De Sitter Constructions from String Theory

de Sitter Constructions from String Theory

Fernando Quevedo (discussion with Shamit Kachru) University of Cambridge Strings 2021 ICTP-SAIFR June/July 2021

C. Crino, FQ, R. Valandro 2010.15903 S. AbdusSalam, S. Abel, M. Cicoli, FQ, P. Shukla 2005.11329 M. Cicoli, FQ, R. Savelli, A. Schachner, R. Valandro 2106.04592 M. Cicoli, I. Garcia-Etxebarria, FQ, A. Schachner, P. Shukla, R. Valandro 2106.11964 de Alwis, Muia, Pasquarella, FQ 1909.01975, de Alwis, Céspedes, Muia, FQ, 2011.13936 De Alwis, Cicoli, Maharana, Muia, FQ 1808.08967 Two related string challenges

• Moduli stabilization

it to a height greater than the height of the barrier, see Fig. 1. In typical KKLT-type models this leads to vacuum destabilization if the added energy density V (φ)/σn,whichis 2 2 responsible for inﬂation, is much greater than the height of the barrier Vbarrier ! 3m3/2MP . Since H2 ∆V (φ, σ)/3, this leads to the bound (1.1) (see [3] for a more detailed discussion ∼ • De Sitterof this issue, while a similar problem in a slightly diﬀerent context was also found in [4]). V

1 Σ 100 150 200 250

Figure 1: The lowest curve with dS minimum is the potential of the KKLT model. Thesecond V (φ) one shows what happens to the volume modulus potential when the inflaton potential Vinfl = σ3 added to the KKLT potential. The top curve shows that when the inflaton potential becomes too large, the barrier disappears, and the internal space decompactifies. This explains the constraint

H ! m3/2.

scale. Therefore if one believes in standard SUSY phenomenology with m3/2 ! O(1) TeV, one should find a realistic particle physics model where the nonperturbative string theory dynamics occurs at the LHC scale or even lower (the mass of the volume modulus in the KKLT scenario typically is not much greater than the gravitino mass), and inflation occurs at a density at least 30 orders of magnitude below the Planck energy density [3]. For a recent analysis of this issue see e.g. [5] and for a discussioninthecontextoftheheterotic string see [6]. This problem is quite generic. For example, recently a new interesting mechanism of moduli stabilization was proposed, which is based on the models with compacification on Nil manifolds with negative curvature [7]. This mechanism presents a significant modification of the compactifications on flat Calabi-Yau spaces, as suggested by the assumption of the

low scale supersymmetry. And yet, the same constraint H ! m3/2 remains valid for the inﬂationary models in this scenario [8]. The situation becomes even trickier in the large volume models of vacuum stabilization [2]. In such models the height of the barrier is much smaller, V m3 M .Inthis barrier ∼ 3/2 P case, the constraint that the inﬂaton potential should not bemuchgreaterthantheheight

–2– it to a height greater than the height of the barrier, see Fig. 1. In typical KKLT-type models this leads to vacuum destabilization if the added energy density V (φ)/σn,whichis 2 2 responsible for inﬂation, is much greater than the height of the barrier Vbarrier ! 3m3/2MP . Since H2 ∆V (φ, σ)/3, this leads to the bound (1.1) (see [3] for a more detailed discussion ∼ of this issue, while a similarDine problem-Seiberg in a slightlyProblem diﬀerent context was also found in [4]). V Dine, Seiberg 1985 4

3 Only fully trust runaway part (swampland conjecture, Vafa et al 2 See Valenzuela’s, Torroba’s talks)

1 Σ 100 150 200 250

Figure 1: The lowest curve with dS minimum is the potential of the KKLT model. Thesecond � at weak coupling and large volume. V (φ) one shows whatV happens to the volume modulus potential when the inflaton potential Vinfl = σ3 added to the KKLTIf quantum potential. Thecorrections top curve shows lead that to whenanother the infl minimumaton potential it becomes too large, the barriermost disappears, probably and the be internal at strong space decompactcoupling,ifies. unless… This explains the constraint H ! m3/2.

In KKLT-based models, it therefore seems that for a gravitinomassm 1TeV the 3/2 ∼ Hubble constant during the last stages of a string theory inflation model should be quite low, H ! 1TeV,whichistenordersofmagnitudebelowtheoftendiscussed GUT inflation scale. Therefore if one believes in standard SUSY phenomenology with m3/2 ! O(1) TeV, one should find a realistic particle physics model where the nonperturbative string theory dynamics occurs at the LHC scale or even lower (the mass of the volume modulus in the KKLT scenario typically is not much greater than the gravitino mass), and inflation occurs at a density at least 30 orders of magnitude below the Planck energy density [3]. For a recent analysis of this issue see e.g. [5] and for a discussioninthecontextoftheheterotic string see [6]. This problem is quite generic. For example, recently a new interesting mechanism of moduli stabilization was proposed, which is based on the models with compacification on Nil manifolds with negative curvature [7]. This mechanism presents a significant modification of the compactifications on flat Calabi-Yau spaces, as suggested by the assumption of the low scale supersymmetry. And yet, the same constraint H ! m3/2 remains valid for the inflationary models in this scenario [8]. The situation becomes even trickier in the large volume models of vacuum stabilization [2]. In such models the height of the barrier is much smaller, V m3 M .Inthis barrier ∼ 3/2 P case, the constraint that the inflaton potential should not bemuchgreaterthantheheight

–2– String Compactifications

• No free parameters in string theory • But many discrete parameters after compactification: (topological numbers of compact manifold, ranks of gauge groups, dimensionality of spacetime, fluxes of different forms,…)

