One-loop masses of Neumann-Dirichlet open strings and boundary-changing vertex operators

We derive the masses acquired at one loop by massless scalars in the Neumann-Dirichlet sector of open strings, when supersymmetry is spontaneously broken. It is done by computing two-point functions of"boundary-changing vertex operators"inserted on the boundaries of the annulus and M\"obius strip. This requires the evaluation of correlators of"excited boundary-changing fields,"which are analogous to excited twist fields for closed strings. We work in the type IIB orientifold theory compactified on $T^2\times T^4/\mathbb{Z}_2$, where $\mathcal{N}=2$ supersymmetry is broken to $\mathcal{N}=0$ by the Scherk-Schwarz mechanism implemented along $T^2$. Even though the full expression of the squared masses is complicated, it reduces to a very simple form when the lowest scale of the background is the supersymmetry breaking scale $M_{3/2}$. We apply our results to analyze in this regime the stability at the quantum level of the moduli fields arising in the Neumann-Dirichlet sector. This completes the study of Ref. [32], where the quantum masses of all other types of moduli arising in the open- or closed-string sectors are derived. Ultimately, we identify all brane configurations that produce backgrounds without tachyons at one loop and yield an effective potential exponentially suppressed, or strictly positive with runaway behavior of $M_{3/2}$.


Introduction
Superstring-theory models based on two-dimensional conformal field theories of free fields have the advantage of allowing, at least in principle, string amplitudes to be computed exactly in string tension α by including all worldsheet instantons. Backgrounds whose internal spaces are Z N -twist orbifolds of tori are of particular interest since their numbers of spacetime supersymmetries are reduced in a "hard way" compared to the case of toroidal compactifications. In this framework, twisted states in the closed-string Hilbert space are mandatory for modular invariance to hold, which implies "twist fields" to exist in the conformal field theory to create them [1]. String amplitudes involving external states in the twisted sectors are based on correlation functions of twist fields, which are notoriously difficult to handle.
Indeed, the seminal work of Ref. [2] presents results only for the case of twist fields creating ground states in the closed-string sector.
In open-string theory, the consistency of orbifold models also implies the presence of distinct D-brane sectors. For instance, in the type IIB orientifold on T 4 /Z 2 [3][4][5], open strings have either Neumann (N) or Dirichlet (D) boundary conditions in the orbifold directions, and are thus attached to D9-or D5-branes. In particular, strings with Neumann boundary conditions at one end and Dirichlet conditions at the other end populate the ND sector. In string amplitudes involving external states of this type, a conformal transformation maps the legs of the diagram to vertex operators localized along the worldsheet boundary. The key point is that the nature of an ND-sector state implies that the worldsheet boundary condition changes from Neumann on one side of the vertex to Dirichlet on the other side. Hence, vertex operators creating states in the ND sector involve "boundary-changing fields" [6] dressed by other objects encoding the quantum numbers.
It turns out that twist fields and boundary-changing fields have identical OPE's [2,6], up to the fact that the former are inserted in the bulk of the worldsheet and the latter on the boundary. Combining this with the method of images which defines surfaces with boundaries as Rienmann surfaces modded by involutions [7,8], correlation functions of boundarychanging fields can be related to those of twist fields. In the literature, this point of view was applied for computing amplitudes with external states of ND sectors in supersymmetric theories at tree level [9][10][11][12] and one loop [13][14][15][16][17][18], while other approaches were followed in Refs. [19][20][21].
In the present work, we consider the type IIB orientifold model of Refs. [3][4][5] compactified on T 2 × T 4 /Z 2 , when N = 2 supersymmetry is spontaneously broken to N = 0. The implementation of the breaking consists of a string version [22][23][24][25][26][27][28][29] of the Scherk-Schwarz mechanism [30,31] along one direction of T 2 . In this case, the supersymmetry breaking scale, M 3/2 , is a modulus inversely proportional to the size of the compact direction involved in the mechanism. Moreover, the free nature of the bosonic and fermionic fields defining the worldsheet conformal field theory is preserved and the results of Ref. [2] apply. An effective potential which depends on all moduli fields is generated by quantum corrections and the question of their stability must be addressed. Assuming the string coupling to be in perturbative regime, loci in moduli space where the one-loop effective potential is extremal with respect to all moduli fields except M 3/2 have been determined in Ref. [32], up to exponentially suppressed terms. At these points, the potential reads n F − n B = 0, implying M 3/2 to be a flat direction (for other theories, see Refs. [33][34][35][36][37][38][39][40][41][42][43]). For arbitrary n F − n B , though, stability of all remaining moduli fields can be analyzed from different points of view.
In Ref. [32], the mass terms of all moduli fields in the NN and DD open-string sectors were derived by direct computation of the potential for arbitrary backgrounds of these scalars.
The untwisted closed-string sector contains three types of moduli fields: Firstly, since the internal metric components do not show up in the dominant term of Eq. (1.1) (except the combination M 3/2 ), they parametrize flat directions up to the suppressed terms. Secondly, heterotic/type I duality was used to show that the Ramond-Ramond (RR) two-form moduli are also flat directions. Finally, the same conclusion definitely applies to the dilaton at one loop. The twisted closed-string sector contains 16 blowing-up modes of T 4 /Z 2 among which 2 to 16 are absorbed by anomalous U (1)'s, which become massive vector fields thanks to a generalized Green-Schwarz mechanism. In this regard, the present work can be seen as a companion paper of Ref. [32], as it provides a derivation of the mass terms generated at one loop by the remaining moduli fields, namely those belonging to the ND+DN open-string sector. This will be done by computing two-point functions of boundary-changing vertex operators of massless scalars in the ND+DN sector, on the annulus and Möbius strip.
In Sect. 2, we review the description of the type IIB orientifold model with broken N = 2 supersymmetry, which involves D9-and D5-branes. Alternative T-dual pictures are also introduced for describing the NN-and DD-sector moduli as positions of D3-branes in the internal space. Sect. 3 defines the string amplitudes we are interested in. Sect. 4 presents all correlators needed to calculate these amplitudes on the double-cover tori of the annulus and Möbius strip. In particular, we review the derivation of Ref. [2] of the correlation function of twist fields that create ground states in the twisted sectors of closed strings. Following the method introduced in Refs. [13][14][15][16][17], we extend the result to the case of "excited twist fields" i.e. operators appearing as higher order terms in the OPE of ordinary twist fields.
In Sect. 5 we compute the two-point functions of interest. While the formulas can be used to extract the one-loop corrections to the Kähler metric and masses of the classically massless scalars of the ND+DN sector, they turn out to be rather cumbersome and obscure.
For this reason, we derive in Sect. 6 a simplified expression of the squared masses at one loop that is valid when M 3/2 is lower than all other non-vanishing mass scales present in the background, precisely in the spirit of Eq. (1.1) which holds in this regime.
In Sect. 7, we apply this result to the last two models highlighted in Ref. [32], which presented all brane configurations that are tachyon free (or potentially tachyon free) at one loop 1 and satisfy n F − n B ≥ 0. The outcome of the two papers is that among the O(10 11 ) non-perturbatively consistent brane configurations, there exist 2 tachyon free setups with n F − n B = 0, and 5 with n F − n B > 0. A third configuration with n F − n B = 0 and ND+DNsector moduli is tachyon free at one loop, up to 2 blowing-up modes of T 4 /Z 2 for which we have not computed the quantum mass terms.
Finally, our conclusions can be found in Sect. 8, while technical points are reported in three appendices.

