Stability and vacuum energy in open string models with broken supersymmetry

We construct type I string models with supersymmetry broken by compactification that are non-tachyonic and have exponentially small effective potential at one-loop. All open string moduli can be stabilized, while the closed string moduli remain massless at one-loop. The backgrounds of interest have rigid Wilson lines by the use of stacked branes, and some models should have heterotic duals. We also present non-tachyonic backgrounds with positive potentials of runaway type at one-loop. This class of models could be used to test various swampland conjectures.


Introduction
This paper explores new (geometric) methods of constructing string theories with spontaneously broken supersymmetry that have enhanced stability, and conceivably naturalness, as a possible route to a string embedding of the Standard Model.
In the past decades, in order to discover important properties and ingredients of string theory such as dualities and branes, exactly supersymmetric models have been the focus of attention, and have been considered from various points of view. Compactifications on Calabi-Yau manifolds [1], orbifold models [2], fermionic constructions [3][4][5], or Gepner points in moduli space [6,7] have all been analyzed extensively. A key point in all these studies is that supersymmetry guarantees stability and all-orders consistency in a Minkowski background, synonymous with the fact that the cosmological constant is precisely zero in the vacuum.
Supersymmetry must however be broken to make contact with particle phenomenology and cosmology, and it is natural to consider performing this already at the string level, rather than postponing it to the supersymmetric effective field theory. In a theory that has supersymmetry broken at the string scale M s without tree-level tachyons [8][9][10], the quantum effective potential in D dimensions is naturally of order M D s [11]. This may imply the existence of scalar field tadpoles responsible for the destabilisation of the initial background, with either runaway behaviour or perhaps attraction to AdS-like vacua associated with very large negative cosmological constants [12] (it is not clear if the latter are supersymmetric or not). Consequently there has been continued interest in non-supersymmetric theories where firstly the effective potential happens to cancel at leading or even higher order [13][14][15][16][17][18], and secondly where supersymmetry breaking is under parametric control because the theory lies on an interpolation from an entirely supersymmetric theory [19][20][21]. Recently there has been a resurgence of interest in the fact that there is a large class of theories of the latter kind that have exponentially suppressed 1-loop effective potential [22][23][24][25][26][27][28][29][30][31][32][33].
In the present work, we consider these questions in open string models, where geometric reasoning makes the physical picture much clearer. In particular this will allow us to focus on an issue that has been somewhat neglected in the literature, namely that, even in theories that have vanishing or exponentially small effective potentials, some of the moduli will generically acquire tachyonic masses at 1-loop [26,27,32]. By developing a global geometric picture of the potential, one can decide which theories are not unstable: as we shall see, systems that have no such tachyons at 1-loop exist, but are extremely constrained.
We will study string models where supersymmetry (with 16 supercharges, implying all moduli to be Wilson lines) is spontaneously broken by coordinate-dependent compactification, which is nothing other than the Scherk-Schwarz mechanism [34] of field theory, upgraded to string theory: it was developed for closed strings in [35,36] and for open strings in [20,37,38]. In this context, the scale M of supersymmetry breaking is of order M s /R, where R is the characteristic radius 1 of the internal space involved in the mechanism. When R is moderately large, the leading contribution to the 1-loop effective potential V is Casimirlike, and arises from the Bose-Fermi non-degeneracy of the shift in the Kaluza-Klein (KK) towers. As a rule-of-thumb, assuming that the lightest mass scale of the background is M, the effective potential is, up to exponentially suppressed terms 2 , (1.1) This dominant contribution arises from the n is not a replacement for the full potential -it corresponds to just the first term in a Taylor expansion in WLs about the critical point -but it can be used to compare the potential energy at different critical points. Criticality relies on the next term vanishing i.e., denoting the gauge group by G and the WLs by a I r , 3 ∂V ∂a I r = 0 , I = D, . . . , 9 , r = 1, . . . , rank G . (1.2) In Ref. [39], this is achieved at points of enhanced symmetry, where states with non-trivial Cartan charges are massless. 4 In our context, such a massless state belongs to a KK tower 1 Throughout this paper, the radii and moduli are dimensionless. All dimensionful quantities are dressed with appropriate powers of M s . 2 These terms are O(e −cMs/M ), where c = O(1). When M is at least 3 orders of magnitude lower than M s , they are (much) lower than 10 −120 M D s , which is the order of magnitude of the observed cosmological constant in dimension 4. Therefore, they have no observational consequences and can be safely omitted. 3 Contrary to Eq. (1.1), there is an equal sign in Eq. (1.2) that follows from an exact symmetry, which is a gauge symmetry. This exact vanishing is expected to be valid to all order in perturbation theory [39]. 4 Denoting Q r the charge operator associated with the r-th Cartan U (1), V contains at linear order in a I r of modes with masses 2kM, k ∈ Z, and identical spin, while the superpartners have masses (2k + 1)M. When an extra massless state acquires a mass by switching on WL's, we leave criticality till the mass reaches M. The reason for this is that one mode in the KK tower of superpartners is now massless. The latter having non-trivial Cartan charges, we regain criticality. 