a tour of flux compactification...

Dynamic Flux

ARF

Outline

Motivation

General

Volume

Axion

Postlude

A Tour of Flux Compactification Dynamics

Andrew R. Frey

McGill University

Wisconsin 3/3/2009

0810.5768 with Torroba, Underwood, & Douglasand previous/ongoing work

Dynamic Flux

ARF

Outline

Motivation

General

Volume

Axion

Postlude

Outline

1 Motivation

2 General Dimensional Reduction in Warping

3 The Universal Volume Modulus

4 The Universal Axion

5 Postlude: Other Questions and Summary

Dynamic Flux

ARF

Outline

Motivation

Review

Problem

Importance

General

Volume

Axion

Postlude

Motivation

1 MotivationBrief Compactification ReviewThe Problem of DynamicsImportance and Applications





Dynamic Flux

ARF

Outline

Motivation

Review

Problem

Importance

General

Volume

Axion

Postlude

Brief Compactification Review

ds2 = e2A(y)ηµνdxµdxν + e−2A(y)gmndymdyn

Start with CY, orientifold by O3 or O7

Add branes and 3-form flux G3 = −i ?6 G3

Warp factor

∇2e−4A = −eφ

2|G3|2 + · · ·

Throats develop near singularities

Classically Minkowski

Moduli Stabilization

Axio-dilaton & complex structure fixed

In unwarped limit m ∼ α′/R3

No-scale structure

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The Problem of DynamicsGravitons are Simple

Gravitons: universal and pretty simple (RS; Greene, Schalm, & Shiu)

Take ηµν → gµν(x, y)For gµν(x, y) ≡ gµν(x), EOM simply Rµν(g) = 0So clearly identifies massless mode

Massive modes satisfy gµν = hµν(x)Y (y) (see Shiu et al)

∇2Y (y) = e−4Aλ2Y (y)

That’s not so bad

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The Problem of DynamicsTrue Scalars are Simple?

Our example: the axio-dilaton

Kinetic terms are simple

∇2δτ = e−2Aηµν∂µ∂νδτ + e2A∇2δτ

Without stabilization, same eigenvalue problem

But fluxes mix KK modes

ηµν∂µ∂νδτ = −e4A∇2δτ +12e8A |G3|2

We’ll return to this

Other Issues

Warp factor fluctuates with dilaton

Also 3-form EOM get linear component

These hint at our general problems

Dynamic Flux

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Review

Problem

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The Problem of DynamicsTensor Fields are Hard

An illustration: gauge fields (ARF & Polchinski)

On T 6/Z2, Bµm and Cµm massless

But EOM include warp factor (through ?) and F5

Introduces chirality

Gµνm = i(?4G)µνm trivialGµνm = −i(?4G)µνm nonharmonic and not explicit

Also requires F5 fluctuations with fluxes on

Kinetic mixing of KK modes

Metric moduli & axions face similar issues, as we’ll see in detail

Kahler vs Complex Structure

Kahler are massless due to no-scale structure

But C4 axions can feel flux

Complex structure also have flux-induced KK mixing (as τ)

Dynamic Flux

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Review

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Importance and Applications

Controls effective theory through Kahler potential

Major issue in most stringy inflation embeddings

SUSY interactions for phenomenology

Spectrum of cosmological constants when SUSY breaking

Wavefunction itself important for interactions

Determining SUSY breaking couplings

Particularly couplings to brane matter

Higher derivative couplings

KK modes as dark matter candidates, etc

Going beyond 4D theory

SM throats strongly deformed by high-scale inflationMaybe stringy or 10D effects observable (ARF, Mazumdar, & Myers)

Clearing up confusion about no-go theorems(Kodama & Uzawa; Steinhardt & Wesley)

Dynamic Flux

ARF

Outline

Motivation

General

Compensators

Interpreting

Volume

Axion

Postlude

General Dimensional Reduction in Warping

1 Motivation

2 General Dimensional Reduction in WarpingSolving Constraints with CompensatorsInterpreting Compensators




Dynamic Flux

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Compensators

Interpreting

Volume

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Solving Constraints with CompensatorsConstraint Equations

General SUGRA equations are highly coupledUsual CY compactifications highly simplified

Only metric fields, so inhomogeneous terms vanish

Linearized EOM factorize into internal and external parts

Nontrivial backgrounds (including flux and warp factors)