• Not only gs and 1/V expansions but many expansions if there are many moduli. • Never have full control but may lead to weak enough coupling and large enough volumes In IIB string theory flux compactifications [125, 126] naturally fix the value of all the In IIB string theory flux compactificationscomplex [125, 126 structure] naturally moduli fix theUa and value the of dilaton all the S and reduce the number of vacua from a con- complex structure moduli Ua and the dilatontinuumS and reduce to a discrete the number but of large vacua set from of points a con- determined by the quantised three-form fluxes. tinuum to a discrete but large set of points determinedIn both DRS by the (Dasgupta, quantised Rajesh, three-form Sethi) fluxes. ([125]) and GKP (Giddings, Kachru, Polchinski) In both DRS (Dasgupta, Rajesh, Sethi) ([125[126]) and] we GKP have (Giddings, flux stabilisation Kachru, of Polchinski) the complex structure moduli and the dilaton of a con- [126] we have flux stabilisation of the complexstruction structure involving moduli a and Calabi-Yau the dilaton orientifold of a con-X with internal G3 fluxes. While in both cases struction involving a Calabi-Yau orientifold Xthewith (static) internal solutionG3 fluxes. requires While that in both the fluxes cases are ISD (imaginary self-dual i.e. G = iG ) ⇤6 3 3 theIn (static) IIB string solution theory requires flux compactifications that the fluxeswhich are[125 ISD, 126 is (imaginary] compatible naturally self-dual fix with the the value i.e. Hodge of6G all3 decomposition= theiG3) G3 (2, 1) (0, 3). Supersymmetry is ⇤ 2 complexwhich structure is compatible moduli withU theand Hodge the dilaton decompositionS andpreserved reduceG3 only the(2 number, if1) there(0 of, 3). is vacua no Supersymmetry (0 from, 3) component a con- is as considered in DRS. a 2 tinuumpreserved to a discrete only if there but large is no set (0, of3) points component determined asKähler considered by the moduli quantised in DRS.Ti are three-form not stabilised fluxes. by the fluxes nor any perturbative e↵ect. The In bothKähler DRS (Dasgupta, moduli Ti Rajesh,are not Sethi) stabilised ([125 by])reason theand fluxes GKP behind (Giddings, nor this any is perturbative the Kachru, fact that Polchinski) e↵ thereect. The exists a Peccei-Quinn synmetry Ti Ti + ici with In IIB string theory flux compactifications [125, 126] naturally fix the value of all the ! complex[126 structurereason] we have behind moduli flux thisU stabilisationa and is the the fact dilaton of that theS thereand complex reduce exists structureconstant the a Peccei-Quinn number modulicis of that vacua and synmetry together from the a dilaton con-T withi T of thei + aic con- holomorphicityi with of the superpotential forbids any Ti ! tinuumstruction toconstant a discrete involvingci buts that large a Calabi-Yau together set of points with orientifold determined the holomorphicityX bywithdependence the internal quantised of theG of three-form3Wfluxes. superpotentialto all While fluxes. orders in forbids in both perturbation cases any Ti theory. However these moduli are the gauge In both DRS (Dasgupta, Rajesh, Sethi) ([125]) and GKP (Giddings, Kachru, Polchinski) thedependence (static) solution of W requiresto all orders that in the perturbation fluxes arecouplings ISD theory. (imaginary However for matter self-dual these fields moduli i.e. localised6G are3 = the iniG gauge D73) branes and therefore standard non-perturbative [126] we have flux stabilisation of the complex structure moduli and the dilaton of a con-⇤ whichcouplings is compatible for matter with fields the Hodge localised decomposition in D7 branese↵ectsG and3 generate therefore(2, 1) a(0 superpotential standard, 3). Supersymmetry non-perturbative for these is fields. The total superpotential for closed string struction involving a Calabi-Yau orientifold X with internal G3 fluxes.2 While in both cases preservede↵ects generate only if there a superpotential is no (0, 3) component for these fields. asmoduli considered The is total in superpotential DRS. for closed string the (static) solution requires that the fluxes are ISD (imaginary self-dual i.e. G = iG ) moduli is ⇤6 3 3 which is compatibleKähler moduli with theTi Hodgeare not decomposition stabilised byG the(2 fluxes, 1) (0 nor, 3). any Supersymmetry perturbative is e↵ect. The 3 2 preservedreason only behind if there this is no is (0 the, 3) fact component that there as considered exists a Peccei-Quinn in DRS. synmetry Ti WTi +=icWi fluxwith(S, U)+Wnp(S, U, T ). (2.4) W = W (S, U)+W (S, U, T ). ! (2.4) Kählerconstant modulicisT thati are together not stabilised with theby the holomorphicityflux fluxes norThe any source ofnp perturbative the of superpotential non-perturbative e↵ect. The forbids e↵ects any areTi Euclidean brane instantons and non-perturbative reasondependence behind this isof theW factto all that orders there in exists perturbation a Peccei-Quinn theory. synmetry HoweverTi theseTi + moduliici with are the gauge The source of non-perturbative e↵ects are Euclideandynamics brane in the instantons! field theory and non-perturbative of D7 or D3 branes such as the condensation of gauginos in the compute the structure of V . Itconstant takes schematicallyc s that together the with form the holomorphicity [37]: of the superpotential forbids any T couplingsdynamicsi for in matter the field fields theory localised of D7 in or D7 D3 branes branes and such therefore as the condensation standard non-perturbative ofi gauginos in the dependence of W to all orders in perturbation theory. Howevergauge these sector moduli of the are D the brane. gauge In the past decade there has been substantial progress in the e↵ects2 generate a superpotential for these fields. The total superpotential for closed string couplingsV gauge forW matterK sector+ fieldsW of0 the localisedW D brane. in D7 In branes the past and decade thereforeunderstanding there standard(2.11) has been non-perturbative and substantial computational progress control in the of Euclidean D brane instantons [127]. Gaugino moduli0 is e↵ects generate/understanding a superpotential and computational for these fields. control The total ofcondensation, Euclidean superpotential D brane being for closed instantons a dynamical string [127]. e↵ Gauginoect, has been well understood from the standard 4d condensation, being a dynamical e↵ect, has been well understood from the standard 4d If there were only one single expansionmoduli is parameter and if, as usual, W0 We↵andectiveK field theory (EFT) but it is more dicult to study from the full 10d e↵ective W = Wflux(S, U)+Wnp(S, U, T ). (2.4) W (since perturbative terms dominatee↵ective over non-perturbative field theory (EFT) terms but it at is weakmoreaction di couplings),cult and to the study full from string the theory. full 10d e↵ective Theaction source and of non-perturbative the fullW string= Wflux theory.( eS,↵ects U)+ areWnp Euclidean(S, U, T ). brane instantons and non-perturbative(2.4) the first term would be the leading order term. It would lift the potential but would K 1 The sourcedynamics of non-perturbative in the field theory e↵ects of are D7 Euclidean or D3 branes brane such instantons as the and condensation non-perturbative of gauginosV = e in theKa¯b DaWD¯bW 0 (2.5) K 1 give rise to a runaway behaviour,dynamics unless in the di↵ fielderent theory order of D7 terms or D3 compete branesV = e such toK as thegiveD condensationWD rise¯W to a of0 gauginos in the (2.5) gauge sector of the D brane. In the past decadea¯b The therea starting hasb been point substantial of the 4D progress EFT isin the the⇣ F-term 4d supergravity⌘ scalar potential for arbi- gauge sector of the D brane.compute In the past the decade structure there ofhas beenV . It substantial takes schematically progress in the the form [37]: minimum which would happen onlyunderstanding if theModuli perturbative and computational expansion Stabilisation control breaks⇣ of downEuclidean and D the⌘ brane in instantons IIB [127]. Gaugino ¯ understandingThe starting and computational point of the control 4D EFT of Euclidean is the F-term Dtrary brane 4d superpotential instantons supergravity [127]. scalarW Gaugino(M potential) and Kähler for arbi- potential K(M , M¯ ) in units of Mp: corresponding expansion parametercondensation, is not small. being This a is dynamical the Dine-Seiberg e↵ect, has been problem well understood[61]. ¯ from the standard 4d compute the structure of V . It takes schematicallycondensation,trary the being superpotential form a [ dynamical37]: W e(↵ect,M ) andhas been Kähler well potential understoodK(V fromM , W theM¯2) standard inK units+ W 4d ofWMp: (2.11) 0 0 K 1 2 Flux compactifications in IIB allowe two↵ective ways field to theory overcome (EFT) this but issue. it is First, more di incult the KKLTto study/ from the fullV 10d= e e↵ectiveK D W D W 3 W (2.6) e↵ective field theory (EFT) but it is more dicult to study from