The N = → N = 0 open-string model
In this section, we review the open-string model considered in Ref. [32,44], which realizes at tree level the spontaneous breaking of N = 2 supersymmetry in four-dimensional Minkowski spacetime. Our goals are to fix our notations, list the massless spectrum at genus-0, and specify the moduli fields whose masses will be computed at one loop in the sections to come.

The supersymmetric parent model
At the supersymmetric level, our starting point is the type IIB orientifold model constructed in six dimensions by Bianchi and Sagnotti [3], as well as by Gimon and Polchinski [4,5].
Compactified down to four dimensions, the full gravitational background becomes and where the Z 2 -orbifold generator is defined as g : (X 6 , X 7 , X 8 , X 9 ) −→ (−X 6 , −X 7 , −X 8 , −X 9 ) . (2.3) The background also contains orientifold planes and D-branes. First of all, there is an O9-plane and 32 D9-branes spanned along all spatial directions. Second, there is an O5plane localized at each of the 16 fixed points of T 4 /Z 2 , and 32 D5-branes transverse to T 4 /Z 2 .
Open strings with one end attached to a D9-brane have Neumann boundary conditions in all spacetime coordinates, while those stuck to a D5-brane have Dirichlet boundary conditions along the directions of T 4 /Z 2 (and Neumann along R 1,3 × T 2 ).

Moduli fields:
• On the worldvolumes of the 32 D5-branes, the gauge bosons can develop vacuum expectation values (vev's) along T 2 , which are Wilson lines. T-dualizing the two-torus, the 32 D5-branes become D3-branes whose positions alongX I , the coordinates along the T-dual torusT 2 of metricG I J ≡ G I J , are nothing but the Wilson-line moduli of the original description [45]. Because in the T-dual picture a D3-brane at (X I , X I ) is transformed under Ω, the orientifold generator, into an "orientifold-mirror" D3-brane located at (−X I , −X I ) [45], there are 64 fixed points in this description, all supporting one O3-plane. 2 Moreover, at genus-0, there are only 16 independent positions along T 2 , which are associated with the brane/mirror brane pairs.
• The locations of the 32 D3-branes (T-dual to the D5-branes) in T 4 /Z 2 are also allowed to vary, provided this is done consistently with the symmetries generated by g and Ω. Indeed, a D3-brane sitting at (X I , X I ) must be paired with an image brane under stretched between one D5-brane and one D9-brane can also lead to moduli fields. The present paper is devoted to the study of these moduli. To be specific, we will derive the masses they acquire at one loop, when supersymmetry is spontaneously broken and their vev's vanish. When these moduli condense, the backgrounds can be described in terms of brane recombinations or magnetized branes [46][47][48][49]. 2 In addition, the initial D9-branes become D7-branes. 3 The initial D5-branes also become D7-branes in this alternative T-dual picture.
Dir ect ion of Sch erk -Sc hw arz (a) Configuration of D3-branes associated with D5branes (orange) and D9-branes (green) in T-dual pictures. In this example, all D3-branes sit on O3planes (blue dots).
Dir ect ion of Sch erk -Sc hw arz (b) Labelling of the fixed points i ∈ {1, ..., 4} along the directions ofT 2 , and schematic labelling of the fixed points i ∈ {1, ..., 16} along the directions of T 4 orT 4 . i = 1 or 3 correspond to points atX 5 = 0, while i = 2 or 4 correspond to points atX 5 = π, whereX 5 is the coordinate T-dual to the direction along which the Scherk-Schwarz mechanism is implemented.

Geometric picture:
In order to specify a particular set of vev's for the moduli arising from the DD and NN sectors, we will use a pictorial representation [32], as shown in Fig. 1a.
We represent the fundamental domain ofT 2 × T 4 /Z 2 modded by the involution (X I , X I ) → superpose the two boxes, keeping in mind that the resulting picture combines information from two distinct T-dual descriptions of the same theory.
In the schematic example of Fig. 1a, all D3-branes are located on O3-planes. Indeed, it has been shown in Ref. [32] that in presence of supersymmetry breaking (to be introduced in the next subsection) these configurations are of particular interest, since they yield extrema of the one-loop effective potential with respect to the moduli arising from the NN and DD sectors (except for M 3/2 when n F = n B ), up to exponentially suppressed terms (see Eq. (1.1)).
Therefore, from now on, we will consider background values of the moduli in the DD and NN sectors corresponding to stacks of D3-branes all located on corners of the six-dimensional boxes. To this end, we label the 64 corners by a double index ii , where i ∈ {1, . . . , 16} refers to the fixed points of T 4 /Z 2 (orT 4 /Z 2 ), and i ∈ {1, . . . , 4} is associated with those in thẽ T 2 directions. Hence, the coordinates of corner ii are captured by a two-vector 2π a i and a four-vector 2π a i , whose components satisfy 1b shows how the labelling looks like when the fixed points i ∈ {1, . . . , 16} are schematically arranged linearly along a vertical axis. In these notations, we will denote by D ii and N ii the numbers of D3-branes T-dual to the D5-branes and D9-branes that are located at corners ii of the appropriate boxes.
In the model based on the background (2.1), we make the choice to implement the Scherk-Schwarz mechanism along the periodic direction X 5 only, and to use the fermionic number F as conserved charge. In practice, F = 0 for the bosonic degrees of freedom and F = 1 for the fermionic ones. Denoting m the two-vector whose components are the Kaluza-Klein momenta (m 4 , m 5 ) ∈ Z 2 along T 2 , the lattices of zero modes appearing in the one-loop partition function are shifted according to the rules 4 m + F a S for closed string , where we have defined As a result, the two gravitino masses are corners ii and jj , regardless of whether they are dual to D5-or D9-branes. 5 Because of the particular role played by the directionX 5 , which is T-dual to the Scherk-Schwarz direction X 5 of the original picture, it is convenient to specify our labelling of the fixed points along the directions ofT 2 . We will denote by i = 1 and 3 those located atX 5 = 0, and by i = 2 and 4 those located atX 5 = π (see Fig. 1b).