5 In other words, Eq. (1.2) is valid provided there is no massive particle with mass less than M. Under this assumption, when the background satisfies non-supersymmetric Bose-Fermi degeneracy at the massless level, n WLs then provide a simple method for increasing the potential. We can exemplify this in the simplest realisation, which is the nine-dimensional case. Let us add a Wilson line on a D9-brane that corresponds, after a T-duality on the circle of radius R 9 , to a D8-brane sitting in the other fixed point πR 9 ≡ π/R 9 of the orientifold operation Ω ′ = ΩΠ, where Π is the parity on the dual circle coordinate. In the absence of supersymmetry breaking, open strings stretched between the D8-branes at the origin and the one at πR 9 have masses, before T-duality, given by (m 9 + 1 2 )M s /R 9 . Supersymmetry breaking adds an additional shift of 1 2 in the fermion masses, such that the fermions stretched between the stack of branes at the origin and the one at πR 9 become massless. Compared to the case where all D8-branes are at the origin with maximal gauge group SO (32), this new configuration, with a gauge group that we denote SO(31) × SO(1) has a higher scalar potential. The configuration is also stable, as will be shown in the next sections. Note the well-known fact that this Wilson line is in O (32) but not in SO (32). It is also not a continuous Wilson line, but rather a discrete one, which signifies that the brane at πR 9 is not a regular brane, in the sense that it has no position moduli. It is rather a half-brane stuck at the fixed point, with no associated gauge a contribution Q r a I r , where the sum is over the massless spectrum. Vanishing of this tadpole follows from the fact that any state of charge Q can be paired with a state of charge −Q. 5 Tadpoles can be analyzed by switching on WLs one by one. See e.g. Eq. (2.3), with a β = 0 and let a α varying from 0 to 1 2 . An extra massless boson (m 9 , F ) = (0, 0) at a α = 0 is replaced by an extra massless fermion (m 9 , F ) = (−1, 1) at a α = 1 2 . When 0 < a α < 1 2 , we leave criticality and a mass scale exists in the range (0, M ).
group. In nine dimensions, beside the SO(32) case, this is the only stable configuration and both of them have negative scalar potential. However, by suitable further compactification and by distributing other D-branes at different orientifold fixed points, one can construct stable configurations with zero or positive effective potentials, with Wilson lines in either SO (32) or O (32).
Of course, to achieve vanishing of the effective potential (at 1-loop, and up to exponentially suppressed terms) and stability of the background, the mass-squared terms of the WLs must be non-negative. However, finding backgrounds satisfying V ≥ 0 without tachyonic moduli proves to be rather delicate. This is due to the fact that the WL masses m I r depend on the difference between the Dynkin indices T R B and T R F of the representations of the massless bosons and fermions [26,27,32] : and the T a 's are Hermitian generators in the representation R. As a result, one can see that generally the more positive V is, the more unstable the background is, because the massless fermions that contribute positively to the potential energy, also contribute negatively to the WL mass-squared. Therefore it is non-trivial that such stable backgrounds exist. Furthermore, the notion of stability of the universe itself can be addressed from a cosmological point of view. As shown in Refs [31][32][33], it turns out that flat, expanding universes are way more natural when n (0) B ≥ 0, due to an attractor mechanism towards a so-called "Quantum No-Scale Regime", which is characterized by evolutions converging to those found for This paper develops a systematic and geometric approach to constructing backgrounds with effective potential vanishing at critical points and non-tachyonic, at the 1-loop level and up to exponentially suppressed terms. This is done within open string theories, which in principle contain the duals of heterotic theories. We also find backgrounds with n (0) B > 0 that are non-tachyonic at 1-loop and where M slides to low supersymmetry breaking scale along its positive potential. One may ask why constructing such models with positive sign of the scalar potential, or with leading term absent, and tachyon free at 1-loop may be relevant. The question is valid since, even when n • Models with positive scalar potential could be a starting point for constructing quintessence models in string theory. The examples provided in the present work are probably not realistic since the leading potential term is too steep and does not lead to an accelerating universe. It is however reasonable to believe that in more refined models with several contributions to the scalar potential, some regions in fields space are flat enough and could lead to quintessence.
• Models with vanishing leading term in the 1-loop scalar potential could be a starting point for deriving a suppressed cosmological constant. By adding several perturbative contributions to the scalar potential, it is in principle possible to stabilize M = O(M s /R) at a small value. At such an extremum, these contributions of the potential will have similar values, therefore of the order of the 1-loop contribution, which is exponentially suppressed for large enough radius. It would be clearly of great interest to construct explicit phenomenological models along these lines. This is however beyond the scope of the present paper.
We would like to re-emphasize that all previous constructions of this type in the literature have tachyonic instabilities at 1-loop. Eliminating these instabilities is the main result of our paper. The fate of such models beyond 1-loop is currently unknown.
In Sect. 2, we discuss the simplest setup in nine dimensions and the emergence of massless fermions for specific values of the Wilson lines. We also show that the only stable configurations are SO(32) and SO(31) × SO(1).