the full 10d e↵ectiveF MN M M 2 compute the structureK 1 of V . It takes schematically2 the form [37]: | | scenario the big discrete degeneracyV •Waction0 Kaction of and+ W flux the0 andW full vacua the string full is theory. used string in theory. suchVF = ae way(2.11)K thatDMWW Dis tunedW 3 W (2.6) / Moduli S, Ti, UIfa there were only oneMN single0 expansionM | parameter| and if, as⇣ usual, W0 W and K ⌘ 2 The tree-level Kähler2 potential for the Kähler moduli satisfies the celebrated no-scale prop- to W0 W = Wnp.ThisthenrequiresW terms toW be(since also included perturbative⇣ in the terms expansion dominateV overW⌘K non-perturbative+ W W terms at weak couplings),(2.11) If there were only one single expansion parameterThe and tree-level if, as usual, KählerW0 potentialW and for theK Kähler moduli1 satisfies the0 celebrated0 no-scale prop- ⇠ Wtree = WWfluxtree(U,= SW)fluxerty(U, SK)i|¯ KiK/|¯ = 3 whichNo (2.5)-scale is just a consequence (2.5) of the homogeneity of . Using this and W stabilising(since perturbative the Ti fields terms when dominate they over compete non-perturbative with1 the termsW0theW at firstterms. weak term couplings), Notice would that be the in this leading limit order term. It would lift the potential but would V erty Ki|¯ KiK|¯ = 3 whichK is just1 a consequencethe fact of the that homogeneity the flux superpotential of . Using this does and not depend on the Ti fields, it implies a positive V = e K DK WD¯W1 0 (2.6) the first term would be the leading order term.3 It would lift thegiveIf potentialF there riseV weretoF but= a¯b e runaway only woulda K oneb D behaviour, singleaWD¯W expansion unless0 parameter di↵erentV order and if, terms as (2.6) usual, competeW0 to giveW and rise toK a the first term in V above is of order Wthe factand that is then the flux subdominant. superpotential Justifying doesa¯b notdefinite depend neglectingb scalar on the potentialTi fields, forit impliesS and aU positiveand stabilises them supersymmetrically by solving give rise to a runaway behaviour, unlessThe di↵ startingerent order point terms of the compete 4Dminimum EFTW (since tois the give⇣ which F-term perturbative rise would⇣ to 4d a supergravity⌘ happen termscompute dominate only⌘ scalar the if potential the structure over perturbative for non-perturbative arbi- of V expansion. It takes terms breaks schematically at weak down couplings), and the the form [37]: quantum corrections to the Kähler potential.Thedefinite starting scalar point potential of the 4D for EFTS and is theU F-termand stabilisesD 4dUa W supergravity= themDSW supersymmetrically= scalar 0. As potential long as for these by arbi- solving equations have solutions for di↵erent values of the compute the structureminimum of V which. It would takes happen schematically only if the perturbativethe form [37 expansion]: breaks downFix S,U and but the T arbitrary¯ trary superpotentialcomputeD W = theDW structureW(M=corresponding)the 0.and As first Kähler of long termV potential as. It expansion these would takesK equations( be schematicallyM parameter, theM¯ ) have leading in units solutions¯ is theof notorderMp form small.: for term. di [37↵ Thiserent]: It would is values the Dine-Seiberg liftof the the potential problem but [ would61]. In LVS the fact that there are moretrary thanU superpotentiala oneS expansionW (M parameters) and Kähler plays potential thequantisedK key(M role., fluxesM¯ ) in they units will of generateMp: the huge number2 of solutions that define the string land- correspondingcompute the structure expansion of parameterV . It takes is not schematically small. This the is form the Dine-Seiberg [37Flux]:give compactifications rise problem to a runaway [61]. in behaviour, IIB allow two unless ways di↵ toerent overcome order thisterms issue. competeV First,W to inK give the+ W KKLTrise0 toW a (2.11) quantised fluxes they willK generate1 the huge numberscape2 but of solutions at this stage that definethe Kähler the string moduli land-Ti have0 a completely flat potential that vanishes FluxIn compactifications this case the two in IIB terms2 allow in two equation ways to overcome (2.10) can this compete issue.VF = e First, withK in theD eachM KKLTW D otherM W to3 W provide a (2.7) / V W K + W0W MNK (2.11)1 | 2 | 2 0 • 2 scape but at thisscenario stageminimumV theF = the Kählere which bigK moduli discrete wouldVDMTWiW degeneracy happenforhaveD0 allKW a values+ completely only3WW of0 of if fluxW the flat vacua fields perturbative potential is even used forthat in expansion those vanishessuch(2.7) that a way breaks break that(2.11) supersymmetryW down0 is tunedand the D W K W = 0. scenario the big discrete/ degeneracyV of fluxWQuantum0 vacuaK + isW used0W in corrections such a way that⇣ W is tunedMN(2.11) M ⌘ T T 0 minimum as long as each comes/ fromThe tree-level a di↵erent Kähler expansion. potential for In the this Kähler0 case moduliK satisfies/WIf0 thereW thewhich celebrated were| | only no-scale2 one prop- single expansion parameter and if, as usual,⇠ W06 W and K 2 for all values of thetocorresponding fieldsW0 evenW for= expansionW thosenp.Thisthenrequires that breakparameterTwo supersymmetry main is notscenariosW small.termsDT haveW This to emerged beK isT alsoW the0 = included toDine-Seiberg 0. fix the in Kähler the problem expansion moduli: [61]. the original KKLT [15] to W0 W = Wnp.Thisthenrequiresa⌧ W terms1 to be also included in⇠ the expansion⇣ ⇠ a⌧ ⌘ erty K K K = 3 which is just a consequence of the homogeneity of . Using this and⇠ 6 If there were only oneIf therefor single⇠ wereK expansion only1/ oneand singleW parameter expansione implies• parameter andTheIfi|¯ that if, there tree-leveli as and theTwo|¯ usual, were if, volume main as Kähler only usual,W scenariosstabilising is0Flux potential one exponentiallyW0 compactifications single haveW for theWand emerged expansiontheandTi Kählerfields largeKK to when modulifix in parameterandW the IIBe they the(since Kählersatisfies. allowRunaway: Here compete Large and two moduli: perturbative the Volume if, ways celebratedDine with as the- theto usual,Seiberg [36 original overcome,W no-scale370 terms]W Wproblem (LVS)0 KKLTterms. prop- this dominate scenarios.W [ issue.15 Noticeand] First, that overK Both in in non-perturbative start this the limit from KKLT the flux superpotential, terms at weak couplings), stabilising the⇠ Ti fieldsV when they⇠ compete withThree the W 0options:W terms. Notice that in this limit V ⇠ V the fact that the1 flux superpotential does not depend on the Ti fields, it implies3 a positive W (since perturbativeW⌧(sinceis terms usually perturbative dominate a blow-up terms over mode dominate non-perturbative that3 over getsertyW non-perturbativeand stabilisedK(sincei|¯ theKi LargeK terms perturbative|¯ to= Volume3values termsthe whichscenario at first weak of at [ is36 terms orderweakjustterm, the37 couplings),] a (LVS)big couplings),in consequence 1 dominate/gV discretes scenarios.abovewhichthe over degeneracy is of is of the large Both first ordernon-perturbative homogeneity start at term ofW flux fromand would vacua of the is then. flux terms be Using is superpotential,subdominant.used the thisat in leading weak and such couplings), a Justifying orderway that term. neglectingW is It tuned would lift the potential but would the first term in V above is of order Wdefiniteand is scalar then subdominant.potential for S Justifyingand U and neglecting stabilises them supersymmetrically by solvingV 0 the first term would be the leading order term.the fact It that would the lift flux the superpotential potential but does would not depend on the Ti fields,2 it implies a positive the first term wouldquantum beweak the corrections string leading coupling to the order Kählergs and term. potential. thereforeD ItW wouldthe= theD first volumeW lift= term 0. the As is wouldlong exponentially potentialquantumto asW these be0 the corrections equationsW but large. leading= W would havenp to.Thisthenrequires order solutions thegive Kähler term. rise for di potential. to↵ Iterent a would runaway valuesW liftterms of the the behaviour, to potential be also included unless but would di in↵ theerent expansion order terms compete to give rise to a Ua S Fix T-modulus: KKLT giveIn rise LVS to the a runaway fact that behaviour, there are more unless than di one↵definiteerent expansion order scalar terms parameters potential competeIn for plays LVS⇠S to theand give the keyU fact riseand role. that to stabilises a there are them more supersymmetrically than one expansion by solving parameters plays the key role. give rise to a runaway behaviour,In summary unless KKLT di requires↵erentquantised ordertuninggive fluxes terms of rise the they to fluxes compete will a runaway generate forstabilisingW to the0 behaviour, give huge1 the whereas number riseTi fields to unless of LVSsolutions a whenminimum di works↵ theyerent that competedefinefor order which the stringterms with would