Partition function:
The one-loop partition function can be divided into four contributions Z Σ , which can be derived from path integrals on worldsheets whose topologies are those of a torus (T ), Klein bottle (K), annulus (A) and Möbius strip (M). These contributions can also be expressed as supertraces over the modes belonging to the untwisted and twisted closed-string sectors, as well as over those in the NN, DD, ND and DN open-string sectors.
For the closed strings, we have In the closed-string sector, this is the only modification in the untwisted sector of the extra generator that implements the Scherk-Schwarz breaking in orbifold language. 5 For the ND and DN sectors, our description in terms of "stretched strings" is somewhat abusive since the corners ii and jj are to be understood in distinct T-dual descriptions.
where τ is the Teichmüller parameter of the worldsheet torus with real and imaginary parts denoted τ 1 and τ 2 > 0, while for the open strings we have In these formulas, L 0 ,L 0 are the zero-frequency Virasoro operators.
In order to give explicit expressions of Z A and Z M , we first define four-vectors m and n whose components are the Kaluza-Klein momenta m I ∈ Z and winding numbers n I ∈ Z along the directions of T 4 . The lattices of zero modes (to be shifted by Wilson lines) of the bosonic coordinates are then given by for the ND and DN sectors , in all open-string sectors.
In the annulus contribution to the partition function, the actions of the neutral group element 1 and generator g on the Chan-Paton indices can be represented by matrices acting on each Neumann or Dirichlet sector ii [4,5], where I k is the k × k identity matrix while for k even Actually, the precise dictionary between the above matrices and those defined in Refs. [4,5] can be found in Appendix A. To be specific, by labelling the branes with Greek indices, the actions of G = 1 or g are represented in the NN sector as follows: Similar expressions apply to the DD sector for G = 1, g, as well as to the ND and DN sectors for G = 1. There exists only one subtlety in the ND and DN sectors for G = g, where one has to multiply all Neumann matrices by signs in the transformation rules, where the index j refers to the fixed point of T 4 /Z 2 where the stack of D5-branes sits. This is explained in Ref. [5] and translated into the notations of our paper in Appendix A. Moreover, the worldsheet fermions associated with the directions X 2 , . . . , X 5 on the one-hand, and those associated with the directions X 6 , . . . , X 9 on the other hand, yield contributions expressed as characters of the SO(4) affine algebra. The latter are associated with a singlet (O), vectorial (V) and two spinorial (S and C) conjugacy classes [55][56][57] In the the Möbius-strip contribution to the partition function, the actions of Ω and Ωg on the Chan-Paton indices can be represented by matrices associated with each Neumann or Dirichlet sector ii , (2.17) Notice the inverted roles of Ω and Ωg in the Neumann and Dirichlet sectors. The precise actions of ΩG for G = 1 or g on the NN sector are [4] ∀α ∈ {1, . . . , (2.18) and similarly for the DD sector. As compared to Eq. (2.14), note the reversal α β → β α in where h is the weight of the associated primary state and c the central charge of the Verma module. With these notations, one obtains where the arguments of all hatted characters are (1 + iτ 2 )/2, and the superscript T stands for the transposition of the matrix to which it applies.
For completeness, the closed-string sector contributions to the partition function Z T and Z K are displayed in Appendix B.

Spectrum:
The classical massless spectrum can be read from the partition function. To this end, it is useful to evaluate the traces over the Chan-Paton indices in the open string sector, which yields where we use the fact that the matrix J k for k even has equal number of eigenvalues i and −i.
From Z A + Z M , one finds that the massless bosonic degrees of freedom are the low-lying modes of the combinations of characters  To proceed the same way for the fermions, it is convenient to define a new double-primed index i ∈ {1, 2} and write i = 2i or 2i − 1. The massless fermionic degrees of freedom extracted from Z A +Z M are then identified as the low-lying modes of the following characters,

Two-point functions of massless ND and DN states
In Ref. [32] the masses at one loop of the open-string moduli arising from the NN and DD sectors were derived by using the background field method. However, in the case of the moduli in the ND+DN sector, the partition function for arbitrary vev's of these scalars is not known and this approach cannot be applied. Therefore, we will derive in Sects. 5 and 6 the one-loop masses of all classically massless scalars in the bifundamental representations of unitary groups supported by D9-and D5-branes by computing two-point correlation functions with external states in the massless ND and DN bosonic sectors. This will be done by applying techniques first introduced in classical open-string theories in Refs. [9][10][11], and at one loop in Refs. [13][14][15][16][17]. For now, we define the relevant vertex operators and open-string amplitudes.

