JHEP01(2020)169 Springer October 22, 2019 January 16, 2020 January 28, 2020 : December 19, 2019 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP01(2020)169 ]. 5 , 2 [email protected] , ]. A large set of supersymmetric Minkowski vacua with 1 ]. Stability of the resulting dS vacuum is inherited from the 4 , 3 . 3 ]. The second step involves a parametrically small downshift to a super- 1910.08217 2 The Authors. c Supergravity Models, Superstring Vacua, D-branes

A three-step procedure is proposed in type IIA string theory to stabilize mul- , [email protected] 6-brane contribution [ Stanford, CA 94305, U.S.A. E-mail: Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: symmetric AdS vacuum, whichThe can third be step achieved is by anD a uplift to small a change dS ofstability vacuum of the with the a superpotential. original positive supersymmetric cosmologicalin Minkowski constant vacuum dS using if vacuum the the is supersymmetry parametrically breaking small [ localized stable supersymmetric Minkowski vacuum,be or done, a for discrete example, set using ofmodulus, two such as non-perturbative vacua. exponents in It in can thestrongly the KL stabilized superpotential moduli for model each is [ protectedof by flat a directions theorem [ on stability of these vacua in absence Abstract: tiple moduli in a dS vacuum. The first step is to construct a progenitor model with a Renata Kallosh and Andrei Linde Mass production of type IIA dS vacua JHEP01(2020)169 ]. The 8 8 19 15 12 ], a large volume compactification 6 8 14 = 1 supergravity N 22 – 1 – 23 25 22 12 11 10 16 6 5 12 6= 0 and supersymmetric AdS minimum 5 W 1 20 ], and the earlier constructions of dS vacua in a non-critical string theory in [ A.3 A large uplift without downshift A.1 A veryA.2 small downshift and A uplift significant downshift and uplift 5.2 Masses of5.3 scalars and fermions Supersymmetric in 5.4 Minkowski minimum Downshift to 5.5 Uplift to dS minimum 5.1 A lesson from WZ model 2.1 A general2.2 theory The basic KL model 3.1 Finding a3.2 supersymmetric Minkowski vacuum Multifield KL scenario 7 KKLT construction, in thedure. effective 4d First, supergravity using approach, stringysuperpotential, involves perturbative a one and can two-step non-perturbative stabilize proce- the contributionsvacuum. volume to of an Secondly, extra effective dimensions withter in the a vacuum supersymmetric with help AdS a of negative an cosmological anti-D3-brane, constant to one a can de uplift Sitter this vacuum with anti-de a Sit- positive 1 Introduction The problem of constructiondimensions of is important de and Sitterthe complicated. vacua KKLT construction from Some in string type ofin IIB theory the [ superstring compactified solutions theory of [ to this four problem include 7 Conclusions A Some illustrative examples 6 String theory embedding of the new 4d supergravity models 4 An example: STU model 5 Stability of dS vacua 3 Multifield supersymmetric Minkowski vacua and their dS uplift Contents 1 Introduction 2 A single field scenario JHEP01(2020)169 ]. A (1.1) 23 – 9 ]. The basic 1 ) are only possible . 1.1 a 0 z ]. However, recently a = a 37 z – 27 at , , where both the superpotential and its = 1 supergravity, this second step can be = 0 0 a z N W a – 2 – ∂ = W a ]. It involves KKLT-type nonperturbative superpotentials 4 ]. ,D 26 – ) = 0 24 a z ( The resulting potentials have a set of isolated minima without flat di- W 1 ]. Most of the properties of these potentials, including the positions of their 5 , 2 ]. It was explained there that by adding pseudo-calibrated anti-Dp-branes wrapped ] was confirmed there by a direct numerical analysis. Thus one may argue that the A single-field realization of this mechanism, called KL model, was originally developed In particular, these vacua remain stable after a small downshift to AdS and a small In this paper we would like to suggest a procedure which allows mass production of Meanwhile for many years the situation in the type IIA string theory remained unsat- In the KKLT type single-exponent models, Minkowski solutions of equations ( 3 4 1 rections, with high barrierslarge separating [ them, and with moduli massesat that infinity can of be thepotential moduli very has a space, supersymmetric whenthe Minkowski parameters. the minimum exponent at vanishes. finite values In of the the two-exponent moduli, KL-type for models, a the broad range of as a specific versionidea of of the the KL KKLTsupersymmetric scenario construction Minkowski is in vacuum that using type instead racetrack IIBpotentials of superpotentials, to string searching and dS. for theory then a uplift [ stable these AdS vacuum, one finds a supergravity is to startbe with relatively a easy set to ofset find. of stable the Minkowski Then stable vacua,dS which, AdS one vacua. vacua, as should and we proceed As consequently willMinkowski with uplift we vacua see, a them if will can to supersymmetry small show, breaking an downshift these in even to vacua dS larger a vacua retain set is larger stability of sufficiently stable of small. the original supersymmetric We additionally require theremain absence stable under of sufficiently flat small directions. deformations of Such the original vacuauplift theory. are to stable, dS. and The they main idea of the scenario of mass production of dS vacua in type IIA dS vacua in thestable type supersymmetric IIA string Minkowski theoryvacua vacua. based depends on on The modelscovariant existence existence which derivatives vanish, have of of as some supersymmetric their points Minkowski progenitors for each of thein moduli. [ Thesituation existence with of dS a vacua stablethe in dS type the IIB state type case. in IIA the string STU theory model became presented similar to the situation in possibility of obtaining metastablein de Sitter [ vacua in typeon IIA supersymmetric supergravity cycles, was and, proposed effective four-dimensional more supergravity specifically, derived anti-D6-branes, froma string one theory nilpotent can in multiplet. a generalize An way the theory that explicit it has KKLT-like includes two-step been construction presented in in the [ type IIA string described by a non-linearlydetailed realized derivation of supersymmetry the and KKLTpresented a construction recently nilpotent of in de [ multiplet Sitter [ vacua from ten dimensionsisfactory, as was illustrated by numerous no-go theorems, see e.g. [ cosmological constant. In four-dimensional JHEP01(2020)169 , 2 χ / 2 3 hc m (1.2) (1.3) m + b = χ a 2 | χ W ab | m K 1 2 e − where 2 | W | K e 3 − . transforms the original Minkowski 0 . = > cb W = 0 in a theory with multiple chiral ¯ m cb AdS V ¯ ac m Mink m ac V m = . | – 3 – = W Mink ∆ ) | ) with b Mink a ) b V 1.1 a ( V ( ] where also the importance of the absence of flat directions of the 2 , and the chiral fermion mass term in the action is b ¯ b is not degenerate. Thus, the first goal of our proposal is to find a class ]. The second derivatives of the potential over scalars form a strictly pos- b V g a b ¯ 38 we will show that the downshifted AdS state remains a minimum of the V D , up to higher orders in 5 a W D ∆ = b ≈ a V W In section In this paper we will construct a broad class ofThe multi-field second models stage of which our satisfy proposal all is the downshift to a supersymmetric AdS. A change Below we give a short description of the three stages we will use here to build a stable A particular example of the scenario of mass production of stable dS vacua developed of the superpotential by addingvacuum into to a it supersymmetric a AdS small vacuumand with term ∆ potential under the condition that the scale of the smallest (non-vanishing) eigenvalue flat directions were presented. required conditions mentioned above.are This still is required. the basis of our work, but two further steps of supergravities where such properties are realized, This was explained in [ potential was discussed, and