Hagedorn-like transition at high supersymmetry breaking scale

We consider phase transitions occurring in four-dimensional heterotic orbifold models, when the scale of spontaneous breaking of N=1 supersymmetry is of the order of the string scale. The super-Higgs mechanism is implemented by imposing distinct boundary conditions for bosons and fermions along an internal circle of radius R. Depending on the orbifold action, the usual scalars becoming tachyonic when R falls below the Hagedorn radius may or may not be projected out of the spectrum. In all cases, infinitely many other scalars, which are pure Kaluza-Klein or pure winding states along other internal directions, become tachyonic in subregions in moduli space. We derive the off-shell tree level effective potential that takes into account these potentially tachyonic modes. We show that when a combination of the usual tachyons survives the orbifold action, it is the only degree of freedom that actually condenses.

number of KK (winding) modes are therefore tachyonic. In fact, even when the volume of the extra directions is of order 1 in string units, 3 it turns out that non-trivial KK or winding modes can be tachyonic. In the descendent orbifold model, because invariant combinations of such potentially tachyonic states survive, there is always a phase in moduli space where a condensation takes place.
In the present work, we focus on the simplest case, where a single combination of the usual tachyonic states considered in the literature -i.e. with non-trivial quantum numbers along S 1 (R) only -survives the orbifold action. We take into account other scalars, with identical charges along S 1 (R) but non-trivial momenta or winding numbers along another internal direction. These modes can be tachyonic in more restricted regions in moduli space.
The question we ask is whether there exists a multiphase diagram associated with various patterns of condensations, and associated with different stable vacua. We find that all of the condensation is actually supported by the tachyon that has trivial quantum numbers along the directions transverse to S 1 (R). In other words, there is a unique Hagedorn-like phase, which is delimited by the usual boundary R = R H . Note that this assumes that the Scherk-Schwarz direction is a factorized circle in the internal space. When the internal metric and antisymmetric tensor are generic, the boundary of the Hagedorn-like phases are much more involved. Moreover, when the orbifold action forces all potentially tachyonic states to have non-trivial quantum numbers in the directions transverse to the Scherk-Schwarz circle, the boundaries of the Hagedorn-like phases as well as the properties of the associated vacua are drastically different. However, these generalizations will be analyzed in subsequent work.
To figure out phase transitions between string models defined in first quantized formalism, the suitable framework should be string field theory [31][32][33][34]. However, such an analysis being equivalent to describing the vacuum structure of the theory, the problem may be tackled within an effective field theory, valid at low energy. Such a description can be determined from our knowledge of the phase associated with the initial orbifold compactification. The latter describes a super-Higgs mechanism in Minkowski space, with an arbitrary scale m3 2 of supersymmetry breaking. Hence, it is a no-scale supergravity [35], which takes into account all light and potentially tachyonic degrees of freedom. The key point is that the supergravity action is valid off-shell. Therefore, it captures other vacua characterized by 3 Throughout this paper we take α = 1.
non-trivial condensates developed in regions in moduli space where tachyonic instabilities take place. Notice that our use of the word "vacuum" is cavalier in the sense that the tachyon condensation lowers the potential of the theory to negative values, which yields a dilaton tadpole. As a result, the new supergravity phase may describe a non-critical string at low energy, with linear dilaton background [9,10].
In Sect. 2, we consider as a starting point the heterotic string compactified on T 2 ×T 2 ×T 2 .
A Scherk-Schwarz mechanism responsible for the N = 4 → N = 0 spontaneous breaking of supersymmetry in 4 dimensions is implemented along one direction, X 4 , of the first internal [17][18][19][20][21]. We determine the regions of the plan (R 4 , R 5 ), where scalars with non-trivial momentum and/or winding numbers along X 4 and X 5 are tachyonic. We then introduce a Z 2 ×Z 2 orbifold action and analyze the conditions for a tachyonic mode with trivial quantum numbers along S 1 (R 5 ) to survive. We stress that the latter is accompanied by an infinite number of potentially tachyonic KK (or winding) modes propagating along (or wrapped around) S 1 (R 5 ). In Sect. 3, we derive the tree level effective potential that depends on all of these scalars. This may be done in the framework of N = 1 supergravity [36,37].
However, because all degrees of freedom of interest arise in the untwisted sector of the Z 2 ×Z 2 orbifold action, we find convenient to derive the potential by applying a suitable truncation of N = 4 gauged supergravity [38][39][40][41][42][43]. In this formalism, the gauging is determined by imposing the mass spectrum in the no-scale supergravity phase to reproduce its string counterpart. The off-shell tree level bosonic action is found to be invariant under the modified T-duality R 4 → 1/(2R 4 ). This is consistent with the fact that this transformation (accompanied with a change of chirality for the fermions) is a symmetry of the 1-loop partition function of the initial string model, and thus a symmetry of the on-shell 1-loop effective potential, in the no-scale phase. It is straightforward to minimize the tree level potential to find that in the case under study, the only mode that condenses in the Hagedorn-like phase is the tachyon that has quantum numbers along the Scherk-Schwarz direction S 1 (R 4 ) only. Because the new vacuum lies at the self-dual radius is not spontaneously broken. Finally, our conclusions and perspectives can be found in Sect. 4.
