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�� ������� ���������� �� �������� ���������� �� ������� ��������� �� ���������� ������ Lecture 2 Stabilization of moduli in superstring theory

„ Fluxes „ NS (perturbative string theory) „ RR (non-perturbative string theory)

„ Non-perturbative corrections to superpotential and Kahler potential Start 2008

Higgs?Higgs? MSSM?MSSM? SplitSplit susysusy?? NoNo susysusy?? IFIF SUSYSUSY ISIS THERETHERE…… The significance of discovery of supersymmetry in nature, which will manifests itself via existence of supersymmetric particles, would be a discovery of the fermionicfermionic dimensionsdimensions ofof spacetimespacetime ItIt willwill bebe thethe mostmost fundamentalfundamental discoverydiscovery inin physicsphysics afterafter EinsteinEinstein’’ss relativityrelativity

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 moduli space.

„ 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 StabilizationStabilization ofof modulimoduli isis necessarynecessary forfor stringstring theorytheory toto describedescribe thethe effectiveeffective 44--dimensionaldimensional particleparticle physicsphysics andand cosmology.cosmology.

This was a long-standing problem. Recently a significant progress was achieved.

Flux compactification

Non-perturbative corrections lead to stabilization of moduli

LANDSCAPE of String Theory with de Sitter vacua Starting point for inflation, provides the exit stage and current acceleration FirstFirst proposalproposal

TowardsTowards cosmologycosmology inin typetype IIBIIB stringstring theorytheory

Dilaton and complex structure stabilization Giddings, Kachru and Polchinski

Volume stabilization, KKLT Kachru, R.K., Linde, Trivedi construction of de Sitter space 2003

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

INFLATION Moduli fields in Calabi-Yau compactification of string theory „ Axion-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). Shape moduli fixed by fluxes

Total volume VV not 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 supergravity „ The 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 RunawayRunaway moduli:moduli: aa dilatondilaton andand thethe totaltotal volumevolume

No-scale supergravity has non-canonical kinetic terms

Add gravity, solve Friedmann eqs. Too steep for dark energy and inflation FLUXFLUX COMPACTIFICATIONCOMPACTIFICATION andand MODULIMODULI STABILIZATIONSTABILIZATION INTEGERINTEGER FLUXESFLUXES smallsmall numbersnumbers inin stringstring theorytheory forfor cosmologycosmology

„ Best understood example: resolved conifold

K and M are integer fluxes The throat geometry has a highly warped region

Redshift in the throat Flux compactification and moduli stabilization in IIB string theory (supergravity + local sources)

The potential with respect to dilaton and volume is very steep, they run down and V vanishes, the space tends to decompactify to 10d and string coupling tends to vanish, unless both are stabilized at some finite values. Dilaton stabilization

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

This equation fixes the shape of CY and the dilaton- axion 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

* Solve equations

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

TheThe overalloverall volumevolume stillstill hashas aa runawayrunaway potentialpotential since W from fluxes does not depend on overall volume modulus NoNo--scalescale PotentialPotential isis PositivePositive-- DefiniteDefinite NoNo--scalescale

Runaway potential for the volume moduli. Dilaton and shape moduli are generically fixed in Minkowski space! KahlerKahler modulimoduli problemproblem (in(in particular,particular, overalloverall volume)volume) KKLT proposal i) non-perturbative superpotential from Euclidean D3-branes wrapped on special 4-cycles ii) non-perturbative superpotential from pure SYM on a stack of D7’s on EffectiveEffective theorytheory forfor thethe volumevolume modulimoduli

SolveSolve

the volume is stabilized in AdSAdS critical point in the regime of validity of calculations! VolumeVolume stabilizationstabilization BasicBasic steps:steps:

Warped geometry of the compactified space and nonperturbative effects (gaugino condensation, instantons) lead to AdS space negative vacuum energy) with unbroken SUSY and stabilized volume

Uplifting AdS space to a metastable dS space (positive vacuum energy) by adding anti-D3 brane at the tip of the conifold (or D7 brane with fluxes)

