精准入门模板

1ModuliStabilizationinStringGas

CompacticationUsingDoubleField

TheoryasaBackground

弦气宇宙模型中以双场理论作为背景的模稳定答辩人：项海波导师：朱传界中国人民大学May.20,20152OutlineIntroductiontostringgascosmologyDoublefieldtheory:T-dualityinvariant4-deffectiveactionModulistabilizationinstringgascompactificationConclusion3Introductionto

stringgascosmology4StandardPictureoftheUniverseDarkenergy73%Darkmatter：WMAP23%

Inflaton

isalsonecessarytoexplaincurrentobservations.4-duniverse

describedbygeneralrelativityStandardmodelparticles4%DominantcomponentsProblemsincosmology…generalrelativityissufferingfromthesingularityproblem.Darkenergy(cosmologicalconstant)DarkmatterInflaton

horizonproblemfinetuneproblem5Withtheseproblems,weturnto6StringGasCosmology!(forearlyuniverse)78AnaturalpictureoftheuniverseemergesIntheconventionalstandardcosmology,itisassumedelementalyparticlesoccupytheuniverse.Astheeveryparticlescanberegardedasmodesofastring,itisnaturaltoimaginetheuniversefilledwithastringgas.10-duniverse

9problemstobesolved:CosmologicalSingularity

SuperstringTheory

10-dimensionsDimensionalityproblemModulistabilizationandmore...Keypoint:T-dualityandBrandenberger-Vafamechanism10Spacetimebackground:Initially,allof9spatialdimensionsaresmallandtoroidallycompactified.And,matterpart:theuniverseisfilledwithaclosedstringgas.Stringswindingaroundthecirclepreventexpansion.StringGasCosmologyBrandenberger&Vafa(1989)GeneralPictureNocosmologicalsingularityT-dualityminimallengthT-duality:

Momentummodes:En=n/R

Windingmodes:Em=mR

T-Duality:R→1/R,when(n,m)→(m,n)111213Pairannihilationofwindingscannotoccuriflargespatialdimensionsaremorethan4.Apossiblesolutionofdimensionalityproblem

Brandenberger-Vafamechanism4-dspacetimeeslargeduetoannihilationofwindingmodes.14

whether6-dinternaldimensionsarestableornotduringthecosmologicalexpansion.ifwecanstabilizethe

dilatoninthisstringgascompactification.()stringcouplingMainresearchquestions:BesidestoverifythevalidityoftheB-Vmechanism,weneedtoinvestigateModulistabilization!1516Doublefieldtheory:

T-dualityinvariant

4-deffectiveaction17

NewdegreesoffreedomoscillatorymodeswindingmodesMaximaltemperature:HagedorntemperatureNewsymmetryT-duality1819WhyDFT?HiddensymmetryT-dualityallowsfortheexistentofnewgeometriesStringyfeaturebeyondSUGRA2021ActionforaString

stringscaleworldsheetinconformalgauge22graviton2-formdilatonLetusconsiderastringinageneralbackground.LowenergyeffectiveactionLowenergyeffectiveaction2324T-dualityinvariant4-deffectiveactionT-dualitytransformationMatrixnotation25ActionforastringgasWecalculatethemassspectrumofastringwithconstantbackgroundfields.Afterthecalculationwereplacedthembyfunctionsofspacetimecoordinates.

Letbethecomovingnumberdensityofthestringgas.Astheenergyofastringisgivenby,weobtainT-dualityinvariant4-dmomentumComovingnumberdensityofastringgas261shapemoduliIdentifytheoppositesidesAmodelofCompactfication

Wecananalyzeeachtorusseparately.27volumemodulifluxmodulishapemoduliModuliforthecompactification

dilaton284-deffectiveaction:EinsteinframevolumefluxshapeShifteddilatonMassofastringEffectivepotentialArealtheory:Thevaluesofthemodulusminimizethevalueoftheeffectivepotential!2930Modulistabilization

instringgascompactification31PreviousworksinstringgascompactificationNumericalevidenceofstabilityofthevolumemoduliisshown.Butthedilatonisrunninglogarithmically.

Watson&Brandenberger2003Usingthe4-deffectiveaction

approach,itisshownthatthedilatonandtheradioncannotbestabilized.This4-deffectiveactionisnotmanifestlyT-dualityinvariant.

Battefeld&Watson2004Theimportanceofmasslessmodesisstressedandthestabilityofthevolumemoduliisprovedanalytically.

Watson2004,Patil&Brandenberger200432NumericalanalysisrevisitedToverifythestabilityforthesimplestcase,weperformednumericalcalculation.ScalefactorStabilized!4-disexpanding6-dinternalspace4-duniverse33ShapemoduliinstringgascompactificationThestabilityofshapemoduliispartiallyanalyzedusingthemasslessmodesattheself-dualpoint.

Brandenberger,Cheung,andWatson2005Here,weintendtogiveacompletestabilityanalysisofallmoduliforasimplecompactification

byusingtheT-dualityinvariant4-deffectiveaction.Wewillalsoclarifywhythedilatonisstabilizedinournumericalresult.

34T-dualityandSelf-dualpointSelf-dualpointT-dualitytransformation1135Massofastring36ModuliStabilizationIFlatdirectionThefirstkindofstringgasconsistingofmodeswhicharemasslessattheself-dualpoint.MassformulaforthiswrappedstringgasEffectivepotential37FlatdirectionThesecondkindofstringgasconsistingofmodeswhicharemasslessattheselfdualpoint.MassformulaforthiswrappedstringgasModuliStabilization

IIThesetwodirectionsareexactlyintersectedat3839Volume,shape,andfluxmoduligetstabilizedattheselfdualpoint!

StablecompactificationLetusconsiderbothcontributionstogether.

Asthewould-beflatdirectionsareorthogonaltoeachother,theflatdirectiondisappearsattheendoftheday.Thepotentialdisappears!EffectivepotentialSowecometoourconclusion:Intheframeofstringgascosmology,usingT-duality,i.e.makingT-dualitymanifestinth