Ds Vacua and the Swampland JHEP03(2019)134 ], Which 7 , 6 to Account for ] May Not Have a 120 7 , − 1

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JHEP03(2019)134 Springer March 16, 2019 March 22, 2019 : : January 24, 2019 : c Accepted Published Received , and Marco Scalisi b Published for SISSA by https://doi.org/10.1007/JHEP03(2019)134 [email protected] Evan McDonough , a [email protected] , . 3 Andrei Linde, 1901.02022 a The Authors. c Superstring Vacua, Supergravity Models

In this note we revisit some of the recent 10d and 4d arguments suggesting that , [email protected] [email protected] Providence, RI 02903, U.S.A. Institute for Theoretical Physics, KatholiekeCelestijnenlaan Universiteit 200D, Leuven, B-3001 Leuven, Belgium E-mail: evan Stanford Institute for Theoretical PhysicsStanford, and CA Department 94305, of U.S.A. Physics,Department Stanford of University, Physics, Brown University, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: may occur for someof extreme dS values of vacua within theKKLT parameters, the scenario they range based on do of a not validity racetrackwhich of prevent superpotential is the requires the not parametrically formation affected 4d small by KKLT uplifting, flattening.gravity models. conjecture We show for The that a KL this broadour scenario version choice is analysis of of compatible do the with parameters not the of support weak the the KL recent model. swampland conjecture. Thus, the results of Abstract: the uplifting of supersymmetric AdS vacuathe leads formation to of a dS flattening vacua. ofand We the explain potential, requires why preventing the considerable corresponding modifications. 10d approach is We inconclusive also show that while the flattening effects Renata Kallosh, dS vacua and the swampland JHEP03(2019)134 ], which 7 , 6 to account for ] may not have a 120 7 , − 1. However, this is 6 10 − ] are already disfavored . = 6 | or more dS vacua in string w dS V | 500 9 5 ]. 8 – 1 – ], but other mechanisms of dS vacuum stabilization 1 3 7 . 120 − 10 . | dS V | 1 14 ]. This allows to use the anthropic constraint 5 – 1 One could argue that even though the new swampland conjecture [ This basic idea makes the string landscape scenario very robust. It cannot be inval- The most important part of dS constructions in string theory is the enormous multi- during the last 15 years, no such proof isderivation or available [ proof, it canhas be validated equation if future of observations state demonstratenot that different the dark case. from energy the First of dS all, equation all of models state of dark energy proposed in [ idated by arguments ofthat naturalness, radiative by corrections the may weakto affect gravity prove conjecture, dS a or vacua. no-go by theorem, Totheory the that disprove cannot would possibility this state exist, scenario that as one allwe suggested would of will recently have the in call 10 the the new no-dS swampland conjecture. conjecture [ Despite many attempts to prove such no-go theorem energy density prior tomove the away calculation from of thebut quantum anthropic many corrections, range other vacua then when will many quantumvacuum enter states of this corrections even range. them are though Thus may the taken onewith precise into should the vacuum be energy account, precision able in to these find states many cannot suitable be calculated are also available. plicity of vacuum states into the theory another and [ the possibilitythe to incredible tunnel smallness from of one the of these vacuum energy: states if there were some vacua with the required The string theory landscape scenariotaneously: emerged to in provide an a attempt mechanismand of to to de solve solve Sitter two the problems (dS) cosmologicaltheory simul- vacuum is constant stabilization based problem. in on the string One KKLT theory, of scenario [ the most popular versions of this 6 Conclusions 1 Introduction 3 KKLT model and its4 consistent generalizations Flattening effects in KKLT 5 The KL model and the weak gravity conjecture Contents 1 Introduction 2 Gaugino condensation in 10d JHEP03(2019)134 . ] ]. 2 11 19 ] demon- 17 , ] were forced 16 ] is lacking evi- 11 ]. It is based on 19 18 ] that a certain version 11 ] suggests that these diver- ]. ] is inconsistent, whereas all ] contained an error corrected 17 14 ] is that, during the uplifting, , – 11 11 11 13 15 , 8 ] and references therein. ]. Secondly, the models of dark energy 9 , 10 , ] that there might be some tension between 8 ] was supposed to confirm the 10d approach [ ] and are fully compatible with the existence 8 7 11 σ 19 , , – 2 – 6 5 . 12 ] show that a more detailed explanation of the ] consists of two parts: a detailed but very com- 12 11 ]. We present a short discussion of this issue in section 12 , ] are pretty straightforward and can be easily verified. However, 11 ] make its results inconclusive [ 17 , , and demonstrate by an explicit example that dS vacua do appear 12 16 , 4 11 we discuss the KL version of the KKLT mechanism [ 5 and 1 can be constructed in the string theory landscape scenario with dS vacua. ]. The original analysis of these divergences in [ 3 − 11 6= ]. Moreover, a recent examination of this issue in [ ]. Recently Van Riet et al. argued [ w In section The results of [ The 4d analysis of the KKLT model in [ Because of the complexity of the 10d investigation, the authors of [ The investigation performed in [ Nevertheless, since this issue is extremely important, one should carefully examine any 12 11 a particular version of thein racetrack [ superpotential; it isthis free class of from models the and problemsHowever, a discussed the specific particular version of version the of weak the gravity conjecture weak introduced gravity in conjecture [ used in [ recent comments onrelation these between issues uplifting, in flattening,in and [ dS sections vacua isfor warranted. proper We discuss choice these offlattening issues of parameters the even potential. if one assumes extremely strong backreaction and and the absence ofstrated dS that vacua. the However, modifiedknown a consistent 4d more 4d versions KKLT detailed of model investigationsupport the in the proposed KKLT scenario, recent [ in swampland in conjecture the [ of [ domain dS of vacua their in validity, do string not theory. gences do not appear atwhich