A Tour of Flux Compactification...
Transcript of 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
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
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
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
Dynamic Flux
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
Review
Problem
Importance
General
Volume
Axion
Postlude
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
3 The Universal Volume Modulus
4 The Universal Axion
5 Postlude: Other Questions and Summary
Dynamic Flux
ARF
Outline
Motivation
General
Compensators
Interpreting
Volume
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Compensators
Interpreting
Volume
Axion
Postlude
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
ARF
Outline
Motivation
General
Compensators
Interpreting
Volume
Axion
Postlude
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
ARF
Outline
Motivation
General
Compensators
Interpreting
Volume
Axion
Postlude
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
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
The Universal Volume Modulus
1 Motivation
2 General Dimensional Reduction in Warping
3 The Universal Volume ModulusLinear SolutionKinetic TermsNonlinear Solution of 10D SUGRA
4 The Universal Axion
5 Postlude: Other Questions and Summary
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
Linear SolutionConstraints & Solutions
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
Linear SolutionConstraints & Solutions
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
Linear SolutionConstraints & Solutions
Example: the KS throat
AdS approximation:
∂cgµν ∼ (r/R)6 vs δgµν ∼ (r/R)2
∂cgrr = 0 vs δgrr ∼ 1
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Linear
Kinetic
Nonlinear
Axion
Postlude
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
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
The Universal Axion
1 Motivation
2 General Dimensional Reduction in Warping
3 The Universal Volume Modulus
4 The Universal AxionLinear SolutionKinetic Terms
5 Postlude: Other Questions and Summary
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Linear
Kinetic
Postlude
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
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Postlude
Kahler
Flux
Future
Summary
Postlude: Other Questions and Summary
1 Motivation
2 General Dimensional Reduction in Warping
3 The Universal Volume Modulus
4 The Universal Axion
5 Postlude: Other Questions and SummaryThe Kahler PotentialModes Stabilized by FluxFuture Directions
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Postlude
Kahler
Flux
Future
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?
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Postlude
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
ARF
Outline
Motivation
General
Volume
Axion
Postlude
Kahler
Flux
Future
Summary
Modes Stabilized by Flux
Dynamic Flux
ARF
Outline
Motivation
General
Volume
Axion
Postlude
Kahler
Flux
Future
Summary
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