Universal dS vacua in STU-models

Stable de Sitter solutions in minimal F-term supergravity are known to lie close to Minkowski critical points. We consider a class of STU-models arising from type IIB compactifications with generalised fluxes. There, we apply an analytical method for solving the equations of motion for the moduli fields based on the idea of treating derivatives of the superpotential of different orders up to third as independent objects. In particular, supersymmetric and no-scale Minkowski solutions are singled out by physical reasons. Focusing on the study of dS vacua close to supersymmetric Minkowski points, we are able to elaborate a complete analytical treatment of the mass matrix based on the sGoldstino bound. This leads to a class of interesting universal dS vacua. We finally explore a similar possibility around no-scale Minkowski points and discuss some examples.


Introduction
It remains a great challenge to construct meta-stable de Sitter (dS) solutions in string theory.
A useful framework is provided by N = 1 supergravity specified through the choice of a real Kähler potential K and a holomorphic superpotential W . This is phenomenologically motivated, sufficiently restricted to provide non-trivial constraints, and general enough to allow for interesting solutions. To obtain a dS solution, supersymmetry needs to be broken, either through explicit breaking, e.g. by adding an anti-brane, or through spontaneous supersymmetry breaking when F-terms are turned on.
Many of the most studied models in the literature, including those inspired by string theory, include exponential contributions to the superpotential motivated by non-perturbative effects [1]. These terms play a crucial role through their contribution to the stability of the models. This is the case in [2,3], where attempts were made to find stable dS in a constructive way through spontaneous breaking of supersymmetry. The starting point for the constructions is either no-scale or supersymmetric Minkowski (Mkw).
In no-scale theories supersymmetry is partially broken through an F-term, and the vacuum energy guaranteed to be zero thanks to a translational invariance of the superpotential.
The theory is then deformed so that the two or three flat directions at the Mkw point become massive, while the vacuum energy is lifted to a positive value. These theories have a natural separation between the scale of SUSY-breaking (as given by the gravitino mass and the value of the superpotential), and the possibly small lifting to the positive cosmological constant of the dS.
A possible disadvantage of this scenario is that the same separation of scales also occurs between the gravitino mass and the much smaller uplifted masses of the no-scale directions.
A way to remedy this is to turn to supersymmetric Mkw. There, the gravitino mass is small from the start, and the challenge is to break SUSY in such a way that all scalar masses become larger than the gravitino mass, while the cosmological constant remains small.
The generic supersymmetric Mkw is fully stable and lacks flat directions. Unfortunately, as proven in [2], any small deformation of the superpotential (or the Kähler potental) just yields another supersymmetric vacuum -Mkw or AdS. Hence, one can never get a dS in this way. The proposed solution of [2] is to add a new field, referred to as a Polonyi field, to generate an additional, flat direction. The Mkw minimum can then be uplifted to a stable dS. Through appropriate choices of parameters, in [2] it is argued that all scalar masses can be made larger than the gravitino mass, and that the cosmological constant can be set at a still much smaller scale.
A drawback of the non-perturbative models is that the exact form of the corrections is not known in general, and it may become difficult to blindly trust the results. One is therefore motivated to try to do without such non-perturbative effects and generate dS solutions perturbatively. This has turned out to be remarkably complicated in simple string motivated examples, and various no-go results have been obtained in the context of geometric compactifications of type II theories [4,5,6].
These models have an isotropic sector with three complex moduli (S, T and U ), while the full non-isotropic theory has 7 complex moduli. Different theories are specified through various kinds of fluxes due to the metric and gauge fields living on the manifold. In order to find stable dS solution it has turned out to be crucial to turn on non-geometric fluxes, and explicit, successful examples can be found in refs [14,15,16].