theland- competeW happen0W terms. to only give Notice if rise the to that perturbative a in this limit expansion breaks down and the minimum which would happen only if the perturbativeDUa W = DSW expansion= 0. As breaks long⌧ as down these and equations the have solutions for di↵erent values of the In this case the two terms in equation (scape2.10) but can at compete this stage with the KählereachIn this other moduli case toT theprovidei have two a a completely terms in equation flat potential (2.103 that) vanishescan compete with each other to provide a standard values of W0 (1 100)minimum (as it is found which in wouldthe concrete first happen term examples in onlyV above [if117 thecorresponding, is131 perturbative of]) order but W expansion expansionand is then parameter breaks subdominant. down–5– is and Justifying not the small. neglecting This is the Dine-Seiberg problem [61]. minimum which wouldminimumcorresponding happen as long expansion only as each if comes parameter the from perturbative a is difor↵ noterent all small.quantised values expansion. expansion This of the fluxes is In fields the this they breakseven Dine-Seiberg case will forK generate those downW that problemW the and breakwhich huge [ the61 supersymmetry number]. of solutionsD W thatK defineW = 0. the string land- ⇠ O minimum as0 long as eachK comes fromT a di↵erentT 0 expansion. In this case K W0W which depends more on thea⌧ perturbative corrections to K. Noticequantum that⇠ corrections froma⌧ the e to–5–factor theFlux Kähler incompactificationsFix the T potential.-moduli:⇠ LVS6 in IIB allow two ways to overcome this issue. First, in the KKLT corresponding expansionforFluxK compactifications parameter1/ and W is ine not IIBimplies allow small. twothat This ways thescapecorresponding volume to is overcome thebut is exponentiallyat Dine-Seiberg this this stage expansion issue. the large First, Kähler problem parameter ine moduli the. Here KKLT [61T isi ].have nota a⌧ small. completely This flat is potential the Dine-Seiberg that vanishes problem⇠ [61]. a⌧ ⇠ V ⇠ 4 for2 KIn LVSV41⇠/ theand factW thatescenario thereimplies are more the that big than the discrete volume one expansion is degeneracy exponentially parameters of large flux plays vacua thee . key is Here used role. in such a way that W is tuned ⌧scenarioisgeneral usually the a expression big blow-up discrete mode for degeneracy thatV the gets order of stabilised flux of vacuaforFluxV to0 allis valuesis compactificationsvaluesV used0 of of inM order the suchp / fields 1 a/g ways evenwhichM in⇠ thats IIB forwhereasV isW those large allow0 is tuned that at in two⇠ LVS break ways the supersymmetry to order overcomeD thisT W issue.KT W First,0 = 0. in the KKLTV ⇠ 0 Flux compactifications in IIB allow two ways to overcome this⇠ issue.V⌧ First,is⇠ usually in the a blow-up KKLT mode that gets stabilised to⇠ values of6 order 1/gs which is large at weakto W stringW coupling= W .Thisthenrequiresg and2 4 therefore3 the2 volumeW2 2 terms is exponentially4 to be also includedlarge.In this incase the the expansion two termsto inW equationW (=2.10W) can.Thisthenrequires compete with each otherW 2 toterms provide to a be also included in the expansion of0 V is V np Ws Mp / Ms m scenarioMs . the Having big discreteV0 vanishing degeneracy at the of minimum flux vacua0 and is used innp such a way that W0 is tuned scenario the big discrete⇠ degeneracy of0 flux vacua is3 used/2 in such a wayweak that stringW couplingis tunedgs and therefore⇠ the volume is exponentially large. stabilisingIn summary the4T KKLTfields⇠ when requires theyV tuning⇠ compete of the with fluxes⌧ the W forWW0 terms.1 whereasminimum Notice LVS that as0 works in long this as for limit each comes from a di↵erent expansion. In this case K W0W which V M supportsi the validity of using the EFT0 ⌧ at scales below M . stabilising2 the Ti fields when they compete with the W0W terms. Notice that in this limit s 2 3 to W0 W = Wnp.ThisthenrequiresIn summary–5–s KKLT requiresWa⌧ terms tuning to of be the also fluxes included for W0 in the1 whereas expansion LVS⇠ worksa⌧ for to W0 W = Wnpstandardthe.Thisthenrequires first⌧ term values in ofVWabove0 (1 is ofW100) orderterms (asW it isand to found beis then alsoin concrete subdominant. included examplesfor in JustifyingK [the1171,/ expansion131 neglectingand]) butW e implies that the volume is exponentially large3 e . Here ⇠ O ⇠ the first term in V above is of⌧ order W and is then subdominant. Justifying neglecting ⇠ depends more on the perturbative corrections tostabilisingK. Notice the thatT fromi standardfields the wheneK⇠ valuesfactorV they in ofcompete theW0 ⇠ (1 with100) the W (as0 itW isterms. found in Notice concrete that examples in this limit [117V, ⇠131]) but stabilising the Ti fieldsquantum when corrections they compete to the Kähler with potential. the W0W terms. Notice⌧ is that usually in a this blow-up limit⇠ modeO that gets stabilised to values of order 1/gs which is large at 2.2.2 Advantages 4 2 4 depends more on the perturbativequantum3 corrections corrections to K to. Notice the Kähler that from potential. the eK factor in the generalIn LVS expression the fact for thatV the there3 order are of moreV0 is thanV0 the oneM first expansionp / termMs parameters inwhereasweakV above in string plays LVS is the the ofcoupling order key order role.–5–gWandand therefore is then the subdominant. volume is exponentially Justifying large.neglecting the first term in V above is of order2 4 3W and2 2 is then4 subdominant.⇠ V ⇠ Justifying neglectings 4 2 4 of V is V W M / M m M . Having V0 vanishinggeneral at the expression minimum and for V the orderIn LVS of V theis V factM that/ thereM arewhereas more than in LVS one the expansion order parameters plays the key role. In thisWe casewould⇠ the like two0 herep termsV to⇠ in emphasises equation3/2 ⌧ ( several2.10s quantum) can advantages compete corrections with of type each toIn otherIIB the summary constructions: Kähler to provide KKLT potential. a requires tuning0 of the0 fluxesp for W s 1 whereas LVS works for 4 2 4 3 2 2 4 ⇠ V ⇠ 0 quantum correctionsV to theMs supports Kähler the potential. validity of using the EFT at scales below Ms. In this case the two terms in equation⌧ (2.10) can compete with each other to provide a minimum⌧ as long as each comes from a di↵erent expansion.In LVS the In this factof case thatV isK thereV W0 areWW0 whichM morep / thanM ones m3 expansion/2 Ms . parameters Having V0 vanishing plays the at key the role. minimum and standard⇠ values⇠ of W0V ⇠(1 100)⌧ (as it is found in concrete examples [117, 131]) but In LVS the fact that1. thereControlled are more flux than backreactiona⌧ one expansion: Background parameters fluxes canV plays beM turned4 thesupports keya on⌧ role. to the generate validity⇠minimumO of a po-using as long the EFT as each at scales comes below fromM a. di↵erent expansion. In this case K W W which 2.2.2for K Advantages1/ and W e implies that theIn volume this case is exponentially the two terms larges in equatione . Here (2.10) can compete with each other tos provideK a 0 ⇠ V ⇠ depends⌧ moreV ⇠ on the perturbative corrections to K. Noticea⌧ that from the e factor in the ⇠ a⌧ In this case the two⌧ is terms usuallytential in a equation blow-up for the mode ( moduli2.10 that) gets can ina stabilised compete controlled to values with way ofsince each order their other 1/gs backreactionwhich to provide is large on at a thefor internalK 1/ and W 4 e2 implies4 that the volume is exponentially large e . Here We would like here to emphasise several advantagesminimum of type as IIB long constructions:2.2.2 asgeneral each Advantages comesexpression from for a diV ↵theerent order expansion.