all was if one neglected takes in intoThis account [ is the 4-fermion just non-derivative one coupling, investigation of of the [ many reasons to believe that the uncertainties involved in the 10d to make various assumptions,texts. or In rely particular, upon somecontain results divergences, results which of required previously the regularization obtained and 10djectures interpretation in [ investigation based of on different various gaugino con- con- condensationin in [ [ the potential flattens because offorms. the The backreaction authors to effectively theders uplifting, propose of and that magnitude dS stronger the minimum that backreactionproposition, never the to which basic should the effect be uplifting of carefully is uplifting. examined. many This or- is a very unconventional support of the no-dS conjectureof was the based KKLT on scenario thevacuum does suggestion to not of a lead [ metastable to dS a vacuum. consistent uplifting ofplicated the 10d supersymmetric analysis, AdS andtake into a account 4d backreaction. investigation of The basic the argument KKLT of model, [ which is modified to with Moreover, the possibility toconstruction roll of down quintessence towards models, the see asymptotic e.g. dS [ state maypiece simplify of the evidence against the string landscape scenario. One of the main arguments in at a statistical significance higher than 4 JHEP03(2019)134 - i ], is κ λλ 25 T h – (2.1) (2.2) ] the ] by a 23 30 – 19 26 , where does not have aT − λ Ae equations of motion, coincident D7 branes c.c. = order in the fermions, 3 c.c. + N G np + coincident D7 branes, J ] deduce from [ W J ∧ ] using the gauge-fixed N ∧ 12 quartic J , 30 J Ω – 11 · Ω 26 · 3 ]. 3 ], where gaugino condensation was G ] demonstrated that one can satisfy G 14 30 D , 20 – Z D 13 ) Z ] for a broad set of parameters of the KL 26 ˙ and wraps the four-cycle D of the internal α ˙ ]) an assumption was made that replacing α ¯ 4 λ ¯ 20 λ ˙ α 12 ˙ α X ¯ λ ¯ λ x – 3 – 4 Tr( d x 4 4 ] is that for many coincident D7 branes, the 3-form d X 4 Z 30 X ] (see also [ 4 s – Z i πl 11 27 4 s 32 i πl couples only to the fermion on a single D7, 3 32 G ⊃ − ]. ], ⊃ − ] relies on earlier papers [ 19 30 Maxw , D7 , 12 YM D7 , S 29 S ] incomplete, as pointed out in [ when describing many D7 branes is the only generalization required. The 12 , 11 ˙ ] and is most suitable for constructing realistic inflationary models [ α 12 11 ¯ λ , ] evaluated the corresponding 4-fermion terms using ˙ α 22 , 11 ¯ λ 11 ] the issue of the 4-fermion coupling on the surface of the 21 , and the authors provided no suggestions for how one could justify it. Moreover, an 8 30 1 confirms and strengthens the results of [ by Tr – ˙ The next step proposed in [ Technically, the computation was performed in [ 5 α We are grateful to Westphal, McAllister and Rudelius for emphasizing this point to us. 27 ¯ 1 λ ˙ α ¯ was not studied. Meanwhile, inλ [ authors of [ and found divergences, which required regularization. couples to the trace of the fermion bilinears on the surface of In [ The D7-brane fills the four-dimensional space manifold. Note that here a non-Abelian gauge group index. symmetric Dp-brane action. Itthe was approximation explicitly quadratic stressed in there fermions. thatin This the the action result is case is supersymmetric of only andwas a is valid presented in known single in only Dp-brane [ with an Abelian vector multiplet. The relevant coupling studied at quadratic ordergaugino in condensate contribution the to fermions. thebut 4d Meanwhile, effective neglect action [ the at intrinsicanalysis 4-gaugino of interaction [ on D7-branes. This latter fact makes the Gaugino condensation refers to the formationin of a the non-vanishing fermion theory. condensate Tr since It it plays provides a athe non-perturbative significant volume part role of of in the the10d superpotential the 4-cycle analysis wrapped KKLT of construction by [ condensing of D7 de branes. Sitter vacua, The fermionic part of the metry [ does not seem tomentioned in suffer [ from any problems with uplifting2 and stabilization of Gaugino dS condensation vacua in 10d investigation of this issueeven by Blanco-Pillado the et unconventional al. versionproper [ of choice the of weak parameterstion gravity of conjecture the KL introducedmodel. model. in Thus, [ this Our model, investigation which allows of strong this vacuum stabilization problem protected in by supersym- sec- dence, JHEP03(2019)134 ) ]. 2.1 14 (2.5) (2.3) (2.4) , 13 ] the issue ] and shown 30 – 30 – 26 26 ] for type IIB theory λ . 2 µ 13 ] and of the underlying , , D λ 2 µ 38 B – λ µνρ λ ¯ λγ L Γ 36 P ¯ λ Tr µνρ A 1 2 Γ ¯ λ Tr λ 3 4 − ¯ ] were recently discussed in [ βα φ 2 g )Tr ] is that their derivation of eq. ( ) ¯ β 12 2 , 11 30 24 ], the 4d supergravity action has a term √ x YM AB – µν 11 ¯ ( f F − 26 35 1 4 , c δ = 0. Therefore, even if in [ + 34 Tr ( − YM 2 µνρ 1 4 ) . And since the on shell value of the auxiliary – 4 – F 11 W 2 λ | ¯ β − 3 2 F µνρ 2 ∇ | µνρ − G Γ φ αβ ¯ λ φ g 3 4 g µ φ 2 ∂ √ − ] that the existence of the perfect square term in 10d action K/ x e R 14 11 9 − 1 2 16 d − − = Z α = − ] for an M-theory compactified on a one-dimensional interval. The F L 33 1 – -symmetric Abelian D7-brane action, which is supersymmetric upon − κ e . ] the dependence on non-Abelian gaugino, derived in [ 31 2 | ] has a perfect square term ), was assumed to be valid without any additional gaugino-dependent terms 12 -symmetric Abelian D7-brane action. F , | 33 κ – 2.2 11 31 ] it was stressed that, as shown in [ (0) singularities cancel due to the perfect square structure in the action, which δ 14 ] a proposal was made that the 4-fermion terms should be present on the D7 brane, But in [ The reason why this was not discussed in [ It was also pointed out in [ In [ The issues with the 4-fermion 10d analysis in [ 13 gaugino interaction vanishes sinceof ( the non-derivative 4-fermionbased coupling on would the be raised, the answer wouldhere be in eq. negative, ( up to the Born-Infeldon type the terms coincident with D9coincident higher branes D7 derivatives. branes must Therefore suggests have that gauginos a they which 4-fermion also live coupling. must havewas a Dimensional based 4-fermion reduction on coupling. to the gauge-fixing. For the Abelian vector multiplet which lives