Gravity on an extended brane in 6D warped flux compactifications

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Gravity on an extended brane in6D warped flux
compactifications
Recent developments in branes and cosmology _at_
APC

• Tsutomu Kobayashi (Tokyo Inst. Tech.)
• with M. Minamitsuji (Arnold Sommerfeld Center for
  Theoretical Physics)

Based on hep-th/0703029
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This work studies

• Braneworld models
• with an internal 2-dimensional space
• compactified by a flux
• My main interests
• Behavior of gravity on a brane, cosmology, black
  holes,

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This work studies

• Braneworld models
• with an internal 2-dimensional space
• compactified by a flux
• My main interests are
• Construction of models
• Behavior of gravity on a brane

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This work studies

• Braneworld models
• with an internal 2-dimensional space
• compactified by a flux
• My main interests are
• Construction of models
• Behavior of gravity on a brane

• Warped flux compactification
• Simple setup to study aspects of braneworld
  models in string theory, in which moduli are
  stabilized by fluxes
• Toy model in 6D Einstein-Maxwell system

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Evading the problem in codimension 2 defects

• 6D bulk with extend branes
• One dimension compactified on KK circle
• Arbitrary energy-momentum tensor on a brane
• Closely related to Peloso et al. hep-th/0603026
  and Papantonopoulos et al. hep-th/0611311
• Other ways out Charmousis and Zegers (2005)
  Kaloper (2004)
• Various massless scalar modes potentially
  contribute to gravity on a brane
• Can 4D Einstein gravity be reproduced on a brane?

c.f. Only pure tension can be accommodated on
codimension 2 branes in Einstein gravity
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Background model
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Background model

• Action (Einstein-Maxwell and Branes)
• where
• Bulk solution

Mukohyama et al. Peloso et al. Papantonopoulos et
al.
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Derivation of the background solution

• double Wick rotated RN-dS solution

Suitable redefinition of coordinates
Period
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Quantization condition for flux

• Gauge field
• They are related via gauge transformation
• phase of brane Higgs field

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Background model
Peloso et al. Papantonopoulos et al.

• Regular bulk with no deficit angle at poles
• Put branes at
• (1) Continuity of induced metric
• (2) Israel conditions
• (3) Maxwell junction conditions
• EOM for
• (1) Continuity of induced metric

integer
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Junction conditions

• (2) Israel conditions
• Energy-momentum on brane
• (3) Maxwell junction conditions

where
and Brane position
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• Single brane model
• Two-brane model

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Comparison with previous study

• In Peloso et al. hep-th/0603026
• No warping
• Z2 symmetry across the equator
• Some of the perturbation modes are projected out
  by Z2 parity consideration
• This does not happen in the warped model of
  Papantonopoulos et al. hep-th/0611311

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Linear Perturbations
We focus on axially symmetric perturbations
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Perturbations in an arbitrary gauge

• Perturbations are split into scalar, vector, and
  tensor modes under the Lorentz group in the
  external Minkowski spacetime
• Metric
• Gauge field
• Field strength

Vector
Tensor
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Tensor perturbations

• Tensor mode is gauge-invariant
• EOM
• Zero-mode solution
• The regularity at the poles and the source-free
  Israel conditions require
• both for single brane and two-brane models

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Vector perturbations

• 3 gauge invariant variables
• Einstein eqs.
• Maxwell eq.
• Israel conditions
• Other boundary conditions show that
  anddo not couple to matter on the brane, and so
  we ignore these modes

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Vector perturbations

• Only the zero mode is present
• General solution in the bulk
• In the absence of sources, the regularity at the
  poles and the Israel condition require
• both for single brane and two-brane models

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Scalar perturbations gauge choice

• Analog of longitudinal gauge
• Suitable for solving bulk
• Brane displacement (bending) should be taken into
  account

Metric
Gauge field
where
Garriga and Tanaka
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Another choice of gauge

• Gaussian-normal gauge is suitable for imposing
  boundary conditions at a brane
• Gauge transformation between the two gauges
• Perturbations of induced metric

etc.
etc.
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Bulk equations

• Einstein eqs.
• Maxwell eqs.

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Bulk master equations

• sector simply reduces
  to
• These 4 variables can be obtained from

Yoshiguchi et al.
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C and Fw modes

• Boundary conditions require that these modes
  vanish everywhere in the bulk both for the single
  brane and two-brane models

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Boundary conditions (not related to brane matter)

• Regularity at the poles
• Continuity of induced metric
• Maxwell junction condition

2 for each pole
Sendouda et al. Yoshiguchi et al.
3 for each brane

etc.
1 for each brane
where
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Boundary conditions (related to brane matter)

• Israel conditions
• Trace of equation

2 for each brane
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Zero-mode solution

• Analytic solution for zero mode
• Single brane model
• 8 integration constants 2 brane bending scalars
  10 to be determined
• of B.C. (independent of matter source) 22
  (poles) 4 (brane) 8
• ? remaining 2 scalar d.o.f.
• Two-brane model
• 12 integration constants 4 brane bending
  scalars 16 to be determined
• of B.C. 22 (poles) 42 (branes) 12
• ? remaining 4 scalar d.o.f.

4 integration constants for each part of bulk
2 (at each brane)
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Gravity on a brane
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Combining bulk and junction equations

• Israel condition
• Bulk eq. and B.C. are combined into a single eq.
  with source
• This can be solved using a retarded Green function

where
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Green function

• Green function is given by
• where are a complete set of
  eigenfunctions of
• normalized as
• We are interested in zero mode
• where

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Zero-mode truncation

• .
• where
• 4D Ricci tensor for the induced metric
• where

Garriga and Tanaka
matter
brane bending
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Recovery of 4D tensor structure

• Brane bending is crucial, as in Garriga and
  Tanaka
• Gravitational coupling at each brane

4D energy-momentum tensor
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Correction from scalar mode

• mode implies a scalar-tensor theory of
  gravity
• Is the correction from this mode suppressed?
• Key quantities
• A combination of energy-momentum tensor
• Perturbed volume of internal space
• Perturbed circumference of each brane

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Single brane model

• Solving a system of linear equations ( boundary
  conditions) shows
• Scalar mode is suppressed at long distances
• This scalar mode controls the volume of
  theinternal space and the circumference of the
  brane

decouples
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Two-brane model

• Situation is similar the effect of the scalar
  mode is suppressed
• Perturbations of volume and circumferences

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Summary

• Recovery of 4D Einstein gravity on a extended
  brane in 6D warped flux compactifications
• Effect of KK modes
• Cosmology
• Analysis in supergravity (Salam-Sezgin model)

Outlook