In Sect. 3, we discuss compactifications to lower dimensions. By distributing frozen half-branes on more orientifold fixed points, there are more possibilities to obtain massless fermions, because of the interplay of the Scherk-Schwarz supersymmetry breaking with the Wilson lines: the net effect is an increase in the effective potential. Our main result is that a variety of stable brane configurations exist in various dimensions, for which the models are non-tachyonic at 1-loop, even when the scalar potential is positive or vanishing. This statement, which is valid up to exponentially suppressed terms, includes all moduli: while open string WLs acquire masses at the quantum level, the NS-NS and RR moduli arising from the closed string sector are massless at 1-loop (except M which is running away when n (0) F − n (0) B = 0). In particular, we find 1-loop marginally stable backgrounds with exponentially small effective potential in 4 dimensions. Notice that the NS-NS moduli (except M and the dilaton) and the RR moduli are expected to be stabilized at 1-loop [26,27,32,40,41] in the dual heterotic models [42]. This is due to the fact that in heterotic string, the internal metric and antisymmetric tensor (dual to the NS-NS metric and RR moduli) are treated on equal footing with the SO(32) Wilson lines present in both theories. In type I, the stabilization of the NS-NS metric and RR 2-form moduli should arise at points in moduli space where nonperturbative D-strings (dual to the heterotic F-string) wrapped in the internal space yield extra massless bosons (see Ref. [43] for an analogous effect due to finite temperature).
Hence, type I models with n In Sect. 4, we discuss nonperturbative aspects of these models, by identifying whether the Wilson lines belong to SO (32) or O (32). Since the latter do not have heterotic duals, they do not exist nonpertubatively.
Finally, our conclusions and perspectives can be found in Sec. 6. They are followed by rather extensive appendices, collecting the conventions and notations and the main techniques used to calculate the scalar potentials used throughout the paper.
2 No WL stability with n (0) In order to find non-tachyonic configurations in non-supersymmetric open string theory at 1-loop, we will analyse toroidal compactifications of type I down to D dimensions and implement a Scherk-Schwarz mechanism. As a warm up, the present section focuses on the simple case of D = 9, with supersymmetry broken spontaneously along S 1 (R 9 ), the internal circle of radius R 9 .

General setup
In order to avoid a Hagedorn-like tachyonic instability, we assume R 9 to be larger than the Hagedorn radius R H = √ 2 . Restricting further to values moderately larger than R H greatly simplifies the expression for the effective potential, which takes a universal form dominated by the contributions of the pure KK modes. In the closed string sector, at zero winding number n 9 along the compact direction X 9 , as well as in the open string sector, the stringy Scherk-Schwarz mechanism induces a shift of the KK masses according to the fermionic number F , which defines the scale M of supersymmetry breaking, In the open string sector, WL deformations along S 1 (R 9 ) can be introduced, which spon- It is convenient to T-dualize S 1 (R 9 ) to switch to a geometrical setting in type I', with D8-branes and two O8-planes located atX 9 = 0 andX 9 = πR 9 , whereR 9 = 1/R 9 . In this picture, the 32 1 2 -branes are located at 2πa αR9 . The allowed configurations consist of p 1 ∈ 2N 1 2 -branes sitting on the O8-plane at a = 0, p 2 ∈ 2N 1 2 -branes coincident with the second O8-plane at a = 1 2 , and stacks of r σ branes each located at some a ∈ (0, 1 2 ), together with their mirrors at −a. The gauge symmetry is U(1) 2 G,C × SO(p 1 ) × SO(p 2 ) × σ U(r σ ), where the mutiplicities are constrained by the RR tadpole condition p 1 + p 2 + 2 σ r σ = 32, and U(1) G , U(1) C are the Abelian factors generated by the dimensionally reduced metric and RR 2-form, G µ9 , C µ9 .
Ultimately, we wish to let the branes move anywhere and find their final stable configurations, possibly with positive or vanishing effective potential. As sketched in the introduction, a sufficient condition for the effective potential V to be at a (local) minimum, maximum or saddle point is that the configuration does not yield masses M such that 0 < M < M.
Hence we expect configurations with branes located at a = 0 and a = 1 2 to be "attractors", as Eq.
acquiring a mass M. It turns out that there is another interesting location, a = ± 1 4 (see Fig. 1). The reason why is shown explicitly in Appendix A.2 but it can be understood qualitatively. When strings are stretched between q branes at a = 1 4 and their mirrors at a = − 1 4 , they also yield massless fermions. On the other hand, strings with end points at a = 0 or 1 2 and a = 1 4 or − 1 4 yield bosons and fermions that have "accidentally" degenerate masses M/2. This "fake supersymmetry" holds for arbitrary winding number n 9 (i.e. for the whole KK tower in the original type I picture). As the leading term in the effective potential is a supertrace, the contributions of these modes to V cancels. This effect is identical to that described in the heterotic context in Ref. [32]. Eq. (1.1) is still valid in these special cases, even though as we will see V has no reason to be generically critical at such a point.

