremely messy ...

Giddings, Maharana

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Compensators

ds2

10 → ds2 10 + 2∂µ∂νSαe2AKα(y)dxµdxν + 2e2ABαm(y)∂µSαdxµdym .

Promoting moduli from parameters to 4D fields:

δB2 = Sα(x)δαB2 + dSα(x) ∧ Rα δC2 = Sα(x)δαC2 + dSα(x) ∧ Tα .

metric compensators Similarly, flux compensators: Inclusion of compensators makes the EOM hopelessly difficult to solve ...

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Linearized Einstein Equations

δGµ

ν =δµ ν uIδI

  • e2A
  • −2 ˜

∇2A + 4( ∇A)2 − 1 2 ˜ R

  • + e−2A

∂µ∂νuI − δµ

ν

uI (4δIA − 1 2δI˜ g) +

  • ∂µ∂νuI − δµ

ν

uI e2A ˜ ∇p(BIp − ∂pKI) + e−2Af KδKG(4)µ

ν

− 1 2

  • δKgµ

ν − δµ ν δKgλ λ

  • e2A ˜

∇2f K , (A.14) δGµ

m = δRµ m =e−2A∂µuI

  • 2∂mδIA − 8∂mAδIA − 1

2∂mδI˜ g + ∂mAδI˜ g − 2∂ ˜

pAδI˜

gmp + 1 2 ˜ ∇pδI˜ gmp − 1 2 ˜ ∇p e4A ˜ ∇pBIm − ˜ ∇mBIp

  • + 2(∂mABIp − ∂pABIm) ˜

∇pe4A + 1 2e8ABIm ˜ ∇2e−4A − e4A ˜ Rn

mBIn

  • ,

(A.15) δGm

n =uIδI

  • e2A

˜ Gm

n + 4(

∇A)2δm

n − 8∇nA ˜

∇mA

  • − 1

2e−2A uI˜ gmkδI˜ gkn + δm

n e−2A uI(−2δIA + 1

2δI˜ g) uI

  • 1

2e−2A ˜ ∇m e4A (BIn − ∂nKI)

  • + ˜

∇n

  • e4A

B ˜

m I − ∂ ˜ mKI

  • − δm

n ˜

∇p e2A (BIp − ∂pKI)

  • + 1

2δKgµ

µ

  • −1

2e−2A ˜ ∇m e4A∂nf K + ˜ ∇n

  • e4A∂ ˜

mf K

+ δm

n ˜

∇p e2A∂pf K − 1 2δm

n f Ke−2AδKR(4) .

(A.16) δT µ

ν = −δµ ν

1 4κ2

10

  • uIδI
  • e−6A(

∇α)2 − 2e−6A uISIm∂ ˜

mα − 2 uIKIe−6A(

∇α)2 , (A.37) δT µ

m =

1 2κ2

10

∂µuIe−6A [∂mSIp − ∂pSIm + ∂mαBIp − ∂pαBIm] ∂ ˜

pα ,

(A.38) δT m

n = −

1 2κ2

10

uIδI

  • e−6A
  • ∂nα∂ ˜
mα − 1

2δm

n (

∇α)2

  • + e−6A

2κ2

10

uI

  • SIn∂ ˜
mα + ∂nαS ˜ m I − δm n SIp∂ ˜ pα + 2KI
  • ∂nα∂ ˜
mα − 1

2δm

n (

∇α)2

  • .

(A.39)

Giddings, Maharana

δGM

N = κ2 10δT M N

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Dimensional Reduction: Ansatz

  • Gauge redundancies: Remove pure gauge d.o.f
  • Constraint equations: Use 10-d relations for d.o.f.

General analysis too messy, but can simplify greatly using: Analogous to Gauss’s Law in E&M:

∂i∂iA0 − ∂i∂0Ai = 0

  • r in GR:

No second order time derivatives, satisfied by any consistent solution at all time.

G0i = T0i

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Dimensional Reduction:

Constraint Equations & Gauge Redundancies

  • “Axial” Gauge:
  • Constraint equations:

Similarly, a set of gauge choices and constraint equations for flux sector.

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Dimensional Reduction:

Constraint Equations & Gauge Redundancies

  • “Axial” Gauge:
  • Constraint equations:

Unwarped background: Warped background:

Transverse-traceless metric fluctuations are inconsistent with equations of motion Use constraint equations to simplify effective action.

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Dimensional Reduction: Ansatz

  • Gauge redundancies: Remove pure gauge d.o.f
  • Constraint equations: Use 10-d relations for d.o.f.

General analysis too messy, but can simplify greatly using: Derive field space metric for moduli (except universal Kahler), including KK modes. KK mass terms, flux induced moduli/KK masses.

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Dimensional Reduction:

Field Space Metric

Dimension reduction of gravity action: Field space metric: Not 6D diffeomorphism invariant Because of the gauge choice and constraint equations, we have chosen a unique representative.

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Dimensional Reduction:

Field Space Metric

Diagonal in indices: no KK-moduli kinetic mixing! (KK orthogonality relation) Extra contributions due to warping!

General structure and explicit expression for the conifold will be discussed in Torroba’s talk.

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Dimensional Reduction:

Effective Theory

Kahler potential remains diagonal in moduli, contains warping and KK mode corrections Flux-induced masses contain mixing between moduli & KK modes!

KK corrections to fluxed induced superpotential.

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Warped EFT: Summary

Many subtle issues need to be taken into account for strong warping - all important and coupled. Calculate warping and KK corrections to 4D EFT, Kahler potential differs from previous proposals. Future direction (in progress): universal Kahler modulus in strong warping. Important for many phenomenological & cosmological applications.

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Summary

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Summary

Warped backgrounds phenomenologically interesting

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Summary

Warped backgrounds phenomenologically interesting We need better understanding of warped vacua/effective theory!