In this section, our aim is to characterize regions in moduli space where one or several generically massive states become tachyonic for sufficiently large supersymmetry breaking scale. The resulting condensation phenomenon will be discussed in Sect. 3.

Towers of KK or winding tachyonic states
In the present work, we consider the heterotic string compactified on the orbifold T 6 /(Z 2 × Z 2 ). For simplicity, the analysis is restricted to the case where the internal T 6 is of the form i.e. with first 2-torus factorized into two circles of radii R 4 and R 5 . The spontaneous breaking of N = 1 supersymmetry is implemented along the compact direction X 4 , by a stringy version [20,21] of the Scherk-Schwarz mechanism [27,28]. The zero point energy in the twisted sectors being non-negative, tachyons can only arise in the untwisted sector.
Before Z 2 × Z 2 projection, the associated 1-loop partition function takes the following form, where C α; n 4 β ;m 4 = (−1)m 4 α+n 4 β+m 4 n 4 . (2.2) In our notations, τ is the Teichmüller parameter of the genus-1 Riemann surface and our definitions of the Jacobi modular forms θ[ α β ] and Dedekind function η are as follows, where N ∈ Z. The lattices of bosonic zero-modes associated with S 1 (R 5 ), the T 2 's and the extra right-moving coordinates are denoted Γ (p,q) , while that associated to S 1 (R 4 ) is written in Lagrangian form, where n 4 ,m 4 ∈ Z. The conformal blocks arising from the left-moving worldsheet fermions depend on the spin structures α, β ∈ {0, 1}. The latter are coupled to the S 1 (R 4 ) lattice of zero modes by the "cocycle" C α; n 4 β ;m 4 [21]. To see that this sign breaks spontaneously supersymmetry, one can switch to a Hamiltonian formulation obtained by where we denote for I ∈ {4, 5} and κ = 0, 1 while SO(8) affine characters are defined as Comparing the lattice dressing of the characters V 8 and S 8 , the supersymmetry breaking scale (or gravitino mass) in σ-model frame is found to be If the sign (−1) n 4 β present in the cocycle reverses the GSO projection in the odd n 4 winding sector, the associated characters O 8 , C 8 yield states heavier than the string scale when satisfied by Z, the characters O 8 and S 8 also lead to very heavy modes when R 4 1.
However, the leading term of the q,q-expansion can yield tachyonic scalars, when R 4 is of order 1. Denoting m I , n I the momentum and winding numbers along the internal directions X I , I ∈ {5, . . . , 9}, the level matching condition at this oscillator level reads The physical states that can be tachyonic in the parent model realizing the N = 4 → N = 0 spontaneous breaking of supersymmetry turn out to have quantum numbers 2m 4 + 1 = −n 4 = , m 5 n 5 = · · · = m 9 n 9 = 0, (2.11) where = ±1. They have non-trivial momentum and winding numbers along the Scherk-Schwarz direction, but are pure KK or winding modes along the remaining internal directions.
For instance, the squared masses in σ-model frame of those having m 6 = n 6 = · · · = m 9 = n 9 = 0 are The largest tachyonic domain in the plane (R 4 , R 5 ) is obtained for m 5 = n 5 = 0, where R H is the Hagedorn radius encountered in heterotic string at finite temperature.
However, subregions where additional states are tachyonic also exist, since (2.14)  are tachyonic in the parent model, which realizes the N = 4 → N = 0 spontaneous breaking. The remaining quantum numbers of these states are 2m 4 = −n 4 = and m 6 = n 6 = · · · = m 9 = n 9 = 0. as R 5 decreases, while more and more KK modes propagating along S 1 (R 5 ) become tachyonic as R 5 increases. By analogy with the finite temperature case [6][7][8][9][10], a Hagedorn-like phase transition is expected to occur when R 4 enters the range (2.13). An instability of the initial no-scale model vacuum [35] should be signalled by the condensation of, at least, the tachyonic modes 2m 4 = −n 4 = , m 5 = n 5 = · · · = m 9 = n 9 = 0. As seen on Fig. 1, a multi-phase diagram may however exist, with different vacua characterized by various condensed modes.