on the brane the 4-fermion Type IIB theory inonly presence 1/2 of of the calibrated maximalsupergravity Dp-branes with 10d coincident and supersymmetry. D9-branes is Oq-planes Therefore an the as action of effective local the action Einstein-Yang-Mills sources supergravity, describing IIB has 10d superspace geometry. The completedepending 10d on supergravity gravitino action, and where dilatino we suppress but terms keep all terms with gaugino and bosons, is one finds that the bilinearin and the quartic action, dependence on gaugino’s comes via a perfectis square a general feature of the Einstein-Yang-Mills supergravity [ with the square offield the involves auxiliary gaugino field, and the requires the presence ofwith the coincident 4-fermion compactified term. D7 branes Ato has proposal reach additional a made features, nice which, in in correspondence [ particular, with allow the 4-fermion terms in 4d action. In [ which might resolve the disagreementrelated with set up the in 4d [ action analysis. in They [ suggested to follow the JHEP03(2019)134 ) ], ], is c 13 2.5 17 (3.4) (3.5) (3.6) (3.1) (3.2) (3.3) , , and c 16 and b ], the value of . (A similar problem in the superpotential, 11 , we studied the gen- b S > b ], this makes the use of | . aT − 17 | , cA | 2 | . Therefore, following [ cA 16 b bS . cAe | b . + + ¯ S, ¯ S S the model can be equivalently rep- ¯ S S aT bS. S S − aT S + + . According to [ − − b ] about the absence of 4-fermion terms ¯ → cAe T ¯ aT T | cAe − 19 S + , ) S, + − | b + T Ae 12 ¯ T + T ) inconsistent for + cA – 5 – aT + | + aT − 0 3.4 aT − T 3 log − W Ae 3 log . This would imply that the backreaction is much Ae − b − = + cAe + = 0 = 0 W 3 log ] introduces an extra term W K | K as well as W − 11 = b cA = = | describing effects of backreaction W in the model ( K c W ) is a perfect square, which is positive everywhere except the point | S cA 3.6 ] that | 11 ) cannot be valid, and therefore the whole 10d analysis needs to be ), where it vanishes. As emphasized in [ ]. 2.2 16 , cA/b 15 − ]. The Kählerpotential can be either in the nilpotent multiplet representing anti-D3 brane responsible for the uplift- ln( 41 – 1 a S 39 After Kählertransformation ( Instead of debating the reliability of the assumption = 0 The denominator in ( T the nilpotent multiplet check whether it may have dS vacua. resented as eral case, including exponentially sensitive to variouslandscape parameters it of may compactification, take so manywe different in will values the analyze for string any the theory given potential as a function of two independent parameters It was argued ingreater [ than the mainmotivated [ effect. This is an unusual proposition which does not seem well or The modification proposed in [ with an extra parameter Here ing [ concentrate on the 4d analysis of the KKLT construction. 3 KKLT model andThe its KKLT consistent model generalizations in the 4d supergravity formulation can be described by a superpotential it follows that the bilinearterm. in This gaugino terms means mustin that be the the accompanied assumption action by in thereconsidered. ( [ quartic It gaugino is likelyand the that consistency it with will 4d reduce physics in to this the aspect construction will of be restored. the kind In what given follows in we [ will even for non-Abelian vector multiplets on D7. From Einstein-Yang-Mills supergravity ( JHEP03(2019)134 , ]. 4 ), 0. − 17 3.6 (3.8) (3.7) 10 − β > = 0 from 0 to 1, W c 1, ] are confirmed . strictly positive 16 ¯ S = 0 S ) a ¯ T K + � = 0 uplifted by increase of T , ( = 1, b a ] is rather subtle; it man- ] for any choice of − A ) 11 e ¯ T 42 2 , + 200 ) practically coincides with the A T ) for 2 ) practically coincides with the ( 11 ) is outside the physical domain a ) and potential Kähler 15 3.7 − β c ¯ 3.7 S ¯ e S ]). The inconsistency disappears for 3.5 2 S + | ( S cA/b 180 17 c 2 | | − b W + ], which we reproduce here for convenience ln( + ). 2 | ]. 1 a . 16 b ]; see [ 160 | 5 aT 3.1 16 − = 42 . − − – 6 – 0 T ], which we reproduce here for convenience. The ¯ T T cAe = 10 16 ) (multiplied by 10 | ]. This immediately makes 140 + b 3.7 − = Re 17 T t ¯ T ). This removes the pole in the denominator in ( + 120 3.6 T 3 log the potential in the model ( from 0 to − b β in ( the potential in the model ( = 2 | β � 3 log b 1.5 1.0 0.5 K -0.5 -1.0 -1.5 -2.0 − + = in the model proposed in [ aT . The main part of the uplifting is not due to the large change of − K = 0. The second (yellow), line shows the potential at c 5 c − 0. In that case, the effects of backreaction are exponentially suppressed, and cAe = | b = 10 > . Thus our results about the existence of dS vacua contained in [ is some positive number [ b . The potential of the theory ( 1. The blue (lower) line shows the potential with a supersymmetric AdS minimum prior to , because in that case the point 1 β = 1. Finally, the upper (red) line shows the potential with a dS (nearly Minkowski) minimum < b = 1, Yet another consistent model is described by In particular, for small To avoid the inconsistency problem altogether, one can add a positive number or Note that the inconsistency of the 4d model proposed in [ c | c < β to cA potential shown in one ofpotential the is figures in shown [ as a function of KKLT potential shown in one of thein figures figure in [ for a large rangeIn of particular, parameters for for small consistent generalizations of the KKLT scenario [ which makes the theory consistent. For example, one may consider the Kählerpotential where definite, which avoids all inconsistencies of the models of [ ifests itself only iftemporarily ignores one this considers issue the andory calculates does full the contain supergravity bosonic a model potential, large one family including finds of fermions. that dS the vacuafunction If [ the- to one appears at large | with Re T dS uplifting occurs as in the original model ( 0 uplifting, at c for but due to the tiny change of Figure 1 JHEP03(2019)134 . As we 5 ] leads to ]. − ]. Thus, in 17 10 12 ], the authors , 14 × , 11 11 ], as the authors 13 77 . 11 ], which commented = 1 12 b ]. ] is inconclusive. More- ], is model-dependent; it 42 12 12 , , 6 for . 