Brane configurations
One can deduce general expectations for the dynamical behaviour based on the above formula in Eq. (1.1), without performing any detailed calculation. Consider for a moment a configuration of D8-branes with p 1 + p 2 + 2q = 32, as shown schematically on Fig. 1. On geometrical grounds and using Eq.(2.3), the number of massless bosonic and fermionic degrees of freedom is where the RR tadpole cancellation condition (in this case simply p 1 + p 2 + 2q = 32) has been used to eliminate q. The lowest value n (0) F − n (0) B = −4032 is reached for (p 1 , p 2 , q) = (32, 0, 0) or (0, 32, 0), and corresponds to a critical point of the potential with respect to the WLs (because q = 0, implying that no mass scale below M is introduced, as discussed in the introduction). As a result, we expect that the configuration where the full SO(32) gauge symmetry is restored yields stabilized WLs. 6 The negative potential remains a source for the motion of M (see Refs [31][32][33] for the associated cosmological solutions in flat space).
The type I theory admits a second moduli space, disconnected from the one we have just been discussing. In the T-dual picture, this family of models is realized by freezing one 6 We say "expect" because we have not shown that all critical points of V satisfy Eq. (1.1). Moreover, we have not shown that all configurations compatible with Eq. (1.1) involve branes at a = 0, 1 2 and ± 1 4 . with only 15 independent degrees of freedom [52]. Restricting as before to configurations with p 1 1 2 -branes located at a = 0, p 2 1 2 -branes at a = 1 2 , and q branes at a = 1 4 with their mirrors at a = − 1 4 , the full gauge symmetry is U(1) 2 G,C × SO(p 1 ) × SO(p 2 ) × U(q), but now with p 1 , p 2 ∈ 2N + 1. The expression (2.5) is unchanged from earlier, and its minimum B . This will be of central importance in the next section.

Effective potential
In order to confirm the above geometrical expectations, let us now present the expression for the 1-loop effective potential, valid for arbitrary WLs. The calculation is carried out in detail in Appendix A.2. In the limit where M is low compared to the string scale, we find where N 2n 9 +1 is a function that gets contributions from the torus, Klein bottle, annulus and Möbius strip amplitudes as follows, where the total number of dynamical a r 's is N = 16 or 15. Let us analyse this potential for the special cases of ( 1 2 )-branes located only at a = 0, 1 2 and ± 1 4 : • At such a point in moduli space, N 2n 9 +1 turns out to be independent of n 9 , and we obtain, as anticipated, where in this case (2.11) • For r = 1, . . . , N, the first derivatives are given by for a r = 0 or 1 2 , , for a r = 1 4 , and where ζ is the Hurwitz zeta function. Thus, the potential is at a critical point with respect to the WLs only when q = 0 or p 1 = p 2 , otherwise the branes at a r = 1 4 are attracted to the largest stack located at a = 0 or 1 2 . Notice that for p 1 = p 2 , the brane configuration respects an additional exact symmetry a → 1 2 − a. As a result, we actually expect the generically exponentially suppressed terms in Eq. (2.12) to be entirely absent when p 1 = p 2 (see Footnote 3).
• The N × N matrix of second derivatives is block diagonal : where I d is d×d identity matrix, ⌊x⌋ is the integer part of x, A is the q×q matrix A rs = δ rs −1, and where Hence a stable brane configuration must satisfy whose compatibility implies p 1 or p 2 to be 0 or 1. When q = 0, this shows that the SO (32) and SO(31) × SO(1) configurations are the only stable ones. It turns out that q ≥ 1 does not yield other solutions. To see this, note that the vanishing tadpole condition implies (p 1 , p 2 , q) = (0, 0, 16) or (1,1,15). However, the eigenvalues of A being 1 (with degeneracy q − 1) and −(q − 1) (with degeneracy 1), we conclude that in U(q) = U(1) × SU(q), even if the q − 1 WLs of SU(q) are massive, the WL of U(1) is tachyonic.
Note that in the quest to find stable (up to exponentially suppressed terms) vacua one might, motivated by Eq. (1.1), have naïvely looked for solutions to n

Algebraic stability conditions
The stability conditions of the WLs can also be derived from pure Lie algebra considerations.
In quantum field theory, the 1-loop effective potential can be written as a Schwinger integral (equivalent to the first quantized formalism), where M is the classical mass operator. We are interested in models where the spectrum arises from massless N 10 = 1 superfields in 10 dimensions, compactified on the Scherk-Schwarz internal circle S 1 (R 9 ). By allowing a WL background, we may have non-trivial n (0) F and n (0) B massless states. In a Scherk-Schwarzed theory, the existence of full towers of KK modes guarantees that V is finite even if the domain of integration of the Schwinger parameter contains the UV region τ 2 → 0.
Up to the exponentially suppressed terms arising from string modes heavier than the supersymmetry breaking scale M, the stringy computation yields an identical expression.