Why supergravity
As soon as R 4 enters the range (2.13), implying M 2 ( ,0,0) to be negative in the parent model that realises the N = 4 → N = 0 breaking, the 1-loop effective potential, which is nothing but the partition function (2.2) integrated over the fundamental domain of SL(2, Z), diverges. The fact that the quantum potential is ill-defined does not signal some fundamental inconsistency of the theory. Indeed, this means that perturbative quantum corrections should be computed around a new, true vacuum [8][9][10][11]. In the latter, the derivation of the mass spectrum using the initial string background is not legitimate anymore. Hence, once the modes 2m 4 + 1 = −n 4 = , m 5 = n 5 = · · · = m 9 = n 9 = 0 have already condensed, we should consider as a possibility, rather than a prediction, the condensation of other KK or winding modes along, for instance, S 1 (R 5 ). This is the reason why we will derive in Sect. 3 the off-shell low energy effective potential associated with all of these potentially condensing degrees of freedom, in order to figure out which of them actually develop non-trivial expectation values.
In presence of the cocycle C [21] in the partition function (2.2), the GSO projection being reversed in the odd n 4 winding sector, the (common) statement that the non-supersymmetric model arises as a spontaneous breaking of a supersymmetric theory (i.e. with no cocycle) is not obvious. To see this is the case, let us recall the initial formulation of the stringy Scherk-Schwarz mechanism as a "coordinate-dependent compactification" [18,20]. In our case of interest, this amounts to coupling the lattice Γ (1,1) (R 4 ) to the boundary conditions of 2 (among 8) real left-moving worldsheet fermions. Before implementation of the orbifold action, the choice of these fermions is arbitrary since they all have identical boundary conditions. This leads where the generalized momenta in this formulation take the following form, We see that both the GSO projection (the β-dependent phase) and the level-matching condition are independent of e, since Therefore, there is a one-to-one correspondence between the states of the supersymmetric and deformed theories. The mass spectrum, however, depends on e. For instance, the masses of the 4 gravitini (or their surviving combination after Z 2 × Z 2 projection) are m3 However, consistency of the heterotic worldsheet theory imposes the supercurrent to be conserved, which forces the deformation to be quantized [18,20,21]. 4 Taking e = 1, not only the gravitino mass (2.7) is recovered, since the properties of the Jacobi modular forms can be used to rewrite the conformal blocks (2.15) in the following form, Im τ |m 4 +n 4 τ | 2 θ α β e −iπe(m 4 α−n 4 β+em 4 n 4 ) , (2.19) i.e. with the cocycle introduced in Eq. (2.2). Because the consistent quantum field theories of massive spin 3 2 particles are supergravities realizing the super-Higgs mechanism, the low energy effective field theory associated with the untwisted sector of the Z 2 × Z 2 model can be described by an N = 4 gauged supergravity [38][39][40][41][42][43], with suitable truncation.

Z 2 × Z 2 projection
We would like now to implement the orbifold action whose generators are defined as (2.20) We are going to see that there are conditions for the coordinate-dependent compactification to be implemented consistently and that the surviving set of potentially tachyonic modes is Let us first implement the Z 2 action generated by G 1 , which breaks N = 4 to N = 2.
Because the boundary conditions of the left-moving worldsheet fermions ψ 6 , ψ 7 , ψ 8 , ψ 9 are initially identical, we may deform those of any two of them to implement the coordinatedependent compactification. For instance, choosing ψ 6 , ψ 7 , the relevant conformal blocks where we have written explicitly the dependence on the quantum number H 1 ∈ {0, 1}, which labels the untwisted and twisted sectors of the Z 2 modding, as well as the dependance on G 1 ∈ {0, 1}, which signals the insertion G 1 G 1 in the supertraces. In the above formula, ξ = 0 or 1 defines two distinct choices of discrete torsions, which are allowed by modular invariance. For e = 1, the expression can be rewritten in terms of a modified cocycle With the cocycle prescription C alone, we have seen that the potentially tachyonic modes are untwisted, H 1 = 0, and have odd winding number n 4 . In such a sector, because C reduces to (−1) (1−ξ)G 1 , the presence of C in the blocs (2.22) modifies the Z 2 projector as follows, which has important consequences.
To see this explicitly, let us use the notations of Eq. (2.17), where the potentially tachyonic modes have quantum numbers m 4 = −n 4 = −Q = , where Q ≡ −N + α 2 are the eigenvalues of the generator of the SO(2) affine algebra associated with the normal-ordered conserved current : ψ 6 ψ 7 :. In complex notations, the affine generator is defined as where σ 1 is the coordinate along the string. The remaining non-trivial quantum numbers of the potentially tachyonic modes satisfy m I n I = 0, I ∈ {5, . . . , 9}. For instance, those with vanishing winding numbers along S 1 (R 5 ) × T 2 × T 2 have equal left-and right-moving Under the Z 2 action, they transform as 5 where |0 NS and |0 are the left-moving NS and right-moving vacua. For the sake of simplicity in the notations, we have set R 4 = 1/ √ 2 in the above expressions, which yields p 4L = 0, We see that there are always invariant linear combinations of states surviving the Z 2 projection. However, the modes with trivial momenta and winding numbers along the twisted directions X 6 , X 7 , X 8 , X 9 (and whose masses are given in Eq. (2.12)) exist in the orbifold model only if ξ = 1. In that case, we will see in the next section that these states condense in the range (2.13). On the contrary, these modes are projected out when ξ = 0, and the properties of the Hagedorn-like phase and its boundary in moduli space must be drastically different. In fact, all potentially tachyonic modes surviving the Z 2 projection when ξ = 0 are pure KK modes (or pure winding modes) along one or more directions of T 2 × T 2 (and possibly along S 1 (R 5 )).
Let us apply G 2 on the potentially tachyonic modes arising in the N = 4 → N = 0 model. For instance, those with vanishing winding numbers along S 1 (R 5 ) × T 2 × T 2 are transformed as (2.26) 5 In the following, e ip IL X I L +ip IR X I R |0 NS ⊗ |0 stands for |p L NS ⊗ | p R , and X I = X I L + X I R .
Notice that the images are not potentially tachyonic states, but only some of their nondegenerate (and massive) bosonic superpartners, which have Q = . In other words, G 2 is not a symmetry of the parent N = 4 → N = 0 model. In order to construct a consistent Z 2 × Z 2 orbifold model, it is however possible to implement the coordinate-dependent compactification with the SO(2) affine generator associated with ψ 6 , ψ 8 instead of ψ 6 , ψ 7 .
Remember that from the point of view of the first generator G 1 , this is nothing but an equivalent conventional choice, so that Eq. (2.25) becomes However, applying G 2 on the same states, we find (2.28) The images are now potentially tachyonic in the parent N = 4 → N = 0 model, and the latter admits a Z 2 symmetry generated by G 2 [20,21,29]. Hence, a consistent model realizing the N = 1 → N = 0 spontaneous breaking of supersymmetry is obtained by modding by Z 2 × Z 2 . Because the affine SO(2) generator involved in the coordinate-dependent compactification rotates the directions 6 and 8, 6 moduli of the tori T 2 × T 2 associated with the directions 6, 7 and 8, 9 are lifted. This has been analyzed in supergravity in Ref. [29], while in the string theory context of Refs [20,21], the moduli of T 2 × T 2 take specific values, at fermionic points.
From now on, we consider the above model with ξ = 1. 7 The potentially tachyonic states are the linear combinations of modes invariant under the mappings (2.27), (2.28), as well as their counterparts with winding numbers rather than KK momenta along some of the internal directions X 5 , . . . , X 9 .
Having identified the states potentially tachyonic, our goal is to derive their off-shell tree level potential and its minima, in order to figure out all different phases.
Gauged N = 4 supergravity in four dimensions is a theory that couples a gravity multiplet to an arbitrary number k of vector multiplets [38][39][40][41][42][43]. The scalar content is a complex scalar S (related to the string theory axion χ and dilaton φ dil ) and 6k real fields that realize a non-linear σ-model with target space .
Properties of such manifolds are briefly reviewed in the appendix. To describe the potential of the theory, it is however convenient to consider the group quotient SO(6, k)/SO(k) [38,39].
The latter can be parameterized by real scalars Z S a , S = 1, . . . , 6 + k, a = 1, . . . , 6, satisfying 8 where η = diag(−1, . . . , −1, 1, . . . , 1), with 6 entries −1. Hence, the Z S a 's describe all physical field configurations, with an SO(6) redundancy. Given the supermultiplet content, the model is further characterized by the gauging, which implements the non-Abelian nature of the vector bosons arising from the vector multiplets and/or the 6 graviphotons. The gauging amounts to switching on structure constants f RST that are totally antisymmetric in their indices. By supersymmetry, the following potential in Einstein frame is generated [39,40], where the Z RU 's are SO(6)-invariant combinations, 4) and the overall factor is related to S as follows (see appendix),

Supersymmetric case
Let us first review how the above framework can be used to describe the effective field theory of an exactly N = 4 supersymmetric heterotic model in Minkowski space [46,47]. Some of the quantum numbers characterizing the spectrum are generalized momenta p = (p L , p R ), which take values in a Narain lattice Λ (6,22) . The latter is a moduli-dependent, even, selfdual and Lorentzian lattice of signature (6,22) [48,49]. There are 22 generically massless vector multiplets satisfying p = 0, which realize the U (1) 22 Cartan sub-algebra of the gauge group generated by the right-moving bosonic string. Other vector multiplets satisfying the level matching condition −p 2 L + p 2 R = 2 become massless at enhanced symmetry points in moduli space, where their momenta satisfy p 2 L = p 2 R − 2 = 0. These states admit towers of pure KK or winding modes along internal directions (for the level matching condition −p 2 L + p 2 R = 2 to remain valid), which may be light (but not massless) far enough from the core of the moduli space. As a result, the low energy effective supergravity valid everywhere in moduli space should take all of these vector multiplets into account, implying k to be infinite [46,47]. In that case, the off-shell action would be invariant under the full T-duality group O(6, 22, Z) [46,47,50,51]. However, in any finite region in moduli space, only a finite number of vector multiplets satisfying −p 2 L + p 2 R = 2 are lighter than some given cutoff scale. In such a domain, the effective supergravity may be restricted to a finite number k of light vector multiplets, with all others integrated out.