The stable dS in [14,15,16] were all found to be close to Mkw vacua with no special properties. The aim of our work is to investigate the existence of examples close to either no-scale or fully supersymmetric Mkw. In this paper we will see that this is indeed the case. In fact, we will be able to show that all vacua generated through the addition of non-perturbative terms can be captured through the presence of non-geometric fluxes. The key to this is the fact that all the relevant properties of the vacua, including the properties of the mass matrix only depend on up to third derivatives of the superpotential. As we will see, the number of real degrees of freedom in the complex superpotential up to third order derivatives is 32, which coincides with the number of real fluxes in the most general duality invariant superpotential. From this point of view the non-perturbative terms do not introduce anything new beyond the polynomial form.
In case of no-scale, we will give examples of non-perturbative potentials admitting stable dS solutions approaching Mkw points with two as well as three massless directions. In both cases we provide the equivalent polynomial superpotential that gives the same values for the cosmological constant and the masses. We also provide the first example where one of the squared masses go through zero and changes sign at the no-scale point. The existence of such a possibility was argued for in [3] but no explicit example was provided. It is crucial that our example is of a more general type than those accessible through the exponential.
In case of supersymmetric Mkw, our analysis more or less automatically selects the restricted class of points that have flat directions. Hence, we can find deformations that take us to dS vacua, and with some care these can be made stable. Our construction does not need the introduction of ad hoc Polonyi fields.
The paper is organised as follows. In section 2 we review STU-models arising from type IIB compactifications, and the duality-covariant form of their superpotential induced by generalised fluxes. In section 3 we outline a systematic way of solving the equations of motion, while keeping track of the stability of the theory close to Mkw or no-scale points.
In section 4 we have collected all our explicit examples. Section 5 contains our conclusions and some outlook. We also provide two appendices that classify all supersymmetric Mkw points, as well as no-scale points, for the full polynomial superpotential.
2 Type IIB compactifications with O3/O7-planes Type IIB compactifications on T 6 / (Z 2 × Z 2 ) with O3/O7-planes and D3/D7-branes as well as all dual backgrounds, can be incorporated within a class of N = 1 supergravity theories a.k.a. STU-models. Such 4D effective descriptions enjoy, in their isotropic incarnation, an SL(2) 3 global symmetry which can be interpreted as string duality relating dual tendimensional backgrounds to each other.
Isotropic STU-models of this type arise from the coupling between the gravity multiplet and three chiral multiplets. These theories are then free of vectors and possess a scalar sector containing three complex scalars denoted by Ψ α ≡ (S, T, U ) spanning the SL(2) Adopting the type IIB language, the S modulus is the one that contains the ten-dimensional dilaton, whereas the T and U moduli are interpreted as Kähler and complex structure moduli respectively.
The kinetic Lagrangian can be derived from the following Kähler potential Note that we are adopting a set of conventions where the imaginary parts of the complex fields are those ones carrying geometric meaning and hence they appear in the above Kähler potential and need to be strictly positive. The real parts, on the contrary, just represent axionic scalars and have no sign restriction. As a consequence, our choice for the origin of moduli space is given by In this paper we will analyse different mechanisms giving rise to scalar potentials for the (S, T, U ) moduli. Since there are no vector fields available, a potential cannot be induced by means of a gauging procedure. However, some other massive deformations turn out to be consistent with minimal supersymmetry. Such deformations are controlled by an arbitrary holomorphic function W (S, T, U ) called superpotential, which induces a scalar potential In the context of isotropic STU-models, sets of stable dS critical points have been found both with pertubatively [14,15,16] and non-pertubatively [17,18]   The main goal of this paper will be that of testing STU-models with polynomial superpotentials induced by generalised fluxes when it comes to searching for dS solutions allowing for full analytical treatment. We will follow the approach of [2] in order to construct simple examples of stable dS vacua both close to SUSY Mkw and close to no-scale points. All of this will be achieved within a string-inspired STU-model with a duality-covariant superpotential.

Perturbatively-induced superpotentials
In the type IIB duality frame of our interest, gauge fluxes of F In the following subsections we will see that superpotentials which are purely induced by gauge fluxes are unable to generate a dependence on the Kähler moduli. In order to obtain such a dependence in this context, one can add generalised fluxes obtained by acting on the geometric ones with a duality chain.