11 17 , . The dotted lines show further = 0 should be suppressed in order to 5 16 . The discussion was based on the c , − c b 11 4, 10 . . However, as we already mentioned in limit. In this sense, investigation of the ]. However, as we already discussed in ) constructed in our paper [ × and c β = 0 c 11 3.7 ], we concentrate on the 4d KKLT models. 77 c . . It also has dS vacua for a broad choice of } ] missed important terms [ 17 2, ]. Thus there are consistent models where the . ] are very short and somewhat confusing. We = 1 ] emphasized a problematic issue related to the b, c 12 – 7 – , 11 { b 12 . = 0 12 11 2 c = 0, shows the dS vacuum with a small positive vacuum ], in the small c 2 11 ]. dS vacua can be found in all of these models, even for from 0 to 12 ), see figure , c leads to destabilization of the dS minimum. 11 3.4 c ] provides correct information about the uplifting in the consistent , and they adequately describe the flattening effects, which indeed ], we believe that the 10d analysis in [ . c ], their 10d analysis, as well as the one in [ 11 β 17 imply the absence of dS vacua, contrary to what was conjectured and ], which we reproduce here, with some modifications, using the original , 11 ]. 12 , despite the flattening of the potential. 15 c 17 ] makes a conjecture that if the parameter ) at small can be large, and the flattening effect is real. However, as we are going to explain, does not 12 c , investigation of this issue in [ 3.7 2 the uplifting, during the further increase of ] do, since this model is inconsistent at large ] do something different: they instead study destabilization of dS vacua, which occurs The black solid line in figure However, while it indeed makes sense to study uplifting in the model of [ Strictly speaking, one should not study these issues in the model of [ Ref. [ This conjecture consists of two incorrect parts. In the present paper we constructed Therefore in the present paper, following [ With respect to 10d, the authors of [ 12 12 notations for the model ( energy obtained by upliftingdS with uplift due to increasesee, of the last increase of bosonic potential of [ model ( of [ after figure 2 given in [ extremely large of [ the previous section, the fullythe consistent same model bosonic ( potential as in [ valid for any value of take place in these modelsconstant as expected in [ flattening many times repeated in [ have a well-defined nilpotent description,model then the the flattening nilpotent description effects, is which not was adequate the to main focusseveral of consistent investigation 4d in generalizations [ of the original KKLT model. These models remain does not apply atis all provided to not the by version gluino of condensate the but KKLT by scenario theThe where Euclidean comments the D-brane on moduli instanton these effects. stabilization feel models that given some in clarifications [ are in order. divergences which appear insection the calculations of [ agreement with [ over, according to [ its parameters [ 4 Flattening effects inRecently KKLT there was a newon paper the on 10d this and subject, 4d by analysis Van of Riet et the al. KKLT scenario [ in [ This model is consistent for any choice of JHEP03(2019)134 . . , , c ], 4 5 c 20 − − = 1 17 c 10 10 log up to − × 02 c = 10 . 2 = c 77 c . It is the . 0 15 2 from 0 to 1 W = 1 c 1, b . 6, and 1. The last . 0 = 0 , a 4 . 0 � , 2 = 1, . , because it would imply a 220 to from 0 to A b ]. = 0 b 1, all the way up to . Moreover, as argued in [ c 16 5 ) for c − 200 15 , one finds a stable dS minimum, the c is too large, so one should choose 5 − b to 10 5 from 0 to 180 = 10 − c b 10 = 0, if to × . To present our results for such a large range 5 c 160 − 3 – 8 – 77 . , and in figure 2 in [ . These results give a simple 4d counter-example 10 b 1 ) (multiplied by 10 × in the previous section. . It is sufficient to take a slightly smaller value of 5 3.4 140 1 = 0 uplifted by increase of 77 . − from 1 c ]. However, this effect has no relation whatsoever to the b 10 = 1 , see figure 18 4 b × 120 − ]. Instead of coming to this obvious conclusion, the authors 77 10 . from 12 − , b 100 = 1 = 11 (1), which is 5 orders of magnitude greater than the required value � 1.5 1.0 0.5 3.0 2.5 2.0 0 b O W = 1, c . = 1, c ]. This effect occurs even for = 0 leads to destabilization of the dS minimum. The red line shows that if one takes 18 a c . The potential in the model ( , which leads to a stable dS minimum shown by the red line in figure 1 in a single figure, we rescale the potential by multiplying it by 10 5 − = 1, ] argue that this scenario is problematic because if one continues increasing But what if The lesson that follows from this calculation is that the increase of Indeed, it was shown long ago that an excessively strong uplifting leads to dS desta- This clearly shows that there is a large family of dS vacua in this theory, even for , still does not seem large enough to convince some authors? To address this question, A 12 c b regime only to show that even in this case the consistent dS uplift is possible. is many orders of magnitude stronger than the original impact. We considered the large = 10 = 1, this leads to destabilization of the uplifted dS vacua. of we now consider an incrediblyfor broad range of valuesof of by the less than 50%there decrease is not of much reasonphysically to unreasonable even situation consider where the the regimeb backreaction with to the smallc impact provided by minimum at b same red line that is shown in figure practically does not affect the uplift: the effect of this increase is completely compensated c bilization [ the uplift parameters carefully [ possibility to uplift from AdS to dS. It is very easy to cure the destabilization of the dS many orders of magnitude greaterfor than the statements ofof [ [ Figure 2 The black line showsThe the dotted potential lines at show furtherincrease uplift of due to increase of and decreases the value of same that is shown by the red line in figure JHEP03(2019)134 , 2 ? c 3 c 3 ∼ shifts c + 20 log τ in figure c = , which could 2 . Thus the true t µ c ) is exponentially for 3.4 c in ( log τ c , so the flattening effect is 02 . ]. The destabilization may c . But we also see that this 220 2 c c . 