Whether k is finite or not, it is convenient to define the index S to take the values 1, . . . , 6 associated with the Abelian generators of the U (1) 6 gauge symmetry generated by the graviphotons, or 7, . . . , 28 for the U (1) 22 Cartan sub-algebra, or finally any "generalized root p ∈ Λ (6,22) of squared length equal to 2" that yields a light vector multiplet in the moduli space region under interest. The use of the word "root" is justified by the fact that in a Cartan-Weyl basis, the components of p = (p L , p R ) are the charges of the associated vector multiplet under U (1) 6 × U (1) 22 [46,47]. Therefore, the p's are the data needed to describe the gauge interactions i.e. the structure constants f RST . Notice however that to write the off-shell effective supergravity valid in a given region in moduli space, we may choose any background value of the moduli fields in this region to compute the momenta p [9].
Consistently, the effective field theory must be independent of this choice. To be specific, up to permutations of the indices, the non-vanishing structure constants are [46,47] f Spp = p S+3,L δ p+p ,0 , S ∈ {1, . . . , 6}, while ε(p, p ) are suitable signs, and brackets · · · stand for background values.
To set these ideas on a simple example, we can consider the N = 4 model obtained by compactifying on the background (2.1), when no Scherk-Schwarz mechanism is implemented.
The associated partition function is given in Eq.
are respectively light for large enough R 5 , or low enough R 5 , when R 4 sits in the vicinity of 1. Therefore, an effective description valid in the region R 4 1 for arbitrary R 5 can be constructed by including both towers of vector multiplets. Of course, when R 5 1, the degrees of freedom wrapped along S 1 (R 5 ) are very heavy and must be set to 0, while for R 5 1 it is the KK states propagating along S 1 (R 5 ) that must be frozen at their trivial background values. For R 5 1, the modes m 5 = n 5 = 0 are the only ones dynamical. In that case, the non-vanishing structure constants to be considered for an effective description valid for arbitrary R 5 are [9] where we set = 1 in the expressions of p 4L , p 4R . If the effective action is independent of the choice of background R 4 around 1, taking R 4 = 1 (with R 5 arbitrary) is particular in the sense that the SU (2) structure constants become explicit, since p 4R = √ 2 are the SU (2) roots, and p 4L = 0.
In the present case, where N = 4 supersymmetry is exact, the non-trivial structure constants in Eq. (3.8) have one index R ∈ {1, 2, 7, 8}, i.e. in the second coset of (3.9). Therefore, the scalar degrees of freedom labelled p and −p must sit, say, in the third and fourth cosets, respectively. In that case, W and thus V do not vanish identically, which allows the scalars with quantum numbers given in Eq. (3.7) to have non-trivial masses in the supergravity description, as R 4 and R 5 vary [9]. Moreover, because the potential we are interested in involves scalars arising from the untwisted sector only, it can either be computed by using the N = 1 formula in Eq. (3.10), or the N = 4 result given in Eq. (3.3). 9 Finally, due to the orbifold action, the complex scalars with quantum numbers p and −p have to be identified.

Non-supersymmetric case
In the background (2.1) modded by Z 2 × Z 2 , the spontaneous breaking of N = 1 supersymmetry we consider is realized by coupling the lattice of zero modes associated with S 1 (R 4 ) to the boundary conditions of the worldsheet fermions ψ 6 , ψ 8 . For simplicity, we will describe the potential of the effective supergravity for a minimal set of degrees of freedom. We restrict to the dilaton φ dil , the radii R 4 , R 5 , and the potentially tachyonic real scalars described at the end of Sect. 2.3, which have vanishing momenta and winding numbers along T 2 × T 2 .
However, our analysis could be generalized to include more moduli and potentially tachyonic modes. The relevant σ-model to start with is based on the target space which deserves comments. The coordinates of the second coset are associated with the metric and antisymmetric tensor moduli fields of the internal 2-torus of coordinates X 4 , X 5 , which we will actually take in factorized form. The third coset is parameterized by the potentially tachyonic real scalars whose quantum numbers are together with an equal number of real superpartners that will be massive after gauging.
They realize the bosonic parts of k − = +∞ chiral multiplets. Due to the action of G 2 (see Eq. (2.28)), the states (3.13) and (3.14) for given m 5 , n 5 have to be identified.
Using Eq. (A.14), we have They mix non-trivially m 4 , n 4 with Q = (Q, Q 2 , Q 3 , Q 4 ), so that Λ (6,22) should be extended to a larger Lorentzian lattice, whose vectors (p L , Q , p R ) have norm −p 2 L − Q 2 + p 2 R . However, in our parent model of interest, the N = 4 → N = 0 spontaneous breaking implies all scalar superpartners of the states A andÃ to be massive. After implementation of the Z 2 × Z 2 orbifold action, restricting the dynamics to the potentially tachyonic modes only, the only relevant Cartan charges p L , p R are therefore those associated with the potentially tachyonic real scalars A andÃ, which are identified. Hence, we switch on structure constants where A = (+, 0, 0), (+, m 5 , 0) or (+, 0, n 5 ). (3.19) Notice that the level matching condition now reads − p L 2 + p R 2 = 1. In the notations of the above formula, it is understood that A andÃ are not independent indices in the sense that states A andÃ always have opposite momenta and winding numbers. However, the structure constants (3.19) being formally similar to those encountered in the supersymmetric case, Eq. (3.8), they cannot induce any super-Higgs mechanism. In particular, the real and imaginary parts of the ω A 's and ωÃ's would be treated on equal footing and have degenerate masses.
In order to break spontaneously N = 4 → N = 0 in the parent supergravity, a non-Abelian structure among the 6 graviphotons must be implemented. However, the generated potential should admit a phase compatible with Minkowski space-time. This has to be the case since the underlying string theory is a no-scale model [35], provided R 4 sits outside the range (2.13). By definition, structure constants satisfying these conditions define the socalled "flat gaugings" [28]. After Z 2 × Z 2 truncation, in N = 1 supergravity language, for the superpotential W and thus the potential V to be affected by the non-Abelian interactions of the graviphotons in the parent theory, additional structure constant f RST having one index in each of the last three cosets in (3.9) must be considered [9,10]. Because our choice of implementation of the stringy Scherk-Schwarz mechanism involves the left-and right-moving quantum numbers along S 1 (R 4 ) only, we switch on 20) where e L , e R andẽ L ,ẽ R have to be determined for the mass spectrum in the no-scale supergravity phase to match that of the underlying string model. Notice that if only e L , e R have been considered in Refs [9,10], we will see thatẽ L ,ẽ R play an important role for matching an enlarged spectrum.

Effective potential in the non-supersymmetric case
In order to write the potential V that involves only untwisted states of the N = 1 → N = 0 orbifold theory, we find easier to use the N = 4 supergravity expression (3.3) rather than its N = 1 counterpart, Eq. (3.10). The link between the constrained fields φ S in Eqs (3.15) and (3.17) and the N = 4 variables Z S a are provided by the relations (see Eq. (A.11)) with all other Z S a ≡ 0. In that case, Eq. (3.2) reduces consistently tô It turns out that the computation of V is greatly simplified by the introduction of indices correlated as follows: For A = 3, 4 or A, we defineÃ = 5, 6 orÃ, respectively. (3.24) With this convention, the structure constants with one index equal to 1 or 7 can be unified in a single notation, . In total, the potential takes the where the sums are indicated explicitly for clarity. In the above formula, we have defined In the present work, because we impose the first internal 2-torus to be S 1 (R 4 ) × S 1 (R 5 ), we can restrict the moduli T and U to be purely imaginary. We define where the precise relation between the real variables R 4 , R 5 and the worldsheet moduli R 4 , R 5 will have to be determined. From the relation (3.23) for S, T ∈ {1, 2, 7, 8}, we obtain where we have defined v(R, x, y) = 1 4 (3.30) The expressions of the Z AB 's and ZÃB's involve in terms of which we have To proceed, we make some remarks on the expansion in ω A , ωÃ of the potential: • At zeroth order, i.e. with no tachyon condensation, only Z 33 , Z 55 and Z 44 , Z 66 are nontrivial. As a result, up to the overall dressing by |Φ| 2 , V reduces to a constant expressed in terms of e L , e R andẽ L ,ẽ R . Since this configuration should describe the no-scale phase characterized by a vanishing cosmological constant, this constant must vanish.
• At next order, V contains quadratic terms in Re ω A , Re ωÃ, which depend on e L , e R but not inẽ L ,ẽ R . V also contains quadratic terms in Im ω A , Im ωÃ, which depend onẽ L ,ẽ R but not in e L , e R . Hence, it is a matter of convention to choose e L , e R rather thanẽ L ,ẽ R to reproduce the tachyonic mass terms (in Einstein frame) of the underlying string model. In that case, Re ω A , Re ωÃ are the associated degrees of freedom and Im ω A , Im ωÃ are massive superpartners to be set to 0.ẽ L ,ẽ R can then be tuned to satisfy the above mentioned cosmological constant constraint. In fact, for these statements to be true, the kinetic terms should also be block-diagonal in Re ω A , Re ωÃ, and in Im ω A , Im ωÃ. This turns out to be the case, since the scalar kinetic terms of the truncated supergravity are determined by the Kähler metric [52], and take the following form at quadratic order, (3.33) From now on, we thus take which yields In the present work, we consider a deformation of the boundary conditions of the worldsheet fermions ψ 6 , ψ 8 , which leads to the potentially tachyonic spectrum described at the end of Sect. 2.3. This amounts to identifying suitably the degrees of freedom ω A and ±ωÃ, where the choice of sign turns out to be a matter of convention. This follows from the fact that the transformation ωÃ → −ωÃ is equivalent to Z 5Ã → −Z 5Ã , and that the latter can be compensated by a flip (e L , e R ) → −(e L , e R ). Making the choice the potential takes the following form, A Ω 2 A + C with constant coefficients defined as As explained before, in order to identify the structure constants responsible for the N = 1 → N = 0 spontaneous breaking, we use our knowledge of the cosmological constant and masses given in Eq. (2.12), which are valid in the no-scale supergravity phase. To work in Einstein frame, we combine the string theory axion χ and dilaton field φ dil into the axio-dilaton scalar In these notations, the conditions to be imposed take the form which are Kähler transformations. Therefore, γ S S, γ 4 γ 5 T , (γ 4 /γ 5 )U are as good variable as S, T , U. In the matching of supergravity with string data, we thus identify with coefficients to be determined.
Notice that the constraints on the masses of the modes with non-trivial momentum or winding number along S 1 (R 5 ) yield, in particular, for A = (+, 0, n 5 ) : which imply 1 γ 2 5 R 5 2 = γ 2 This has several consequences. Firstly, γ 5 is related to the choice of background, γ 5 = 1/ R 5 . Secondly, because of the level matching condition, which fixes − p 4L 2 + p 4R 2 = 1, we have −e 2 L + e 2 R = −2, and then −ẽ 2 L +ẽ 2 R = 2 for C (0) to vanish. We stress that had we considered only the pure momentum (or pure winding) states, γ 5 would have only been related to This is the reason why in Refs [9,10],ẽ L ,ẽ R are not introduced (or set to 0). It is therefore important to take into account both momentum and winding states along S 1 (R 5 ), because this fixes the r.h.s. of Eq. (3.43) to 1. In the end, the constraints on C (0) and C (2) A admit 2 solutions, where σ = ±1. In fact two more solutions exist, but it can be shown that when both the real and the imaginary parts of the moduli T , U are taken into account, only the solutions (3.44) reproduce correctly the mass spectrum arising for arbitrary metric and antisymmetric tensor backgrounds in the internal directions X 4 , X 5 .
It would be interesting to see whether σ and/orẽ 2 L = 2 +ẽ R could be fixed by taking into account more degrees of freedom in the supergravity action.
We are ready to display the final expression of the low energy tree level effective potential, • The quartic terms in Ω A 's being positive, at fixed φ dil , V is bounded from below.
• When R 4 sits outside the range (2.13), all quadratic terms in Ω A 's are positive as well.
Hence, all vacua are degenerate, with vanishing cosmological constant: This is the "no-scale" phase, where m3 2 E ≡ e φ dil /(2R 4 ) is arbitrary. • According to the initial string theory mass spectrum, the quadratic terms in ω A 's are invariant under the T-duality transformations R 4 → 1/(2R 4 ) and R 5 → 1/R 5 . It turns out that the full potential V and thus the full tree level bosonic action respect this T-duality. 11 • When R 4 sits in the tachyonic range (2.13), degrees of freedom can condense. From the first line of Eq. (3.46), we see that extremizing V with respect to R 4 and Ω (+,0,0) fixes R 4 = 1/ √ 2 (the self-dual radius) and the total sum A Ω 2 A = 1/4. To figure out which of the potentially tachyonic modes A actually condense, it is enough to note that all other terms in the second and third lines of Eq. (3.46) are non-negative. Hence, V is minimal when all scalars with non-trivial momentum or winding number along S 1 (R 5 ) vanish. In other words, all of the 11 The kinetic terms are invariant under the T-duality transformations valid in the supersymmetric case, R 4 → 1/R 4 and R 5 → 1/R 5 , as well as the Kähler transformation R 4 → 2R 4 . Hence, they are invariant under R 4 → 1/(2R 4 ). condensation is supported by the tachyon having m 5 = n 5 = 0. At fixed φ dil , there are 2 branches of minima, which are reached for the backgrounds Ω (+,0,0) = ± 1 2 , Ω (+,m 5 ,0) = Ω (+,0,n 5 ) = 0, m 5 , n 5 = 0, (3.48) The Ω A 's and R 4 are stabilized, R 5 is a flat direction, and the T-duality transformation is not spontaneously broken. The new mass spectrum can be found by expanding V in small perturbations around the expectations values,  Strictly speaking, the word "mass" is a misnomer, since the dilaton has a tadpole. In fact, in terms of the string frame metricĝ µν = e 2φ dil g µν , the tree level action involving the Ricci scalar, dilaton and potential reads As proposed in Refs [9,10], this suggests that the condensed phase of the effective supergravity may describe the low energy physics of a non-critical string theory, with linear dilaton background.