⇠ of VV0 is V In0 thisM⇠ casep / KMs Wwhereas0W which in LVS the order V ⇠ weak string coupling gs and therefore the volume is exponentially large. ⇠ V ⇠ minimum as long as each comesgeometry from just a di renders↵erent the expansion. compactification In this manifold case K conformallyW0a⌧W Calabi-Yau.which2 4 ⌧3 is usuallyThere-2 2 a blow-up4 mode that⇠ getsa⌧ stabilised to values of order 1/gs which is large at for K 1/ andofWV iseV impliesW Mp that/ theM volumes m is exponentiallyMs . Having V large0 vanishinge at. the Here minimum and 1.InControlled summary flux KKLT backreaction requires: tuning Background of the fluxes fluxes can for beW0 turned1 whereas on to⇠ generate LVS works a po-0 for 3/2 forea⌧ the understanding of the underlying⇠ moduliV ⌧ We space would⇠ is4 better like⇠a here⌧ than to emphasise inV otherweak⇠ string string several coupling⌧ advantagesgs ofand type therefore IIBV constructions:⇠ the volume is exponentially large. for K 1/ and W e implies that the volume is⌧ is exponentially usually a blow-up largeV modeMs supportse that. Heregets the stabilised validity of to using values the of EFT order at scales 1/g which below M iss large. at ⇠ V standard⇠tential values for the of W moduli0 in(1 a controlled100) (as it way is found since in their concrete backreaction examples⌧ onV [⇠ the117, internal131]) but s theories. Some⇠ O progress has been made recently in computingK the form of theIn Kähler summary KKLT requires tuning of the fluxes for W0 1 whereas LVS works for ⌧ is usually a blow-updepends modegeometry more that on just the rendersgets perturbative stabilised the compactification corrections to valuesweak to manifoldK of. string Notice order conformally coupling that 1/g froms1. Calabi-Yau.whichg theControlleds ande isfactor therefore largeThere- flux in thebackreaction at the volume: Background is exponentially fluxes large. can be turned on to generate a po- ⌧ potential including the e↵ect of warping4 2 [62–694 ].2.2.2 Notice Advantages that the warpingstandard induces values of W0 (1 100) (as it is found in concrete examples [117, 131]) but weak string couplinggeneralgs foreand expression the therefore understanding for V thethe of volume order the underlying of V is0 is exponentiallyV0 moduliMp / space isM large. betters whereas thantential in otherLVS for the string the order moduli in a controlled way since their backreaction on the internal ⇠ In summaryV ⇠ KKLT requires tuning of the fluxes for W0 1⇠ whereasO LVS works for K of Vtheories.is V SomeW 2M progress4/ 3 hasM 2 beenm2 madeM recently4. Having in computingV vanishing the at form the of minimum the Kähler and depends more on the⌧ perturbative corrections to K. Notice that from the e factor in the In summary KKLT requirescorrections0 tuningp to the of the definitions 3 fluxes/2 ofs for theW correct0 1 moduli whereasWe coordinateswouldgeometry LVS like works just here which renders to for emphasise are the however compactification several advantages manifold of type conformally IIB constructions: Calabi-Yau. There- 4 ⇠ V ⇠ ⌧ standard0 values of W0 (1 100) (as it is found in concrete examples [117, 131]) but 4 2 4 V potentialM supports including the validity the e↵ect of of using warping the EFT [62–69 at]. scales⌧ Notice below thatM thes. fore warping⇠ O the understanding induces general of the expression underlying moduli for V spacethe order is better of thanV is inV otherM string/ M whereas in LVS the order ⌧ snegligible at large volume. K 0 0 p s standard values of W0 corrections(1 to100) the (as definition it is of found the correct independs concrete moduli more coordinates examples on the1. which perturbativeControlled [117 are, 131 however]) flux corrections but backreaction to K: Background. Notice that2 fluxes4 from3 can the bee2 turnedfactor2 on in to the4 generate⇠ aV po-⇠ ⇠ O theories.K Some progressof hasV is beenV4 made2 W recently0 M4p / in computingMs m3 the/2 formM ofs . the Having Kähler V0 vanishing at the minimum and depends more on the2.2.2 perturbativenegligible Advantages at large corrections volume. to K. Noticegeneral that expression from the foreVtentialthefactor order for in the of the moduliV0 is V in0 aM controlled/ ⇠ wayM sincewhereasV their⇠ in backreaction LVS the⌧ order on the internal 2. Suppressed scalar potential scale: The starting pointpotential of dS models including is the theV classicale↵ectM of4 psupports warping [62s the–69 validity]. Notice of that using the the warping EFT induces at scales below M . 4 2 4 2 4 3 2 2 4 ⇠ s V ⇠ s general expression forWe2.V wouldSuppressedthelow-energy like order here scalar to of emphasiselimit potentialV0 is ofV type scale0 several: IIBM The advantagesp string/ startingof V theoryMis ofpoints typeV compactifiedwhereas of IIB dSW constructions: models0 M inpcorrections/geometry LVS on is the an theM classical orientifolds justm to order3 the/ renders2 definition ofM a thes⌧ Calabi-. compactificationHaving of theV correct0 vanishing manifoldmoduli at coordinates the conformally minimum which Calabi-Yau. and are however There- ⇠ V ⇠ 4 ⇠ V ⇠ ⌧ 2 4 low-energy3 2 limit2 of type IIB4 string theoryV compactifiedM supports on an orientifold thefore validity of the a Calabi- understanding of using the EFT of the at underlying scales below moduliMs space. is better than in other string of V is V W0 Mp1./ ControlledYauM threefolds fluxm3/ backreaction2 X.M Thiss .: Background Having is a controlledV0 fluxes⌧vanishing procedure cans be turned at if the the onnegligible compactification minimum to generate at large a and po- volume. volume2.2.2 Advantages 4 ⇠ VYau⇠ threefold X. This⌧ is a controlled procedure if the compactificationtheories. volume Some progress hasV been⌘ made recently in computing the form of the Kähler V M supports the validitytential for of the using moduli the in a EFT controlled at scales way since below theirM backreaction. on the internalV ⌘ s 2.2.2 Advantagess 2. Suppressed scalar potentialWe would scale like: The here starting to emphasise point of dS several models advantages is the classical of type IIB constructions: ⌧ geometry just renders the compactification manifold conformally Calabi-Yau.potential including There- the e↵ect of warping [62–69]. Notice that the warping induces low-energy limit of type IIB string theory compactified on an orientifold of a Calabi- 2.2.2 Advantages fore the understanding of the underlyingWe moduli would space like is here better to than emphasisecorrections in other several tostring the advantagesdefinition1. Controlled of of the type correct flux IIB backreaction constructions: moduli coordinates: Background which are fluxes however can be turned on to generate a po- Yau threefold X. This is a controlled procedure if the compactification volume theories. Some progress has been made recently in computing thenegligible form of the at Kähler large volume. V ⌘ –7– tential for the moduli in a controlled way since their backreaction on the internal We would like here to emphasisepotential including several the advantages e↵ect of warping of type [1.62––7–Controlled69 IIB]. Notice constructions: flux that backreaction the warping induces: Background fluxes can be turned on to generate a po- 2. Suppressed scalar potentialgeometry scale: The just starting renders point the of compactification dS models is the manifold classical conformally Calabi-Yau. There- corrections to the definition of the correct modulitential coordinates for the moduli which are in a however controlled way since their backreaction on the internal 1. Controlled flux backreaction: Background fluxes can be turned on tolow-energy generate limit a po- of type IIBfore string the theory understanding compactified of theon an underlying orientifold of moduli a Calabi- space is better than in other string negligible at large volume. geometry just renders the compactification manifold conformally Calabi-Yau. There- Yau threefold X. This istheories. a controlled Some procedure progress if the has compactification been made recently volume in computing the form of the Kähler tential for the moduli in a controlled way since their backreaction on the internal –7– V ⌘ 2. Suppressed scalar potential scale: The startingfore point the of understanding dS models is the of the classical underlyingpotential moduli space including is better the than e↵ect in other of warping string [62–69]. Notice that the warping induces geometry just renderslow-energy the compactification limit of type IIB string manifold theory compactified conformally on an Calabi-Yau. orientifold of a There- Calabi- theories. Some progress has been made recentlycorrections in computing to the definition the form of of the the Kähler correct moduli coordinates which are however fore the understandingYau threefold of theX underlying. This is a controlled moduli procedure space is if better the compactification than in other volume string potential including the e↵ectV of⌘ warpingnegligible [62–69]. Notice at large that volume. the warping induces theories. Some progress has been made recently in computing the form of the Kähler corrections to the definition of the correct moduli–7– coordinates which are however potential including the e↵ect of warping [62–69]. Noticenegligible that at the large warping volume. induces 2. Suppressed scalar potential scale: The starting point of dS models is the classical corrections to the definition of the correct moduli coordinates which are however low-energy limit of type IIB string theory compactified on an orientifold of a Calabi- –7– negligible at large volume. 2. Suppressed scalar potential scale: The startingYau threefold point ofX dS. This models is a is controlled the classical procedure if the compactification volume low-energy limit of type IIB string theory compactified on an orientifold of a Calabi- V ⌘ 2. Suppressed scalar potential scale: The starting pointYau of dSthreefold modelsX. is This the is classical a controlled procedure if the compactification volume V ⌘ low-energy limit of type IIB string theory compactified on an orientifold of a Calabi- Yau threefold X. This is a controlled procedure if the compactification volume V ⌘ –7–