some examples of local KL-type Minkowski minima without Here see for example [ itive definite matrix under aand condition the that matrix all fermion mass eigenvalues are non-vanishing general class of modelsproperty with of stable Minkowski all minima Minkowskimultiplets without is vacua flat that ( directions. thethe holomorphic-anti-holomorphic The fermion mass basic mass matrix matrix, of scalars is a square of cussed in this paper. Inprotecting the the more vacuum conventional version state of typically the disappears KKLT mechanism, in the the barrier limitdS of vacuum small state SUSY in supergravity breaking. inKL general, type, and inspired in particular by in type supergravity IIA models of string the theory. The first stage of our proposal it to find a moduli consists of awe sum can of find KL-type a superpotentialsculation, stable for which supersymmetric yields each Minkowski a of vacuum strictlywithout the state positive flat moduli. by definite directions. a mass In These simple matrix thisthe results analytical uplifted for case remain cal- dS all practically if string unaffected the theory resulting andis SUSY moduli, ensure the breaking stability special in of the property uplifted of dS state the is KL not too construction large. and This the of set of more general models dis- broad range of parameters. in this paper is basedbilization on in a the generalization type of IIAa the string significant KL theory. simplification scenario While to occurs many the in of problem our the of results models moduli are where sta- model-independent, the superpotential of string theory minima, the masses of the moduli at the minimum, etc., can be found analytically for a JHEP01(2020)169 ) 1.6 (1.4) (1.5) (1.6) ), ( 1.5 and therefore the for general choice 5 120 − is small enough. Thus 10 ) so that W ∼ 2 1.4 F , 2 χ . m ]. The dS potential has a minimum 0 and in section 4 ,  4 3 > , ), the uplifted dS vacua inherit their 2 . ) the stable Minkowski vacuum remains 2 / 2 3 F 2 / 2 3 if the term ∆ / / 2 3 3 3 would have m m m W 3 and 120 −  – 4 – − , m 2 W | χ 2 χ F m | m = 10 is small, in addition to ( in supergravity [  dS 2 | Λ = V 2 X ) are known as conditions of strong moduli stabiliza- | F | 1.4 DW | is much greater than the scale of the gravitino mass K the result is derived using an explicit expression for the second (and, consequently, e ac 5 = W m 2 F ]. Note that the mass matrix of the strongly stabilized Minkowski 41 – in Planck units. We only require that the susy breaking should be smaller 39 15 . In section , − 5 W , = 0 to the dS state with 2 , 1 and V The reason we want to start with Minkowski rather than AdS is that the discrete One may wonder why would we need to start with a supersymmetric Minkowski vac- Finally, the third stage of our proposal is the uplift to dS, using the anti-D6-brane, or K supersymmetric Minkowski state. The downshift iswith required because the without observable it, value the vacuum ofwith the cosmological constant upliftedSUSY breaking from would the be Minkowski too state form small. the smallness A of controllable downshift the disentangles cosmological SUSY constant. breaking dard version of theMinkowski vacuum KKLT before construction? upliftingthe to And Minkowski dS, vacuum? why whereas one do can we make want the to upliftsupersymmetric directly downshift Minkowski from from state the This is feature always continue a protecting minimum stability of and the can AdS be and strongly dS stabilized. vacua originating from the which is 10 than the typical scalewhich can involved be in extremely the high in supersymmetric supergravity. Minkowski vacuumuum construction, and then downshift to AdS, instead of starting directly with AdS, as in the stan- of derivatives of the potentials inare supergravity. always If met thestability required for property conditions from sufficiently are their small met Minkowskisupersymmetry progenitor. (and ∆ they breaking Note that must this occur does on not a mean that very the small scale, such as the electroweak scale and the dS potential is positive if A general analysis ofis stability of performed the in uplifted multifield dS KL vacuum, models based in on sections conditions ( at the position offor moduli dS slightly state to shifted be fromnilpotent a direction, the minimum ones requires that they the had total in supersymmetry AdS. breaking, including The a condition for sufficiently small ∆ a stable AdS after thevalues downshift. of In the this gravitino way mass. we construct AdS model withan many equivalent different nilpotent multiplet Constraints of thetion type [ ofminimum ( practically does not depend on ∆ of the fermion matrix JHEP01(2020)169 if 0 T (2.7) (2.1) (2.2) (2.4) (2.5) (2.6) = T we will present re- . . A ) T = 0 ( 0 1 . T W = ) T T + (  1 0 ) dT . T W , ( dW 1 2 dT ) ) = 0 (2.3) + . ) = 0 with 0 ) = 0 dW T ) T T 0 ( ( T  i θ T = 0 − 1 ( ( 0 ) for its derivative is satisfied at a localized ¯ 1 0 T + + T W T t W ) W = + 2.5 0 + – 5 – T = + | T 0 T using the string theory landscape scenario. 0 ( ,W ) = ( 0 0 ) = 1 ¯ T W ) T 0 T W W ( W 120 T T = + 3 ( − W + 0 + − ) as a simple condition W T 10 0 W T = 2.4 W ≈ 3 3 ln( − DW − = . = A K ) such that equation ( DW , one can certainly obtain many of them by an uplift directly from the T ( dS V W ). For example, one may consider a superpotential 0 T ( ), one can represent ( , or at a series of points, rather than along a flat direction. Then one should take W 2.3 0 − T Our investigation will be mostly analytical, but in appendix Thus, in the mass production scenario one can construct a large variety of supersym- But if a general goal is to study a possibility to construct stable dS vacua with dif- = 0 which satisfies the required conditions This corresponds to a supersymmetric stable Minkowski vacuum state. Our strategy of findingpotential a having a theory localized with minimum consists afind of the supersymmetric a following function Minkowski steps. vacuum Firstpoint state of all, with one a should W Using ( This theory has a supersymmetric Minkowski vacuum at some value of the field and 2.1 A generalConsider theory first a theory of a single field In this theory one has sults of a numerical analysismain of conclusions. a particular string theory motivated model illustrating2 our A single field scenario states, depending on many differentof ways stable of dS the vacua downshifting, obtained andmany either an possible after even the supersymmetric greater downshift, number Minkowski or vacua.on by multiple The a AdS vacua direct three-step and uplift procedure many versions fromof allows of one the uplifts. to of cosmological rely This the constant may Λ help to address the smallness ferent values of supersymmetric Minkowski vacuum. Wedownshift present in an appendix example of such an upliftmetric without Minkowski vacua, a an even greater variety of stable supersymmetric downshifted AdS JHEP01(2020)169 . 0 t (2.8) (2.9) = (2.10) (2.11) (2.12) (2.13) ]. For 0 5 appears T 2 m = 0 [ to can be achieved 2 V 2 dt d DW . One can describe AdS is real, i.e. X V 0 . is negligibly small, and i θ 3 0 to the Kahler potential, t 4 0 t X, + µ ¯ relating 2 24 X 0 . 2 0 µ t t 3 . X 2 = 2 ) 2 + = |