As already mentioned, in such backgrounds, where all modes lighter than M are massless, we may switch on WL deformations. Denoting the normalized WL by y r , we can write where Q r is the charge under the r-th Cartan U(1) of the gauge group G = κ G κ . By viewing the supertrace as where R B , R F are the representations of the bosonic and fermionic massless states before deformation, Eq. (2.17) yields the following at second order in WLs : In this expression, the linear term # 1 Q r y r in the brackets is absent, due to the sum over the weights. 7 By splitting any representation R of G into a direct product of representations . As a result, the squared masses of the WLs y r are determined by In the brane configurations of Sect. 2.2, we noticed that expression (2.20) remains true when the undeformed background contains branes located at ± 1 4 , thus generating a U(q), q ≥ 1, gauge group factor, provided p 1 = p 2 . In these circumstances, or when q = 0, the  Table 1, gives the required Dynkin indexes, from which we find for the SU(q) WLs .
(2.23) 7 Moreover, # 1 turns out to vanish, due to the sums over the KK momentum and F = 0, 1. This can be seen after Poisson resummation over m 9 . Table 1: Dimensions and Dynkin indexes of representations of simple Lie groups. By convention, the Dynkin index in the fundamental representation is fixed to 1.
As expected, these results are in agreement with the stability conditions found from the explicit computation of the potential. 8 3 Non-tachyonic models with n (0) We concluded in the previous section that there are no brane distributions in nine dimensions that are simultaneously stable with respect to the WLs and yield a non-negative potential. However, between the two stable brane configurations with gauge groups SO(32) and SO(31) × SO(1), we did note that the latter yields a higher effective potential because of the lower dimension of its gauge group and the presence of extra massless fermions stretching between the two O-planes. This brane setup was stable because the SO(1) factor comes from a frozen 1 2 -brane. It seems reasonable to suppose that upon compactification to lower dimensions, where there are more O-planes in type II orientifolds, stable configurations might exist in which 1 2 -branes are frozen to different O-planes, decreasing the dimension of the gauge symmetry and increasing the number of massless fermions even further, and raising 8 Comparing the eigenvalues of the (mass) 2 matrix (2.14) with ∂ 2 V/∂y 2 r | y=0 , one finds an additional factor of 2 for the SU (q) WLs. This is because contrary to our convention in the algebraic computation, the Dynkin indices of the fundamental representations of SO(p) and SU (q) in the string partition function differ by a factor of 2.
the effective potential even more. The hope is that there are then configurations which, apart from being stable with respect to the brane positions, also have n

Geometric and algebraic picture
To explore this possibility, we will use a compactification on a torus T 10−D , with internal metric G IJ , I, J = D, . . . , 9, and Scherk-Schwarz action always taken to lie in the 9-th direction. As a result, the scale of supersymmetry breaking is The massless spectrum is derived in Appendix A.3, Eq. (A.30), but again its counting can be inferred from geometrical arguments : Besides the contribution from the closed string sector, massless bosons arise from strings attached to a single stack of 1 2 -branes, with the bosonic part of vector multiplets arising Dir ecti on of Sch erk-Sch war z Next, the generic algebraic derivation of the WLs stability condition in nine dimensions can also be generalised. Denoting again the gauge symmetry group as G = κ G κ , the r-th Cartan U(1), r = 1, . . . , rank G, admits WLs denoted y I r along the internal directions I = D, . . . , 9. 9 Taylor expanding the potential, one obtains [26,27,32]   Returning to our specific type II orientifold setup, when all 1 2 -branes are located at the corners of the internal box, the effective potential is critical, and we are looking for the configurations satisfying (i) These conditions admit many solutions. For instance, Table 2     in the fundamental. Thus, it is only by careful study of the effective potential in the next subsection that we will be able to conclude that they also do not induce instabilities.
Finally, we should mention that the reader interested only in stable brane configurations irrespective of the sign of n

Effective potential
The conclusions made above on geometric and algebraic grounds can of course be recovered directly from the 1-loop effective potential, which is derived for any spacetime dimension Notice that the ε I α 's are not all dynamical degrees of freedom. In fact, ε I α ≡ −ε I β when α and β label branes images of one another. Moreover, if p A is odd, in addition to the pairs of such 1 2 -branes, there is always one left over, say the α-th, which is frozen at the corner, ε I α ≡ 0.
In this subsection we will also choose the internal metric G IJ , I, J = D, . . . , 9, so that all KK and winding mass scales are greater than the supersymmetry breaking scale. We also take the latter to be lower than the string scale, in order to avoid any Hagedorn-like tree-level instability. In total then, we assume Under the above assumptions, we find In this expression, L is the set of pairs (α, β) such that α and β are 1 2 -branes in the neighbourhood of either of the corners 2A − 1 and 2A, for some A = 1, . . . , 2 10−D /2. The sectors (α, β) yield light strings stretched between these 1 2 -branes that generate the bosonic adjoint and fermionic bifundamental representations of SO(p 2A−1 ) × SO(p 2A ). In our notation, F is the fermionic number of these modes. Moreover, we have defined an effective inverse metriĉ for the internal space transverse to the larger Scherk-Schwarz direction, X 9 . The index l 9 is obtained by Poisson resummation over the KK momentum m 9 (of the initial type I picture) along X 9 . Finally, the function H ν is defined in Eq. (A.11). From this result, it is natural to parametrise the NS-NS moduli space by (Ĝ ij , G 9i , G 99 ). Some further remarks are in order : • In the initial background, where all 1 2 -branes are located at corners, we have ε I α = 0, α = 1, . . . , 32, I = D, . . . , 9, implying thatN 2l 9 +1 becomes l 9 -independent there, (3.14) Hence, as expected, the effective potential satisfies the rule-of-thumb, and Taylor expandN 2l 9 +1 (ε, G) to quadratic order in ε I r : (3.17) In the above expression, we have defined a single brane at position ε I α , and its mirror at ε I β ≡ −ε I α . Extracting the contributions of these branes fromN 2l 9 +1 , we find that they induce a term in V of the form which is independent of the degrees of freedom ε I α . In fact, all cosines and H functions with non-trivial ε I α -dependance cancel one another, as expected from our previous arguments. We can proceed the same way for (p 2A−1 , p 2A ) = (3, 1) or (1,3). In these cases, there is one dynamical brane, its mirror and one frozen 1 2 -brane at one corner, and another frozen that from the adiabatic argument of Ref. [53] applied to the heterotic / type I duality (see Sect. 4), one expects that some closed string moduli may be stabilized, though nonperturbatively from the type I point of view [43]. Finally, let us stress that the above results were derived assuming Eq. (3.10) is satisfied, which guarantees that ultimately the NS-NS metric components live in a (very) large plateau of the effective potential. The goal of the next section is to see which configurations remain (marginally) stable at 1-loop if one ventures outside the region defined by Eq. (3.10).

Extension of the domain of validity
For the backgrounds presented so far to be marginally stable at 1-loop, we have imposed that Under the above assumption, the 1-loop potential, which is computed at the end of Appendix A.3, takes the form To understand this result, first consider all 1 2 -branes to be coincident with the O(D − 1)planes, p A of them sitting at the A-th corner, at position parameterized by a A ≡ (a D A , · · · , a 9 A ), a I A ∈ {0, 1 2 }, A = 1, . . . , 2 10−D , I = D, . . . , 9. In that case, we have The interpretation of such dressing functions is that the exponential terms in the potential that we have so far been neglecting may become large when G ii ≫ G 99 (i.e. R i ≫ R 9 in an untilted torus). Indeed on an untilted torus such terms are best evaluated by Poisson resumming direction 9 only and making a saddle point approximation, which yields a contribution proportional to e −2π √ G 99 /G ii = e −2πR 9 /R i . The physical meaning of such factors, which can be important when R i > R 9 , is that the KK modes in the i-th direction (with masses going like 1/R i in the type I setup) have to traverse the entire Scherk-Schwarz direction 9 before they can feel the supersymmetry breaking, so they contribute to the potential with the typical Yukawa factor. As R i increases in size the KK modes become light enough that this is no longer a suppression, and the contribution can no longer be neglected.
The rule-of-thumb then is that a direction is allowed to become large (in the original type I picture) as long as the Scherk-Schwarz breaking is Bose-Fermi degenerate or absent for its KK modes. This is equivalent to considering brane configurations such that Nl( W) is l ′ -independent, which is the case when all but one pair of corners (2B − 1, 2B) satisfy Up to a relabelling, we will take the remaining couple of corners to be A = 1 and 2. In fact, for such backgrounds to be marginally stable at 1-loop at least in region (3.10) of the moduli space, we must also impose condition (iii) in Eq. (3.7) : In this case, the dynamical open string WLs are those associated to SO(p 1 ), and the corresponding degrees of freedom can be defined as , 0 for p 1 odd .

(3.26)
In these variables, we obtain the ε I r 's are not assumed to be small. To discuss the stability of the backgrounds where all branes are located at corners (except when p 1 = 2, 3, for which they can sit anywhere), it is however enough to Taylor expand Nl( W), which leads to where the massless spectrum counting reproduces Eq. (3.4), and where Of course, the mass terms are absent for p 1 = 0, 1, 2, 3. For p 1 ≥ 4, the WLs have positive definite r-independent squared masses, if ∆ IJ , I, J = D, . . . , 9 is itself positive definite. This is easily seen to be the case, since V I ∆ IJ V J for an arbitrary vector V I , yields Minimizing the potential by setting these terms to zero, we then have

Nonperturbative analysis of the models
There are consistency conditions on string backgrounds of a nonperturbative nature that are invisible in string perturbation theory. One of them is the fact that, whereas in perturbation theory the ten-dimensional gauge group of the type I theory looks to be O(32) rather than SO(32), at a nonperturbative level the part disconnected from SO(32) cannot be defined [54].
This is consistent with the fact that the dual heterotic string has a gauge group that is Spin(32)/Z 2 , which contains in particular spinorial representations under the gauge group.