Models with only gauge fluxes
Type IIB compactifications with only F (3) & H (3) fluxes were originally studied in ref. [19].
where all superpotential couplings are consistently chosen to be real thanks to our conventions (2.3) for the origin of moduli space.
The N = 1 supergravity defined by the above superpotential has a so-called no-scale symmetry due to the absence of the Kähler modulus T . This implies that its real and imaginary parts respectively appear as a completely flat and a run-away direction in the scalar potential. The latter in turn implies that the only maximally symmetric solutions that these models can have must be Minkowski.

Models with generalised fluxes
Starting from a geometric STU-model, one can start acting with SL(2) 3 transformations to obtain dual models. In this way, it becomes natural to conjecture the existence of a completely duality-covariant superpotential [21] containing all possible STU-terms up to linear in S and up to cubic in T & U . We will now summarise here the correspondence between generalised isotropic fluxes and superpotential couplings appearing in the N = 1 effective 4D description. The complete generalised flux-induced superpotential can be written as where 2 the couplings in are introduced and explained in table 1, whereas the details of the couplings in quadratic and cubic behaviour in T and represent additional generalised fluxes which do not even admit a locally geometric description. These were first formally introduced in ref. [22] as dual counterparts of the unprimed fluxes.

Non-perturbatively-induced superpotentials
Starting again from the geometric superpotential (2.5) induced by gauge fluxes, one could instead introduce non-perturbative effects in order to generate a T -dependence in the superpotential, which is typically exponential. Following the idea of [23], these could be used in order to further fix T in perturbatively-constructed dS solutions where it appears as a  Table 1: Mapping between unprimed fluxes and couplings in the superpotential in type IIB with O3 and O7. The six internal directions are split into " − " labelled by i = 1, 3, 5 and " | " labelled by a = 2, 4, 6.
run-away direction. Such non-perturbative effects can thus be seen as an alternative way of breaking the no-scale symmetry of (2.5) w.r.t. generalised fluxes.
As mentioned earlier, amongst possible mechanisms to generate this type of superpotentials, we find the phenomenon of gaugino condensation within the gauge theory sector living on D7-branes or D-brane instanton effects. In particular, the line of including gaugino condensation [24] has been considered in the literature as a possible mechanism to further stabilise the Kähler moduli (see e.g. refs [25,26]).
A fairly generic prototype of non-perturbatively-induced superpotential can be written as W non-pert. = (P F − P H S) couplings Type IIB fluxes where P Z is a priori an arbitrary holomorphic function of S & U and α > 0 is some characteristic constant that depends on the physics of the explicit non-perturbative phenomenon in question 3 . Unfortunately, not much is known about the explicit form of the function P Z (S, U ) since it would be extremely difficult to compute from stringy principles. In the construction of ref. [1], an argument was presented that would allow one to ignore such an (S, U )-dependence in P Z , thus reducing the prefactor in front of the exponential to a constant, at least in the large volume regime.
In order to solve the field equations for the six real scalars in the STU-model of our interest and find maximally symmetric solutions, we combine two crucial observations that transform the extremality conditions for the scalar potential (2.4) into an algebraic system of linear equations in the superpotential couplings.
The first fact that we use is that a general non-compact SL(2) 3 duality transformation takes any point in moduli space to the origin (2.3). This statement holds for any supergravity model in which the scalars span a homogenous space, just as e.g., SL (2) SO (2) 3 . Due to the general form of the scalar potential, the formulation of its extremality conditions in the origin is just given by a set of six algebraic quadratic equations in the superpotential couplings.
Not only does this simplify the problem significantly, but such a restricted search for critical points can even turn out to be exhaustive if one moreover includes a set of superpotential terms which happens to be closed under non-compact duality transformations [11].
In our case, for a pertubatively-induced superpotential, this is in particular true if one keeps the complete duality-invariant superpotential given in (2.6).
It may be worth mentioning that, for the case of non-pertubatively induced superpotentials of the type in (2.9), such a search for solutions will generically imply a loss of generality.
This is specifically related to the fact that the SL(2) T symmetry appears to be broken by the presence of the exponential term (2.9), unless a non-trivial compensating transformation under SL(2) T is allowed for the constant α appearing there.
The second prescription that we make use of, is that of treating the derivatives of the superpotential w.r.t. the three complex fields as independent quantities. This line of investigation was first pursued in ref. [15] where the so-called SUSY-breaking parameters were introduced. By looking at the form of the F-terms in the origin of moduli space, one can reexpress six of the fluxes as functions of the above constants. Upon doing so, one can use the remaining fluxes to easily solve the field equations, since they can appear there at most linearly.
However, it was later noted in ref. [2] that such a method is nothing but an overconstrained case of the more general situation where all derivatives of the superpotential evaluated in the origin up to third order take part in a various-step-procedure. Specifically, if one starts out with the following arbitrary cubic superpotential 4 and all W derivatives appear as arbitrary complex numbers, one can: • Choose W 0 in order to fix the gravitino mass scale, • Choose W α in order to fix the SUSY-breaking scale, • Fix part of the W αβ 's in order to solve the field equations (where they only appear linearly), while suitably choosing the remaining ones, • Fix W αβγ such that the mass matrix (where, as well, they only appear linearly) be positive definite.
The mass matrix for the six real moduli fields at a critical point has the following general form where V αβ etc. just denote ordinary derivatives of V w.r.t. the complex fields Ψ α . In order to significantly simplify the problem of getting all positive eigenvalues and hence construct proper minima of the potential, it might be very useful in some cases to observe [2] that the mass matrix becomes block-diagonal upon imposing V αβ = 0. Such a condition, which can be easily solved in terms of the W αβγ , also turns out to enforce a pairwise organisation of the mass spectrum, which will in such cases only possess three distinct eigenvalues.
The positivity of the cosmological constant, instead, is here controlled by the sign of the following quadratic combination which means that such a procedure can systematically produce analytical stable dS solutions provided that all the different complex coefficients in the (3.2) can be fixed independently.
Due to the form of the Kähler potential (2.2), we should restrict to those complex coefficients that do not generate superpotential terms with degree higher than one w.r.t. S, i.e. The final crucial observation is now that such a parameter space is in one-to-one correspondence with the complete set of 32 real superpotential couplings collected in tables 1 and 2 describing the most general duality-invariant superpotential for our STU-model. The mapping relating generalised fluxes to complex superpotential derivatives is, in fact, linear and invertible. This in particular implies that, whenever a stable dS solution is found for a certain superpotential derivative configuration, this will always admit an STU-realisation in terms of 32 generalised perturbative fluxes.

