Flux Compactifications Accelerating Universes And All That

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Flux Compactifications Accelerating Universes
And All That

• Sandip Trivedi
• Tata Institute of Fundamental Research, Mumbai,
  India
• Goa Jan. 07

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• Introduction Motivation
• Type IIB Strings with Fluxes
• Supersymmetry Breaking and de Sitter Universe.
• The Landscape and Conclusions

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Some References

• Flux Compactifications
• Giddings Kachru, Polchinski, hep-th/0105097
• Douglas and Kachru, hep-th/0610102
• Kachru, Kallosh, Linde, S.P.T., hep-th/0301240

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Some References

• The Landscape
• Bousso and Polchinski, hepth/0004134
• Susskind hep-th/0302219
• Susskind The Cosmic Landscape

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Introduction

• Flux Compactifications
• String Theory Requires 10 dimensions.
• Extra 6 dimensions are small. curled up.
• Many Possibilities, e.g. Calabi Yau manifolds.
• Any particular Calabi Yau manifold can take many
  different sizes and shapes.

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Introduction

• These are examples of Moduli
• O(100) Moduli is common.
• In a Field Theory description these are Flat
  directions.

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Moduli Stabilisation

• Typically Many Flat Directions in String
  Compactifications. (100)
• Different Sizes and Shapes.

Physical Parameters e.g., G_N, alpha, vary along
these directions
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Moduli and their Problems

• These flat directions are a problem.
• Typically they will be lifted once supersymmetry
  is broken.
• Cosmology
• However if mass is small they will overclose the
  universe (Mgt10 Tev)

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String Theory Typically lead to run-away
situations. Not stable vacua.
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Flux Compactifications
Main Advantage Flux gives rise to controlled
ways of stabilising the Moduli and Breaking
Supersymmetry.
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Flux Compactifications

• Important In Phenomenology
• Calculate Standard Model Couplings
• b) Supersymmetry Breaking

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• Important In Cosmology

Positive Vacuum Energy DeSitter Universe
Slowly Varying Potential Inflation
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Flux Compactifications

• Another Advantage
• Concentrated Flux gives rise to large Warping.
• Natural way to constructed models of Randall
  Sundrum (or large extra dimension) type.

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6 Dim Calabi- Yau
Warped region with throat
Brane or String weighs less inside due to
gravitational red-shift
Warp Factor
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Flux Compactifications

• Curl Up Extra Dimensions while turning on Flux in
  these directions.
• Flux generalisation of magnetic flux (including
  higher forms F_2,F_3,F_4, )
• Gives extra potential energy depending on size
  and shape of compactification

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Internal directions
Non-compact directions
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FLUX COMPACTIFICATION
Why Does Flux Help?
Any Value of R1,R2 Allowed Moduli
R2
R1
Torus Is Flat, Curvature Vanishes.
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Flux Compactifications
Size Modulus Shape Modulus
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Turn On Magnetic Field
R2
AR1 R2
R1
Dirac Quantisation
Extra Cost In Energy
E
A
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Type IIB String Theory Promising Corner to
Begin Giddings, Kachru, Polchinski Fluxes
Three-Forms Five-Form Branes D3
(fill 31 dimensions), D7, 03,07. (N0
5-Branes/Planes)
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• Type IIB String Theory
• must be closed.
• Such closed and non-trivial fluxes lie in a
  vector space. Its dimensionality is a
  topological invariant, .
• Fluxes are also quantised.

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Fluxes
Total Number of allowed Fluxes Exponential in
is finite, determined by tadpole condition

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Fluxes

• For reasonably big the total number of
  allowed fluxes can be very large.
• is quite common.
• This gives rise to an exponentially large number
  of vacua.

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• Type IIB String Theory
• Take String Coupling to be small,
• Work in Supergravity approximation. To be valid,
  the curvature should be small compared to .
  Also volume should be big compared to .

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Basic Strategy

• We will assume these approximations are valid and
  self-consistently find vacua where the required
  conditions are met.

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Basic Strategy

• Two Steps
• First find Susy preserving vacua with all moduli
  stabilised.
• Next break supersymmetry and find
• (metastable) vacua.

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I)
II)
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• I) Moduli Stabilisation With Susy
• Internal Space E.g. Calabi Yau with
  identifications (Orientifold 3 and 7 planes).
• Flux gives rise to a relatively minor
  modification of the internal space a warp
  factor.