–7–

–7– Contents

1 Introduction 1

2 Boson Stars 4 2.1 Fermion stars 5 2.2 Boson stars 5 2.3 Oscillatons, Minisclusters and Axion Stars 7

3 Boson Stars from Closed Strings 8 3.1 Axion Stars 9 3.2 Moduli Stars 11

4 Q-balls and Boson Stars 20 4.1 Q-Balls from Open Strings 23 4.2 PQ-balls 24

5 Axitessence 28

6 Formation Mechanisms 30

1 Introduction

M = M 3/2 (1.1) PlanckV

4 a 2/3 ⇤ M e V (1.2) ⇠ Planck M f Planck (1.3) ⇠ 2/3 V 2 MPlanck a 2/3 69 M M e V 10 GeV (1.4) ⇠ m ⇠ Planck

a 2/3 m AdSM e :V N=1, 4D Effective(1.5) Field Theory a ⇠ P 2 MP 2/3 M MP (1.6) V ⇠ m ⇠ V V 2 MP a 2/3 Ma = MP e V (1.7) ⇠ ma 0.05

10 20 30 40 50 -0.05 W0 = G3 ⌦ (1.8) ^ -0.10 Z -0.15 e.g. KKLT -0.20 -0.25

–1–

e.g. LVS IIB Features

• Fluxes imply (warped) Calabi-Yau • No-scale structure

• Scales m3/2<< MKK<

• Two sets of 3-fluxes F3, H3 (allows `tuning’) • GVW Superpotential W(S,U) not renormalised!

• Many loop (gs) and �’ corrections to K computed

• Kahler moduli gauge couplings Wnp(T) de Sitter?

NS Fluxes

• Anti D3 brane Wrapped D7 Brane

Throat

RR Fluxes • D+F terms in EFT or T-branes Anti D3 Branes Figure 1: Description of a deformed conifold with 3-form fluxes (a KS throat) embedded in a compact geometry, with anti-D3-branes trapped at the tipofthethroat.Beyondthe throat, the compactifications may include other ingredients, like D7-branes wrapped on 4-cycles, etc, which are not relevant for the generation of the warp factor on the throat, but may lead to other interesting effects (like non-perturbative superpotentials).

