N=2 ->0 super no-scale models and moduli quantum stability

We consider a class of heterotic N=2 ->0 super no-scale Z_2-orbifold models. An appropriate stringy Scherk-Schwarz supersymmetry breaking induces tree level masses to all massless bosons of the twisted hypermultiplets and therefore stabilizes all twisted moduli. At high supersymmetry breaking scale, the tachyons that occur in the N=4 ->0 parent theories are projected out, and no Hagedorn-like instability takes place in the N=2 ->0 models (for small enough marginal deformations). At low supersymmetry breaking scale, the stability of the untwisted moduli is studied at the quantum level by taking into account both untwisted and twisted contributions to the 1-loop effective potential. The latter depends on the specific branch of the gauge theory along which the background can be deformed. We derive its expression in terms of all classical marginal deformations in the pure Coulomb phase, and in some mixed Coulomb/Higgs phases. In this class of models, the super no-scale condition requires having at the massless level equal numbers of untwisted bosonic and twisted fermionic degrees of freedom. Finally, we show that N=1 ->0 super no-scale models are obtained by implementing a second Z_2 orbifold twist on N=2 ->0 super no-scale Z_2-orbifold models.


Introduction
In string theory, even when starting classically in a flat four-dimensional background, the vacuum energy induced at the quantum level is hard to reconcile with the present cosmological constant. When supersymmetry is hardly broken, the 1-loop effective potential is generically of order M 4 s , where M s is the string scale, which is far too large. On the contrary, if supersymmetry is exact, the quantum potential vanishes identically at least at the perturbative level, or leads non-perturbatively to an anti de Sitter vacuum with restored supersymmetry. A priori more promising, the no-scale models [1] consist somehow of an intermediate situation. At the classical level, these backgrounds realize in flat space a spontaneous breaking of supersymmetry at a scale m 3/2 , which is a flat direction of the tree level potential. However, if the order of magnitude of the quantum effective potential is dictated by m 3/2 , it happens to be generically too large. Moreover, the quantum potential induces tadpoles for the classical moduli fields, including the dilaton and the "no-scale modulus" parameterized by m 3/2 , which are responsible for a destabilization of the flat background.
Some exceptions however exist, at least at the 1-loop level, when the spontaneous breaking of supersymmetry arises via "coordinate dependent compactification" [2,3], the stringy version of the Scherk-Schwarz mechanism [4]. This total breaking of supersymmetry can be implemented on initially N = 4, 2 or 1 heterotic or type II orbifold models, as well as on orientifold theories [5], or on marginally deformed fermionic constructions [6,7]. In this framework, some theories referred as super no-scale models [8][9][10][11] induce an exponentially suppressed 1-loop vacuum energy, whose order of magnitude can easily be of order (or lower than) the presently observed cosmological constant. Type II [12] and orientifold [13] theories with exactly vanishing vacuum energy at 1-loop even exist. In all known examples, the models arise at extrema of the quantum effective potential, with respect to all directions that are lifted. However, the question of the stability of the non-flat directions must be addressed. In other words, does the model sit at a minimum, maximum or saddle point of its potential ? This problem has been addressed for the super no-scale models realizing the N = 4 → 0 spontaneous breaking of supersymmetry [10] and is reconsidered for less symmetric theories in the present work, such as models implementing an N = 2 → 0 breaking.
In Ref. [10], one considers the N = 4 → 0 no-scale models for given internal metric, antisymmetric tensor and Wilson lines background expectation values. Supposing the point in moduli space is such that no mass scale below m 3/2 exist, we denote cM s the lowest mass scale above m 3/2 . In this case, the 1-loop effective potential takes the form [14] V N =4→0 1-loop = ξ(n F − n B ) m 4 3/2 + O c 2 M 2 s m 2 3/2 e −cMs/m 3/2 , (1.1) where the gravitino mass m 3/2 scales inversely to the volume involved in the stringy Scherk-Schwarz mechanism and n F , n B are the numbers of massless fermionic and bosonic degrees of freedom. The m 4 3/2 dominant contribution arises from the light towers of Kaluza-Klein states, whose masses are of order m 3/2 , while ξ > 0 depends on moduli other than the dilaton and m 3/2 . For the quantum vacuum energy and tadpoles to be exponentially small, one can focus on the models satisfying the super no-scale condition n F = n B [8][9][10]. In this case, the 1-loop vacuum energy is of the order of the observed cosmological constant, provided the gravitino mass m 3/2 is about 2 orders of magnitude smaller than the scale cM s . However, switching on small marginal deformations, collectively denoted Y , around the point in classical moduli space we started with, one induces new mass scales lower than m 3/2 . Some of the n F +n B initially massless states acquire small masses. When the mass scales YM s reach the order of m 3/2 , the exponential contributions in Eq. (1.1) are O(m 4 3/2 ), thus correcting n F , n B which now take new integer values. In other words, n F , n B are effectively functions of Y which interpolate between different integer values corresponding to distinct massless spectra.