3 0 ) 0 W t bT 0 ¯ 3 T 8 t AdS − X, ( V 00 | X 3(∆ Be . ) to a dS state with a nearly vanishing W = 2 − ) ) + 2 2.9 ≈ − 0 ¯ ) ( T t , the required uplifting does not affect the 9 aT W 2 . As a result, the vacuum energy becomes 3 0 2 ) t − W + W 8 W AdS W = . T Ae (∆ – 6 – V 5 2 V (∆ + = 3 (∆ = 2 0 dt K 3 ln( 4 d to the superpotential, the Minkowski vacuum state . An extra coefficient 2 e ) 0 µ − W 3 2 0 t requires the uplift parameter t 3 W − W 2 = = ) = = 120 (∆ = K T − is given by ∆ K 2 e 10 AdS T,X m W ( V = ∼ to the superpotential and a term 2 KL / at the minimum of the potential is given by dS 3 2 X is not canonically normalized. W V 2 , one finds that the shift of the field from T m µ T = 0 after calculating all quantities of interest, such as the potential. W X ], is to use the K¨ahlerpotential 1 ) = 0 becomes a stable supersymmetric AdS vacuum, with 0 t ( V If one adds a small constant ∆ For simplicity, we will consider the theories where and the racetrack superpotential with two exponents which can arise, for example,ogous in 4-cycles. the presence Gaugino of condensation two on stacks the of D7 first branes one wrapping is homol- responsible for the KKLT-type 2.2 The basicThe KL simplest model string theoryproposed related in realization [ of the general mechanism described above, strong stabilization of theapplicability original of Minkowski these vacuum state. conclusionsmass and An should the approximate be estimates criterion much is of detailed smaller the discussion than requirement of other that this mass issue the parameters in gravitino describing section the potential, see a The gravitino mass after uplifting is given by Once again, for sufficiently small values of ∆ and then taking One finds that the upliftcosmological of constant the state with An uplifting of this vacuum statedue to to a the dS anti-D3-brane state contribution with represented ait by nearly by a vanishing nilpotent adding field a term due to the fact that with sufficiently small ∆ the leading contribution to The mass of the field Here all derivatives are taken at JHEP01(2020)169 , ], A i θ 44 and – 0 + (2.17) (2.18) (2.19) (2.14) (2.15) (2.16) . This t 42 W = CB T → B , at the supersym- . T CA . The parameters b B a A 2 . ), separated from each 0, and → satisfying the condition ln 2 branes in the first stack, ) b b A . π/N 1 + − 2.16 . a a W N  − a = 2 b (∆ −  nπ , b b 3 2 a, A, b, B  i ! bB aA a, b, A, B >
. + and  b , but it increases the value of ) = 0 b B a A ) 0 0 1  bB aA t b ) is periodic in the Im T direction with −  T b B a A ( . Meanwhile the simultaneous rescaling T a − B ( 2 . If there are b B a A ln π/N = 0 shown in ( a ln C V + ( bT

ln = 0 to  − V a 3 8  = 2 b a − V b b − a Be 1 − – 7 – makes this minimum AdS, with the value of the  aAbB 1 − − a = 0 ,DW to span large range of possible values, due to the 2 9 a 2 W b B a A = ) B = = 0 = 0 is given by  3 0 by a factor 2 to W t ) = 0 , and it increases the height of the barrier stabilizing the 0
iθ 8 0 A t 0 V ) exists for any values of ) t t C and 0 W ( + − t 3(∆ ( 0 A 00 t = W 2.17 − 0 = W = W 0 0 T t 9 2 AdS by a factor V = T . 0 2 t = 0. Looking for the solutions in the full complex plane m 2 µ under the simultaneous rescaling of the parameters . It has a series of minima with , the second one for the term 0 b 2 t π aA > bB − 2 aT a m = 0, branes in the second one, one has − depend on the values at which the complex structure moduli are stabilized [ 2 and W Ae N B It is useful to understand the properties of the KL potential and the mass of the volume Adding a small correction ∆ As a first step of our analysis, we look for supersymmetric Minkowski minima, which In what follows we will consider the models where = 0 and modulus rescaling does not affect the positionthe of mass the of minimum the field minimum in the KL potential at The supersymmetric vacuum ( a > b potential at its minimum downshifted from θ The mass squared of bothmetric real Minkowski minimum and with imaginary components of the field The last term reflects the facta that period the potential other by high barriers.could be Importantly, destabilized the after potential smallit does deformations is not of sufficient the have potential. to any flat limit Without loss directions our of that investigation generality, to the supersymmetric Minkowski vacuum with for ∆ one finds can be found by solving equations and and and therefore one maylarge expect variety of vacua in the string theory landscape. term JHEP01(2020)169 ), 3.2 (3.6) (3.2) (3.3) (3.4) (3.1) , and 3 c ), ( 3.1 ]. Some aspects of 45 ]. , by a factor of 46 0 2 t 40 , , but in this section and in 5 m . For the theory ( ) (3.5) i 5 0 . n , independent equations one should impose the conditions
. , n 0 n i . ) , . . . , n , ,...,T T i T . Note that this result is valid inde- ) 0 , 1 for each ]. Thus one has a significant freedom 0 T i n n ( T 5 = , increases 0 T i [ = 1 i c ) = 0 i i T c U( n ) = 0 W T , with a general K¨ahlerpotential = , 0 n 0 i = i − is very simple. First of all, one should find n =1 ,...,T T i ,...,T T i ) X 1 1 0 ), leads to T n i T T = + T ( – 8 – ,...,T , . . . , n , i 1 3.3 0 ,...,T 1 2 T T W 0 = , 1 T ( W i T = ,...,T ( T W = 1 = 1 K i T W T by a factor of i W ∂ = W , 0 i t T ]. Importantly, the height of the barrier in this scenario K = U( 41 – W 39 , ) which has an extremum at this point. Then the theory with the n decreases 5 , cb 2 , 1 [ ,...,T → 1 2 T b ) = 0, which should be satisfied at W and we will concentrate on superpotentials W i 4 3 (∆ T ca D We will analyze a general class of such models in section In this context, the procedure of constructing a theory which has a supersymmetric Now we will generalize the single-field models discussed above to the multi-field This minimum can be easily uplifted to dS while remaining strongly stabilized by taking > → 4 pendently of the choice of the K¨ahlerpotential. section a function U( superpotential has a supersymmetric Minkowski vacuum at which should be satisfied at some point Minkowski vacuum at a given point To find a supersymmetric Minkowski vacuum at and this last condition, in combination with ( and a superpotential 3 Multifield supersymmetric Minkowski3.1 vacua and their Finding dS a uplift Consider supersymmetric a Minkowski theory vacuum of the fields potential can be stronglyespecially stabilized suitable by for a being properimplementation choice a of of part this the of scenario parameters, the in which inflationary string makes theory theory it are [ discussedconstructions in capable [ of stabilizingdS spaces. many string theory moduli in AdS, Minkowski and increases the height of theto barrier change the by properties a of factor the of KL potential. µ is not related to supersymmetry breaking and can be arbitrarily high. Therefore, this a JHEP01(2020)169 , as = 0. j (3.7) (3.9) i (3.14) (3.11) (3.12) (3.13) (3.10) V dθ θ 2 i d , dθ 0 i t = and i t j V dt . In other words, 2 i j d , and to a separate dt i , and by adding to V dθ t 2 i d X dθ 2 , the uplift of the AdS µ 4 and . , ), but it shifts the minimum j n , k 2 V . dt ) . 2 i j  d n 3.12 j dt one should have , N . T 2 i ) − 0 W V i dθ i ) ) = 0 2 i 2 + T dt T 0 j 0 i d d t = 0 n dθ i T = + T 2 i ( j (2 i 2 i ( t i N i i t · V N T T 2 n dθ =1 ¯  W 2 i Y j X d j 2 = 0 (3.8) ... = ln( dt n X ) =1 N · i i i X j − 1 W ) N W k V + dt – 9 – . The supersymmetric vacua that we are going j i ) i 2 = 0 one has i t T 0 1 d i n , =1 ∂ iθ dt T i (2 θ X 3 (∆ W i 2 i + t n + − =1 N = 0 Y j 2 i 1 t ) = i = ≈ − 4 T 0 = i ( = dθ dV T 2 i K i ( to the superpotential does not affect the vacuum stability = 2 = AdS T m 2 i V W K W m as ∆ . i i θ T ) in the minima with = 0. A useful general result simplifying our investigation is that in the are equal to each other, and are given by 3.6 i i θ θ ), ( . We will consider the K¨ahlerpotential 0 i 3.9 and T i t = As we will see in the example of the STU model in section Adding a small term ∆ Moreover, at the supersymmetric Minkowski vacuum, the matrices i T 6= 0, and i vacuum can be achievedthe by K¨ahlerpotential adding an uplifting to term, the which superpotential can a be term generally represented as and makes only a smallto contribution the to AdS the state mass with matrix the ( negative cosmological constant The final result is where the whole expression should be evaluated at the Minkowski minimum well as the mass matrix of theare canonically diagonal. normalized fields at The the minimumfields masses of the of potential, the fields of the canonically normalized counterparts of the The only non-zero second derivativesthe of corresponding the matrix potential is are block-diagonal.vacuum state This by simplifies reducing investigation it of toanalysis stability the with of analysis respect the of to stability with respect to to explore, ast well as the vacuamodels ( after the downshifting and uplifting, are positioned at at and represent the fields and To find a supersymmetric Minkowski vacuum at JHEP01(2020)169 X (3.24) (3.19) (3.21) (3.23) (3.18) (3.20) (3.22) (3.15) (3.16) (3.17) X. . 