18 15 20 200 = 10 for large . Thus, our results do not reveal any c 2 3 to − ], it was recognized that combining this 180 c 2 1 ] was based on an appealing but incorrect 42 = 10 , . We find a set of dS minima for each value of c 160 – 9 – c ) multiplied by 10 . This conjecture was not based on any actual , and the potential is flattened by more than 40 b 12 c , see figure , c 3.4 related to the gravitino mass and the strength of 11 0 20 log 140 c W if − t b + 20 log = τ 120 τ . The minima of all of these potentials are at the same point = . That is why the position of the minimum at large 5 t t − = 100 10 1.5 1.0 0.5 3.0 2.5 2.0 T in the broad range from × c 112, with the same curvature at the minimum, and approximately the 5 . 1 ≈ forbid dS vacua. and c ≈ 5 − bc 10 . The potential in the model ( 20 log × 5 does not − . 1, for t 1 The opposite statement made in [ One may wonder how can one see the promised flattening effect at large ≈ = is 45 orders of magnitude smaller than model with inflationdestabilize effectively the leads volume to modulusoccur an in at the additional a very large contributionis early Hubble to universe constant proportional [ because to thesupersymmetry the height breaking, of square which the was of barrier often in considered small. the KKLT scenario orders of magnitude. 5 The KL model andA the year after weak the gravity conjecture invention of the KKLT model [ suppressed at large towards greater values of relation between the flattening effectdemonstrates and that the well-formed existence metastable ofb dS dS minima vacua. may In exist particular, even figure if one assumes that very real, even thougheffect it occurs only for unreasonably large conjecture that one caninvestigation. ignore It is incorrect, because the term proportional to τ same height of the barrier, making the dS vacuumThe metastable. answer is that thein height order of to the reveal barrierheight the in of similarity this of the figure the was barrier enhanced shape decreases by of approximately the as the factor potential for various bc and plot it asc a function of Figure 3 JHEP03(2019)134 , 0 t and (5.3) (5.4) (5.1) (5.2) = A ) have T 5.3 ]. We should ]. Importantly, 22 , 22 21 , 6= 0, this minimum µ . We assume, without 21 , 2 0, and branes in the first stack, 18 1 S, . 2 π/N N µ a b − = 2 + , b b bT depend on the values at which the − a, b, A, B > b B ) = 0 a A and . B 0 Be t 1 ], and therefore one may expect ( − B b B a A . If there are and 44 π/N , aT + bT ln A − − a b a 43 − = 2 , b 1 Ae − Be a – 10 – 29 a − ,DW + 0 = b B a A makes this minimum AdS. For W 0 t 0 ) = 0 0 A W t ) = or more versions of string theory compactification, after ( − does not lead to vacuum destabilization [ W = 500 T,S ) has a stable supersymmetric Minkowski minimum at 120 ( 0 . The parameters T − ( W KL 10 V W ) with the racetrack superpotential is just one of the many possible a > b ∼ 5.1 ]. , the second one for the term ], is to change the superpotential to the racetrack potential with two exponents, 25 – aT 18 − branes in the second one, one has 23 2 = 0, the potential Ae N The KL model ( Adding a small correction to In what follows we will consider the models where This problem disappears if supersymmetry breaking in this theory is sufficiently high, µ to span large range of possible values, due to the large variety of vacua in the string emphasize that for ourrequire purposes flatness (the of the theory potential,etc. of large vacuum We excursions only stabilization) of need we theis to do field parametrically make not smaller away sure from than need that the theminimum. to squares the vacuum This state, of depth can the of masses be the of achieved AdS all in (nearly moduli the Minkowski) in KL minimum the construction. vicinity of the ways to findwhich nearly may supersymmetric emerge vacua. amongall 10 quantum One corrections may arewhere consider taken supersymmetry general into is account. superpotentials unbrokena solution), in If then there Minkowski the space iscosmological tiny constant (e.g. a uplift point where required for in equations describing the ( dS moduli space space with an incredibly small the height of the barrierbe in arbitrarily this high. scenario is Therefore, notof this related the potential to can parameters, supersymmetry be which breakingtheory strongly and makes [ stabilized can it by especially a suitable proper for choice being a part of the inflationary which yields can be easily uplifted to dS while remaining strongly stabilized [ For which can be found by solving two equations, B theory landscape. ogous 4-cycles. Gauginoterm condensation on the firstand one is responsibleloss for of the generality, that KKLT-type complex structure moduli are stabilized [ but there are severalposed other in ways [ to stabilize the KKLT potential. The simplest one, pro- which can arise, for example, in the presence of two stacks of D7 branes wrapping homol- JHEP03(2019)134 0 t . for a 1 (5.6) (5.7) (5.8) (5.5) − b − a ], one might 46 , 45 . 1 . Thus the KL model B < 2 b ] that the KL model may b − & a 19 A , and is smaller than 10 B ] for a detailed technical discus- b B a A . 2 1 . , by considering sufficiently small 47 2 and 1 B < . ) as follows: = a/ 0 0 A > > 1 5.5 & b t b t 0 and − < b − A A 2. Then one can check that the strongest B 2 , as in natural inflation. For such axions, there a/ 2 , e – 11 – = , b t ) , e 2 a 1 = 1 0 1 f are strongly suppressed even for moderately large b t b < > < − − 0 0 0 e b 2 | b t a − a t φ a − a t | − e ( e ∼ b B a A and 0 a t = − 0 e , a t 0 sufficiently large, which is necessary for consistency of the supergravity − e b t 0 t ), we can represent the conditions ( and ) and the weak gravity conjecture for the simplest axion inflation models 0 5.4 5.7 a t . It becomes even easier to satisfy all conditions mentioned above for . ), ( b B 5 Then where does the statement of Moritz andTo Van Riet understand [ it, let us discuss the relation between the conditions of the type Consider, for example, the case Using ( In analogy with axion scalar potential generated by instantons, and with the knowl- The status of such models can be probed by applying simple consistency tests. Just If the number of possible vacua in the landscape is large enough, the number of nearly 5.5 and contradict the weak gravity conjecture come from? of ( with the Higgs-type potential constraint originates from the second inequality, This expansion parameter isA smaller > than 1/4 for All these conditions are satisfied,is for expected example, to if befor consistent a and broad reliable choice in