To study the local stability of an N = 4 → 0 super no-scale model [10] around a point in moduli space characterized by integer n F and n B , one has to expand these two functions at quadratic order in Y . Due to the underlying N = 4 structure, the moduli deformations Y are Wilson lines along T 6 . The result is that those which are associated to non-asymptotically free gauge group factors become tachyonic at 1-loop. They condense, break spontaneously the associated gauge symmetry which enters a Coulomb branch, and induce a destabilization of the vacuum. On the contrary, the Wilson line associated to asymptotically free gauge group factors become massive and are dynamically attracted to Y = 0. The Wilson lines associated to conformal groups remain massless.
In Sect. 2, we consider N = 2 → 0 super no-scale theories realized as Z 2 -orbifolds of N = 4 → 0 no-scale models. At the level of exact N = 2 supersymmetry, the twisted hypermultiplets introduce new moduli fields living on a quaternionic manifold. We show that the implementation of the stringy Scherk-Schwarz mechanism can always be chosen so that all twisted moduli acquire a tree level mass of order m 3/2 and are no more marginal in the non-supersymmetric theory. In these models, the super no-scale condition amounts to having classically equal numbers of massless untwisted bosonic and twisted fermionic degrees of freedom. Moreover, the tachyons that appear at tree level in the parent N = 4 → 0 theory [3] when m 3/2 is of the order of M s are automatically projected out in the N = 2 → 0 models. In other words, when m 3/2 is large and local perturbations of other moduli are allowed, no Hagedorn-like instability occurs.
In Sect. 3, we study the local stability of the untwisted moduli in this class of super no-scale models. The analysis generalizes that of Ref. [10] by taking into account, in the 1-loop effective potential V N =2→0 1-loop , the contributions arising from the twisted fermions. The expression of V N =2→0 1-loop , which we determine at second order in moduli fields, is distinct in each branch of the gauge theory along which the classical background can be deformed. To be more specific, we derive the quantum potential as a function of all moduli fields in the pure Coulomb branch, as well as in some mixed Coulomb/Higgs branches. Moreover, we show that in this class of models, because all moduli fields are untwisted, the number of marginal deformations in any branch of the gauge theory is universal once the model is compactified down to two dimensions.
In Sect. 4, we show that N = 2 → 0 super no-scale Z 2 -orbifold models can automatically lead descendent N = 1 → 0 super no-scale theories, by implementing a second Z 2 orbifold twist. However, we argue that the analysis of the background stability must be generalized to include new twisted moduli deformations.
A summary of our hypothesis and results can be found in the conclusion, Sect. 5.

A class of N = → 0 super no-scale models
In this section, we construct N = 2 → 0 super no-scale backgrounds, keeping in mind the goal of Sect. 3, which is to study their stability at the quantum level. In the framework of heterotic Z 2 -orbifold compactifications, at the exact N = 2 level, the models admit special Kähler moduli belonging to vector multiplets arising in the untwisted sector. Quaternionic deformations belonging to hypermultiplets also exist and occur generically in both untwisted and twisted sectors. In the following, we highlight a particular class of models characterised by an implementation of the stringy Scherk-Schwarz supersymmetry breaking that lifts classically all moduli of the twisted sector. These models are generic in the sense that both types of moduli, special Kähler and quaternionic, are allowed. However, they are also particular, since the complex structure of the internal space cannot be deformed away from the orbifold point, as follows from the non-existence of twisted deformations [15].
We consider heterotic no-scale models on T 2 × T 4 /Z 2 , where the N = 2 → 0 spontaneous breaking of supersymmetry is implemented by a coordinate dependent compactification on T 2 . We denote the spacetime, T 2 and T 4 coordinates as X 0,1,2,3 , X 4,5 and X 6,7,8,9 , respectively. For notational convenience, we restrict to the case where the stringy Scherk-Schwarz mechanism is implemented along the compact direction X 4 only, which is supposed to be large, for m 3/2 to be lower than M s . Moreover, even if it is not necessary, we will quote our results in the case where the second direction of T 2 is also large. Our aim is twofold : • First, we want models that develop a super no-scale structure. In terms of the 1-loop partition function Z, the effective potential can be expressed as an integral over the funda- where τ = τ 1 + iτ 2 is the Techmüller parameter. As long as a model sits at a point in moduli space where no mass scale is lower than m 3/2 , the untwisted and twisted sectors both yield contributions as shown in Eq. (1.1), so that In the above expression, n u F , n u B are the numbers of massless untwisted fermions and bosons, while n t F , n t B are their counterparts in the twisted sector. For a model to be super no-scale, we require • Second, we want the precise implementation of the coordinate dependent compactification to imply the twisted moduli present at the N = 2 level to be lifted at tree level.