Stable dS vacua close to SUSY Mkw points
Let us now apply the above prescription in order to find interesting dS vacua obtained by starting from a SUSY Mkw solution and subsequently breaking SUSY by small amounts. The details concerning the most general supersymmetric Mkw solution within our STU-model can be found in appendix A.
In order to apply the above formalism, we first need to understand our Mkw points in the language of W derivatives. These are identified by the following choice: whereas all the other 12 complex higher derivatives stay completely arbitrary. One can check that the above choice already automatically implies the field equations and guarantees the (semi-)positiveness of the mass matrix.
Now we need to construct a consistent Ansatz to break SUSY by small amounts and hence move away from the SUSY Mkw critical point. To this end, we introduce a small parameter , i.e. such that | | 1, that can be taken to zero whenever one wants to go back to the supersymmetric situation. Here we can make use of the no-go theorem in ref. [2], which guarantees that our construction automatically selects points close to a SUSY Mkw with two real massless directions. Without flat directions the theorem only allows for deformations into other SUSY points, and any dS critical point is excluded.
Such a deformation Ansatz reads where (κ 0 , κ S , κ T , κ U ) are arbitrary complex numbers of O(1). By plugging the (3.6) into the field equations, the second derivatives of W can only appear there linearly. Hence, one can use six out of the ten real independent parameters in W αβ in order to find extrema of the potential. The other four of them can be arbitrarily chosen.
As far as the mass matrix is concerned, the remarkable feature of these dS points close to a SUSY Mkw with two flat directions is that the light directions always coincide with the two real sGoldstini i.e. the superpartners of the Goldstone fermions responsible for the SUSY-breaking mechanism. This implies that these are the only directions that can turn into tachyons in the small limit.
At this point it is useful to impose the pairwise degeneracy condition for the mass spectrum, i.e. V αβ = 0, in order for the two sGoldstini directions to become degenerate. Such a situation turns out to be very special since it produces two sGoldstini directions with the same mass given by their average, which happens to be subject to a universal geometric stability bound [28,29] where γ ≡ V 3 e K |W | 2 andR ≡ Rᾱ βγδ gᾱḡ β gγḡ δ denotes the sectional curvature of the Kähler manifold along the g α plane. As already noted in ref. [15], the sectional curvature can be rewritten asR = 2 n eff , where  In section 4, we will present an explicit example of dS vacuum close to a SUSY Mkw point within our class of STU-models inspired by type IIB compactifications with generalised fluxes, which was obtained by following the above procedure.