More generally, there are nonperturbative consistency conditions of K-theory origin [54,55], which can also be understood with simpler methods in terms of consistency of gauge theories on various D-brane probes [56] from the viewpoint of local and global [57] anomaly cancelations.
Let us discuss which Wilson lines are allowed from this nonperturbative point of view.
Starting from SO (32) and cannot therefore be defined nonperturbatively.
The natural question is then to ask which of the brane configurations/Wilson lines in Table 2   There seems to be enough freedom in the models of Table 2 to satisfy these constraints by suitably distributing the minus signs among the discrete WLs.
Finally, another potential constraint comes from adding D5-brane probes into our models, which have USp(2n) gauge groups, and then checking potential global Witten anomalies [57]. However since the corresponding spectra are non-chiral after compactification to four dimensions, we did not find any additional constraints.

Comments on swampland conjectures
One natural application of the class of models we constructed in this paper is to test the various recent swampland conjectures [44][45][46][47][48][49][50][51]. In this section we make preliminary remarks and leave a full study to future work.
• One of the swampland conjectures is that |V ′ | > CV, where C is a constant of order 1 [48]. For the models with potentials that are not exponentially suppressed, since the potential is of runaway type in the supersymmetry breaking radius, this is always satisfied.
The models with exponentially small effective potential, V ∼ e −R , where R is the typical Scherk-Schwarz radius, are somewhat different. The canonically normalized field is of the form σ = log R. Then |V ′ |/V ∼ |RV R |/V ∼ R which becomes arbitrarily large, easily satisfying any constraint for large enough R. At higher-loop orders one may need an additional condition at each loop to cancel the leading contribution to the vacuum energy, so presumably at some loop order the potential will become polynomial in R and therefore • Another swampland conjecture is that the only possibility for the dark energy in string theory is quintessence [49]. However, whereas one can (relatively) easily find stable string models with positive (exponential for canonically normalized fields) potentials and runaway rolling vacua, they do not generically lead to accelerating cosmologies. The reason is that the exponent of the exponential is larger than the critical value (equal to √ 2 in Planck units in four-dimensions) needed to generate an accelerating universe. It would be interesting to check if in more sophisticated compactifications with supersymmetry breaking, the universe is accelerating.
• It would be worth investigating whether the generic nonperturbative instability of the non-supersymmetric Kaluza-Klein vacua [58] takes place in our models. The latter possessing massless fermions, it is unclear a priori if the instability persists.
• Finally, it would be interesting to study the weak gravity conjecture coming from branebrane interactions, and the quantum corrections to the D1-branes tensions and charges in our class of models with positive scalar potential, by generalizing the framework recently discussed in [59].

Conclusions and perspectives
In this paper, we presented a large class of models with exponentially small or positive effective potential in type I string theory, at the 1-loop level. The models are based on simple toroidal compactifications, with discrete deformations corresponding to 1 2 -branes stuck on orientifold fixed points (in a T-dual language). The great advance over previous works is that these models are (marginally) stable at 1-loop with respect to all moduli fields, except the supersymmetry breaking scale and dilaton when the potential is non-vanishing (up to  [53]. An interesting exercise which we leave for future work would be to construct these stable heterotic duals explicitly. As the class of models we constructed relies heavily on 1 2 -D-branes at orientifold fixed points, with no associated gauge group, the largest possible gauge symmetry we can obtain is rather small : for a stable brane configuration with zero or positive scalar potential it is SO(5), which is obviously not large enough to accommodate the Standard Model gauge group. It is therefore an important question to find ways to enhance the available gauge symmetry without re-introducing Wilson line instabilities. One obvious way to do this would be to compactify on orbifolds. In this case, additional orientifold planes (O5-planes in type I string, which are of three different types) would be generated and corresponding D5-branes would have to be added, for consistency with the RR tadpole cancellation conditions. In such a construction, the Standard Model gauge group would then be realised on the D5branes, with the D9-sector we have been focussing on in the present paper playing the role of a hidden sector generating the observed dark energy.
Finally, the class of open string models we have considered extends that found in a heterotic context, and can be considered from a cosmological viewpoint. It turns out that whatever n (0) B is, a flat, homogeneous and isotropic universe can always enter into an ever-expanding "Quantum No-Scale Regime" [31][32][33]. What is meant by this is that the evolution approaches that found for n Str Ωq One passes from one picture to the other by Poisson resumming, Characters : Our definitions of the Jacobi modular forms and Dedekind function are At v = 0, it is standard to denote θ 0 0 = θ 3 , θ 0 1 = θ 4 , θ 1 0 = θ 2 , θ 1 1 = θ 1 , in terms of which the SO(8) affine characters can be written as For the amplitudes T , K and A, the useful modular transformations are For the Möbius strip amplitude, it is convenient to switch from any character χ to a real "hatted" characterχ defined by [10] where h is the weight of the associated primary state and c is the central charge. The transformation from the open to the closed string channel, called the P-transformation, then takes the form Limiting behaviours : In the final expressions of the amplitudes, we display the dominant contributions arising from light states (compared to the supersymmetry breaking scale) and are more schematic about those associated with heavy modes. For this purpose, we will use where K ν is a modified Bessel function of the second kind. At large and small arguments, it has the following behaviour : A.2 Massless spectrum and potential in 9 dimensions We are interested in the orientifold projection of the type IIB theory in 9 dimensions, with Scherk-Schwarz spontaneous breaking of supersymmetry implemented along the internal circle S 1 (R 9 ) of radius R 9 . The torus amplitude contribution to the effective potential V is where the lattices depend on G 99 = R 2 9 . The orientifold projection leads to the overall normalization factor 1 2 , as well as to the Klein bottle contribution where the argument of the characters is 2iτ 2 . As explained in Sect. 2.1, the open string sector can be described either in type I or type I' language, obtained by T-dualizing S 1 (R 9 ).