Vertex operators and amplitudes
In the T-dual pictures, let us consider two corners i 0 i 0 and j 0 i 0 on which are located N i 0 i 0 ≥ 2 and D j 0 i 0 ≥ 2 D3-branes T-dual to D9-branes and D5-branes, respectively. As seen in the third line of Eq. (2.22), the open strings "stretched" between these stacks give rise to 2n i 0 i 0 d j 0 j 0 massless complex scalars (depicted as solid strings in Fig. 2b). In the initial description in terms of D9-and D5-branes, we are interested in correlation functions of vertex operators in ghost pictures p and −p of the form where z 1 , z 2 are insertion points on the boundary of a worldsheet whose topology is either that of the annulus or Möbius strip, Σ ∈ {A, M}, and In the above definitions, we use the following notations: • k µ is the external momentum satisfying on-shell the condition k µ k µ = 0.
• φ(z) is the ghost field encountered in the bosonization of the superconformal ghosts [60].
• λ is the matrix where Λ 1 , Λ 2 are arbitrary n i 0 i 0 × d j 0 i 0 real matrices [4]. It labels the states that transform as the ( • From now until Sect. 6, we restrict our analysis to the case where the internal metric is diagonal, for some radii R I , R I . In this case, the formalism of Ref. [2] applies without having to generalize it. Denoting ψ µ (z), ψ I (z), ψ I (z) the Grassmann fields superpartners of the bosonic-coordinate fields X µ (z), X I (z), X I (z), we define a new basis of degrees of where H u are scalars introduced to bosonize the fermionic fields. Because the external legs bring quantum numbers (λ α 0 β 0 , ) and (λ T β 0 α 0 , − ) of the ND and DN sectors, they must be attached to the same boundary of the annulus. Therefore, the second boundary is sticked to another brane labelled γ, which can be any of the 32 D9branes (in green) or 32 D5-branes (in orange). On the center and right panels, the same three diagrams are displayed, with the open-string worldsheets seen as fundamental domains of the involution These definitions apply to a Euclidean spacetime. In the Lorentzian case, replace (X 0 , ψ 0 ) → i(X 0 , ψ 0 ). 8 Characters O 4 S 4 would yield states in the spinorial representation of positive chirality of SO(4), whose weights are ( 1 2 , 1 2 ).
acting on double-cover tori of Teichmüller parameters [7,8,58,61] τ dc = i τ 2 2 for the annulus and In this description, the external legs are conformally mapped to points z 1 , z 2 , where vertex operators change the boundary conditions of the worldsheet fields vanish, which shows that the backgrounds we consider, i.e. where no brane recombination is taking place [46][47][48][49], imply the effective potential to be extremal with respect to the scalars in the ND+DN sector.

OPE's and ghost-picture changing
In order to treat symmetrically both vertex operators when computing the correlation functions (3.1), we switch to the ghost picture p = 0. This is done by applying the formula where T F is the supercurrent given by To this end, we display all necessary operator product expansions (OPE's). First of all, for the "ground-state boundary-changing fields", we have which introduces "excited boundary-changing fields" τ u , τ u . Moreover, the other fields satisfy [13,14] Using these relations, we obtain for = +1 where we have defined while the expressions for = −1 are obtained by exchanging all subscripts and superscripts 3 and 4. Because we are interested in states massless at tree level, the Kaluza-Klein momentum in the T 2 complex direction u = 2 is set to 0 in the "external" parts of the vertex operators.
In the "internal" parts, notice the appearance of "excited boundary-changing operators" Given the above definitions, the correlation functions (3.1) split accordingly into external and internal pieces. The former, are useful to derive the one-loop corrections to the Kähler potential of the ND+DN sector massless scalars. Note that in order to bypass the issue that on shell 1 u=0 |K u | 2 ≡ k 2 /2 = 0, we may have kept the Kaluza-Klein momenta along T 2 arbitrary. On the contrary, the internal parts,

Genus-1 twist-field correlation functions
The main difficulty in computing the two-point functions in Eqs.

Instanton decomposition of correlators
In closed-string theory compactified on T 2 × T 4 /Z N where N ∈ N * , the complex fields defined in Eq. (3.5) depend on holomorphic and antiholomorphic worldsheet coordinates, Z u (z,z). Moreover, upon parallel transport, the internal Z u undergo some Z N rotations and translations, where the shifts v u and v 2 implement the T 4 and T 2 periodicities.
The twist fields create the states in the twisted sectors of the closed-string Hilbert space.
For some given κ ∈ {1, . . . , N − 1} and u ∈ {3, 4}, let us denote by σ u (z,z) the one that creates the ground state in the κ-th twisted sector. The requirement that positive frequency modes in the expansions of ∂Z u and ∂Z u annihilate the twisted ground state determines the OPE of ∂Z u (z) and ∂Z u (z) acting on σ u (w,w) as z approaches w, In the right-hand sides, τ u , τ u ,τ u ,τ u create excited states in the κ-th twisted sector. The OPE's capture the local behavior corresponding to the rotations of the coordinates Z u but do not carry information about the global translations v u . This data is recovered by imposing global monodromy conditions which describe how Z u (z,z) and Z u (z,z) change when they are carried around a set of twist fields with vanishing total twist. Splitting the coordinates of T 2 and T 4 into background values and quantum fluctuations, the whole global displacements arise from the classical parts Z u cl (z,z). With this decomposition, the correlators of interest on a Riemann surface Σ of genus where the total twist is trivial, A κ A = 0 modulo N , for the result not to vanish [1]. In this expression, the sum is over instantons with worldsheet actions In the following, we first compute the "quantum parts" of the correlation functions and then derive the classical actions.

Stress-tensor method
To determine for a given u ∈ {3, 4} the quantum part of the correlator (4.4), Ref. [2] uses the stress-tensor method. It consists in exploiting the OPE's between the stress tensor T u (z) and the primary fields To this end, one considers the quantity in terms of which we may write and exploits the OPE to obtain (4.11) To summarise, the stress-tensor method amounts to determining the Green's function g(z, w), then deduce T u (z) , and finally integrate the differential equations (4.8).