New terms in linearized EOM: for 2-form gauge fields

G3 sources ∼ δGµν ∧6 F5 + G3 ∧6 δFµν

New components of EOM nontrivial: in our example

dδF5 ∼ δG3 ∧G3

No new dynamical information, so constraints

These are endemic to any dynamical fields

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Compensators

Interpreting

Volume

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Solving Constraints with CompensatorsCompensators

To solve constraints, need more components: compensators(ARF & Polchinski; Gray & Lukas; Giddings & Maharana)

Form field fluctuations

Warp factor changes harmonic conditionSimilar to introducing new componentsFlux requires mixing between 3-forms and 5-form

Metric Fluctuations

Warp factor requires components δgµm (Rµm equation)Generally must vary warp factor δAAll this before stabilization

No-compensator Gauge

Warped harmonic (Shiu et al)

∇n

(δgmn −

12gmnδg

)− 4∂nAδgmn = 0

No simple Hodge decomposition

Dynamic Flux

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General

Compensators

Interpreting

Volume

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Interpreting Compensators

Constraints and compensators related to gauge invariance(Douglas & Torroba following Arnowitt, Deser, & Misner)

ds2 = e2Ae2Ω(u)dxµdxµ + 2∂µuηmdxµdym + e−2Agmndymdyn

Constraint equations are really gauge constraintsAs usual Hamiltonian constraint of gravity

Compensators ηm enforce constraints

Defines natural field space metric

Canonical momenta orthogonal to gaugeGive gauge invariant definition of fluctuations

“Shift” compensators to internal or external componentsAlso naturally appear in ADM framework

I’ll display familiar EOM but refer to this framework

Dynamic Flux

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Outline

Motivation

General

Volume

Linear

Kinetic

Nonlinear

Axion

Postlude

The Universal Volume Modulus

1 Motivation


3 The Universal Volume ModulusLinear SolutionKinetic TermsNonlinear Solution of 10D SUGRA



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Linear SolutionWhat It’s Not

Not the usual rescaling gmn → e2u(x)gmn

Due to Laplace equation, e2A(y) → e2u(x)e2A(y)

Internal metric e−2Agmn is invariant

Einstein-frame factor (from action)

e2Ω = VCY /Vw , Vw =∫

d6y√

ge−4A ∝ e2u

External metric e2Ae2Ωηµν is invariant

Comments

Constraints satisfied trivially, but...

This rescaling is pure gauge

Replacing e2Ω = e−6u does not solve constraints

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Linear SolutionConstraints & Solutions

For clarity, call modulus c(x) (ARF, Torroba, Underwood, & Douglas)

As before, Einstein-frame factor

e2Ω = VCY /Vw

Rµm = 0 becomes

∂m∂µe−4A(y;c) ⇒ e−4A(y;c) = e−4A0(y) + c(x)Rµν = 0 implies

ηm = e2A(y;c)e2Ω(c)∂mB(y) , ∇2B = V 0W /VCY − e−4A0

Gauge Constraints

Follow from DMπMN = 0 Hamiltonian constraints

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The metric becomes

ds2 =(e−4A0 + c

)−1/2 (c + V 0

w/VCY

)−1 [ηµνdxµdxν

+2∂µc ∂mBdxµdym] +(e−4A0 + c

)1/2gmndymdyn

gmn does not fluctuate

As expected F5 = (1 + ?10)d(e4Ωe4Ad4x

)Valid to linear order in ∂µc, ∂µ∂νc

Gauge-Invariant Fluctuations

External δgµν = 2e2A+2Ωηµν (δA + ∂cΩ)Internal δgmn = −2∇(m

(e4A+2Ω∂n)B

)Warp factor δA = ∂cA− e4A+2Ω∂mA∂mB

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Example: the KS throat

AdS approximation:

∂cgµν ∼ (r/R)6 vs δgµν ∼ (r/R)2

∂cgrr = 0 vs δgrr ∼ 1

Dynamic Flux

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Kinetic Terms

Plug into action

Get cancellations due to compensator

Gcc =34e4Ω =

34

(c + V 0

w/VCY

)−2

Hamiltonian Inner Product

Gcc =∫

d6y√

ge−4A

[δπmnδπmn − 1

8δπδπ

]Canonical momentum πMN = c δπMN

δπMN = δgMN − gMNδg

Generalizes to show orthogonality

Dynamic Flux

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Nonlinear Solution of 10D SUGRASolution

ds2 = e2Ae2Ωgµνdxµdxν + e−2Agmndymdyn

gµν = gµν(x)− 2(∇µ∂νc + e2Ω∂µc∂νc

)B(y)

gµν a pp wave

Same shifted warp factor and compensator B(y)External Einstein equation also gives 4D Einstein equationIndependent derivation of kinetic term