of the parameters. vicinity of the minimum of its potential superpotential should be under control near the minimum of the KL potential if Note that the terms values of potentials in the KL model.be applied The to precise gaugino sense condensationEuclidean in is M5 which via this brane the intuition instantons dualsion from in description of instantons M-theory of can related (see gaugino issues). condensation e.g.would Putting as expect [ aside that the higher technical order details non-perturbative of contributions the to origin the of gaugino such condensate terms, one compactification interpretation. One cana achieve it, for any edge that D-brane instantonsanticipate enter higher the order theory non-perturbative via contributions the to superpotential the [ gaugino condensate super- supersymmetric vacua should beproblem also of uplifting extremely in large, suchthe which vacua. models should However, where it address it is may the always happen nice general in to a have explicit controllablelike way. examples in of the original version of the KKLT scenario, one should check that the radius of JHEP03(2019)134 a is 1. f (5.9) i α some- 0 + t 1. ]. Here t 48 < = 1. One can 0 1, where . The theory T a t > 1. This is the 0 2, to the weak . − t . Therefore all / 0 e 0 < 3 a t ∼ f a t a t p ∼ f , see e.g. [ . t , the periodicity of the α ,... ) π 2 bα b , + 3) − 1, i.e. if cos a bt = 1 ( bt < direction in canonical variables bt − n 1 2 e T S − 0 ) cos( − e e 2 abt bB bBW 3 ), where 1 ) + 2 − b nS + ( aα ). The requirement of the exponentially large + 3) + a O – 12 – at 3.1 cos ( being the position of the minimum. Just as in the ( = (3( t at 0 at ) t n 2 b in the KKLT model appears as a consequence of the − bS − S + e e a a , and indeed we do not expect that the supergravity 0 + ( 2 f 0 1, but this is not our immediate concern as long as t − aT aA . − 1, with aAW t 1, to ensure that the supergravity approach to string theory 2 ABe ]. Thus, just as in the usual axion scenario, the smallness of t t < 1 Ae 6 +3 − 0 19 − [ 1. Therefore, while the requirement that higher order nonper- a t 0 = 0 t 0 − t e W KL = n > V = 1 becomes less relevant. T W < in the natural inflation scenario only if 1 ) for S a 1 − for Re e nS n/f − 0 α t ( e 3 2 O 1 may or may not be required for the existence of dS vacua in KKLT. ∼ , after potentially significant quantum corrections are taken into account. However, q Finally, we return to the KL scenario. The full potential as a function of Note that the most important features of the KKLT potential, such as the position However, already at this level the situation is somewhat different from the one in nat- Now let us return for a moment to the KKLT scenario with a superpotential with a On the other hand, if, like in the string theory landscape, we do not care about natural > = is the axion decay constant, which describes, up to a factor 2 0 = n a of the AdS minimum,barrier the responsible position for of theconsistency the vacuum constraints stabilization dS mentioned are minimum above concentrated after aremay at imposed uplifting, become on and unreliable the the theory for approximation potential is at adequate for it is important tois have adequate. To beeasily on find the parameters safe which side, satisfya one t both may of also these impose conditions, but the in condition general, the condition turbative corrections should bespeculative. suppressed seems Secondly, there reasonable, is at nowe this inflation do stage in not it the need is axionic towe somewhat direction know already in the discussed, the exact KKLT the form model,where main of so requirement the is KKLT potential that in it the should axion have direction. a As dS minimum the effective axion decay constant requirement that the higher order nonperturbative effects are suppressed, ural inflation. FirstS of all, we are unaware of any higher order KKLT instantons with requires the condition natural inflation scenario, this requirementgravity is conjecture equivalent, up requirement to that adescribes the factor the effective periodicity of axion the decay KKLTθ potential constant in the Im inflation, about quantum corrections and theand shape of only the want potential in to theconstraint angular make direction, sure that the potential has somesingle minimum exponent somewhere, then the suppression of nonperturbative effects in a vicinity of the minimum of the KKLT potential f axion potential. One cantions trust the axion potentialcondition ignoring that higher makes order natural instanton inflation correc- so difficult to implement. is a series of instantons with action JHEP03(2019)134 ] ) 19 the 5.10 . , and a (5.12) (5.11) (5.10) ), and 0 /b 1 ]), which 5.5 /bt 1 ≈ b 19 , then they 0 ) is canceled ∼ t − /a ], b 1 ] imposed their a ∼ f 5.10 43 t ∼ , 19 ], to introduce an 0 t 19 ) in ( /at b 1 ) are exponentially sup- − ∼ a ) (eq. (16) of [ 5.1 | a f B | , | A | ) is also reliable. No higher order ) are satisfied at , , we see that the potential has three . 1 5.9 . α 1 min( ] imposed an additional condition that 5.10 > 3 > 0 & 19 t . Thus the authors of [ 1 2 | 0 ) ˜ ˜ µ µ t 0 b ) the coefficient ( 0 b B a A t W − ) and ( | ( ). = ln a – 13 – O (

5.5 5.5 /b ∼ 1 = ∼ b . 1 b B a A t 0 b −

1 t ≈ − a ). All important features of the KL potential appear − − f a & ). It could be tempting, following [ f /a t 1 5.10 are known to us. Thus, in our opinion, the consistency 1, as in ( 5.7 0 ∼ t < ), and the strong form of the weak gravity conjecture ( ), or require that the corresponding instanton-type expansion ) 1 ( t b 0 t − ) 5.4 > 5.6 a b ( 1 , which is not required in the KL model; for example, it was used − − a a − b e ( ]. That is exactly what we did formulating the conditions ( f ]. Secondly, the authors of [ − ) in ( , at the steep potential wall, very far away from the minimum. This ) must be smaller than the two previous terms for e b 0 49 23 ) is reliable because of the exponential suppression of the nonperturba- 1, t . However, this interpretation can be misleading. b ), one finds that for 5.9 − 0 t > 5.1 − ), one may simply ignore it. ) a ] appears only if one makes two other steps. First of all, the authors of [ ] is satisfied for a broad range of parameters such that ) seriously. As we will see, this is not a problem as well. The problem ) imply that 