Ground-state twist field quantum correlators on the torus
Let us specialize to the case where Σ is a genus-1 surface. We will denote its Teichmüller parameter as τ dc for future use, when we see the genus-1 Riemann surface as the double cover of open-string surfaces. 11 In order to derive the quantum part of the correlator (4.4) for a given u ∈ {3, 4}, , the starting point is to write the most general ansätze for g(z, w) and the companion Green's function satisfying the following properties: • Double periodicity z → z + 1, z → z + τ dc and w → w + 1, w → w + τ dc (and similarly forz in h).
• Local monodromies consistent with the OPE's given in Eq. (4.2). For instance, when z is transported along a tiny closed loop encircling some z A , g must transform as • A double pole for g(z, w) as z → w dictated by Eq. (4.10), and finiteness of h(z, w) as This can be done by defining cut differentials [2] which form a basis of holomorphic one-forms on the torus that possess suitable monodromy behaviors as their arguments approach each of the insertion points z A . Denoting which takes some value in the set {1, . . . , L − 1}, and following the notations of Ref. [2], such a basis is given by where the second argument at τ dc in the modular forms is implicit. In these formulas, we have defined the functions (4.15) and denoted to the variable z. 13 Given these notations, the Green's functions may be expressed as where C AB and B AB are "constant coefficients." 14 The function g s (z, w) is doubly periodic in z and w and handles the double-pole structure of g(z, w) as z approaches w. It can be expressed as [2] (4.18) where explicit knowledge of the function P (z, w) is not required in the computation of correlation functions of ground-state twist fields. However, it does matter for correlators of excited twist fields, as will be seen in Sect. 4.5. We will come back to this issue at that stage.
The next step is to implement the global monodromy conditions, which by definition are "trivial" for the quantum fluctuations Z u qu (z,z) (see below Eq. (4.3)). In practice, this implies that The subscripts "N − κ" and "κ" are "names". They do not refer to varying indices. Moreover, the indices α 1 , . . . , α L−M and β 1 , . . . , β M here should not be confused with labels of branes also denoted by Greek letters elsewhere in our work. 13 Note that no periodicity condition is imposed for the individual variables z A (which are kept implicit in the cut differentials). However, double periodicity in z implies double periodicity of the whole set of points z A , when they are moved together.
14 They depend only on the insertion points and τ dc .
where {γ a , a = 1, . . . , L} is a basis of the homology group of the genus-1 surface with L punctures. 15 To solve these equations, it is convenient to define an L × L cut-period matrix W a A as follows, Indeed, it is easily checked that the expressions satisfy the global monodromy conditions.
Finally, the correlator can be found by applying the stress-tensor method to find the holomorphic dependence on the z A 's, and then a second time using the Green's functions where f (τ dc , κ 1 , . . . , κ L ) is a function arising as an "integration constant". The latter can be determined by coalescing all insertion points, since the left-hand side reduces in this case to 1 , which is the partition function.

Instanton actions
In the OPE's (4.2), the actions of the background parts ∂Z u cl (z) and ∂Z u cl (z) on σ u (w,w) for u ∈ {3, 4} are trivial multiplications. Hence, for the monodomy properties to be satisfied as z is transported along a tiny closed loop encircling any z A , the doubly-periodic ∂Z u cl (z) and ∂Z u cl (z) must be linear sums of cut differentials. To determine the coefficients, one imposes the global monodromy conditions where the v u a 's are displacement vectors. The solution of these equations can be expressed in terms of the inverse cut-period matrix, where in the present context the index A is summed over 1, . . . , L − M , and A is summed Moreover, we have redefined in the above formulas With the definition of the Hermitian product of one-forms on the torus, the classical action for a single complex coordinate u ∈ {3, 4} reads

Useful correlators on the torus
In this subsection, we consider all correlators involved in the open-string amplitudes of Eqs. (3.14) and (3.15), but display their values computed on a genus-1 surface. In this case, they have holomorphic and antiholomorphic dependencies.
For the OPE's of the twist fields to match those of the boundary-changing fields we are interested in, we now consider the case where Because κ 1 = κ 2 , we can omit from now on the subscripts A of the twist fields. Using Eq. (4.22), we obtain for u = 3, 4 The 2 × 2 cut-period matrix W a i defined in Eq (4.20) involves only one cut differential, to be integrated on the cycles of the genus-1 surface Σ, γ 1 : z → z + 1 and γ 2 : z → z + τ dc , which yields In these notations, the background action written in Eq. (4.27) reads for u =∈ {3, 4} where v u a , a ∈ {1, 2}, are the displacements introduced in Eq. (4.23). In our case of interest, given an instanton solution, the real-coordinate background X I cl (z,z), I ∈ {6, . . . , 9}, winds n I times and l I times the circle S 1 (R I ) as z is transported along γ 1 and γ 2 , so that For the T 2 coordinate u = 2, which is not twisted, the above formula apply with cut differentials that induce trivial local monodromies. In other words, replacing ω(z) by 1, the relevant cut-period matrix becomes 1 1 Defining displacements v u a for u = 2 exactly as those given in Eq. (4.33), one obtains which is the well know result for the instanton action on a two-torus [62]. The sum over instantons appearing in Eq. Correlator τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) qu : To derive the correlator of excited twist fields, we will follow the technique described in Refs. [13,[15][16][17]. Thanks to the OPE's between ∂Z u , ∂Z u and the ground-state twist fields given in Eq. (4.2), and using the splitting defined Eq. (4.3), we may divide accordingly this correlation function for u ∈ {3, 4} into two pieces, where we have defined To derive part (1) of the correlator, we use Eq. (4.24) which becomes Remember that due to their local monodromy behaviors, these expressions diverge at the insertion points. Hence, the limits defined in Eq. (4.37) contribute a finite result which is where we denote Using the Green's function g(z, w) defined in Eq. (4.9), part (2) of the correlator can be expressed as In the present case, Eqs. (4.17) and (4.21) become where g s (z, w) is defined in Eq. (4.18). The latter involves a function P (z, w) derived in Ref. [2], and whose expression is given by (4.43) The right-hand side is written in terms of a unique function γ (see Eq. (4.15)) as well as Notice that in the above formula, we adopt the notations of Ref. [2] but it turns out that z). Computing the limits in Eq. (4.41), we find (4.46) Moreover, using in the derivation the second expression in Eq. (4.42), an explicit expression for the term linear in C is obtained, Adding the pieces (1) and (2) of the correlator, we obtain for u ∈ {2, 3} Correlator τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) qu : Proceeding the same way, and using the fact that F 2 (z 2 , z 1 ) = F 1 (z 1 , z 2 ), we obtain the identity

Bosonic correlator:
The propagator of the spacetime coordinates X µ is given by which leads to (4.51)

Full amplitudes of massless ND and DN states
We are now ready to use all ingredients introduced in Sects. 3 (5.1)

Useful correlators on the annulus and Möbius strip
Let us first collect the correlators presented in the previous section now evaluated on the open-string surfaces Σ = A and M, and to be used to express the amplitudes A α 0 β 0 extΣ and Correlator σ u (z 1 )σ u (z 2 ) qu : In Ref. [13], the method of images was applied on the Green's functions of Sects. 4.2 and 4.3 to define their open-string counterparts. The latter were used to derive the correlator between two ground-state boundary-changing fields by using the stress-tensor method. The result amounts essentially to take the "square root" of the closed-string result, i.e. for u ∈ {3, 4}, where f op is a normalization function. Notice that the product σ 3 σ 3 qu σ 4 σ 4 qu involves ϑ 1 (z 1 − z 2 ) − 1 2 , which is well defined up to a sign. We will see in the next subsection how such ambiguities can be lifted.
The instanton actions can also be derived from the closed-string result given in Eqs.
|v u 2 | 2 for NN and N , where the displacements and their T-dual counterparts are given by Correlators τ u (z 1 )τ u (z 2 ) qu and τ u (z 1 )τ u (z 2 ) qu : On the genus-1 surfaces, the twist fields τ u (z,z) and τ u (z,z) are excited only on their holomorphic sides (see Eq. (4.2)).
Therefore, their correlation functions take formally the same forms as those of the excited boundary-changing fields τ u (z) and τ u (z) evaluated on A and M. There is however a subtlety concerning part (1) of the correlator, Eq. (4.39).
When considering the full amplitudes i.e. with the instanton dressings e −S Σ cl , the openstring actions are divided by 2 compared to the closed-string case. Hence, for the full openstring correlators to preserve their interpretations on double-cover tori, one should rescale the displacements in parts (1) as follows, |v u 2 | 2 → |v u 2 | 2 /2 and |ṽ u 2 | 2 → |ṽ u 2 | 2 /2. As a result we have for u ∈ {3, 4} which can be used to derive

Bosonized-fermion correlators:
In Ref. [13], it is shown by applying the method of images on the Green's functions used in the stress-tensor method that the correlators of bosonized-fermion on the open-string worldsheets are identical to those on the double-cover tori. They are given in Eq. (4.52). For q = ± 1 2 , products of two such correlators are well defined up to signs.

Full expressions of the amplitudes
Putting everything together and defining z 12 = z 1 − z 2 to lighten notations, the full expression (3.14) of the external part of the one-loop two-point function of massless bosonic states in the ND and DN sectors is, for Σ ∈ {A, M}, which is independent of the choice of ∈ {−1, +1}. In this expression, we use the following notations: • l stands for (l 4 , l 5 ), and l, l are the four-vectors whose components are l I andl I .
• The normalization factor of the external fermion correlators is given by [62] • C Σ l l ν int andC Σ l l ν int are normalization functions to be determined. They stand for products of the form possibly dressed by signs that may depend on the instanton numbers l, l or l , l . Indeed, as stressed before, pairs of correlators of twist fields as well as pairs of correlators of spin fields yield signs ambiguities. Moreover, for the amplitude computed on the annulus, the normalization functions should contain sums over the free boundary condition denoted γ. Furthermore, C A l 0 ν int takes into account two contributions associated with the NN and ND worldsheet boundary conditions, whileC A l 0 ν int describes those arising from DD and DN boundary conditions. Similarly, the internal piece (3.15) of the amplitude is independent of and can be expressed in terms of the same normalization functions, where the factor 4 accounts for the trivial sum over the external-fermion spin structure ν ext , C is given in Eq. (4.47), and we have defined In the following, we will not consider anymore in Eqs. (5.8) and (5.11) the irrelevant contributions of the external bosonic correlators, [ · · · ] −2α k 2 , which are equal to 1 on shell. We stress again that we could have introduced non-trivial Kaluza-Klein momenta along T 2 to avoid ambiguities in extracting information from the amplitude A α 0 β 0 extΣ .
Normalization functions C Σ l l ν int andC Σ l l ν int : As said in the remark below Eq. (4.22), C Σ l l ν int andC Σ l l ν int may be determined by using the fact that when z 1 and z 2 coalesce, the effects of the ground-state boundary-changing operators compensate each other. Hence, the external part of the amplitude reduces, up to a multiplicative factors, to selected pieces of the open-string contributions to the partition function. To identify precisely which pieces are relevant, Fig. 4 shows what the diagrams in Fig. 3 become when z 12 → 0. In this limit, the cut differential associated with either of the complex directions u ∈ {3, 4} becomes trivial, ω(z) → 1, so that

This leads to
which has to be identified with for A , for M . In these expressions, C is a constant number, 16 while the supertraces are restricted to the open-string modes with ends attached to branes as shown in Fig. 4. For the identification to be possible, one has to switch the T 2 , T 4 andT 4 lattices in the partition functions Z A and Z M from Hamiltonian to instantonic forms, which is done by Poisson summations, where we have defined For the case of the annulus, using the definitions of the characters given in Eq. (B.4), we identify where we have defined On the contrary, all terms associated with the group generator g vanish, due to the fact that the diagonal components of the matrices J k are zero. In the expressions of C A l l 1 andC A l l 1 , we have introduced a coefficient ρ that accounts for the ambiguity arising in their determination, since ϑ 2 ν int = 0 for ν int = 1. This coefficient will be determined in the sequel. Finally, notice that the identification has lifted all sign ambiguities associated with the twist-and spin-field correlators. These signs depend on the instanton numbers l , l, l and the positions of the D3-branes α 0 , β 0 , γ.
To perform the similar computation in the Möbius strip case, note that Z M is expressed in terms of "hatted characters" defined in Eq. (2.19). However, in light-cone gauge, the characters associated with the worldsheet fermions multiplied by 1/η 8 arising from the bosonic coordinates yield low-lying states at the massless level. Hence, all phases e −iπ(h−c/24) appearing in the definitions of the hatted characters cancel each other and we may simply remove all "hats" on the ϑ and η functions when identifying Eq. (5.15) with the amplitude (5.14).
In that case, the normalization functions are found to be For instance, when one restricts to the boundary conditions shown in Fig. 4, the first trace appearing in the expression of Z M yields,

Consistency when supersymmetry is restored:
All normalization functions can be injected back in Eqs. (5.8) and (5.11) to obtain the full expressions of the amplitudes A α 0 β 0 extΣ and A α 0 β 0 intΣ for arbitrary z 1 , z 2 . To analyze their structures in more details, let us focus on the instantonic sums. For the internal part of the amplitude computed on the annulus, we obtain for each given l , l a contribution of the form where the second equality is obtained by applying a generalized Jacobi identities [62] with non-zero first arguments, as well as the specific form of the vector a S . For given l , l , the similar sum for the coefficientsC A l l ν int is obtained by changing α 0 → β 0 and N ↔ D. We are now ready to determine the constant ρ by taking the limit R 5 → +∞ in Eq. (5.11). Indeed, l 5 = 0 is the only contribution in the sum over l 5 that survives in this limit. Hence, A α 0 β 0 intA vanishes when supersymmetry is restored if and only if ρ = 1, since in that case only, Eq. (5.23) projects out all even values of the wrapping number l 5 . Indeed, in the supersymmetric case, the effective potential cannot be corrected perturbatively, which implies ρ to be such that the one-loop corrections to the masses we are computing vanish.
We can proceed the same way for the internal part of the amplitude computed on the Möbius strip. For fixed l , l, we have while for given l , l , the analogous sum forC M l l ν int is obtained by changing v 4 /α 2 → α 2 /v 4 . In the limit R 5 → +∞ where supersymmetry is restored, the amplitude A α 0 β 0 intM vanishes consistently.
As can be seen from Eqs. (5.8) and (5.11), the sums over the spin structure ν int in the external and internal parts of the amplitudes are identical, up to the insertion of the sign (−1) δ ν int ,1 for A α 0 β 0 extΣ . Of course, this does not make any difference in the case of the Möbius strip since the normalization functions for ν int = 1 vanish. On the contrary, for Σ = A, the extra sign amounts to changing ρ → −ρ in Eq. (5.23). As a result, the external part of the amplitude, A α 0 β 0 extA , does not vanish in the decompactification limit, and yields a one-loop correction to the Kähler potential of the massless scalars in the ND+DN sector, even in the supersymmetric case.

Integration over the moduli and vertex positions:
What remains to be done is operator, say z 2 ≡ 1 2 , and integrate over the location of the other. In the case of the annulus, both vertices must be located on the same boundary, so that As a result, denoting the integrated amplitudes by calligraphic letters, the internal part reads the Möbius strip, z 1 must follow the entire boundary. However, the latter being twice longer than the one considered on the annulus, z 1 can actually be parametrized as z 1 = x 1 + iy 1 , where x 1 ∈ {0, 1 2 }. As a result, the internal part of the integrated amplitude is 26) and similarly for the external part.
In these forms, the full two-point functions are not particularly illuminating, while performing explicitly the integrals is certainly a hard task. Hence, our goal in the next section is to extract simpler answers valid in the case where the scale of supersymmetry breaking is low.

Limit of low supersymmetry breaking scale
The analysis of Ref. [32] is In the present section, we would like to find similar results for the masses of the moduli fields present in the ND+DN sector of the theory. This will be done by imposing all mass scales other than M 3/2 to be proportional to M s = 1/ √ α and then taking the small α limit.

Limit of super heavy oscillator states
In order to treat all massive string-oscillator states as super heavy in the Hamiltonian forms of the partition functions, let us rescale the Teichmüller parameters of the open-string surfaces as follows 17 Physically, this amounts to stretching the surfaces along their proper times in order to look like field-theory worldlines with topology of a circle. The main practical consequence of the rescaling is the approximation dc sin(πz)(1 + · · · ) , when |Im z| < Im τ dc , where from now on, ellipses stand for terms exponentially suppressed when α → 0, i.e.
Notice that compared to Eq. (5.1), we impose y 1 , y 2 to be strictly lower than Im τ dc for the above formula to always be valid. More generally, throughout the derivations to come, we will write formulas in their generic forms. Indeed, because in the end all quantities will have to be integrated, taking into account extra contributions arising only at special values of the integration variables results in subdominant corrections for small α . We will come back to this issue at the end of this section.
Keeping this in mind, we redefine (6.4) in terms of which the components of the cut-period matrix can be expressed like The first expression is easily found by integrating over z finite along γ 1 and replacing all sines in Eq. (6.3) by their dominant exponentials when Im τ dc is large. By contrast, W 2 can be derived by integrating z between x 0 and x 0 + τ dc , x 0 ∈ R, using a primitive of ω in its form given in Eq. (6.3). As a result, we obtain that When taking the limit of small α in Eq. (5.23), it turns out that the terms proportional to f A l l α 0 D (and f A l l β 0 N for the formula involvingC A l l ν int ) are exponentially suppressed. Notice that they arise from the ND and DN sectors of the partition function Z A , which therefore cease to contribute to the amplitudes in this limit. In the case of the annulus, we then arrive at the expression while for the Möbius strip we obtain In the above formulas, the limit of small α in C andĈ will be derived in Sects. 6.3 and 6.4.

Limits of large compactification scales
We would like now to have all compactification mass scales other than M 3/2 very large. In practice, this amounts to taking small radii R 4 , R I and dual radii α /R I limits. In order to avoid having to consider very large instanton numbers in such a regime, it is convenient to apply Poisson summations over l 4 , l and l [62], which lead for Σ = A to and for Σ = M (6.10) One may think that considering T 4 to be small would imply having the T-dual torusT 4 large. This is not true, as can be seen by redefining the radii as follows, where r 4 , r I are fixed and dimensionless. Indeed, all radii and dual radii vanish as α → 0.
As a consequence, the limit of small R 4 implies that we may restrict the dominant term in in the ellipsis. In total, we obtain for the amplitude computed on the annulus while on the Möbius strip we have similarly α + 2 ln 4 + · · · (C +Ĉ) + · · · . (6.14)

