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Lecture 2

Stabilization of moduli in superstring

theory

FluxesNS (perturbative string theory)

RR (non-perturbative string theory)

Non-perturbative corrections to

superpotential and Kahler potential

Higgs? MSSM?Higgs? MSSM?

SplitSplit susysusy??

NoNo susysusy??

Start 2008Start 2008

IF SUSY IS THEREIF SUSY IS THERE……The significance of discovery of supersymmetry in nature,

which will manifests itself via existence of supersymmetric particles,

would be a discovery of the fermionicfermionic dimensions of dimensions of

spacetimespacetimeIt will be the most fundamental discovery in It will be the most fundamental discovery in

physics after Einsteinphysics after Einstein’’s relativitys relativity

SUPERSYMMETRY:

GravityGravity Supergravity/StringSupergravity/String

theorytheory

There are no single scalars fields in

susy models!

Scalars come in pairs, they are always complex in supersymmetry. For example, axion and dilaton or axion and radial modulus. Generically, multi-dimensional modulispace.

An effective inflaton may be a particular direction in moduli space.

In string theory scalars often have geometrical meaning: distance between branes, size of internal dimensions, size of supersymmetric cycles on which branes can be wrapped, etc

Stabilization of moduli is necessary for string theory Stabilization of moduli is necessary for string theory

to describe the effective 4to describe the effective 4--dimensional particle dimensional particle

physics and cosmology.physics and cosmology.

This was a longThis was a long--standing problem. Recentlystanding problem. Recently a significant a significant

progress was achieved.progress was achieved.

FluxFlux compactificationcompactification

NonNon--perturbative correctionsperturbative corrections

lead to stabilization of lead to stabilization of modulimoduli

LANDSCAPE of String Theory with de Sitter LANDSCAPE of String Theory with de Sitter vacuavacua

Starting point for inflation, provides the exit stage and curreStarting point for inflation, provides the exit stage and current accelerationnt acceleration

Towards cosmology in type IIB Towards cosmology in type IIB string theorystring theory

Dilaton and complex structure stabilization Giddings, Kachru and Polchinski Dilaton and complex structure stabilizationDilaton and complex structure stabilization Giddings, Kachru and Polchinski

Kachru, R.K., Linde, TrivediKachru, R.K., Linde, Trivedi

Kachru, R. K., Maldacena, McAllister, Linde, Trivedi

Volume stabilization, KKLT

construction of de Sitter space

First proposalFirst proposal

INFLATIONINFLATION

20032003

Moduli fields in Calabi-Yau

compactification of string theoryAxion-dilaton,

Complex structures (shape moduli),

Kahler moduli (size moduli)

In effective four-dimensional theory moduli

are massless scalar fields without a potential

(starting with perturbative string theory

without fluxes).

Total volume not fixed by fluxes Total volume not fixed by fluxes

Shape moduli fixed by fluxesShape moduli fixed by fluxes

How to lift moduli?

Add fluxes

Add non-perturbative corrections to the

superpotential and Kahler potential

One can still have runaway moduli

Effective N=1 d=4 supergravityThe F-term potential depends on the real Kahler potential K

and on the holomorphic superpotential W

In the simplest models in the context of IIB string theory we

sum over the volume-4-form moduli, complex structures

and axion-dilaton

Runaway moduli: a Runaway moduli: a dilatondilaton andand

the total volumethe total volume

No-scale supergravity has non-canonical kinetic terms

Add gravity, solve Add gravity, solve FriedmannFriedmann eqseqs.. Too steep for dark Too steep for dark

energy and inflationenergy and inflation

INTEGER FLUXES small INTEGER FLUXES small

numbersnumbers

in string theory for cosmologyin string theory for cosmology

Best understood example: resolved

conifold

K and M are integer fluxesK and M are integer fluxes

FLUX COMPACTIFICATION andFLUX COMPACTIFICATION and

MODULI STABILIZATIONMODULI STABILIZATION

The throat geometry has a

highly warped region

RedshiftRedshift in the throatin the throat

Flux compactification and moduli stabilization

in IIB string theory (supergravity + local sources)

Dilaton stabilizationDilaton stabilizationDilaton stabilization

Warping fixed by local sources (tadpole condition) and non-

vanishing ISD fluxes

This equation fixes the shape of CY and the dilaton-

axion

The potential with respect to The potential with respect to dilatondilaton and volume is very steep, they run down and V and volume is very steep, they run down and V

vanishes, the space tends to vanishes, the space tends to decompactifydecompactify to 10d and string coupling tends to vanish, to 10d and string coupling tends to vanish,

unless both are stabilized at some finite values.unless both are stabilized at some finite values.

4d description* Susy at scale 1/R(CY) N=1 effective action* Specify the Kahler potential, superpotential and gauge couplings

* At the leading order in and

* Add the superpotential due to fluxes* Add the superpotential due to fluxes

* Solve equationsSolve equations

The complex structure fields and the axion-dilaton are fixed.

The overall volume still has a runaway potentialThe overall volume still has a runaway potentialsince W from fluxes does not depend on overall volume modulussince W from fluxes does not depend on overall volume modulus

NoNo--scale Potential is Positivescale Potential is Positive--

DefiniteDefiniteNoNo--scalescale

Runaway potential for the volume moduli. Dilaton and shape Runaway potential for the volume moduli. Dilaton and shape moduli are generically fixed in moduli are generically fixed in MinkowskiMinkowski space!space!

KahlerKahler moduli problem (in particular, overall moduli problem (in particular, overall

volume)volume)

KKLTKKLT proposalproposali) noni) non--perturbative superpotential from Euclidean D3perturbative superpotential from Euclidean D3--branes wrapped on branes wrapped on

special 4special 4--cyclescycles

ii) non-perturbative superpotential from pure SYM on a stack of D7’s on

EffectiveEffective theory for the volume modulitheory for the volume moduli

SolveSolve

the volume is stabilized in the volume is stabilized in

AdSAdS critical point in the regime of validity of critical point in the regime of validity of

calculations!calculations!

Volume stabilizationVolume stabilizationVolume stabilization

Basic steps:Basic steps:

AdS minimumAdS minimum Metastable dS minimumMetastable dS minimum

Warped geometry of the compactified space and Warped geometry of the compactified space and nonperturbativenonperturbative effectseffects

((gauginogaugino condensation,condensation, instantonsinstantons) lead to AdS space ) lead to AdS space

negative vacuum energy) with unbroken SUSY and stabilized volumenegative vacuum energy) with unbroken SUSY and stabilized volume

Uplifting AdS space to a Uplifting AdS space to a metastablemetastable dS space (positive vacuum energy) by dS space (positive vacuum energy) by

adding antiadding anti--D3 brane at the tip of the conifold (or D7 brane with fluxes)D3 brane at the tip of the conifold (or D7 brane with fluxes)

Uplifting due to the antiD3 brane at the tip of the conifoldUplifting due to the antiD3 brane at the tip of the conifold

The total potential for the Kahler modulusThe total potential for the Kahler modulus

V(AdS) + V(anti-D3)= V(dS)

in KS warped geometry

MetastableMetastable

dSdS vacuavacua

Racetrack model

Same Kahler potential

Two exponents and a constant term in

superpotential

KKLTKKLT--based new ideasbased new ideas

String LandscapeString Landscape SusskindSusskind

Statistics ofStatistics of Flux VacuaFlux Vacua Douglas, Dine, Kachru,Douglas, Dine, Kachru,……

Cosmic Strings Produced by the end of InflationCosmic Strings Produced by the end of Inflation

SplitSplit SupersymmetrySupersymmetry,,

MetaMeta--stablestable vacuavacua withwith susysusy breakingbreaking

Landscape IdeaLandscape Idea

• With account of loop corrections each vacuum will change. However, the total lanscape picture with many vacua will survive

• There are many vacua with negative, vanishing and positive energies

• Somewhere there is our vacuum with

Is there a better idea?Is there a better idea?

where N, the number of where N, the number of vacuavacua, is required to be, is required to be

The number of phenomenologically (or anthropically) acceptable

vacua is smaller than the number of total vacua

Eternal Inflation

Vilenkin, Linde

KKLT

Bousso, Polchinski;

Susskind; Douglas

String Theory LandscapeString Theory LandscapeString Theory Landscape

Perhaps 10100

- 101000

different vacua

Perhaps 10100

- 101000

different vacuaInflationary

slow-roll valleys

RECENT DRAMATIC PROGRESSin moduli stabilization in string theory

GKP-KKLT scenario Giddings, Kachru, Polchinski 2001; Kachru, R. K., Linde,Trivedi, 2003

De Sitter model building Saltman, Silverstein, 2004

Building a better racetrack Denef, Douglas, Florea, 2004

Fixing all moduli in an F-theory compactification Denef, Douglas, Florea, Grassi, Kachru, 2005

Freund-Rubin and G2 vacua Acharya, Denef, Valandro, 2005

Fluxes and gaugings Derendinger, Kounnas, Petropoulos, Zwirner, 2005

Type IIA moduli stabilization DeWolfe, Giryavets, Kachru, Taylor, 2005

Fixing all moduli in M-theory on K3xK3 Aspinwall, R.K. 2005

On de Sitter vacua in type IIA Saueressig, Theis, Vandoren, 2005

Moduli stabilization in type IIB orientifolds Lust, Reffert, Shulgin, Stieberger, 2005

Large Volume Stabilization models Berglund, Balasubramanian, Conlon, Quevedo, 2005

Building a better racetrack Denef, Douglas, Florea, 2004

KKLT scenario realized in specific stringy

constructions.

The model is a The model is a CalabiCalabi--YauYau threefold with 2 threefold with 2 KahlerKahler modulimoduli and 272and 272

complex structure complex structure modulimoduli. The . The modulimoduli space admits an space admits an orientifoldorientifold

action which allows to reduce the action which allows to reduce the modulimoduli space of the space of the CalabiCalabi--YauYau

complex structures to just 2 parameters. The model was studied bcomplex structures to just 2 parameters. The model was studied byy

string theorists for a long time, starting with Candelas, Font, string theorists for a long time, starting with Candelas, Font, Katz and Katz and

Morrison in 1994Morrison in 1994..

The defining equation for the Calabi-Yau 2-parameter

subspace of the complex structure moduli space is

DDFDDF have stabilized the axionhave stabilized the axion--dilatondilaton,,

and the complex structures ,and the complex structures ,

They have also stabilized two Kahler moduli by explicit They have also stabilized two Kahler moduli by explicit

construction of the KKLTconstruction of the KKLT--type nontype non--perturbative instanton perturbative instanton

corrections to the superpotential.corrections to the superpotential.

Stabilization of Stabilization of modulimoduli in this theory is in this theory is

the basis for an inflationary model with the basis for an inflationary model with

the spectral index n=0.95 in perfect the spectral index n=0.95 in perfect

agreement with WMAP3 (next lecture)agreement with WMAP3 (next lecture)

Fixing All Moduli for

M-Theory on K3 x K3

Aspinwall, R. K. Kashani-Poor,

Bergshoeff, Sorokin,Tomasiello

andand IIBIIB string theory on K3*Tstring theory on K3*T22/Z/Z22

D3/D7 model of inflation in string theoryD3/D7 model of inflation in string theory

K3 surface

1810-1893

Eduard Kummer

1906-2000

Erich Kähler Kunihiko Kodaira

1915-1997

K2 (8,611 m) is the

second highest

mountain in the world

and is regarded as

one of the hardest to

climb

The name K3

surface was given

in the 50’s when

the mountain peak

K2 was in the news

Kummer surfaces

In 1864Ernst Eduard Kummar gave

the following real one-

dimensional family of surfaces

of degree four (quartics):

A Kummer surface has

sixteen double points, the

maximum possible for a

surface of degree four in

three-dimensional space

Paul

M-theory compactified on K3xK3:

incredibly simple and elegant

Without fluxes in the compactified 3d theory there are two80-dimensional quaternionic Kähler spaces, one for each K3.

With non-vanishing primitive (2,2) flux, (2,0) and (0,2), each K3becomes an attractive K3: one-half of all moduli are fixed

40 in each K3 still remain moduli and need to be fixed by instantons.

There are 20 proper 4-cycles in each K3. They provide instanton corrections from M5-branes wrapped on these cycles:

Moduli space is no moreModuli space is no more

Aspinwall, R. K.

On K3 x there are 4-cycles which may

or may not lead to non-vanishing instantons

Status inStatus in 20042004

According to the old rules (established before fluxes were introduced), the

relevant M-theory divisors in our model have an arithmetic genus 2. Therefore

one could incorrectly conclude that there are no

stabilizing instantons for our favorite cosmological

model.

KKLT 1: gaugino condensation. Works only for vector multiplets, does

not work for hypers.

KKLTKKLT 2:2: instantonsinstantons from Euclidean 3from Euclidean 3--branes wrapped branes wrapped

on 4on 4--cyclescycles.. May work for vector May work for vector multipletsmultiplets andand hypershypers..

WittenWitten 19961996: in type IIB : in type IIB compactificationscompactifications under certain conditionsunder certain conditions

there can be corrections to the superpotential coming from Euclithere can be corrections to the superpotential coming from Euclideandean

D3 branes. The argument is based on the D3 branes. The argument is based on the MM--theory counting of the theory counting of the fermionfermion

zero modes in the zero modes in the DiracDirac operator on the M5 brane wrapped on a 6operator on the M5 brane wrapped on a 6--cycle of a cycle of a

CalabiCalabi--YauYau fourfour--fold. He found thatfold. He found that such corrections are such corrections are

possiblepossible only in case that the fouronly in case that the four--fold admits fold admits

divisors with holomorphic characteristic equal to divisors with holomorphic characteristic equal to

one,one,

In presence of such instantons, there is a In presence of such instantons, there is a

correction to the superpotential which correction to the superpotential which

at large volume yields the term required at large volume yields the term required

in the in the KKLTKKLT constructionconstruction

We established a We established a new rulenew rule, replacing , replacing

the rule in presence of the rule in presence of

fluxesfluxes

In presence of fluxes there were indications that the

rule may not be necessary

Robbins, Sethi (2004); Gorlich, Kachru, Tripathy and Trivedi (2004); Tripathy and Trivedi

(2005); Saulina (2005); Berglund, Mayr (2005); Gomis, Marchesano and Mateos (2005)

In F-theory compactifications on K3 x K3 one of the attractive K3

must be a Kummer surface to describe an orientifold in IIB, the

second attractive K3 can be regular.

The orientifold on K3 x

Inflation in string theoryInflation in string theoryInflation in string theory

KKLMMT brane-anti-brane inflation

Racetrack modular inflation

D3/D7 brane inflation

Kahler modular inflation