AdS minimum Metastable dS minimum Uplifting due to the antiD3 brane at the tip of the conifold

The total potential for the Kahler modulus in KS warped geometry V(AdS) + V(anti-D3)= V(dS)

Metastable dS vacua Racetrack model

„ Same Kahler potential

„ Two exponents and a constant term in superpotential KKLTKKLT--basedbased newnew ideasideas

StringString LandscapeLandscape Susskind Statistics of Flux Vacua Douglas, Dine, Kachru,…

CosmicCosmic StringsStrings ProducedProduced byby thethe endend ofof InflationInflation

Split Supersymmetry, Meta-stable vacua with susy breaking Eternal Inflation Bousso, Polchinski; Vilenkin, Linde LandscapeLandscape IdeaIdea Susskind; Douglas KKLT

• 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

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

The number of phenomenologically (or anthropically) acceptable vacua is smaller than the number of total vacua

IsIs therethere aa betterbetter idea?idea?

StringString TheoryTheory LandscapeLandscape

100 1000 PerhapsPerhaps 1010100 -10-101000 Inflationary differentdifferent vacuavacua slow-roll valleys RECENT DRAMATIC PROGRESS in 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 Calabi-Yau threefold with 2 Kahler moduli and 272 complex structure moduli. The moduli space admits an orientifold action which allows to reduce the moduli space of the Calabi-Yau complex structures to just 2 parameters. The model was studied by string theorists for a long time, starting with Candelas, Font, Katz and Morrison in 1994.. The defining equation for the Calabi-Yau 2-parameter subspace of the complex structure moduli space is

DDFDDF have stabilized the axion-dilaton, and the complex structures ,

They have also stabilized two Kahler moduli by explicit construction of the KKLT-type non-perturbative instanton corrections to the superpotential. StabilizationStabilization ofof modulimoduli inin thisthis theorytheory isis thethe basisbasis forfor anan inflationaryinflationary modelmodel withwith thethe spectralspectral indexindex n=0.95n=0.95 inin perfectperfect agreementagreement withwith WMAP3WMAP3 (next(next lecture)lecture) Fixing All Moduli for M-Theory on K3 x K3

2 and IIB string theory on K3*T /Z2

D3/D7 model of inflation in string theory

Aspinwall, R. K. Kashani-Poor, Bergshoeff, Sorokin,Tomasiello K3 surface

Eduard Kummer Erich Kähler Kunihiko Kodaira

1810-1893 1906-2000 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 1864 Ernst 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 Aspinwall, R. K. „ Without fluxes in the compactified 3d theory there are two 80-dimensional quaternionic Kähler spaces, one for each K3.

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

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

ƒ There are 20 proper 4-cycles in each K3. They provide instanton corrections from M5-branes wrapped on these cycles: ModuliModuli spacespace isis nono moremore KKLT 1: gaugino condensation. Works only for vector multiplets, does not work for hypers. KKLT 2: instantons from Euclidean 3-branes wrapped on 4-cycles. May work for vector multiplets and hypers.

„ On K3 x there are 4-cycles which may or may not lead to non-vanishing instantons

StatusStatus inin 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. WittenWitten 19961996: in type IIB compactifications under certain conditions there can be corrections to the superpotential coming from Euclidean D3 branes. The argument is based on the M-theory counting of the fermion zero modes in the Dirac operator on the M5 brane wrapped on a 6-cycle of a Calabi-Yau four-fold. He found that suchsuch correctionscorrections areare possiblepossible onlyonly inin casecase thatthat thethe fourfour--foldfold admitsadmits divisorsdivisors withwith holomorphicholomorphic characteristiccharacteristic equalequal toto one,one,

In presence of such instantons, there is a correction to the superpotential which at large volume yields the term required in the KKLT construction 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)

WeWe establishedestablished aa newnew rulerule,, replacingreplacing thethe rulerule inin presencepresence ofof fluxesfluxes The orientifold on K3 x

„ 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. InflationInflation inin stringstring theorytheory

KKLMMT brane-anti-brane inflation D3/D7 brane inflation

Racetrack modular inflation Kahler modular inflation