2 ) µ i T + + . i W | T . i ln( 3 N + ∆ AdS i − 0 i V t | ) uplifts the vacuum state to i . N 0 i b t 0 i = − t X n i =1 e (2 2 j b i i X µ . . − N n =1 B 0 i i i − e i Y 0 − t i i k k ) 2 . T n i 0 j − − ) =1 = B i = i i b t ) ) X 0 i 0 i − 0 B A i i W t t (2 i e + i X i T a b 0 i (2 (2 n =1 t B Y j i + i n n 3 (∆ ln a =1 =1 2 b ¯ i i 0 i Y Y X, ) t − ,K i i 2 2 , are − e b i = a i ) ) i X i W T ) − 0 i i 1 i A N − – 10 – b e W W T ¯ i i T − i − i , a a n e =1 A k i i i , we have the value of the potential at the minimum X − ) are fixed in the supersymmetric Minkowski vacuum = (∆ T i e 3 (∆ 3 (∆ µ B = i ( i t i 2 − ¯ ) X T 0 i X i A > ≈ (2 t n i =1 T X i − X 4 4 i W a and n b =1 µ µ i Y K − − = 4 (∆ e W i 0 i µ T K i ) + B e i W a = T − = n =1 e i i X + 2 min / i A 2 3 − V T i m n =1 T i i X ln( a i − + e N 0 i A W n =1 i X n = =1 . The real field solutions, for each i − X i W = = The gravitino mass after uplifting to a (nearly Minkowski) dS vacuum state is given by K W This finally yields, for both choices of the superpotential, The K¨ahlerpotential and superpotential of the full theory involve a nilpotent multiplet for each Using the prescription outlinedtwo superpotentials above one in finds a applicationdirections stable to supersymmetric for Minkowski all the vacuum moduli. without modelsstate any All flat with at fields their either values of satisfying the these equation 3.2 Multifield KLAs scenario a particular string theorysuperpotential motivated used example, in one the can KL take a scenario theory for with each the moduli: racetrack This state is a dS vacuum with an extremely small cosmological constant for a stable dS state. For small ∆ This corresponds to a stable dS state for A subsequent addition of the nilpotent field contribution JHEP01(2020)169 , 6 i , b W f i , i (4.1) (4.2) (4.3) (4.4) = (3.25) A . 0 , . = 0 The i W )) a  2 2 ) T µ 2 ). A general T + 2 + of our paper. ). = 0, T 3.17 i 2 ( ). Note that this 5 , and, for each T T g , whereas ( 0 i ( W i t ), ( U g ¯ X 3.13 W ) + . 1 X ¯ i i ) + 3.16 X T b b for ∆ ]. The KL generalization 1 ) for + − X T i i 0 i + 4 , a a t 1 3 + − 3.21 is a nilpotent field, such that T and vanish in the absence of = 1  i (( T i i 2 X 3 ( T ) µ A ] and in section B i 3 i − as shown in ( b a 5 T corresponds to , 3 , the minimum of the potential shifts , 2  3 ) ) + 2 3 2 , T 2 2 t W 3 ) 1 T i 1 W q T t b ( + (∆ 3 − + under the logarithm represents the uplifting , and )+ i T − 2 ) 2 2 3 ( a T t 2 p ( T under the conditions ( T  i – 11 – + + = B µ 2 ln 3 i T 6 b T ( i − ¯ g D X X, A ] can be represented, in slightly different notations,
V = 0, as well as for sufficiently small values of these and are proportional to i X 2 )+ 2 4 1 a 3 ln( µ T 2 µ T stands for is some constant. Since q W − + = 2 + ) 1 g 2 ) + 2 T T 2 i ( T , T  proportional to T and ( S + 2 i i 2 ), where one should use equation ( W 2 2 T p W AdS dt 3 ln = 0 and d V 3 3.12  =1 6 brane, − i X 3 ln( W ) D 1 − + T ) ]. The term 1 4 W + T 1 + we mentioned that the large freedom of choice of the parameters T 1 + ∆ T ln( 0 2.2 ln( stands for the field W − , are given by ( 1 − 2 = = T µ Thus, this model belongs to the class of the models studied in the previous section. With an account taken of the small contribution represented by a nilpotent superfield If one takes into account the tiny correction ∆ = = 0, the K¨ahlerpotential can be equivalently represented as gives us a possibility to significantly modify the properties of the potential. K W 2 , the minimum can be easily uplifted to dS. The uplifted vacuum describes a stable i K where both parameters the uplifting. In particular, for ∆ The last term in this expressionwhich is can responsible for be the represented uplifting as contribution to the potential, Here in notations of [ contribution of the X 4 An example: STUAs an model example particularly relevant for thewill dS consider construction here in the the type STUof IIA string model, the theory, extending we STU the analysisas model of follows: considered [ in [ X dS state for sufficientlydiscussion small of ∆ stability conditions can be found in [ In section B to the AdS stategeneral with result does not depend on the choice of the superpotentials masses of all moduli inand the Minkowski vacuum, or in an AdS or dS vacuum with small ∆ This model has a supersymmetric Minkowski vacuum at JHEP01(2020)169 = 1. from (5.4) (5.1) (5.2) (5.3) (4.5) (4.6) P 0 i t M is given by , arbitrary a χ z χ ) ). , unbroken su- = 0 is equal to a M z M ( B . + 2 W = µ ) ∂ 2 A µ . Therefore the second B covariantly holomorphic γ ( + M ¯ χ . 2 2 1 0 A fields. This expression should ( − . , 2 > ) i 4 U g 2 2 ]. We use units in which T ), where one should take 2 i b  B − 47 M , − 4.5 2 ) and e and a Majorana fermion ) i + W 2 i ¯ = a ), we define a 2 . 2 . dt T ¯ a z B B d | 0 i ) , A z t i ¯ AB ( ( a m 2 i = 1 supergravity − 2 t z N = i 2 ( + W m T i A, B  | M i N 3 t ( m a 2 1 dB A = 3 for 3 3 = − V ( t ≡ e 2 – 12 – 3 − 2 i 2 d 3 2 / ) 2 t N 1 3 A W √ Mg 1 B 2 = t M = µ 8 ) ). − K/ 2 ) = i e χ B∂ N = T ) 2 ( µ 3.25 2 i i A, B ∂ B ( and a pseudo-scalar 5 m dA W + V ], originally introduced in [ iγ A 2 2 A d + from ( µ ), and arbitrary holomorphic superpotential ¯ a A field, and 2 ¯ ( i W z A∂ 2 , ¯ χ S µ dt d a ∂ 2 z ( g ( √ 1 2 K , a complex gravitino mass − − m ), and = L 3.21 = 1 for the 1 N We use the notation of [ A particular string theory motivated set of superpotentials for the STU model is given superpotential which is related to the (real) gravitino mass, 5.2 Masses ofConsider scalars a and general fermions case in K¨ahlerpotential of an arbitrary number of chiral matterIn superfields addition to the holomorphic superpotential This simple consequence of supersymmetry isof the de underlying Sitter reason vacua, of which the mass we production will explain below. If the non-vanishing masspersymmetry of requires the that fermion thederivative in of mass a the of single scalar the chiral potentialthe scalar multiplet square at field is of the the is supersymmetric fermion also minimum mass. The WZ model with a scalar as in its stableequation AdS ( and dS descendants, is given by ( 5 Stability of dS5.1 vacua A lesson from WZ model by the KL superpotentials In this set of models, the mass matrix in the supersymmetric Minkowski vacuum, as well vacuum, are given by where be evaluated at the Minkowski vacuum with parameters, the masses of all moduli in the Minkowski vacuum, or in a slightly uplifted dS JHEP01(2020)169 = 0 = 0 (5.5) (5.6) (5.7) (5.8) (5.9) (5.17) (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) (5.10) V W a , see for = D ab . m = W ¯ a a ¯ V . m D a ∂ ≡ = 0 m ) ¯ ¯ m . K  ¯ a a ∂ . ( K 6= 0 W. 2 1 2 , 1 2 | . c b ¯ ¯ b ¯ b ¯ W. ¯ + m D ¯ a m ¯ ¯ a − | m ¯ ¯ 3 c ab a b m ¯ ¯ D m b ∂ Γ 2 ¯ a and that the potential is K¨ahler g − 2 ≡ ∂  2 K/ ab , | ¯ . K/ m ¯ e m = a ≡ . b ¯ m e b W. ¯ ! a b . m hc . = ¯ ¯ ¯ g 6= 0 + m m D − ∂ ¯ ab ab ¯ a ¯ ¯ a a a a = 0 a + = 1 supergravity are equal to ¯ V V ≡ | ¯ ¯ = ∂ ¯ m b D m D = 0 W m 2 b b ¯ ¯ 2 V b a χ b N ¯ | b ¯ ¯ a a a a ¯ m D K/ V V m m m ) χ D e 2 | a a

