Flux Compactifications:
An Overview
Sandip Trivedi
Tata Institute of Fundamental
Research, Mumbai, India
Korea June 2008
• Introduction & Motivation
• A Toy Model
• IIB String Theory with Fluxes
• IIA/I String Theory with Fluxes
• The Landscape And Conclusions
Flux Compactifications
•Curl Up Extra Dimensions.
•Turn on Fluxes Along These
Directions.
•Fluxes are generalisations of
Magnetic Flux in Maxwell Theory.
Internal
directions
Non-compact
directions
Introduction
•Compactifications without Flux:
Unsatisfactory.
•Have unwanted flat directions.
Called Moduli.
•These are absent in Flux
compactifications. With interesting
consequences.
Moduli Stabilisation
•Typically Many Flat Directions in String
Compactifications. (~100)
•Different Sizes and Shapes.
V ( )

Physical Parameters e.g., G_N, alpha, vary along these directions
String Theory:
Typically lead to run-away situations. Not
stable vacua.
Introduction
Turning on Fluxes: Leads To Controlled
Stabilisation of Moduli.
A mimimum which lies in some region of
field space where approximations are valid.
Flux Compactifications
Important In Phenomenology:
a) Calculate Standard Model Couplings
b) Supersymmetry Breaking
Important In Cosmology
Positive Vacuum Energy: DeSitter Universe
Slowly Varying Potential: Inflation
Flux Compactifications
Another Advantage:
• Concentrated Flux gives rise to large
Warping.
• Natural way to constructed models of
Randall Sundrum (or large extra
dimension) type.
A Toy Model
Why Does Flux Help?
S   g Rd x
6
Any Value of R1,R2 Allowed: Moduli
R2
R1
Torus Is Flat, Curvature Vanishes.
Flux Compactifications
Size Modulus:
Shape Modulus:
Turn On Magnetic Field
1
g ( R  F F  )d 6 x
4
S
R2
A=R1 R2
R1
F12 A  N
Dirac Quantisation :
F12 
Extra Cost In Energy:
2
E
A
2
N
A
A
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.
Type IIB String Theory:
Promising Corner to Begin
Giddings, Kachru, Polchinski
Fluxes: Three-Forms:
Five-Form:
Branes: D3 (fill 3+1 dimensions), D7,
03,07.
Type IIB String Theory:
Promising Corner to Begin
Giddings, Kachru, Polchinski
Fluxes: Three-Forms:
Five-Form:
Branes: D3 (fill 3+1 dimensions), D7,
03,07.
(N0 5-Branes/Planes)
Type IIB String Theory:
must be closed.
•Such closed and non-trivial fluxes lie
in a vector space. It’s dimensionality is
a topological invariant, .
•Fluxes are also quantised.
More On Fluxes
Total Number of allowed Fluxes:
Exponential in
is finite, determined by tadpole
condition:
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.
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.
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.
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
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.
This can stabilise the size moduli
Giving rise to a Vacum with
negative Cosmological Constant.
Breaking Susy
•Susy Breaking can be introduced,
e.g. due to Anti-D3 Branes.
•The resulting vacua can then have
a positive cosmological constant.
I)
II)
Spectrum:
String Modes
KK Modes
Shape Moduli
Size Modulus
gravitinio
Mixed Anomaly Moduli Mediation
(Choi, Nilles, et. Al.)
•The F component of the size
modulus:
•The resulting moduli mediated
contribution to soft masses:
•This can be comparable to the anomaly
mediated contribution
•For
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)
Variations on the Theme
•Use Higher Derivative corrections to
stabilise Size Moduli.
•Balasubramanium, Conlon, Quevedo.
•Etc
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…)
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:
Fluxes in IIA String Theory
•Superpotential:
•Depends on both size and shape
moduli
Type IIA Continued
•Taking some fluxes to be large the
volume can be stabilised at a large
value, and dilaton at a small value.
IIA With Fluxes
•The Manifolds are not Calabi Yau
any more.
•Instead they are half-flat manifolds.
Need to be understood better.
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.)
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.)
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).
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
Landscape
Bousso, Polchinski
Susskind
Many different vacua.
Many different directions
Varying cosmological constants.
Transitions between them are possible.
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?
Landscape:
My Views:
•Anthropics: Should be the last resort.
Conventional explanations have testable
consequences.
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 modellike vacua. We don’t know enough about the
theory to answer this yet.
•Also, understanding time, the initial
singularity etc might help.
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.
Landscape
•Statistics: Much maligned.
•My main worry : don’t know enough
about string theory to make reliable
estimates.
•An efficient way to zero in on small
cosmological constant vaccua would
be more useful. Don’t know how to
do this yet.
Ashok, Douglas
The number distribution of vacua for a
small cosmological constant is flat: