Flux Compactifications: An Overview

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Flux CompactificationsAn Overview

• Sandip Trivedi
• Tata Institute of Fundamental Research, Mumbai,
  India
• Korea June 2008

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• Introduction Motivation
• A Toy Model
• IIB String Theory with Fluxes
• IIA/I String Theory with Fluxes
• The Landscape And Conclusions

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Flux Compactifications

• Curl Up Extra Dimensions.
• Turn on Fluxes Along These Directions.
• Fluxes are generalisations of Magnetic Flux in
  Maxwell Theory.

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Internal directions
Non-compact directions
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Introduction

• Compactifications without Flux Unsatisfactory.
• Have unwanted flat directions. Called Moduli.
• These are absent in Flux compactifications. With
  interesting consequences.

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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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String Theory Typically lead to run-away
situations. Not stable vacua.
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Introduction
Turning on Fluxes Leads To Controlled
Stabilisation of Moduli. A mimimum which lies in
some region of field space where approximations
are valid.
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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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A Toy Model
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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Toy Model Continued

• Lesson Flux tends to expand the size of
  directions along which it extends.
• Also it tends to contract the size of directions
  in which it does not extend.
• Balancing these leads to moduli stabilisation.

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

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More on 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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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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Size Moduli Stabilisation And Susy
Breaking Kachru, Kallosh, Linde and Trivedi
(KKLT)

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

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This can stabilise the size moduli Giving rise to
a Vacum with negative Cosmological Constant.
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Breaking Susy

• Susy Breaking can be introduced, e.g. due to
  Anti-D3 Branes.
• The resulting vacua can then have a positive
  cosmological constant.

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I)
II)
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Spectrum
String Modes
KK Modes
Shape Moduli
Size Modulus
gravitinio
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Mixed Anomaly Moduli Mediation (Choi, Nilles, et.
Al.)

• The F component of the size modulus

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• The resulting moduli mediated contribution to
  soft masses
• This can be comparable to the anomaly mediated
  contribution
• For

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Flavour Violations Might be Suppressed

• Flavour structure related to shape moduli.
• Susy breaking related to size moduli.
• In this way the origin of flavour and susy
  breaking are naturally segregated, and flavour
  violation in soft susy breaking terms can be
  small.
• (Choi et. Al., Conlon)

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Variations on the Theme

• Use Higher Derivative corrections to stabilise
  Size Moduli.
• Balasubramanium, Conlon, Quevedo.
• Etc

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Type I Theory

• Use Open String Fluxes to stabilise some of the
  moduli.
• In Type I for example Kahler moduli can be
  stabilised in this way.
• Also (on Torus) complex structure moduli.
• (Bacchas, Antoniadis, Maillard, Kumar)

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Type IIA String Theory

• Both Open and Closed String Moduli can be
  stabilised at Tree-Level.
• (Derendinger, Kounnas, Petropoulos, Zwirner
  deWolfe, Giryavets, Kachru, Taylor )
• Fluxes

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Fluxes in IIA String Theory

• Superpotential
• Depends on both size and shape moduli

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Type IIA Continued

• Taking some fluxes to be large the volume can be
  stabilised at a large value, and dilaton at a
  small value.

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IIA With Fluxes

• The Manifolds are not Calabi Yau any more.
• Instead they are half-flat manifolds. Need to be
  understood better.

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Some More Recent Developments

• Use Fluxes To Study Field Theory Models of
  Dynamical Susy Breaking.
• Fluxes result in Geometrizing some aspects.
• (Diaconescu et. al, Kachru et. al, Verlinde et.
  al.)

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Recent Developments

• Most of our knowledge is restricted to when the
  volume is big and warping is small.
• Attempts to go beyond are underway. Compute
  corrections to Kahler potential and
  superpotential (if any).
• (Giddings, Maharana, Douglas et. al.)

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Recent Developments

• Do not start with a Calabi Yau Manifold.
• Instead consider a manifold with negative
  curvature, e.g. Nil Manifold.
• This can lead to simpler constructions of dS
  vaccua.
• (Silverstein).

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Landscape

• Many many different vacuua
• Exponential large number
• Large number arise because starting with a given
  compactification one can turn on many different
  kinds of fluxes.
• Third Betti number

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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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Landscape

• Statistics Much maligned.
• My main worry dont know enough about string
  theory to make reliable estimates.
• An efficient way to zero in on small cosmological
  constant vaccua would be more useful. Dont know
  how to do this yet.

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

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