3 – 13 – ,D , , ab − = ¯ , D = = K a b ¯ m a − ab = = W, ¯ V m 1 2 + b ¯ m 2 m a = 0 m a b ¯ ¯ a V m ∂ b Mink ab ≡ D − a ∂ ≡ M b m ¯ g 2 ( a m D 2 . Using this notation we can rewrite the standard F-term m ) as follows ab  g m K/ ) 2 ¯ a b | a e m K = K/ b ¯ K e K D m a ∂ W = = V a 1 2 | a ∂ a = 3 a = ( ∂ D ¯ ∂ m − ] 1 2 m ab − V a ¯ a = 38 2 ∂ m + b ¯ | m a 2  g m a ≡ DW ∂ | ( m = ¯ a K e ¯ D m a = D V The scalar mass matrixtakes in the supergravity following at form an extremum of the potential If the extremum is supersymmetric, This condition issupersymmetric sufficient extrema for are also the possible potential if to have an extremum, however, the non- where we have takeninvariant. into Note that account the that potential has an extremum if potential The first derivative of the potential becomes, This expression simplifies at the supersymmetric Minkowski minimum with The K¨ahlermetric is Note that the fermion action has the matter fermion mass terms: It might be useful toexample give eq. also an (18.16) alternative in expression [ for the fermion matrix It means that The complex masses of the chiral fermions in We also define the K¨ahlercovariant derivatives as The complex gravitino mass is covariantly holomorphic, which means that JHEP01(2020)169 (5.22) (5.23) (5.24) (5.18) (5.20) (5.19) (5.21) 0 for any ≥ ¯ c ¯ . In such case Φ ] on the basis of ¯ c 2 c g ). We will present c 5.2 , , ¯ c , ¯ φ Φ ¯ c = 0 , c 2 g | c    . m | 0 3 . = Φ 0 b ¯ Mink − ¯ ab , . ¯ c b ¯ ¯ ¯ V 2 φ > φ ¯ 0 0 χ | m b φ ¯ a ¯ c ]. ¯ c m ≥ > c 2 χ 1 m ¯ g m b b | ¯ ¯ 0 b ¯ should not have zero modes Mink ¯ ¯ ¯ c a φ φ ac c = V > m g m ab b ¯    – 14 – ac b b m ¯ ¯ ¯ = φ Mink Mink a a m = V V a a a φ b ¯ b ¯ is positive definite, so we have Φ Mink ,V φ φ Mink a a ¯ c = V Mink V c ) b ¯ a g = 0 ¯ 2 sc φ φ a M m ( b ¯ Mink a = . It means that the second derivative of the potential is non- V a a ]. The second derivative of the F-term potential in N=1 super- m φ φ 2 is a positive definite matrix, namely, . The matrix 2 ac M the fermion mass matrix m a φ = c In case of only one chiral multiplet with a non-vanishing fermion mass and a canonical An additional requirement we will impose on the class of models of our interest is that Thus, at the supersymmetric critical point in Minkowski space the mass matrix is block K¨ahler metric we have reproducedbelow a the class WZ of feature modelssimplest where shown example Minkowski in of vacua such are eq. case discrete ( and is have the no KL flat model directions. [ A It means that we can guarantee thatnon-vanishing, eigenvalue Minkowski of the extremum fermion is mass a is local minimum. If the smallest, but this analysis was that allof Minkowski flat supersymmetric directions, vacua but can never have only tachyons. positive masses, only models with strictlySitter minima. positive second derivatives are relevant for construction of de for an arbitrary vector negative in all directions in the moduli space. The conclusion reached in [ all its eigenvalues are positive. Indeed, for an arbitrary vector where Φ vector Φ. Thus where diagonal, at each diagonal part of it we have a positive definite matrix, and consequently supersymmetric vacuum. Namely we find that This case is simple and is defined by the conditions that and it was studiedgravity is in particularly [ simple in terms of the fermion masses of the chiral multiplets in the 5.3 Supersymmetric Minkowski minimum JHEP01(2020)169 , m ab m ) and (5.28) (5.29) (5.31) (5.30) (5.25) (5.26) (5.27) in the δz 5.30 + where again z χ = 0 is given by m to a z W. m  W ) + ∆ ∆ can be described in the z 2 K ( e AdS a = z Mink . . , m ¯ W to m    . Therefore first we will find the ... m = 0 + b W ¯ . ) = a AdS AdS a ¯ . ab ab z Mink a g m ¯ ( V V m z ) was absent in Minkowski case, now it 2 ¯ is m = 0 ∆ − b ab AdS δz b ¯ W 5.26 b b ¯ ¯ ¯ K c m AdS AdS a ¯ a a 1 , + ¯ − m V V − D z ¯ c ) c 2 K =    – 15 – 6= 0 ab g e ,W ac = m ) ( = W AdS m ab ¯ z 2 a − K V AdS e = z, m ) = ( = 2 sc a b ¯ AdS a δz m Mink M V ( K z 6= 0 and supersymmetric AdS minimum ) = ¯ z W z, ( AdS K are terms smaller than the one we present, and ∆ ... The off-diagonal term in the mass matrix ( Now we can evaluate the change of the mass matrix in the supersymmetric AdS versus negative, however, it istherefore parametrically will small not comparative change the to positivity the of first the term massis in matrix. present ( The first term here, as beforeof is the positive definite: fermion masses. thefirst scalar masses Therefore term are taking related is into to of thevacua account square remain no the discrete relevance change and from since doof the not the acquire total minimum flat term direction. is is But parametrically positive since small, the definite our shift at vacua of any remain the point, position discrete. if The the second term is the change of the position of the minimum isMinkowski. also The small. mass formula at the point the following formula Here we have neglected smaller terms.is parametrically Thus small we comparative can to see the that smallest if eigenvalue the of mass the of fermion gravitino matrix mass ∆ The unbroken supersymmetry condition can be also given in the form One finds that the shift of the position of the minimum consistent with and the superpotential ofpart the of moduli the fields. superpotentialchange by of We the some would position small of like constant,that the only for ∆ minima to in generic AdS, values change versus of the the one constant in Minkowski. We require now The change of thegeneral position case. of We would the like minimum to from preserve the functional dependence of the K¨ahlerpotential Here we slightly modifythe the potential. superpotential so In that our notation it this does means not that vanish at the extremum of and 5.4 Downshift to JHEP01(2020)169 ) = 5.10 (5.35) (5.36) (5.32) (5.33) (5.38) (5.37) (5.34) aX since the m , X m , . X b ¯ 0 D ¯ m > a 2 = m / 2 3 . is much smaller that − according to eq. ( m ¯ X I 120 m 3 m − . X. m aX − 2 I 10 µ 2 ¯ m m 120 | ¯ I − ≥ + F 2 | bI ¯ a 10 W . = R I . ≥ 2 have to be at or above the scale of = | ¯ − 2 m ¯ 2 | m is the holomorphic-anti-holomorphic | I m dS | , describing also an uplifting anti-Dp- m m abI ¯ F 3 | b m ¯ | . b ¯ dS = 0 in our models. a I 3 a X m 2 − V g / = m − 3 2 W 2 + b ¯ | 2 a 2 | . The value of the potential now is I m X | I g ¯ }  m ∂ I m a b | ¯ m  + ab m ¯ | ¯ c ∂ – 16 – | X,W ) χ ¯ m ¯ m a, X m X = ab ¯ { c ) = − m ¯ c 5.9 m 2 X m | b g ¯ = = since a X, ac g W I I | K 2 dS m ab 3 m ) in case that neither a superpotential nor its K¨ahlerco- V + − ,V etc. Thus for all superfields, including the nilpotent one, − b ¯ 2 ¯ K J | m 5.17 ¯ = 0 m , where = X W 0 ¯ I J I D I V dS a g D | K ( D aI as shown in eq. ( a K | m e D χ = = m = | b ¯ dS dS a is the moduli space curvature and V V ). We also note that the new terms depending on abX ¯ I m bI ¯ , a 5.17 R , will use an index m } X The holomorphic-holomorphic mass formula in an uplifted dS model take the form We do not study the derivative of the potential in the direction of the nilpotent scalar D 2 a a, X have only terms proportional to scalar part of the nilpotent multiplet is not a fundamental scalar but is proportional to fermions. where part of ( the mass matrix. Our goal is to describe dS minima with Λ The general mass formulavariant ( derivative vanish can be given in the following form { Thus, the supersymmetry breaking terms the gravitino. We are looking now for the new extrema of the potential and the values of brane in string theory. We now haveD additional terms to take into account like If we find aAdS class minima of without models tachyons. satisfying this condition, we5.5 are guaranteed to Uplift find to aHere dS class we minimum of add to our model a nilpotent multiplet Thus we need that thematter mass fermion of gravitino is much smaller that the lightest eigenvalue of the from the Minkowski minimum, requires thatpositive the definite corrections in to the theirrelevant mass Minkowski we matrix, minimum, need which was are to small. requirethe that scale To of the make the scale all fermion of of matter the these mass gravitino corrections mass The condition that the supersymmetric AdS extremum remains a minimum, inherited JHEP01(2020)169 W (5.47) (5.43) (5.44) (5.45) (5.46) (5.39) (5.40) (5.41) (5.42) . ]. ].