Note that in the present work, the Z 2 twist is non-freely acting and gives a priori rise to massless states in the twisted sector. This situation is to be contrasted with the simpler one, already studied in Ref. [10], where the Z 2 twist on T 4 also shifts the direction X 5 of T 2 . In this case, there are no fixed points and the twisted states are automatically super massive (the strings are stretched along X 5 ), even at the N = 2 level.

A representative model
The starting point to construct the simplest model that realizes the above goal is the E 8 ×E 8 heterotic string compactified on T 2 × T 4 . The stringy Scherk-Schwarz mechanism can be introduced by implementing a Z 2 orbifold shift along X 4 , while a Z 2 orbifold twist acts on X 6,7,8,9 . The 1-loop partition fonction is (2.5) In our conventions, the spin structures a, b, the twists H, G and γ, δ, γ ′ , δ ′ are integer modulo 2, while our definitions of the Dedekind η and Jocobi θ α β functions can be found in Ref. [16]. The conformal block associated to the T 4 /Z 2 directions is where Γ + 4,4 and Γ − 4,4 are the contributions of the 4-torus zero modes that are even or odd under the Z 2 twist. As explained in the Appendix, they satisfy Γ + 4,4 = Γ − 4,4 . Finally, the T 2 coordinates contribution involves the shifted lattice where h, g are integer modulo 2, q = e 2iπτ and in terms of integer momenta m 4 , m 5 and winding numbers n 4 , n 5 , as well as the internal metric and antisymmetric tensor, through the Kähler and complex structure moduli The N = 2 → 0 spontaneous breaking is implemented by coupling the lattice shift h, g to the spin structure a, b, where a = 0 (a = 1) corresponds to spacetime bosons (fermions). This is done by inserting in the partition function the modular invariant sign [3] S a;h b ;g = (−1) ga+hb+gh . (2.10) Note that the light spectrum must have vanishing winding numbers along the large direction X 4 , which implies h = 0 (see Eq. (2.8)) and S = (−1) ga . If nothing else is introduced in the partition function, the initially massless bosons (a = 0) don't see the breaking, while the massless fermions (a = 1) acquire a mass of order m 3/2 . In this case, the number of massless fermions is always vanishing and the model has no chance to be super no-scale. To remedy this fact, we insert in Z another modular invariant sign [17] which for h = 0 yields SS ′ = (−1) g(a+H) , so that : • In the untwisted sector, H = 0, the situation is as before. The N = 2 → 0 breaking induces a tree level mass m 3/2 to the massless fermions, while the massless bosons are not modified.
• In the twisted sector however, H = 1, the situation is reversed. The N = 2 → 0 breaking induces a tree level mass m 3/2 to the massless bosons, while the massless fermions are not modified.
Therefore, we have n u F = 0 , n t B = 0 , (2.12) and the super no-scale condition (2.3) becomes In other words, in the 1-loop effective potential, we want the contribution of the untwisted massless sector, which is purely bosonic, to compensate that of the twisted massless sector, which is purely fermionic. Note that the consequences of the SS ′ insertion in a partition function extend far beyond the particular example we consider. They are valid in any heterotic Z 2 -orbifold model, where the stringy Scherk-Schwarz mechanism is implemented along the untwisted directions.

The spectrum
In order to see how things work in detail, we write the partition function (2.4) in terms of SO(2N) affine characters (2.14) In the untwisted sector, H = 0, we find where we have defined characters associated to the shifted T 2 as with momentum m 4 redefined as 2k 4 + g in the expressions of p L , p R in Eq. (2.8). Similarly, one obtains in the twisted sector, H = 1, Some remarks are in order : • Due to the large volume of T 2 , the sector h = 1, which yields non-vanishing winding number n 4 + 1 2 , leads contributions of order e −πτ 2 Im T 1 /4Im U 1 ≪ 1 for τ ∈ F . Thus, all conformal blocks proportional to O 2,2 1 0 and O 2,2 1 1 will not be considered explicitly from now on.
• The massless spectrum arises from the conformal blocks proportional to O 2,2 0 0 , for vanishing momenta and winding numbers along T 2 . In the untwisted (twisted) sector, as announced before, it is bosonic (fermionic). It is accompanied by towers of light bosonic (fermionic) Kaluza-Klein states, with momenta 2k 4 and m 5 .