So far valid for null waves (particle-like)

Forms and Other EOM

F5 and Bianchi as usual

3-form and axio-dilaton EOM still trivially satisfied

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Nonlinear Solution of 10D SUGRAImportance

Highly nontrivial check of EFT from linear theoryNo fudge-factors: form precisely needed for consistency

First 10D background with correct nonlinear dynamicsCorrects confusion about stability of dimensional reduction

May generalize to timelike propagation and cosmologyImportant for correcting no-go theorems

A small step toward looking for 10D corrections in inflation

Dynamic Flux

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Outline

Motivation

General

Volume

Axion

Linear

Kinetic

Postlude

The Universal Axion

1 Motivation



4 The Universal AxionLinear SolutionKinetic Terms


Dynamic Flux

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Motivation

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Volume

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Linear

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Linear SolutionStructure

Multiple components required (ARF, Torroba, Underwood, & Douglas)

δC4 = a0 ∧ ω4 + a2 ∧ ω2 , δA2 = −da0 ∧ Λ1

ω2 = J + dK1, ω4 = ?J + dK3 not harmonic

Both components needed for self-duality

δA2 required for G3 EOM

δF5 = da0 ∧ ω4 + da2 ∧ ω2 +igs

2(δA2 ∧ G3 − δA2 ∧G3

)Gauge Issues

C4 transforms under A2 gauge transformations

Requires field redefinition to find globally defined δC4

Second set of constraints requires second compensator

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Linear SolutionConstraints & Self-Duality

Self-duality requires da2 = e4Ω ?4 da0 and

ω4 −igs

2(Λ1 ∧ G3 − Λ1 ∧G3

)= e−4Ae2Ω?ω2

K1 = e4AdK solves both

Wedging with J and integratingDifferentiating with

8d? (dA ∧ dK) = −de−4A∧ J2−igse−2Ω

(dΛ1 ∧ G3 − dΛ1 ∧G3

)Factors of eΩ requiredSame as constraint from F5 EOM

G3 EOM constraint is

d?dΛ1 = −4ie2Ωe4AdA ∧ dK ∧G3

No stress tensor at linear order

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Kinetic Terms

Plug into action again

CS terms vanish due to index counting

Cancellations from compensators and constraints

Gaa =34e4Ω

Factor of 3 from complex structure normalization

Same as volume modulus

Hamiltonian Inner Product

From 5-form and 3-form electric fields

Generalizes to show orthogonality with other modes

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Kahler

Flux

Future

Summary

Postlude: Other Questions and Summary

1 Motivation




5 Postlude: Other Questions and SummaryThe Kahler PotentialModes Stabilized by FluxFuture Directions

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Kahler

Flux

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Summary

The Kahler Potential

Volume modulus and axion consistent with Kahler potential

ρ = c + ia0 and K = −3 ln(ρ + ρ + 2V 0

w/VCY

)Physically K = −3 ln Vw(c)/VCY

Note the difference from unwarped K = −2 lnV (c)/VCY

Can just shift away constant due to no-scale structure

Include D3 positions by holomorphy 2c = ρ + ρ + k(X, X)What’s the big deal, then?

Important check on consistency of inflationary models

Impacts modulus stabilization

Aeaρ →(AeaV 0

w/VCY

)eaρ

No-scale structure not constraining with more moduli?

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Kahler

Flux

Future

Summary

Modes Stabilized by Flux

The harder problem (ARF & Maharana, hep-th/0603233)

Axio-dilaton in AdS5 throat (modeled on KS) plus bulk

Flux mass term constant vs vanishing in throat

Localizes when warped KK mass < bulk flux massMass “redshifts” as well

When are massive moduli in EFT?

Dynamic Flux

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Kahler

Flux

Future

Summary

Modes Stabilized by Flux

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Kahler

Flux

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Future Directions

Multiple Kahler moduli (work in progress)Physical re-intrepretation of moduli

Timelike (even inflationary) nonlinear solutionsSlow progress toward cosmological backgrounds

KK mode mixing and stabilized moduli

Other types of compactifications

Eventually understanding beyond 4D EFT

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Kahler

Flux

Future

Summary

Summary

1 Motivation





a tour of flux compactification...

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