Spectrum : For reasons that will become clear shortly, in order to determine the massless spectrum, we first split the generic configuration as follows : • p 1 1 2 -branes on an O8-orientifold plane located at a = 0, • p 2 1 2 -branes on a second O8-orientifold plane located at a = 1 2 , • q branes at a = 1 4 , with their mirrors at a = − 1 4 , • r σ branes at a = a σ ∈ (0,  Effective potential : We proceed with the derivation of the 1-loop effective potential.
For this purpose, it is convenient to define a WL matrix W = diag e 2iπaα ; α = 1, . . . , p 1 + p 2 + 2q + 2 where ℓ = 2 τ 2 and ℓ = 1 2τ 2 for the annulus and Möbius strip amplitudes, respectively. The arguments of the characters in A and M are iℓ and 1 2 + iℓ, respectively. Even though K vanishes, we may also write it in the transverse channel, where ℓ = 1 2τ 2 and the characters are taken at iℓ. There are no UV divergence as ℓ → +∞ (τ 2 → 0) in K + A + M, when the RR tadpole cancellation condition is obeyed. The latter amounts to setting the coefficient of (S 8 /η 8 )W 0 (or (Ŝ 8 /η 8 )W 0 ) to zero. Besides the sign of M already mentioned, this constrains the number of 1 2 -branes to be In the large R 9 limit, the torus amplitude T is dominated by the level-matched pure

A.3 Massless spectrum and potential in D dimensions
In this subsection, we extend some of the 9-dimensional results to the case of a toroidal compactification on T 10−D . The metric of the internal torus is G IJ , and the Scherk-Schwarz mechanism is implemented along the direction X 9 . The genus-1 Riemann surface amplitude is then where a S is the (10−D)-dimensional vector that implements the 1 2 -shift of the momentum m 9 , while any "primed" vector only has 9 − D entries corresponding to the non-Scherk-Schwarz directions, The Klein bottle contribution is In the second line, we use the RR tadpole cancellation condition, which fixes the number of where G 99 ≫ 1 is understood, in order to avoid tachyonic instabilities.
In the open string sector, the WL moduli can be organized in matrices as W I = diag e 2iπa I α ; α = 1, . . . , 32 , I = D, . . . , 9 , (A. 33) where α labels the 1 2 -branes. At a generic point in moduli space, we will denote by a α the vectors with real entries a I α , I = D, . . . , 9. Of course, not all of them are independent dynamical degrees of freedom, since dynamical branes can freely move only in pairs with their images, while the remaining ones are frozen at O(9 − D)-planes. (l I +n Iτ )G IJ (l J +n J τ ) (−1) l 9 (a+ã)+n 9 (b+b) . (A.44) The last sign, which couples the spin structures (a, b) and (ã,b) to the wrapping numbers n 9 ,l 9 of the worldsheet around the Scherk-Schwarz direction X 9 , is responsible for the spon- Moreover, integration and discrete sums over l 9 , n 9 can be inverted in a suitable way, in order to "unfold" the fundamental domain F . Schematically, we can write [62] F dτ 1 dτ 2 l 9 ,n 9 f l 9 ,n 9 (τ,τ ) = F dτ 1 dτ 2 f 0,0 (τ,τ ) + In total, the effective potential, which combines all four worldsheet topologies, can be found in Eqs (3.11), (3.12).
Effective potential at KK scales lower than M s : To complete this section, we rederive the effective potential for arbitrary WL matrices (A.33). This is done under an alternative assumption on the internal metric compared to the above analysis. Namely, we take all internal directions (in the original type I picture) to be large, in string units, wherel is a vector whose last entry is odd, l ≡ ( l ′ , l 9 ) ∈ Z 10−D =⇒l ≡ ( l ′ , 2l 9 + 1) . (A. 52) By noting that the argument of H 5 is O( √ G 99 ), unless it vanishes when k = 0, we conclude that A + M = 8Γ(5) π 5 √ det G l tr (Wl D D · · · Wl 9 9 ) 2 − tr (W 2l D D · · · W 2l 9 9 ) (l I G IJlJ ) 5 where # = O(1) is positive. Since even l 9 yields supersymmetric and therefore vanishing contributions, we can change l 9 → 2l 9 + 1. By noting that