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String Cosmology ��������� � ������ ������ �� ��������� ��� ������������� ������� �� � �� ���� ���� ������ ��������� ���� � �� ������� ���������� �� �������� ���������� �� ������� ��������� �� ���������� ������ 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
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