• The remaining light spectrum arises from the conformal blocks proportional to O 2,2 0 1 . It is composed of towers of fermionic (bosonic) Kaluza-Klein states, with momenta 2k 4 + 1 and m 5 , which are superpatners of the above mentioned states, with mass degeneration lifted. The mass gap in these sectors is the gravitino mass m 3/2 , which satisfies It vanishes in the large T 2 volume limit, Im T 1 → +∞, U 1 finite, where supersymmetry is recovered.

The untwisted sector
In order to realize a super no-scale model, we first count the massless states in the untwisted sector, H = 0. In a theory where the breaking of supersymmetry is spontaneous, there cannot be any physical tachyon when the order of magnitude of m 3/2 is lower than M s . We thus have where the ellipsis account for the contributions of n u B fermionic superpartners of mass m 3/2 and all more massive states. However, only Z 0 0 needs to be expanded, since the sector h = 0 in Z 0 1 vanishes, as can be seen in the second line of Eq. (2.15). Not that this fact is not specific to the present model. It arises from the supersymmetry breaking sign S and the identity Since Z 0 0 is the partition function of the parent N = 4 → 0 no-scale model, it is more naturally expressed in terms of SO(8) and SO(16) affine characters, using (2.21) Defining G (T 4 ) the gauge symmetry group arising from the T 4 lattice on the right-moving side of the string, and reminding thatŌ 16 +S 16 =Ō In order to find the representations in which the n u B untwisted massless states are organized, we expand N ± are non-vanishing if the Γ ± 4,4 lattices moduli sit at enhanced symmetry points, while 4 is the rank of G (T 4 ) . In these notations, we find   where 0 k denotes a sum of k consecutive 0's. The brackets in the exponents ofq are squares of roots and weights i.e. charge 6-vectors under the U(1) 6 Cartan generators [18]. If the SO(4) affine characters can be expanded in a similar manner in terms of root or weight 2-vectors it is relevant for our purpose to rotate the orthogonal basis of the 2-dimensional Cartan subalgebra through an angle π/4, thus interpreting SO(4) as SU(2) × SU (2) : To summarize, the massless untwisted sector is the bosonic part of N = 2 supermultiplets : 1 gravity multiplet (graviton, graviphoton), 1 tensor multiplet (antisymmetric tensor, dilaton, gauge boson), 1 vector multiplet (gauge boson, complex scalar) in the adjoint [19]. The remaining part of the massless untwisted spectrum, amounts to the bosonic degrees of freedom of N + vector multiplets and 4 + N − hypermultiplets. If for N ± = 0 this yields 4 neutral hypermultiplets, we are going to see that non-vanishing N ± are required for the model to develop a super no-scale structure, which gives rise to enhanced gauge theories with charged scalars.

The twisted sector
In order to find the massless twisted sector, we expand and write  [19]. The number of twisted massless fermionic degrees of freedom is thus The super no-scale condition is obtained at the self-dual point The 4N + and 4N − untwisted bosonic states Since the above solutions yield N + = N − = 4, we have to describe the representations of the spectrum given in Eq. (2.30).
we can write 3q and (3 + 2)q in terms of SU(3) en roots and Cartan generators, where ǫ 6 , ǫ 7 = ±1. This shows the existence of an SU(2) en gauge symmetry in the descendent N = 2 → 0 model, coupled to 2 copies (for ǫ 7 = ±1) of scalar fields in the fundamental representation, and neutral scalars. Moreover, as was the case in the solution (a), the terms 1q and (1+1)q in Eq (2.37) lead an U(1) en gauge symmetry coupled to fields of charges ± √ 2.
In total, the massless spectrum arising in the solution (b) from the enhanced symmetry of the T 4 lattice amounts to the bosonic parts of N = 2 supermultiplets : 1 vector multiplet in the adjoint representation of SU(2) en × U(1) en , 2 hypermultiplets in the [2] SU (2)en , 1 hypermultiplet of charge √ 2 under U(1) en , 1 hypermultiplet of charge − √ 2 under U(1) en and 2 neutral hypermultiplets. The full gauge symmetry is therefore The would-be tachyons  along the direction X 4 of the large T 2 . It takes the form where s 0 denotes one of the n B + n F bosonic or fermionic degrees of freedom of mass below where M high is the lowest mass scale above m 3/2 in the spectrum, which in practice can be very high. The justification of Eq. (3.1) can be found in Ref. [10] but can be summarized as follows : • The existence of an infinite tower of Kaluza-Klein states associated to the direction X 4 involved in the supersymmetry breaking implies the partition function to be integrable over the full upper half strip, −1/2 < τ 1 < 1/2, τ 2 > 0. No ultraviolet divergence occurs as τ 2 → 0.