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More on Moduli Stabilisation

• The moduli of interest are size and shape
  deformations of the Calabi-Yau space.
• These get a mass, ,
• where, , is the Radius of
  compactification.
• Thus the lifting of these moduli can be studied
  in a 4 dim. Effective field theory.

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Shape Moduli Stabilisation

• A superpotential arises at tree-level.
• This depends on the shape moduli and the
  axion-dilaton.
• Generically this fixes all these moduli.

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Shape Moduli Stabilisation
And is the holomorphic-three form on the
Calabi Yau, which depends on the shape moduli.
Gukov, Vafa, Witten Giddings, Kachru, Polchinski
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Spectrum
String Modes
KK Modes
1/R
Shape Moduli
Size Moduli
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Size Moduli Stabilisation

• Non-perturbative Corrections to Superpotential
  can also arise.
• These are dependent on Size moduli and can
  stabilise them.

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Size Moduli Stabilisation

• Two possibilities
• Euclidean D3 Branes wrapping four-cycles of
  appropriate topology (ED3 action depends on size
  of cycle)
• b) Non-perturbative gauge dynamics on coincident
  D7 branes

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Size Modulus Stabilisation

• Euclidean D3 Branes must wrap divisor whose
  Arithmetic genus is unity.
  Witten.
• For D7-Branes wrapping divisors this condition
  need not be met. The Fluxes can lift additional
  zero modes.
• Goerlich, Kachru, Tripathy, Trivedi

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Example One Size Modulus
Shape moduli get a treel level mass and are
heavier. After Integrating them out resulting
effective theory for the size modulus is
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For small we can get a large
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if W_0 ltlt1. With 10(100) vacua this leaves many
possibilities.
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Spectrum
String Modes
KK Modes
1/R
Shape Moduli
Size Modulus
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• II) Supersymmetry Breaking
• Add anti-D3 brane
• Warped geometry allows scale of susy breaking to
  be small

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6 Dim Calabi- Yau
AdS_5 like region Which terminates
Anti D3 Brane at Bottom
Turn on Flux along a cycle of small size. This
concentrates the flux and gives rise to
significant warping.
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Klebanov Strassler
6 Dim Calabi- Yau
AdS_5 like region Which terminates
Anti D3 Brane at Bottom
Warp factor at the Bottom
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Supersymmetry Breaking and Uplifting
The anti D3 brane leads to an extra term

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• Thus we see that all moduli are finally
  stabilised and supersymmetry broken.
• And this allows us to obtain desitter vacua in
  string theory.

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More on deSitter Vacua
Prior to this construction there were some No-go
Theorems which said that deSitter Universes could
not be arise. The obstruction was really that
moduli had not been stabilised and supersymmetry
was being broken, leading to run-away behaviour
and not deSitter vacua.
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Spectrum
String Modes
KK Modes
Shape Moduli
Size Modulus
gravitinio
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• By varying the relative strength of the anti-D3
  brane contribution and we can get
  either a negative or positive cosmological
  constant.
• To get a small cosmological constant these would
  have to very precisely cancel. One would pay a
  price, but since there are many-many vacua to
  begin with this is still possible.

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Ashok, Douglas
The number distribution of vacua for a small
cosmological constant is flat

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The deSitter vacua are metastable. They can decay
by Coleman-Deluccia instantons or Hawking-Moss
instantons. Resultant lifetime
Easily bigger than years Smaller
than Poincare recurrence time
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Bousso, Polchinski Susskind
Landscape
Many different vacua. Many different directions
Varying cosmological constants. Transitions
between them are possible.
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Landscape

• Many Questions
• Is String Theory Predictive?
• Who ordered all the other vacua?
• How do we find the Standard Model vacuum?
• Should we give up on Naturalness?
• The Anthropic Principle?

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Landscape

• My Views
• Anthropics Should be the last resort.
  Conventional explanations have testable
  consequences.

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Landscape

• Too early to conclude that string theory not
  predictive. By inputing some data(
  ) we might be able to predict a lot.
• Key Question In coupling constant space how
  closely spaced are the standard model-like vacua.
  We dont know enough about the theory to answer
  this yet.
• Also, understanding time, the initial singularity
  etc might help.

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Landscape

• What is clear though is that at our present
  level of understanding, String Theory is more
  akin to a general framework than a specific UV
  completion of the standard model.
• So we should use it as a framework for model
  building and for understanding gauge theories.
• This might well be its best use as we lead up to
  the LHC.

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Conclusions
Fluxes can help stablise all moduli. Resulting
Compactifications interesting for
phenomenology. And for Cosmology. Landscape here
to say.