• Complex structure/Dilaton upliftembeds (D it intoU diffWerent possible≠ compactification0, D manifoldSWs. This≠ approach0) separates the local properties of the models, such as the gauge group, the massless matter spectrum, running of gauge coupling, etc, from properties depending strongly on the global features of the compactification, such as supersymmetry breaking, scalar field potentials, etc. AlargeclassoflocalD-braneconfigurationsleadingtochiral 4d world-volume • gauge sectors is provided by D3-branes (or D3-branes) at singularities. It is thus Non critical strings, negative curvaturenatural to combine techniques of model building with D3-branes at singularities with the construction of highly warped throats using deformed conifolds with fluxes. Indeed in this paper we construct explicit geometries containing deformed conifolds, and orbifold singularities sitting at the corresponding 3-spheres. Introduction of an explicit set of suitable 3-form fluxes leads to a warped throat, with the compact 3-cycles and the orbifold singularity at its tip. Finally introducing a set of D3-branes • Kahler uplift and D7-branes (all dynamically trapped at the tip of the throat) at the orbifold

3 • Nonperturbative effects on D3 branes, ... Type IIB flux compactifications provide two ways to overcome this problem. First, in the KKLT scenario the big discrete degeneracy of flux vacua is used to tune W0 to an exponentially small value so that W W .ThisthenrequiresW 2 terms to be 0 ⇠ np np also included in (2.9) stabilising the T -fields when they compete with W0 Wnp terms [15]. Notice that in this limit quantum corrections to the Kähler potential can be consistently neglected since the first term in (2.9) is subdominant given that W 2 K W W W 2 0 p ⌧ 0 np ⇠ 0 for K 1 (this is always the case at large volume since the perturbative e↵ects K are p ⌧ p suppressed by inverse powers of ). V The second case is LVS models where the fact that there is more than one expansion parameter plays the key rôle. In this case the two terms in (2.9) can compete with each other to provide a minimum as long as each comes from a di↵erent expansion. Hence at Preprint2 typeset in JHEP style - HYPER VERSION ⌧s DAMTP-2013-48, HRI/ST/1305 the minimum one has W0 Kp W0 Wnp which, for Kp 1/ and Wnp e ,yields ⇠ ⌧s ⇠ V ⇠ an overall volume of order W0 e . Here ⌧s is a blow-up mode that gets stabilised to 1. Argument fromV Consistency⇠ I:asmallW0 has been required by the following con- values of order 1/gs. It is therefore large for weak string coupling, implying that the CY volume is exponentiallysistency argument. large [47 The–49]. use of a derivative expansion in a supersymmetric effective 2 In summary,field KKLT theory requires indicates a that major there tuning should of also the be fluxes an expansion to obtaininW powersW of ϵ F/M1, 0 ⇠ np ⌧≡ whereas LVS workswhere forF is natural the auxiliary values field of the of the flux relevant superpotential light fields of and orderMWan ultraviolet(1 100) cutoff. 0 ⇠ O (as found in concreteImposing examplesϵ 1impliesthatthesuperpotentialwhichisproportionalto [50, 51]) but depends more on perturbative correctionsF should to be ANoteontheMagnitudeoftheFluxSuperpotential≪ K K. Notice that,very from small the [3, 4].e factor in the general expression (2.5), the order of V0 is 4 2 4 2 4 3 2 2 4 V0 Mp / Ms , whereas in LVS the order of V is V W0 Mp / Ms m3/2 Ms . ⇠ V ⇠ 4 ⇠ V ⇠ ⌧ Having V0 vanishing2. Argument at from the minimum Consistency and II:anaturalvalueV Ms supportsW0 the(1 validity10) has of been the argued EFT at to be ⌧ ≃ O − scales below Mincompatibles. with a four-dimensional effective field theory since it implies background fluxes with an energy density of order the string scale: V (M 4). This is not 2.2.2 Advantages Michele Cicoli,1,2,3 Joseph P. Conlon,4 Anshumanflux ≃ O s Maharana,5 Fernando Quevedo3,6 true sinceAchievements the important quantity to look at is not the scalingofthefluxpotential We would like here to emphasise1 several advantages of type IIB constructions: energy but itsDipartimento vacuum expectation di Fisica value e Astronomia, (VEV). If the Universitàdi dilatonandthecomplex Bologna, 1. Controlledstructure flux backreaction modulivia are Irnerio: fixed Background supersymmetrically, 46, 40126 fluxes Bologna, can then be Italy. turned this VE onVisvanishingatleading to generate a po- 2 • Well defined prescriptionINFN, Sezione exists di Bologna, that4 Italy. includes general tential fororder, the even moduli if it in would a controlled formally scale way assinceMs their.Inordertotrustthefour-dimensional backreaction on the internal 3 stringygeometry ingredients:eff justective renders field theory,Abdus the compactificationbranes, one Salam has ICTP, toorientifolds, check Strada manifold that the Costiera warping, conformally effects 11, used anti Trieste to Calabi-Yau. (T) fix the- 34014,branes, Kähler Italy. There- moduli, 4 perturbative,fore thedevelop understanding non a-perturbative potentialDepartment of the whose underlying effects, VEV of Theoretical satisfies etc. moduli V Physics, spaceM is4 University better. than of in Oxford, other string 1 Kebble St, Oxford, UK. ⟨ ⟩≪ KK theories. Some progress5 Harish has been Chandra made Research recently Institute, in computing Chhatnag the form Road, of the Jhusi, Kähler Allahabad 211 019, India. • W0potential<<1 is includingplausible the6 e ↵dueect of to warping the large [52–59]. number Notice that of the fluxes. warping induces 3. Argument fromDAMTP, Phenomenology University I:asmall of Cambridge,W0 has been Wilberforce argued to Road, be necessary Cambrid alsoge CB3 0WA, UK. • Perturbativecorrectionsfor a to viable theeffects phenomenology.definition in of LVS the In correct thein originalcontrol moduli efforts coordinatesas to the stabili volumese which the Kähler are is however moduli T ,a negligible at large volume. exponentiallynon-perturbative large. term AllW npcomputedwas added to W soflux [5].far In harmless. order to stabilise the T -moduli Abstract: The magnitude of the flux superpotential Wflux plays a crucial rôle in de- 2. Suppressedat values scalar large potential enough scale to trust: The the starting effective point field theory, of dS modelsWnp has is to the be classical of the same −10 • Antibrane:low-energyorder limit nonlinearly as W ofterminingflux type,requiringthelattertobe‘finetuned’tovaluesassmallas1 IIB string the realised scales theory of compactifiedSUSY IIB string (nilpotent on compactifications a CY goldstino orientifold.) after This0 ismoduliin stabilisation. It has