Stable dS vacua close to no-scale Mkw points
Another simple way of constructing Mkw points stable up to flat directions, is to consider the so-called no-scale models, where SUSY is broken by large amounts but only in a single direction (usually T in our notation) in such a way that the corresponding F-term contribution to the potential exactly cancels the negative contribution from −3|W | 2 .
A realisation of a no-scale supergravity model is given by the GKP superpotential (2.   providing the first realisation of the general situation discussed in ref. [3]. In this section we will present explicit analytical examples of (stable) dS solutions obtained by making use of the technical machinery presented in section 3. We will first start by giving a generalised-flux realisation of stable dS close to a SUSY Mkw; in this case we will have two light modes lifted at O( 2 ). Secondly, we will move to discussing dS solutions close to no-scale Mkw points. There we will present both generalised-flux realisations of solutions borrowed from non-perturbative superpotentials and new examples which cannot be interpreted as coming from such exponential superpotentials. Only within these new cases will we be able to find the first realisation of an approximate no-scale dS solution with three light modes, one of which is lifted at linear level, while the other two get lifted at quadratic level. However, such a solution will turn out to be unstable.

Stable dS close to a SUSY Mkw point
This is a simple example of the application of the constraints in (3.10). A particularly simple choice that reduces the number of involved real parameters from 6 to 1 is where λ T can then be seen as a relative scaling between Re[F T ] and Re[F S ] or Re[F U ].
In the language of the prescription given in (3.6), we may additionally pick O( ) terms for W 0 and write