• Moreover, when the volume of this compact direction is large, compared to the string scale, the integral over the region τ 2 < √ 3/2 of the strip is exponentially suppressed. Therefore, when m 3/2 is lower than M s , we can replace up to exponentially suppressed terms the domain of integration over the fundamental domain of SL(2, Z) by the upper half strip.
• The non-level matched states are projected out of this integral and only the physical ones remain.
• Among them, the degrees of freedom heavier than the lowest mass scale M high above m 3/2 yield exponentially suppressed contributions, compared to those arising from the T 2 Kaluza-Klein modes based on the states s 0 whose masses are below m 3/2 . In particular, the oscillator modes at mass level M s and the winding states along T 2 are suppressed. The scalar fields that may parameterize marginal deformations along any phase of the gauge theory can be listed from the massless untwisted spectrum of Eq. (2.26), namely : • The internal metric and antisymmetric tensor of the large T 2 . They can be expressed in terms of those of the initial background we denote with upper indices "(a)" and 2 × 2 deformations, They correspond to the degrees of freedom [4] i.e. T 1 , U 1 , which parameterize the Coulomb branch of the U(1) 2 gauge symmetry generated by the T 2 lattice.
• The internal metric and antisymmetric tensor of T 4 . They can be expressed in terms of the metric of the Cartesian product of four circles of unit radii and 4 × 4 deformations,

Contribution of the untwisted states
We first consider the branch of the gauge theory where U(1) 4 en is in its pure Higgs phase and U(1) 2 grav,ten × U(1) 2 × E 7 × SU(2) × E 8 in its pure Coulomb phase. In this case, the allowed background deformations are those given in Eqs If this is what we will do later in this section, we find convenient to start with the second approach.
Thanks to the identity (2.20), which is valid even in the deformed background, the sector h = 0 in Z 0 1 vanishes. Thus, we can write where V N =4→0

1-loop
is the potential of the parent N = 4 → 0 model, while the suppressed terms arise from winding modes along X 4 . In order to compute the effective potential V N =4→0 1-loop , the massless states s 0 to be considered are those of the parent N = 4 → 0 model, which are charged under SU(2) 4 en × E 8 × E 8 . The left-moving squared masses of their Kaluza-Klein modes along X 4,5 can be written as [22], in terms of generalized momenta   along T 2 are redefined in complex notation, The Kaluza-Klein towers of states yield shifted complex Eisenstein series of asymmetric integer modulo 2 weights g 1 , g 2 , and we have defined the coefficient The total 1-loop effective potential Combining the untwisted and twisted states contributions, Eqs (3.9) and (3.16), the wouldbe dominant term proportional to M 4 s /(Im T 1 ) 2 ∝ m 4 3/2 cancels out, due to the super no-scale condition. In order to simplify the charge-dependent corrections, we use the fact that 1 2 where R is a representation of SU(2), E 8 or E 7 and the sums over I, J run over the corresponding rank, In total, we obtain in the branch where U(1) 4 en is in its pure Higgs phase and U(1) 2 grav,ten × U(1) 2 × E 7 × SU(2) × E 8 in its pure Coulomb phase we find perfect agreement with the analysis based on the parent E 8 × E 8 theory.
We said before that from the N = 4 → 0 viewpoint, the scalars Y ji , j ∈ In the pure Coulomb branch of the gauge theory, the total 1-loop effective potential is therefore Of course, each Abelian factor of U(1) 4 en can be in its own Higgs or Coulomb phase, independently of the others. Thus, there exist a pure Higgs, a pure Coulomb and mixed Coulomb/Higgs branches that realize the spontaneous breaking of the gauge symmetry U(1) 4 en → U(1) k en , k ∈ {0, . . . , 4}. In each case, the 1-loop effective potential takes a form similar to Eqs (3.19) and (3.24). Some remarks are in order : • The pure Higgs branch of the U(1) 4 en gauge theory is of real dimension 4 × 4, parameterized by Y ji , j, i ∈ {6, 7, 8, 9}. The ubiquity of the 4 × 4 dimension in the above branches is not accidental. In two dimensions, the internal space is where the first T 4 refers to the directions X 2,3,4,5 , the second one to X 6,7,8,9 and the last one stands for the right-moving coordinates X In this relation, Adj G is the adjoint representation of G realized by the bosonic parts of N = 4 vector multiplets, with corresponding β-function coefficient b G . R t H is the representation of H realized by the fermionic parts of the twisted hypermultiplets, whose contribution to To derive Eq. (3.28), we use the fact that massless degrees of freedom in a representation R K of any gauge group K contribute to the β-function In Ref. [10], it is shown that in the N = 4 → 0 super no-scale models, the Wilson lines associated to the asymptotically free gauge theories are stable. In this case, the non-Abelian gauge symmetries are expected to confine at low energy. On the contrary, the non-asymptotically free gauge theories yield Wilson line instabilities, which should survive in the infrared. Turning back to the N = 2 → 0 super no-scale models, we find in the background (a) the β-function coefficients while those of U(1) 2 grav,ten ×U(1) 2 are vanishing. Thus, at low energy, the E 8 gauge symmetry should confine, while our description of the enhanced U(1) 4 en and SU(2) gauge symmetries, together with that of the spontaneous breaking of E 7 to a rank 7 subgroup, are expected to be valid.