5.4 ) should be sufficient, and until one provides a good reason for an additional ( b 1 a t / 19 − 19 a − 5.5 5.5 f 5.10 5.10 a ( . If the consistency conditions ( / ), see e.g. [ 0 0 1 t t ( ], but not in [ ∼ & According to the discussion of the racetrack potential in eq. (21) of [ By using ( The main requirement for any model of dS vacuum stabilization in string theory is In our opinion, this last step is unwarranted. Indeed, in the limit However, suppose that one wants to err on the side of caution and take the speculative But it is hard to justify this additional requirement. Indeed, if the expression for the Indeed, nonperturbative terms in the KL superpotential ( O b 18 t − = a at are automatically satisfied at all by the pole 1 proposed in [ consistency of the KL model. to be valid int the vicinity ofeven the the dS speculative minimum condition of ( the potential for some range of values conditions ( conditions at led them to aplayed strong the additional central constraint role in their investigation, but which is not actually required for the condition ( discussed in [ assumed that in [ the last term in ( superpotential ( tive terms, then theeffects expression for suppressed the by lastconditions term ( in ( constraint ( additional condition similar to the conditions ( parameter is small, different periodicities along thethe axion potential direction. by One introducing mayf effective try axion to describe decay this constants property of pressed for Despite the fact that we consider a single axion field JHEP03(2019)134 1, > 0 b t ] exactly 19 1, 2 (including > ] and are fully a/ 2. In this case, 0 ], is compatible 7 ), and even the & , a/ a t 11 ]. 6 5.5 2 the weak gravity = 19 b a/ a > b = ], which is based on a race- b ) introduced in [ 18 5.10 ] against dS uplifting were based 12 ) introduced in [ , . There are no known restrictions on 3 2 11 ˜ µ 5.10 2 N ). Therefore for = . Our results are consistent with the results of ] and in the present paper reveal the existence 5.6 B – 14 – B , 2 17 ]. They may take different values for each stack of , 3 1 ˜ µ & 16 1 44 , A N 29 . These quantities depend on the values of the complex 1 = ˜ . We are unaware of any no-go theorems that would forbid µ / A B 2 3 , µ 2 ] does not lead to any additional constraints on the parameters & 19 A π/N ]. The full 10d theory is expected to provide results compatible with , therefore the additional constraint ( 14 = 2 , b t ) and b 13 a = 1, are satisfied for , 1 t > ) b 0 b t π/N − ) − b a a = 2 − ]. Similar results are valid as well for the more general case a a 20 Thus, the results of our analysis of all presently available consistent generalizations of We also argue that the KL version of the KKLT scenario [ We conclude, that for a broad choice of parameters, the KL version of the KKLT To give some particular examples, let us first consider the case Grant PHY-1720397, and bythe the National Simons Science Foundation and grant.MS Engineering is Research EM supported Council by is the of supported ResearchHorizon Canada Foundation 2020 in – via research Flanders part and a (FWO) innovation by and PDF programmeagreement the under fellowship. No. European the Union’s lodowska-Curie Marie 665501. grant Sk Acknowledgments We would like to thankJakob Sergio Moritz, Ferrara, Shamit Susha Kachru, Parameswaran, Liam Tomcent McAllister, Rudelius, Van Miguel Hemelryck, Eva Montero, Thomas Silverstein, Van Sandip Riet,comments Trivedi, Alexander and Vin- Westphal, and discussions. Timm Wrase The for work helpful of RK and AL is supported by SITP, by the NSF with the weak gravity conjecture for a broad choicethe of 4d parameters KKLT of model thecompatible do KL with not model. the support existence the of recent dS swampland vacua conjecture in [ string theory. the 4d description of theof KKLT the scenario. 4d All KKLT presently modelof available consistent discussed dS generalizations vacua in in [ the KKLT scenario. track superpotential and does not present any problems with uplifting [ 6 Conclusions In this paper weon note a that number the 10d ofimportant arguments terms conjectures [ of and [ an incomplete theory of gaugino condensation, missing having such parameters in the string theory landscape. construction does notspeculative violate form the of the standard weak consistency gravity conjecture constraints ( ( coincides with ourconjecture second proposed constraint in in [ ( of the KL model.and One ( can show thatref. all [ required conditions,the including case the size or signstructure of moduli, the as ratio discussed ˜ inthe [ branes, depending on moduli stabilization. one has ( Here JHEP03(2019)134 12 Phys. , Phys. , D 99 , Phys. ] , Ann. Rev. , JHEP , ]. ]. , , ]. ]. Supergravity D ]. SPIRE SPIRE 10 SPIRE (2007) 733 SPIRE IN ]. IN Phys. Rev. [ IN IN , ][ 79 ][ ][ SPIRE -attractors and IN α SPIRE (2003) 046 ][ IN de Sitter vs Quintessence in arXiv:1808.09440 05 ][ [ ]. models of de Sitter uplift ]. ]. JHEP hep-th/0302219 , D ]. , de Sitter Vacua with a Nilpotent The Landscape, the Swampland and the Dark energy, Rev. Mod. Phys. The Tension between de Sitter vacua in string theory SPIRE arXiv:1712.09693 , IN arXiv:1810.08518 de Sitter Space and the Swampland [ arXiv:1808.08967 SPIRE SPIRE [ [ ][ Brane gaugino condensate in 10d IN IN (2019) 1800075 SPIRE arXiv:1808.09428 ][ ][ IN ]. [ 67 arXiv:1810.05506 ][ Distance and de Sitter Conjectures on the [ Toward de Sitter space from ten dimensions – 15 – Physics of String Flux Compactifications (2018) 041 ]. SPIRE IN 06 ][ (2019) 1800091 (2019) 1800079 SPIRE Swampland Conjectures and Late-Time Cosmology hep-th/0701050 (2019) 180 IN 67 [ Flux compactification 67 ), which permits any use, distribution and reproduction in (2019) 1800068 JCAP Fortsch. Phys. ]. ]. ]. ][ Landscape, the scale of SUSY breaking and inflation , ]. ]. , hep-th/0301240 arXiv:1809.09018 arXiv:1707.08678 67 [ [ [ B 788 SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN [ [ [ (2007) 119 IN IN CC-BY 4.0 ][ ][ 57 Fortsch. Phys. arXiv:1901.02023 Fortsch. Phys. The Statistics of string/M theory vacua This article is distributed under the terms of the Creative Commons , [ , The Anthropic landscape of string theory Phys. Lett. Gaugino Condensation and Geometry of the Perfect Square hep-th/0411011 (2019) 046006 (2018) 046010 (2003) 046005 , Fortsch. Phys. [ , ]. D 99 D 97 D 68 SPIRE IN hep-th/0303194 hep-th/0610102 Superfield Rev. (2004) 004 (2019) 066003 String Theory Rev. and dS Uplifts arXiv:1812.06097 arXiv:1812.10448 large-scale structure surveys Swampland Era of Precision Cosmology [ Nucl. Part. Sci. arXiv:1806.08362 Rev. [ [ M. Raveri, W. Hu and S. Sethi, H. Ooguri, E. Palti, G. Shiu and C. Vafa, Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, F. Denef, M.R. Douglas and S. Kachru, L. Susskind, G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, M.R. Douglas, M.R. Douglas and S. Kachru, S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, R. Kallosh, A. Linde, E. McDonough and M. Scalisi, 4 R. Kallosh and A.D. Linde, R. Kallosh, M. Cicoli, S. De Alwis, A. Maharana, F. Muia andR. F. Kallosh, Quevedo, A. Linde, E. McDonough and M. Scalisi, F.F. Gautason, V. Van Hemelryck and T. Van Riet, Y. Hamada, A. Hebecker, G. Shiu and P. Soler, Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, J. Moritz, A. Retolaza and A. Westphal, [9] [7] [8] [4] [5] [6] [2] [3] [1] [17] [18] [14] [15] [16] [12] [13] [10] [11] Attribution License ( any medium, provided the original author(s) and source areReferences credited. Open Access. JHEP03(2019)134 , ]. , , 01 ]. , B , D 54 SPIRE IN ]. JHEP ][ SPIRE Nucl. Phys. ]. , , IN ][ ]. SPIRE Phys. Rev. SPIRE Nucl. Phys. IN D3-brane , , ]. IN ][ ][ ]. SPIRE IN SPIRE Strong moduli stabilization ][ Dirac actions for D-branes IN Ten-Dimensional hep-th/0504041 SPIRE [ ][ IN ]. ][ Yang-Mills Theories with Local arXiv:1001.5028 ]. [ ]. ]. ]. SPIRE hep-th/0408036 arXiv:1209.0499 IN Racetrack Potentials and the de Sitter [ [ [ SPIRE (2005) 2745 ]. SPIRE SPIRE Analytic Classes of Metastable de Sitter ]. IN SPIRE IN IN ]. IN Supersymmetry and stability of flux vacua 22 ][ (2010) 072 ][ Chaotic inflation and supersymmetry ][ arXiv:1811.05463 [ hep-th/0303209 [ Flux-induced SUSY-breaking soft terms on SPIRE [ 06 SPIRE IN SPIRE arXiv:1106.6025 (2005) 268 IN (1983) 457 ][ Breakdown of Local Supersymmetry Through Gauge ]. (2013) 2268 IN [ – 16 – ][ ]. ][ Fermions, T duality and effective actions for D-branes Planck 2018 and Brane Inflation Revisited JHEP (1983) 105 , (2019) 187 C 73 SPIRE B 708 B 125 (2003) 051 SPIRE Unification of Yang-Mills Theory and Supergravity in IN 01 IN D-brane non-perturbative effects and geometric deformations hep-th/9510209 04 ][ hep-th/9603142 [ ][ arXiv:1406.4866 B 120 [ [ Class. Quant. Grav. Racing through the swampland: de Sitter uplift vs weak gravity (2011) 083519 , ]. Heterotic and type-I string dynamics from eleven-dimensions Eleven-dimensional supergravity on a manifold with boundary JHEP ]. JHEP , Phys. Lett. , Nucl. Phys. arXiv:1012.4018 hep-th/0511042 arXiv:1805.00944 , , [ [ [ D 84 Eur. Phys. J. SPIRE (1996) 506 (1996) 94 , SPIRE IN (2014) 011 Phys. Lett. [ IN , [ 10 hep-th/9608019 B 460 B 475 arXiv:1811.01023 [ Gluino condensation in strongly coupled heterotic string theory [ (2011) 061 (2006) 053 (2018) 099 Phys. Rev. , JHEP (1982) 97 04 05 09 , (1983) 413 Maxwell-Einstein Supergravity, Its Currents andB the 195 Issue of Its Auxiliary Fields Ten-Dimensions Fermion Condensates Supersymmetry: Lagrangian, Transformation Laws and212 SuperHiggs Effect Nucl. Phys. Nucl. Phys. (1996) 7561 Potentials from Fluxes in AdS/CFT JHEP D7-D3 brane systems on backgrounds with fluxes and phenomenology (2019) 008 in bosonic backgrounds Vacua breaking JHEP Swampland Conjectures JHEP E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, G.F. Chapline and N.S. Manton, S. Ferrara, L. Girardello and H.P. Nilles, E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, P. and Hoˇrava E. Witten, P. Hoˇrava, D. Baumann, A. Dymarsky, S. Kachru, I.R. Klebanov and L. McAllister, A. Dymarsky and L. Martucci, andP. Hoˇrava E. Witten, P.G. Camara, A.M. L.E. Ibáñezand Uranga, L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Proeyen, R. Kallosh, A. Linde and Y. Yamada, D. Marolf, L. Martucci and P.J. Silva, R. Kallosh, A. Linde, B. Vercnocke and T. Wrase, R. Kallosh, A. Linde, K.A. Olive and T. Rube, E. Dudas, A. Linde, Y. Mambrini, A. Mustafayev and K.A. Olive, J.J. Blanco-Pillado, M.A. Urkiola and J.M. Wachter, J.J. Blanco-Pillado, R. Kallosh and A.D. Linde, J. Moritz and T. Van Riet, [36] [37] [34] [35] [32] [33] [29] [30] [31] [27] [28] [25] [26] [22] [23] [24] [20] [21] [19] JHEP03(2019)134 ] , ]. JHEP (2014) , 10 ] SPIRE (2008) 483 (1996) 343 IN Fortsch. Phys. and dS , 87 ][ JHEP D3 hep-th/9808087 hep-th/9907026 , [ , B 474 Supersymmetric models with Les Houches ]. arXiv:1404.2601 (1998) 007 , [ Cambridge Monographs on Nucl. Phys. , ]. 09 , arXiv:1411.1121 [ SPIRE IN SPIRE ][ JHEP IN , ][ ]. Gluino Condensation in Superstring Models (2014) 117 ]. 12 SPIRE On uplifts by warped anti-D3-branes IN – 17 – ][ SPIRE ]. Cosmology with Nilpotent Superfields IN JHEP , ][ arXiv:1804.10504 [ ]. Inflation and String Theory hep-th/0607050 SPIRE [ Superpotentials and membrane instantons IN ]. [ ]. SPIRE Emergence of Spontaneously Broken Supersymmetry on an IN ][ SPIRE , Cambridge University Press, Cambridge U.K. (2015) SPIRE IN (2006) 031 arXiv:1809.06618 IN (1985) 55 ][ [ Large Field Inflation/Quintessence and the Refined Swampland Distance ][ 11 arXiv:1502.07627 [ On D3-brane Potentials in Compactifications with Fluxes and Wrapped PoS(CORFU2017)175 B 156 Nonperturbative superpotentials in string theory , JHEP Les Houches Lectures on Constructing String Vacua , ]. ]. ]. arXiv:1408.4096 [ (2015) 058 (2019) 1800098 SPIRE SPIRE SPIRE IN hep-th/9604030 IN arXiv:0803.1194 IN mathematical-physics/inflation-and-string-theory?format=HB [ Conjecture Mathematical Physics http://www.cambridge.org/mw/academic/subjects/physics/theoretical-physics-and- [ [ [ product groups and field dependent[ gauge couplings A. Murugan, D-branes 05 67 Phys. Lett. 143 Anti-D3-Brane in KKLT dS Vacua R. Blumenhagen, D. Baumann and L. McAllister, E. Witten, J.A. Harvey and G.W. Moore, F. Denef, D. Baumann, A. Dymarsky, I.R. Klebanov, J.M. Maldacena, L.P. McAllister and E.A. Bergshoeff, K. Dasgupta, R. Kallosh, A. Van Proeyen andJ. T. Moritz, Wrase, A. Retolaza and A. Westphal, C.P. Burgess, A. de la Macorra, I. Maksymyk and F. Quevedo, S. Ferrara, R. Kallosh and A. Linde, R. Kallosh and T. Wrase, M. Dine, R. Rohm, N. Seiberg and E. Witten, [48] [49] [45] [46] [47] [44] [41] [42] [43] [39] [40] [38]