(4.2)
Hence the cosmological constant takes the form This two-parameter set of solutions can then be run through the inequalities in (3.10) which at O( 0 ) provide the simple bounds The equations in ( In order to be more concise, we present the explicit solution with λ T = 4. The fluxes resulting fluxes are given in table 3. Due to its linear dependence, the limit → 0 can be read quite easily. As it was discussed before, the eigenvalues of (m 2 ) I J are organised by pairs. For λ T = 4 they take the values 6) and the cosmological constant is then simply Stable dS close to no-scale Two examples will be discussed of stable dS close to no-scale points. We will use a special kind of solutions connected to (3.12), which were originally motivated by non-perturbative superpotentials of the kind of (2.9). Due to the simple mapping between 16 complex values of the W derivatives and a 32 fluxes superpotential, it was possible to construct non-geometric polinomial superpotentials that share the same fundamental physical properties. We will provide explicit solutions making emphasis on this equivalence.
For these examples it is sufficient to consider superpotentials with the functional form of (2.9) W non-pert. = (P F − P H S) + P Z e i αT , (4.8) with with α 0 , α 1 , α 2 and α 3 being real parameters. As it was mentioned, conditions (3.5) define a no-scale point. We may also start, as we did In the present examples we will not enforce V αβ = 0. This is partly motivated by the fact that we want to show some explicit realizations of no-scale points with 3 massless modes. But most importantly, in the superpotential (4.8) we have 12 real fluxes which naturally fit with a description in terms of 6 real supersymmetry breaking parameters and 6 real constraints placed by the equations of motion (see [15]). Hence, once picked F α with the required → 0 limit, exact and well defined solutions can be found, with parameterizing a continuous trajectory from the chosen Mkw point. In this case it will be sufficient to break supersymmetry as Solutions corresponding to this choice are then parameterised in terms of , λ U and the exponent factor in the non-perturbative term α. The O( 0 ) behaviour will be naturally of the no-scale type. This is all the background we need to find examples of stable dS close to no-scale points with 2 or 3 massless modes. In analogy with the SUSY case, λ U will help us parameterise the direction in which we approach no-scale points. In particular, it will determine whether we land in a 2 or 3 massless Mkw. In addition, these no-scale points will be of the GKP type. We will present simultaneously their corresponding manifestations as 32 fluxes models, in which, interestingly, only 16 will end up being different from 0. Hence, through this map, we will find for free non-geometric polynomial superpotentials with 16 fluxes that show dS close to a no-scale point.
For instance, if one picks α = 2 5 , very interesting physics occurs. The corresponding solution in terms of the 12 fluxes of the non-perturbative superpotential (4.8) is given in the left part of table 4 for λ U = 2. As explained, we may compute the corresponding W derivatives and then map them into the 32 fluxes non-geometric superpotential. The resulting solution is given in     The superpotential is clearly of the GKP type at O( 0 ).
To see an example of stable dS close to a no-scale wth 3-massless modes one can simple take α = 11 10 and λ U = 1. The solutions in terms of the 12 parameters in the non-perturbative superpotential can be seen in the right part of    exhibits, as expected, a simple GKP form. and the scaling behaviour of the eigenvalues of (m 2 ) I J can be observed in figure 3. Notice that in this case all the massless modes grow quadratically. We will discuss an example with a mass growing linearly with in the next subsection. The O( 0 ) superpotential exhibits, as expected, a simple GKP form.
It is worth mentioning that there exist plenty of approximately no-scale stable dS solutions which can be constructed analytically without the aid of explicit non-perturbative terms by using the procedure sketched in section 3 which may be combined with solving the pairwise degeneracy constraint for the mass matrix V αβ = 0. These solutions generically approach no-scale Mkw points with two flat directions, but of a generalised type (see appendix B).

Conclusions
In this paper we have shown how dS vacua can be constructed close to no-scale as well as supersymmetric Mkw points using a polynomial superpotential with non-geometric fluxes.
As far as the existence of critical points and the features of the corresponding mass spectra are concerned, the superpotential with generalised fluxes covers all possibilities, and explicit non-perturbative terms will not bring in anything new.
The case of supersymmetric Mkw is particularly interesting. In contrast to previous constructions, such as [2], we did not need to add a Polonyi field to be able to break supersymmetry. Instead, we managed to arrange for flat directions already within the STU-model itself. However, whether it is actually possible to tune parameters to obtain the desired hierarchy with V m 2 gravitino m 2 scalars still remains to be seen. The large mass of the Polonyi field of [2] was arranged through tuning of its Kähler potential. Since we have a large number of fluxes available we expect to be able to perform similar tunings in our case.
It would be interesting to try to construct a phenomenologically viable example and we hope to come back to this issue in the future.
In contrast to the non-perturbative terms the generalised fluxes are fully determined by string dualities and the number of parameters limited to 32. Furthermore, a picture is emerging where at least some combinations of the non-geometric fluxes can be given a fully geometric interpretation through compactifications of string/M-theory on other topologies It will be interesting to see whether any of the stable dS vacua that can be constructed using the techniques of this paper can be given a fully geometric interpretation. Which features of the topology will be crucial for success? The advantage of such a solution is that it will be under firm controll from within supergravity and not dependent upon nonperturbative and string-inspired corrections under limited control. sitting in T which remain totally flat. Moreover, upon imposing a special degeneracy condition on the fluxes, it is actually possible to obtain exceptional no-scale models with an extra massless mode. In every no-scale model, SUSY is naturally broken by large amounts in the