41 , ¯ X 39 2 / ) the uplift term is given by X, 2 3 a 2 . K 2 m ) , z , K . ¯ a 1 X e z  . ( , X, = 1 a α 2 χ ¯ = 0 X b β z . K . α . i ∂ AdS 2 X, 4 δz K 4 | a 2 e V µ µ K  X ( iz 1 K a α K uplift m with z e − e + , m AdS ∂ V 4  a α 2 a 4 1 = µ | V z z  | + µ ) are parametrically greater than the scale of I b 2 β – 17 – | z 2 AdS means that there is a small shift of a position of = | = m = ∂ | X a 2 V AdS a a α AdS b β µ z m z m = z V | V | ∂ b β ∂ uplift h 2  z uplift | a α a V α ∂ z V z F a α ∂ a ∂ α z  z ∂ ∂  | − W 2 χ X = m D b β 2 K δz e = X m ] and used in the context of STU moduli stabilization in [ To make sure that our newly constructed dS state is stable we will require that the Using this formula for the shift of the moduli, one finds that in dS stage the value The mass formula at the dS extremum can be related to the one in the supersymmetric 48 supersymmetry breaking is parametrically small,fermions comparative in with the the scale chiral ofgravitino multiplets, scalars is and in parametrically small, addition comparative tochiral with multiplets our the scale earlier of requirement scalars that and the fermions in mass the of smaller than the supersymmetry breaking in the nilpotent direction, This feature was established in the one-moduli case before in [ And since theuplifting, scalar we masses conclude ( that the shift of the minimumof is supersymmetry small. breaking in the unconstrained chiral moduli directions is parametrically whereas the second on is We end up with an expression for the shift of the minimum The new position of the dS minimum is at The first term gives The AdS part of the potential remains the same on moduli. For We can see that parametricallythe small minimum. We canpotential make due this to statement uplifting more terms specific and by using looking real at and the imaginary change fields in the in [ AdS before uplifting, takingminimum into takes account place, that after the theof shift uplift the of dS terms a minima are versus position added. the of AdS Note the minima that is moduli due the at to shift the the of uplifting the terms and position their dependence The related mass formulae were established in the form depending on holomorphic JHEP01(2020)169 it X 120 2 χ − , the b ¯ m (5.48) (5.49) (5.50) a 10 ) which g  ≈ ) there is 2 | 5.37 X 5.38 m  | 2 . And since the susy | to guarantee that all I a 2 m m / | 3 = ¯ ) are negligibly small, and m , particularly motivated by ¯ I I W g has an additional small factor ¯ 5.17 I c m ¯ m , and 0 abc ) has a first term which is positive K > m . , is due to a smallness of the down shift 2 χ 2 / 5.37 2 3 ) comparative to its positive definite first b ¯ Mink ¯ φ m a m AdS ab 3 φ V V 5.37 > ≥ ≈ – 18 – ). We conclude that up to small terms involving b ¯ 2 ). The values of the moduli space metric | b ¯ ¯ dS φ a versus matter fermions. In dS case ( F in the mass matrix ( b ¯ V | 5.47 dS 0. a 5.38 m = 0. The term V dS ab > a V ), ( φ abX and the third holomorphic derivative of the superpotential it is useful to have the exact formulas for the second deriva- m 5.37 ¯ I W bI ¯ a ) remain small comparative to the first term in eq. ( R and . In models where the superpotential has a simple dependence on 5.38 X . K ¯ 2 m ), ( | but it is multiplied by a small value of ¯ directions is much smaller that in the nilpotent direction, the only term we find that X are small due to eq. ( abX a m m b ¯ 5.37 X ¯ c X m 2 ¯ m µ m  | ¯ c is the smallest eigenvalue of the matter fermion at the dS minimum. c 2 | g ). The smallness of the first term χ a ac and m m may indicate what is the required level of smallness of | m a 5.38 As we have shown above, the explicit choices of For unspecified Thus, the off-diagonal terms The holomorphic-holomorphic mass formula in an uplifted dS model is shown in The second derivative of the potential in ( m , abI tives of the potential inmoduli eqs. space ( curvature m terms in eqs. ( is always positive. string theory like the multifielddS KL minima models, is are guaranteed available, inthe by which mass an the formula evaluation mass studied of production here, the of where shift the in conditions the for position its of positivity are the specified. vacuum and of the form since the mass matrix in dS vacuumfrom remains positive the definite, Minkowski without vacuum. flat directions, as inherited eq. ( to AdS defined byan a additional small term involving value a of gravitino third holomorphic mass derivative of thebreaking covariantly holomorphic in all to consider is and therefore where small. Note thatterm all additional termsm in ( or exceeds it for larger values of Λ definite. It differs from its Minkowskihowever, value due by to construction the shifts this ofMinkowski the term vacua position had is of no the positive minimum, flat directions, definite. since the It shifts is of also the positions strictly of positive the if minima are our Since supersymmetry breaking inmeans chiral that directions supersymmetry is breaking veryorder in small, of the gravitino mass direction for of extremely the small nilpotent cosmological superfield constant of is the of order the Λ and the potential is positive JHEP01(2020)169 ] = 49 3 G = 0 in as well as W ], and they 6 4 = F = 0 DW F in the superpotential 0 to be only of the type W 3 for each moduli was used. G has two (or more) exponen- = 0. The relevant equation W 3 ] to string theory is contained = 0, therefore in our type IIA W 4 H 0 , which breaks supersymmetry. 6 F . Once the exponent is added, one F 0 W are included) Minkowski supersymmetry are included), since in KKLT, as opposite W W Ω with imaginary self-dual (ISD) 3-flux which has to be satisfied for each three-cycle , – 19 – ∧ ] is only in the non-perturbative exponents, here 6 3 4 ] a single exponent in D 4 G N ] how it is possible to satisfy this equation. In the R 4 − = ] we had a type IIA solution of the KKLT-type. In the 6 4 D W N + 6 O ] is that the non-perturbative part of ) we notice that we only have 4 N = 0. Supersymmetric solutions require 2 W − flux. Once we add non-perturbative terms, if it is a single exponent, ], with the title ‘Satisfying stringy requirements.’ For example, it is 3) , 4 6 (0 = F G 6= 0, which results in a constant term H 3) 0 , F (0 and G − 3 2 . To compare the GKP setup with the type IIA models (before the non-perturbative τH τ dF To compare with KKLT, we first notice that Minkowski supersymmetry is broken in Other interesting points concerning the relation of these models to string theory are A discussion of the relation of the models developed in [ − R 3 were uplifted to dS minimausing using anti-D3 anti-D6 branes, branes. in a Thuscurrent complete in paper, analogy [ with we KKLTthe added uplift Minkowski a supersymmetric vacua, second followed byuplifted non-perturbative supersymmetric to exponent, AdS dS vacua, which vacua. which were enabled us to get KKLT (before the non-perturbative termsto in GKP, finds AdS supersymmetric vacua. In ouroriginates models from the constant term we find supersymmetric AdS minima, as in KKLT. These were described in [ terms are added to For example, to haveother unbroken conditions. Minkowski supersymmetry But we onemodels do needs not (before have the the non-perturbative Romansis terms mass, always in broken. F (2,1). In GKPof the Klebanov-Strasssler moduli throat. stabilization is Inthe studied supersymmetric vacuum, in Minkowski and solutions the itfield context was of shown the by compact GKP version that it is possible to stabilize the dilaton-axion models. The details will be worked out in futurewith regard studies. to 10d classical background(before on which the the Type introduction IIA string ofinvolve theory the instantons is compactified 4d and superpotential anti-D6 branes). Type IIB GKP models [ is independently. It iscurrent explained paper in the only [ we difference have with [ two ofTherefore them we believe for that each there modulus, are no and problems this with does the tadpole not condition affect in the the current tadpole condition. in section 2.2 ofpointed [ out there thatthe the D6-brane only charges, since non-trivialRomans all Bianchi mass other identity and RR is also as the the well 2- tadpole as and condition NS-NS 4-fluxes for fluxes are are absent, and absent, we have no The main focus of this paperspired was by on type IIA a new string theory. constructionory of Here embedding dS we of minima would like in our to 4d new addressence supergravity supergravity some in- with models. issues the of setup We the in note string [ that the- tial in terms this for each paper moduli, the whereas only in differ- [ 6 String theory embedding of the new 4d supergravity models JHEP01(2020)169 , ]. 