Conclusion
In this work, we have initiated the study of phase transitions occurring in string theory, when the scale of spontaneous breaking of supersymmetry is of the order of the string scale.
Even if they are physically very different from the Hagedorn instabilities developed at high temperature, they share technical similarities about internal or temporal cycles along which bosons and fermions have distinct boundary conditions. Significant differences nevertheless exist.
when the radius R 0 of the Euclidean time circle falls below the Hagedorn radius R H . In the supersymmetry breaking case, even when the Scherk-Schwarz mechanism is implemented along a single factorized circle S 1 (R), the analogous 2 real scalars may be projected out of the spectrum by an orbifold action. When this arises, the "Hagedorn-like region" in moduli space is not the domain R < R H , but a subregion where tachyons with non-trivial momenta or winding numbers along other internal directions condense. This possibility yields interesting new phenomena that will be described elsewhere. Moreover, the instabilities occurring at high supersymmetry breaking scale can be analyzed when the internal metric and antisymmetric tensor are generic. In that case, target space duality transformations imply the Hagedorn-like region to be much more involved, with a fractal structure.
In the present paper, we have considered a Z 2 ×Z 2 heterotic orbifold setup that illustrates the simplest situation. In this example, a real scalar with non-trivial quantum numbers only along the Scherk-Schwarz circle S 1 (R 4 ) survives the modding action, and becomes tachyonic when R 4 < R H . It is accompanied by an infinite number of potentially tachyonic scalars, with momenta or winding numbers along a transverse circle S 1 (R 5 ). We have derived the tree level effective potential that depends on these degrees of freedom. It turns out to be symmetric under the T-duality transformation R 4 → 1/(2R 4 ) and to allow only two phases.
The former is associated with the initial no-scale model, where the cosmological constant vanishes and the supersymmetry breaking scale is arbitrary. In the second phase, the tachyon with non-trivial quantum numbers only along S 1 (R 4 ) condenses, which stabilizes R 4 at the self-dual point 1/ √ 2, as well as the infinity of other scalars at their origin.
It is a long-standing problem to better understand the nature of the condensed phase, with negative potential and dilaton tadpole. In Refs [9,10], it is proposed to be associated with an underlying non-critical string, with linear dilaton background. It would be interesting to compare the gauge symmetries, masses and interactions arising in supergravity and string theory to provide further evidence for such a conjecture. In addition, one may extremize the full effective action instead of the potential, in order to derive a dynamical transition between the condensed and the no-scale phases. Solving the equations of motion in Euclidean time may also yield instantonic transitions.
A question tackled in the core of our work is the determination of the structure constants of N = 4 gauged supergravity that are appropriate for describing the low energy physics of a string theory no-scale model. When N = 4 supersymmetry in 4 dimensions is exact, the constants f RST are related to the charges of all (light) vector multiplets under the U (1) 6 L × U (1) 22 R Cartan subgroup. In other words, they are nothing but the generalized momenta of the Narain lattice, (p L , p R ) ∈ Λ (6,22) [46,47]. However, (p L , p R ) is no longer universal among the degrees of freedom of a vector multiplet, when N = 4 is spontaneously broken. For this reason, we have restricted the effective supergravity to a single (and potentially tachyonic) real scalar in each vector multiplet, in order to avoid any ambiguity in the choice of charge (p L , p R ). The remaining structure constants responsible for the non-Abelian gauging of the N = 4 graviphotons have been determined for the tachyonic mass spectrum of the underlying string model to be reproduced. Clearly, it would be very interesting to generalize the analysis of Refs [46,47] to the case of a (total) spontaneous breaking of N = 4 supersymmetry, in order to identify all structure constants f RST from pure string theory quantum numbers. To show this, we first observe that the dimension of the space of solutions of the above system is d p,q − d 0,q . Next, let us view the M S a 's as the p + q entries of p vectors M a . It turns out that the M a 's can be generated by the action of SO(p, q) modulo SO(p). This can be seen by first defining p vectors v a ∈ R p,q by v S a = δ S a . They are invariant under the action of SO(q) in the following sense: In the following, we specialize to the case p = 2, which is mostly encountered in the core of the paper. Definingφ S = 1 2 M S 1 + iM S 2 , S ∈ {1, 2, . . . , 2 + q}, (A.11) the defining equations (A.7) of SO(2, q)/SO(q) can be written as (φ 2+i ) 2 = 0. with the following definitions [53,54]