Mixed Coulomb/Higgs phases of E 7 × SU (2)
In the background (a), the phases of the gauge symmetry (2.36) that remain to be described are those where the E 7 × SU(2) group is spontaneously broken to some subgroup of rank r < 8. This happens when degrees of freedom in the [56] E 7 ⊗ [2] SU (2) bifundamental representation condense. However, the E 7 × SU(2) gauge theory realized in the N = 2 → 0 Z 2 -orbifold model involves untwisted states only and is therefore obtained by truncation of the parent N = 4 → 0 model. This means that its phase structure is expected to be similar to that presented for the U(1) 4 en gauge symmetry coupled to charged quaternions.
In the parent N = 4 → 0 theory, the right-moving T 16 R in Eq. (3.25) yields the E 8 × E 8 gauge symmetry. In two dimensions, the degrees of freedom, and among them the marginal deformations, are in one-to-one correspondence with the operators On the contrary, the Z 2 action is non-trivial on the characters of the first E 8 , and thus on the associated right-moving operatorsŌ a , a ∈ Adj E 8 ≡ [248] E 8 . Actually, since In this section, we would like to justify that the N = 2 → 0 super no-scale models realized as T 2 × T 4 /Z 2 compactifications with stringy Scherk-Schwarz mechanism along T 2 yield descendent N = 1 → 0 super no-scale models, once a second orbifold twist is implemented.
However, we argue that the resulting 1-loop effective potential requires further study.
Starting with any N = 2 heterotic model compactified on T 2 ×(T 2 ×T 2 )/Z 2 , where the Z 2 twist generator is denoted G, one obtains an N = 2 → 0 no-scale model by implementing a stringy Scherk-Schwarz mechanism along the first T 2 . Contrary to Sect. 2, we do not suppose that all massless bosons in the twisted sector acquire a tree level mass of order m 3/2 , so that twisted moduli may exist. We consider the descendent N = 1 → 0 no-scale model obtained by implementing a Z ′ 2 orbifold twist of generator G ′ , which acts on the first and third T 2 's. In this case, the 1-loop effective potential of the N = 1 → 0 model can be written as where M L , M R are the left-and right-moving masses. In this expression, "untwisted ′ " denotes the spectrum of the parent N = 2 → 0 model, while "twisted ′ " refers to the twisted spectrum, with respect to Z ′ 2 . Since G ′ twists the first T 2 , the twisted ′ states invariant under G ′ have vanishing momenta and winding numbers along the directions X 4,5 . Therefore, their tree-level masses are independent of m 3/2 and are supersymmetric. In other words, the bosons/fermion degeneracy in this sector is not lifted classically. 7 This shows that the Str over the twisted ′ sector with the (1 + G ′ )/2 projector inserted is vanishing. Since the corresponding conformal blocks form an SL(2, Z) modular orbit with the Str over the untwisted ′ sector with G ′ inserted, this second Str is also vanishing. Therefore, which is an exact identity, no matter the scale m 3/2 is, compared to M s . It is valid for 7 The spontaneous breaking of supersymmetry is mediated to the twisted ′ sector by quantum interactions with the non-supersymmetric untwisted ′ sector. arbitrary moduli deformations of the parent N = 2 → 0 model that survive the (1 + G ′ )/2 projection. 8 An important consequence of this equation is that if the parent N = 2 → 0 theory sits at a point in moduli space where it develops a super no-scale structure, then the descendent N = 1 → 0 model is also super no-scale.
Moreover, Eq. (4.2) suggests we may write However, it is important to stress that this is only possible if new moduli of the N = 1 → 0 theory are not switched on. The latter may arise from the twisted ′ sector of the theory, whose tree level mass spectrum is not affected by the Scherk-Schwarz breaking. To be more specific, the group elements G ′ and GG ′ may admit fixed points (copies of the second and third T 2 's, respectively) and thus introduce new massless N = 1 chiral supermultiplets. By switching on vacuum expectations values to their bosons, the gauge symmetry of the N = 1 → 0 model (which arises from the untwisted ′ sector) enters Higgs branches, which are parameterized by moduli having no counterpart in the parent N = 2 → 0 theory. Note that these moduli are complex structure deformations of the first and third T 2 's modded out by Z ′ 2 , or the first and second T 2 's modded out by the diagonal subgroup of Z 2 × Z ′ 2 . Since the masses M L , M R of the initial N = 2 → 0 untwisted ′ sector depend on these moduli, Eq. (4.2) is a quantum potential for these deformations. As a result, the latter acquire positive or negative squared masses at 1-loop, which yield additional conditions for the background to be stable.