0 i 2 T and 2 fluxes. F ) which 3 i T H ( i W and , which plays a 3 3 F H ]. The procedure 1 ) for each of the moduli i . This simple procedure 0 T i ( T i in type IIB, but the W superpotentials 3 F n ], whereas the uplifting role of the 3 ] where single KKLT type exponents 4 is also absent in our type IIA models. In 3 superpotentials H n – 20 – which is dual to , take a sum of all of these superpotentials, and ]. The corresponding uplifting mechanism via the 6 0 i F one should identify 37 T – 0 i T 27 ]. 51 , 50 associated with the integral over a 3-cycle over K independent problems for each of these moduli. To find a supersymmetric n A simple example of string theory motivated supergravity models of this kind involves In our case the final products are the metastable de Sitter vacua, and we have three The main problem in the past was to uplift type IIA AdS vacua to dS, various no- It is also interesting that in type IIB in GKP there are non-vanishing are absent. Moreover, the NS-NS 3-flux , the problem of finding a stable supersymmetric minimum for all of these moduli becomes 4 i the racetrack KL-type superpotentials.vacua The in problem a of theory constructionoutlined of of above stable this automatically Minkowski type generalizessupersymmetric with Minkowski this vacua are a solution possible single in for superstring modulus theory, any and was number they are solved of stable in [ moduli. [ Such Minkowski vacuum at anyhave point a vanishing firstsubtract derivative from at them theautomatically sum produces of a all theory ofin with these absence a superpotentials of stable at flat supersymmetric directions. Minkowski vacuum at stages in the production.without The flat directions first in stage models involves withpresent multiple construction paper scalars. of is The that stable crucial if Minkowski observationT one made vacua in takes the a sumdivided of into Mass production isusing the manufacture assembly of lines. largeadded quantities An as of assembly the standardized lineparts semi-finished are products, is assembly added often a moves in sequence from manufacturing until workstation process the to final in assembly workstation which is where parts produced. the are given in our next papers [ 7 Conclusions for all moduli directions, motivatedThis by the fact U-duality was of also thewere discussed non-perturbative used, in string whereas theory. the in previous theexponents. paper current paper [ we A use further racetrackmass superpotentials investigation production with of of two or the dS more vacua string in theory supergravity, with embedding and of without the non-perturbative terms, mechanism is of go theorems are known,anti-D6-brane see was e.g. discovered [ relativelyanti-D3-brane recently, was in known [ long timeAnother ago reason since why in the typeno-go KKLT construction IIA theorems, is in models that type we we have IIB have now used theory. presented the dS non-perturbative vacua, exponents overcoming in the the superpotential Our type IIA modelsF have a six-flux particular the integer prominent role in Klebanov-Strassler throat, istype vanishing in IIA our appears type to IIA models. be simpler Therefore than type IIB. JHEP01(2020)169 , + W A 2 | ) de- i W T a ( D | W ( K e = and appendix 2 3 F . ] consistent with type IIA 2 / 1 2 3 are described by 4 parameters m i 3 T − to the original superpotential without any loss of generality. Then = i W AdS > b V i a ], corresponding to a nilpotent multiplet in . = 4 , W 3 AdS – 21 – and K show that the conclusions about stability of dS vacua 5 6-branes [ D . More general explicit examples will appear in the future. 4 overcompensates for the effect of the downshift and in this way 2 | W X D . For each such set, one can take | i K . The resulting dS state inherits the stability property of its (grand)parent , in Planck units. Both of these goals can be reached at the next two stages B e , which is not much of a restriction. i 120 120 ≈ − − B and i ) i 2 10 10 | A > b , ≈ ≈ Moreover, instead of adding a small constant ∆ Our results obtained in section Examples of such models involving multiple KL models [ The second stage describes how to preserve the stability of the Minkowski vacuum when The third stage leads to the final product, a metastable dS vacuum, when one adds However, this is not a finished product yet, for two reasons. First of all, we do not live W i i b X , A dS dS i i D qualitative features and consequences. during the downshift stage,pending one on can all add moduli a fields.of small the If function potential this of to function all AdS, iswhich moduli but sufficiently is fields it small, also does it ∆ preserved can notalize by shift affect the the the the mechanism minimum positive subsequent of definiteness small dS of uplift. the vacua production mass Thus matrix, introduced one in can our significantly paper, gener- while preserving its tials. The conditions fornot this have require flat that directions, 1) 2)are the mass parametrically of supersymmetric gravitino small, Minkowski at vacuum comparative stageof does to two production, and the in susy scale Minkowski breakingrestriction vacuum. at of on stage If scalars the three these stabilized functional conditions form at are of the satisfied, first there stage is no further a a stable supersymmetric vacuum can bea constructed for any values of these parameters with at stage three of production are valid for more general K¨ahlerpotentials and superpoten- and STU model inIn section discussing the relationIIA between string the theory parameters onerescaling of without has affecting our to stability. multiple In keep particular,developed KL in in in models the this mind context with paper, of that the the type superpotentials it KL-related scenario for is each easy field to change the parameters via Λ supersymmetric Minkowski vacuum if the resultingis supersymmetry sufficiently breaking small. in dS vacuum string theory inspired supergravity are presented in the paper, see section | many different stable dS vacuaslightly can overcompensates be produced. the downshift, Incosmological those constant. we cases This find where is dS the awhich uplift vacua part only is of with very the necessary very standard string small to theory values landscape account of construction, for the the tiny positive value of the cosmological constant downshifting it to a supersymmetricthe AdS order vacuum with of a the negative gravitino cosmological mass constant squared, of Λ an uplifting contribution of the supergravity. Here we assume that the total SUSY breaking parameter in a supersymmetric world,much so because it we can need violatewould to the like break stability to of SUSY. find the aΛ vacuum But mechanism that we generating we a should constructed. dS not Secondly,of vacuum we break the with it a procedure. tiny too cosmological constant JHEP01(2020)169 , 2 ]. 5 G − = 1, 51 , (A.1) (A.2) (A.3) (A.4) u a 50 = 0 and = 10 6523 are 0563417. . . 