a controlledstring procedure units.been Even ifargued the though compactification thatWflux valuesis determined of volumeWflux from is a large1arepreferred,andevenrequiredforphysicaland combination so that the of integers, following small • Hierarchieshierarchy of scales: is valid: ≪ values of Wconsistencyflux are allowed reasons. in the multi-dimensional This note revisits space these of integer arguments. fluxes. Weestablishthatthecou- Ms 1 1 1/4 Mp E plingsMKK of= heavy Kaluza-KleinMs modes= g to light. states scale(2.10) withtheinternalvolumeas 4. Argument⌧ from Phenomenology1/6 ⌧ II:thestringscale⌘ ` ⌘ p Mssis setp by the Planck scale MP −2/3s 2⇡ ↵0 4⇡ g M V/M 1/2 1andarguethatconsistencyofthesuperspacederivativeV and the internal volumeKK P, Ms MP / ,whereasthegravitinomassdependsalso As mentioned above,∼ at tree-levelV the∼ ≃V dilatonV≪ and the complex structure moduli are on W : mexpansionW M requires/ .ThereforethestandardphenomenologicalpreferenceforgF/M2 m /M 1, where F is the auxiliary field of the light fixed supersymmetrically0 3/2 ≃ 0 at PDVW = D W = 0 via3/2 non-zeroKK quantised G fluxes 16 S U ∼ ≪ 3 Ms MGUTfields10 andGeVM the from ultraviolet unification andcutomff3.Thisgivesonlyamildconstraintonth/2 Msoft (1) TeV in order to efluxsuperpo- arXiv:1310.6694v1 [hep-th] 24 Oct 2013 ≃ ≃ ≃ ≃ O 4 −11 address the hierarchy problem,1 requires/3 (10 )andW0 (10 ). tential, W ,whichcanbeeasilysatisfiedforV ≃ O ≃ O (1) values of W .Thisregime flux ≪ V O flux is also statistically favoured and makes the Bousso-Polchinski mechanism for the vacuum 5. Argument from Statistics:asmall–7–W0 has also been argued to be preferred on statistical grounds.energy In the hierarchically original treatments more e [6]fficient. the magnitude of W0 was argued to be uniformly distributed. More recently, arguments have been given that the statistical

distribution of W0 can peak at zero [7], indicating some preference for a hierarchically

small value of W0.Similarly,recentstatementshavebeenmadearguingthatasmall

cosmological constant requires a small W0 [8, 9].

–2– Challenges to KKLT, LVS,...

• Fluxes under control only in SUSY 10D? (Sethi, Kachru- Trivedi, de Alwis et al…)

• All SUSY breaking part is 4D EFT. Trust EFT?(Carta, et al, Moritz et al, Kallosh, Gautason et al, Hamada et al, Kachru et al.)

• Tuning W0<<1? in KKLT (Demirtas et al, Alvarez-Garcia et al)

• Higher corrections in LVS? (Cicoli et al.)

• Antibranes (non susy, singularity?) (Bena et al, Moritz et al, Cohen-Maldonado et al, Gao et al, Crino et al.)

• Tadpole problem (Bena et al., see Grana’s talk) Quintessence from Strings?

• Need stabilise all moduli except for quintessence field: as difficult as getting de Sitter bounds from fifth-forces [12]. Moreover, if the quintessence field is a string modulus which sets the visible sector• gaugeOr have kinetic function, many a fields rolling modulus rolling would but give slower than rise to a time variation of the coupling constants. This last problem can be avoided simply by considering a modulusquintessence. which is not supporting Difficult. the visible sector stack of D-branes. However, evading fifth-force bounds is more complicated. The volume mode couples democratically• toFifth all fields force with Planckian and varying strength, and couplings so it cannot constraints (e.g. be the quintessence field. This is a direct consequence of the locality of the SM construction. The fact that thevolume volume mode modulus has to couple or to dilaton SM fields canproblematic) be seen by looking at the relation between the physical Yukawa couplings Yîjk and the holomorphic ones Yijk(U) which depend just on the complex structure moduli because e.g. Banks, Dine, Douglas ‘00 of the holomorphicity of the superpotentialYukawa’s and the axionic shift symmetry [139]:

K/2 Yijk(U) Yîjk = e , (3.4) K˜iK˜jK˜k q where K˜i is the Kähler metric for matter fields. Due to locality, the physical Yukawa couplings should not depend on the overall volume, and so the matter Kähler metric K˜i has to depend on the volume mode in order to cancel the powers of in K/2 V V e . Consequently, the volume mode has always a direct Mp-suppressed coupling to SM-fields from expanding the matter Kähler metric in the kinetic terms. The best case scenario is therefore when the quintessence field is a modulus di↵erent from the overall volume which supports a hidden sector stack of branes, while the visible sector is localised on a blow-up mode which does not intersect with the quintessence divisor. This has been advocated in the context of swampland conjec- tures in [9]. However even in this case, one would need to check that no interaction between the quintessence modulus and visible sector fields is induced by kinetic mix- ing between the moduli (see for example the moduli redefinitions in [140–142]induced by non-canonical kinetic terms) or between hidden and visible sector Abelian gauge bosons [143–146]. This issue is currently under detailed investigation [147].

3.2 The swampland and the Higgs As already pointed out in [22], the swampland conjecture is in tension with basic features of the Higgs potential. In fact if h is the standard Higgs ﬁeld and the quintessence ﬁeld, the total scalar potential can be written as:

2 V = V˜ (h)+Vˆ ()withV˜ (h)= h2 v2 . (3.5) The swampland conjecture at the maximum of the Higgs potential for h =0thenimplies:

V Vˆ() Vˆ() |r | & 1 = & 1 . (3.6) V , V˜ (h)+Vˆ () v4 + Vˆ ()

4 However the quintessence potential today has to scale as Vˆ (0)=⇤ . Typical quintessence potentials have the form Vˆ ()=⇤4 e with 0. Hence Vˆ ( ) Vˆ ( )=⇤4,imply- 0 ' 0 ' 0 ing that the ratio in (3.6) violates the swampland conjecture by 57 orders of magnitude!

– 22 – Open Questions

• Control of quantum and �’ corrections

• Realistic phenomenology (de Sitter but no SM?) See A. Schachner’s poster • Moduli Stabilisation in F-Theory

• Populating the landscape (large # of U moduli +

vacuumit to a height transitions) greater than the height of the barrier, see Fig. 1. In typical KKLT-type models this leads toMotivations vacuum destabilization if the added energy density V (φ)/σn,whichis 2 2 responsible for inflation, is much greater than the height of the barrier Vbarrier ! 3m3/2MP . Since H2 ∆V (φ, σ)/3, this leads to the bound (1.1) (see [3] for a more detailed discussion ∼ • ... of this issue, while a similarPopulating problem inthe a slightly String di ffLandscape.erent context was also found in [4]).[Aguirre, Johnsons, Larfors, ’09, ’10] V • HowV is it populated? ϕ 4 Eternal inflation is not enough.

3 • Starting from a given de Sitter, is it possible

to2 up-tunnel?

1 Σ χ 100 150 200 250

nχ Figure 1: The lowest curve with dS minimum is the potential of the KKLT model. Thesecond − V (φ) V = e Mχ V (ϕ) + V (χ) one shows what happens to the volume modulus potential when the inflaton potential Vinfl = σ3 0 1 added to the KKLT potential. The top curve shows that when the inflaton potential becomes too 2 large, the barrier disappears, and the internal space decompactifies. This explains the constraint ϕ2 V (ϕ) = μ4 − 1 0 ϕ 2 H ! m3/2. ( Mϕ )

V χ μ4 e−2χ/Mχ ae−χ/Mχ be−3χ/Mχ In KKLT-based models, it therefore seems that for a gravitinomassm 1TeV1( the) = χ [− + + ] 3/2 ∼ Hubble constant during the last stages of a string theory inﬂation model should be quite low, H ! 1TeV,whichistenordersofmagnitudebelowtheoftendiscussed GUT inﬂation

–2–