To evaluate these quantum masses, one may again apply Eq. (3.1), dressed with an overall factor 1 2 , but with the masses M L now depending of these new moduli. We mention that the dependance of the M L 's on these deformations can be determined in the classical effective N = 1 gauged supergravity at low energy, by following the method applied in Ref. [25] in a similar problem.
A class of N = 1 → 0 super no-scale models To proceed, we focus on the N = 2 → 0 super no-scale Z 2 -orbifold models presented in Sect 2, where all moduli arise in the untwisted sector, and construct descendent N = 1 → 0 super no-scale models. Compactifying down to two dimensions, the internal space is where the first T 2 refers to the directions X 2,3 , the second one to X 4,5 , the third one to X 6,7 and the last one to X 8,9 . As before, the stringy Scherk-Schwarz mechanism is implemented along the second T 2 . At the N = 4 → 0 level i.e. without implementation of the Z 2 × Z ′ 2 action, the right-moving coordinates of the last three T 2 's as well as T 16 R generate a gauge symmetry G or rank 22. Choosing a Cartan subalgebra, and a basis for it which diagonalizes G and G ′ , we can impose the projector on the set of marginal operators of the parent N = 4 → 0 model, and find those which survive in the descendent N = 1 → 0 theory, namely where N ± (2) are the numbers of non-Cartan generators of G    (4.9) and the second ones in the conjugate representations. Those with degeneracy 16 yield chiral families, while those with degeneracy 32 are non-chiral.
In the same spirit, the generators G ′ and GG ′ fix 16 copies of the second and third T 2 's, respectively. The associated twisted sectors are similar to that fixed by G, up to the fact that the tree level masses are not affected by the stringy Scherk-Schwarz mechanism.
Together, they are nothing but the twisted ′ sector of the N = 1 → 0 model and, at the massless level, contain full N = 1 chiral multiplets in representations of E 6 as in Eq. (4.9), with similar U(1) × U(1) ′ charges.
Assuming the moduli of the twisted ′ sector are not switched on, we can apply Eq. (4.3) to derive the 1-loop effective potential of the N = 1 → 0 super no-scale model that descends from the background (a). We have computed the potential of the parent N = 2 → 0 theory when the latter is allowed to be deformed along the branch where U(1) 4 en is Higgsed and the U(1) 2 × E 7 × SU(2) × E 8 gauge symmetry is in its U(1) 18 Coulomb phase. In Eq. (3.19), the moduli that survive the Z ′ 2 projection are the components of the metric and antisymmetric tensor that respect the T 2 × T 2 × T 2 factorization, namely T 1 , U 1 , as well as Y ji , j, i ∈ {6, 7} and j, i ∈ {8, 9}. In particular, as seen in Eq. survive. Thus, we obtain (4.10) In this expression, the fact that the complex scalar deformations that parameterize the Higgs phase of E 6 × U(1) × U(1) ′ do not appear means that these moduli are simply set to zero i.e. that the descendent model sits at the origin of the Higgs phase of the E 6 × U(1) × U(1) ′ gauge symmetry. On the contrary, the complex scalars that span the Higgs phase of the rank 2 group associated to the large T 2 , as well as the Higgs phase of U(1) 4 en , are switched on. Thus, the potential (4.10) is that obtained when the gauge group of the N = 1 → 0 Alternatively, we could have allowed the initial parent background (a) to be deformed along its pure Coulomb phase, i.e. when U(1) 2 × U(1) 4 en × E 7 × SU(2) × E 8 → U(1) 22 .
In Eq. (3.24), the only moduli that survive the Z ′ 2 projection are T 1 , U 1 , whose expectation values Higgs the rank 2 group associated to the large T 2 . The effective potential of the descendent model takes therefore the apparently trivial form However, since none of the complex scalar deformations along the Higgs phase of U(1) 4 en × E 6 × U(1) × U(1) ′ appears, we conclude that the descendent model sits at the origin of their Higgs phase and that the gauge symmetry is enhanced to U(1) 4 en × E 6 × U(1) × U(1) ′ × E 8 . Therefore, Eq. (4.11) does not contain any information that is not already encoded in Eq. (4.10). Nonetheless, it is interesting to note that technically, the two expressions are obtained from different choices of Cartan subalgebras of G = U(1) 2 × SU(2) 4 en × E 8 × E 8 in the N = 4 → 0 initial theory, which fall into distinct equivalences classes.