0 = 1. W W − 0 = 22 s with = u X, 0 X B 2 W µ + and = 5 and S e s 3 0 2 b = 0: required for the existence u − 2 0 e = s µ s W . B ) B S − S + s a , S ], as well as a compactification on = 0 and − ( ¯ 7 X e . manifolds, like K3 fibration models, a 3 − s 50 ) , 3 1 W X and the nilpotent field 5 A U 10 − with T + + × U U X u ( = 10 b 2 and 873 . – 22 – − µ 3 e W U u ) + ), one finds the value & ∆ S B = 1. The parameters 2 s + − µ 3.22 A S U = 1. We will take the following set of parameters: u a 0 ) is shown in figure s − 4 ln( = 3, e = u s − b T,U A S ) ( + U V = 2, + W s U a = 5 and ) and the requirement that the minimum is at + ∆ 0 ]. In these papers we also discuss the issue of tadpole conditions and other 0 3 ln( u 3.21 51 W − = 10, = u = = U A . A more detailed investigation of the string theory embedding of our proposal K W 2, 6 . Now we will study the same potential by including the contribution With these parameters, using ( It would be interesting to find more general explicit models of this kind, which lead = 0 at = 1 2 u and, in the next step, including the term which is justand sufficient by to making compensate the the value of downshifting the the potential potential at by its minimum ∆ positive. To make these small of the supersymmetric Minkowski minimumThe for resulting ∆ potential As an example, weµ will look for ab supersymmetric Minkowski vacuum withfound ∆ using ( A.1 A very small downshift and uplift A Some illustrative examples In this section weplified will STU illustrate model our describing general two approach fields by a numerical analysis of the sim- We are grateful toT. J.J. Wrase for Carrasco, stimulating N. discussionsSITP, Cribiori, and by S. important the US comments. Kachru, National This C. Sciencetion work Foundation Roupec, Origins is grant of PHY-1720397, E. supported the by by Silverstein, the Universeby and Simons program the Founda- (Modern Simons Inflationary Fellowship Cosmology in collaboration), Theoretical and Physics. CICY model, and a multi-holein Swiss M-theory cheese [ model [ global string theory related constraints. We expect more studies ofAcknowledgments this kind in the future. to stable dS vacua inof supergravity, string and can theory. besection We eventually discussed associated the with various issues versions on of a embedding mass our production of modelsWe dS study into vacua there string in a supergravity theory compactification is in on contained specific in our CY next papers [ JHEP01(2020)169 . is X and W (A.5) (A.6) (A.7) S is indeed . One can 2 S / 3 , and when we m 5 and − U = 10 , . . of the nilpotent field 3 3 3 06. Thus change of . − − − W . Thus dS state is stable. 0 X are 2 10 10 10 2 − µ S × × × and are given by , which is 3 orders of magnitude 1398 1386 1387 6 . . . U S − . 5 10 = 2 0563417 to = 2 = 2 . and s 2 4 0 × − U ) with respect to the real parts of the field 26491 have masses from A.2 . , m – 23 – , m , m 0 3 3 3 S ), ( − − − W = 1 10 10 10 A.1 2 and / × × × 3 U m 5512 5488 5489 . . . = 5. U = 3 = 3 = 3 1 3 u = 0 in the Planck mass units. The supersymmetric Minkowski minimum m m m 0036583, to bring 2 . µ 0 = 1 and − S = W = 0 and , and then show how it changes when we introduce ∆ . The potential of the model ( 1 W = 0 is at V As one can see, everything goes as expected: after aIt small is downshift instructive to and compare uplift the we mass matrix of all (canonically normalized) fields in , for ∆ expected on the basis of the investigationA.2 in section A significantTo downshift check and how uplift stable thiswe added result ∆ with respect to the more significant downshift and uplift, In other words,uplift, the and moduli the masses potentialGravitino also do does mass not not in change change thissmaller much, much see state than during figure is other the masses. downshift and Stability of the the dS vacua for small values of mass matrix remains positivelythe definite, real but components requires of diagonalization. the The fields eigenstates for The masses of the imaginary components of the fields the supersymmetric Minkowski state andthe in masses the of uplifted the dS real state. components In of the the Minkowski fields state, and the masses of the imaginary components have the same values. After the uplift, the subsequently uplift the potential by including the contribution have a stable dScheck minimum that with the respect same to is the true real for parts the potential of of the the fields complex components of these fields. Figure 1 U with changes visible, we will zoom atin a very figure small vicinity of the minimum of the potential shown JHEP01(2020)169 . ) u 3 B − = 5. with 3.12 10 U X × and u 658 A , see ( . C = 1 and , which is 3000 4 p S and 1 M 3 11 − − = 0, 10 V × = 0. It has a minimum at 2 by a factor of µ 6712 . U . The upper left figure shows the 1 and with respect to the real parts of the 5 1 − 00001 in the previous example. This . = 0. Thus the white hole in the ground 0 V − = 10 = . It is only about 4 times smaller than the 4 ), Figure W − W . The resulting modification of the potential is A.2 4 10 – 24 – − × ), ( 10 × A.1 835 . 44 . 1 = 4 . For example, one can simultaneously rescale 2 ∼ / 3 2 2.2 = 0, with the supersymmetric minimum at µ m 2 µ 0, which requires including the contribution of the nilpotent field after the mass matrix diagonalization are 2 without changing the position of the supersymmetric Minkowski vacuum. . = 0 and S V > 7 C − W , in a small vicinity of its minimum. It is shown in units 10 10 and U × U . The potential of the model ( and 873 . 3 0. Instead of showing it, we cut the potential at S This model allows lots of freedom in the choice of the parameters, which can change The gravitino mass is & 2 the dS state remains stable. the final results, seeby section some factor However, this change would increase the mass of the field quite significant, and yet ourfields procedure in yields the a dS stable statethe dS fields is vacuum. positively The definite. mass matrix In of particular, all masses of thelightest real moduli components of mass, the downshift and uplift significantly modify the mass matrix, but dS vacuum with µ almost 3 orders of magnitudechange greater requires than uplift ∆ with field times smaller than thepotential height for of ∆ the potentialThe shown upper in right figure figureV shows < the potentialclearly for shows ∆ that the potential below it is negative. The lower figure shows the stable uplifted Figure 2 JHEP01(2020)169 , 8 3 − by a . We 10 dS 1 × V in the Planck 4 − would similarly 10 C × , and a simultaneous 0 = 4 u in this vacuum is 2 we show that by taking 2 µ 3 dS V . However, if one is interested 120 − = 0 and 10 W by the same factor ∼ would increase s dS B u V b ) for ∆ and A.2 and s ]. This shows that the scenario discussed above – 25 – 5 u A [ ), ( a 0 s A.1 one can uplift the Minkowski minimum to a stable dS 4 , and significantly strengthen the vacuum stabilization. − ), which permits any use, distribution and reproduction in S 10 , i.e. close to the height of the barrier shown in figure 8 , in Planck density units. In figure × 8 − − 10 10 = 4 × 2 × would increase µ CC-BY 4.0 s = 2 b This article is distributed under the terms of the Creative Commons is 2 , and verify its stability. dS V 1 1 and . As we mentioned in this paper, in the class of models we consider here, the s U a ). A simultaneous rescaling of . . The potential of the model ( and 1 S 3.25 The height of the potential barrier stabilizing the Minkowski vacuum in the model any medium, provided the original author(s) and source are credited. is (meta)stable with respectproduce to stable all uplifted fields. dS states By with changing evenOpen parameters much Access. greater of vacuum the enerQgyAttribution model, density. License one ( can the uplift parameter minimum with also verified that this minimumfields is stable with respectmixed to real-imaginary the mass imaginary terms components vanish. of the Thus the uplifted dS vacuum shown in figure Minkowski vacuum, one maypossible study instability a of possibility the dS ofwe state a ignore after the very the stage uplift. large ofshown uplift, Therefore in the in limited figure downshift the only to example by studied AdS, below a directly uplift the stableshown Minkowski in vacuum figure A.3 A largeIn uplift the without previous downshift subsectionstremely we small studied value of the the downshift cosmologicalin and constant uplift the to general a structure dS of state string with theory an ex- landscape after uplift from the supersymmetric and ( increase the massMeanwhile of a the simultaneous field decreasedecrease of of is quite flexible. Figure 3 mass units. 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