Actually, the branch of the background (a) that yields the maximum information on the moduli masses of the descendent N = 1 → 0 super no-scale model is that where the rank of the gauge symmetry group is minimal. It is obtained when U(1) 4 en is totally Higgsed and E 7 × SU(2) is in its Higgs branch of maximal dimension. If we had com- in this phase, we may have found instabilities that induce the breaking of E 7 (or E 7 × SU(2)) to subgroups of lower ranks. Using this expression of V N breaking of E 6 × U(1) × U(1) ′ to subgroups of lower ranks.

Conclusion
The super no-scale models [8][9][10][11], which by definition have exponentially suppressed effective potential at 1-loop for low supersymmetry breaking scale m 3/2 , may help to build quantum theories consistent with flat space, as well as to cancel the dilaton tadpole. However, for this to have any chance to work, the backgrounds must be stable at the quantum level. The question of the moduli stability in the N = 4 → 0 super no-scale models was addressed in Ref. [10] and the purpose of the present work is to initiate the analysis of the N = 2 → 0 case.
The particular class of models we focus on are heterotic Z 2 -orbifolds on T 2 × T 4 /Z 2 , where the N = 2 → 0 spontaneous breaking of supersymmetry is implemented via a stringy Scherk-Schwarz mechanism [2,3] along T 2 . We show that a specific implementation of this mechanism induces a mass of order m 3/2 to all initially massless bosonic (fermionic) degrees of freedom arising in the twisted (untwisted) sector. Thus, the super no-scale condition, which amounts to canceling the would-be dominant m 4 3/2 contribution to the effective potential, is fulfilled by adjusting the number of massless untwisted bosons to match the number of massless twisted fermions. An obvious but nevertheless fundamental consequence of this choice of supersymmetry breaking is that all twisted moduli are lifted classically. Moreover, the models do not suffer from classical Hagedorn-like instabilities [20], whatever high m 3/2 may be. Actually, the classical tachyons arising in their parent N = 4 → 0 theories (without In general, the super no-scale structure emerges precisely at such points of extended gauge symmetry. Thus, the representative points in moduli space of these backgrounds are extrema of the 1-loop effective potential [23]. When these extrema are saddle or maxima, the N = 2 → 0 super no-scale backgrounds are destabilized into either a Coulomb, Higgs or mixed Coulomb/Higgs branch. The string computation of the 1-loop effective potential being based on on-shell data at tree level, the result depends on the branch along which one supposes the theory may be deformed. In a representative example of the class of N = 2 → 0 super no-scale models described above, we have evaluated the 1-loop effective potential in the pure Coulomb phase as well as in mixed Coulomb/Higgs branches. It is enough to derive explicit expressions at quadratic order in moduli fields to conclude on eventual destabilizations in classically marginal directions that become tachyonic at 1-loop.
We also show that for any implementation of the Scherk-Schwarz mechanism along T 2 , the N = 2 → 0 super no-scale Z 2 -orbifold models yield N = 1 → 0 super no-scale backgrounds, by implementing a second Z 2 orbifold action. The models are naturally chiral.
Classically, the twisted sector of the second Z 2 remains N = 1 supersymmetric and contains new moduli fields. In the present work, the study of the vacuum structure is only partial, in the sense that these new marginal deformations are not switched on. Considering the above-described stringy Scherk-Schwarz breaking of supersymmetry, the orbifold structure is therefore not deformed into a "non-supersymmetric version of smooth Calabi-Yau compactification". Under these conditions, the pattern of (±1, ±1) eigenvalues with respect to the Z 2 × Z 2 twists can again be found, for any choice of Cartan subalgebra in the underlying "grandparent" N = 4 → 0 model. The resulting structure of classical vacua in the descendent N = 1 → 0 theory is that of a unique Higgs branch, inside of which loci where gauge symmetries coupled to charged complex scalars are restored, when the expectation values of the latter vanish. Note that at a generic point of the Higgs branch, all gauge group factors with no charged complex scalars remain obviously unbroken, but may confine in the infrared.
The models we consider in this work are not studied out of the super no-scale regime, i.e. when m 3/2 is large enough for the 1-loop effective potential not to be exponentially suppressed. At high supersymmetry breaking scale, at early times in a cosmological scenario, the potential may be positive and drive dynamically the model into the super no-scale regime [10,11,27], or be negative and admit an AdS vacuum [21,27]

Appendix
The 1-loop partition function associated to a d-dimensional torus can be factorized into a Γ d,d contribution of the lattice of momenta and winding numbers, and a part arising from the left-and right-moving bosonic oscillators. In this Appendix, we would like to write Γ d,d as a trace over two different basis. We first consider the case d = 1, before generalizing the result to arbitrary d.