Limit α → 0 of U 1 and F 1 (z, z 2 )
In this subsection and the following, our aim is to derive the limits of C andĈ for small α , i.e. the contributions arising from parts (2) of the correlators τ u τ u qu = τ u τ u qu . Because the results can be obtained with no more effort for any Teichmüller parameter, we will keep the real part of τ dc arbitrary, and z 1 , z 2 will be chosen anywhere in the torus represented by P in the complex plane, where P is the parallelogram with corners at 0, 1 and τ dc .
The important thing, though, is that Eqs. (6.1) and (6.4) hold. Hence, our computations of parts (2) are valid for excited twist fields (for closed strings) and excited boundary-changing fields (for open strings). Let us start by deriving the limits of U 1 and F 1 (z, z 2 ), which will be used in the next subsection to derive those of C andĈ.
To see that this definition makes sense, notice that the meromorphic function Ω(U ) is doubly periodic on the genus-1 Riemann surface Σ, with two simple poles at U = 0 and U = Y 1 .
• When − 1 2 Im τ dc < Im Y 1 < 0, we can apply the change of variable (6.21) which yields Hence, assuming that ε is bounded when α → 0, Eq. (6.18) is legitimate for small enough α . However, the first dominant term in the ellipsis is now −2iπ q dc e 2iπ(Y 1 −U 1 ) , 19 and the equation becomes Hence, we obtain that which is equivalent to Eq. (6.20) and leads to ε → 0 when α → 0. The assumption on the boundedness of ε being consistent, we have also found solutions in the present case.
In both instances, ε can be expressed in terms of exponentially suppressed contributions subdominant to those we have taken into account explicitly. Its leading behavior is derived in Appendix C. 19 It can be found from Eq. (C.2) where the function H is defined in Eq. (C.1).
By imposing U 1 to be located in P, we find the two roots of Ω(U ), Since we know that there cannot be other solutions modulo 1 and τ dc , a cross-check of this result is to observe that Y 1 − U 1 satisfies consistently Both possible choices of U 1 yield the same function F 1 (z, w).
What we need to analyze in order to derive the limits of C andĈ is its expression for where z ≡ x + iy ∈ P, and x, y ∈ R. Notice that Eq. (6.2) can be applied to both ϑ 1 functions appearing in the denominator (for small enough α ).

Limit α → 0 of C andĈ
We are now ready to evaluate the limits of C andĈ for small α .

C when α → 0:
Using the relation given in Eq. (4.47), the term linear in C in Eqs. (6.13) and (6.14) can be written as , a ∈ {1, 2} , (6.34) and W 2 is given in Eq. (6.5). We are going to show that F 1 contributes exponentially suppressed terms while F 2 yields a finite result.
In order to evaluate F 1 , we impose the points z of the representative path of the cycle γ 1 to satisfy Im z ≡ 1 2 Im τ dc . The advantage of this choice is that F 1 (z, z 2 ) can be replaced by 1 + · · · all along the path, Omitting the ellipsis, an explicit integration using a primitive of the integrand yields an exactly vanishing result. However, the exponentially suppressed terms in the numerator may be large once multiplied by cos(πz 12 /2). To take them into account, one can find upper bounds valid for given signs of 1 2 Im τ dc − y 1 , 1 2 Im τ dc − y 2 and y 1 − y 2 . As an example, when ( 1 2 Im τ dc − y 1 )( 1 2 Im τ dc − y 2 ) < 0 we obtain cos π 2 z 12 F 1 ≤ 1 0 dx 4 e −π| 1 2 Im τ dc −y 2 | K = · · · , (6.36) where the constant K is any majorant of |1 + · · · | for small enough α . It turns out that in all instances the contributions proportional to F 1 are suppressed.
To compute F 2 , we have to consider two cases. When 0 < 1 2 y 1 + 3 2 y 2 < Im τ dc , Eq. (6.29) allows us to decompose the integral into two pieces, where in the last line we use the fact that sin φ → 1 when α → 0. On the contrary, in case (b), only sin[π(z − z 2 )] can be replaced by a single large exponential. However, it is possible to integrate the dominant term of the integrand, and show as we did for F (1) 2 that the result dominates the integral arising from the ellipsis. Combining both pieces, we find that the conclusion of Eq. (6.42) holds again.
Let us move on to the second case, namely Im τ dc < 1 2 y 1 + 3 2 y 2 < 2Im τ dc , which can be treated by following the same steps as before. The starting point is Eq. (6.31) which leads to the decomposition 2 equals 1 at the lower bound of the integral, it can be shown as before that it yields an extra exponentially suppressed contribution after integration. Therefore, Eq. (6.40) remains valid.
In the integrand of F (2) 2 , it is always safe to replace sin[π(z − z 2 )] by a large exponential. This is also the case for sin[π(z − z 1 )] in case (c), for which we can write (6.46) In the first line, the path of integration for z is the segment that forms an angle φ ∈ (0, π) with the horizontal axis. Integrating the upper bound, we conclude that (6.47) In case (d), only sin[π(z − z 2 )] can be replaced by a large exponential. The integration using a primitive as before shows that the conclusion of Eq. (6.47) is again true.

Integration over τ 2 , z 1 , z 2 and final result
Collecting the contributions of parts (1) and (2) (involving C andĈ) of the correlators τ u τ u qu = τ u τ u qu , the braces in the amplitudes (6.13) and (6.14) reduce to The origin of the terms of order α 2 /R 4 5 will be explained at the end of this section. We are now ready to display the main result of our work. Implementing the correct dimension, the mass squared acquired at one loop by the classically massless state (λ, ) belonging to the ND+DN bosonic sector realized by strings "stretched" between the stack of N i 0 i 0 ≥ 2 D3-branes (T-dual to D9-branes) and the stack of D j 0 i 0 ≥ 2 D3-branes (T-dual to D5-branes) is given by where the supersymmetry breaking scale (2.7) reduces to To conclude this section, notice that Eq. (6.52) guaranties or rules out the stability of moduli fields in the ND+DN sector only when the coefficient in parenthesis is not zero.
When the latter vanishes, one has to compute four-point functions to conclude.

Subdominant contributions:
As announced below Eq.

Stability analysis at one loop
As seen in Ref. [32], most of the brane configurations implying the one-loop effective potential to be extremal and tachyon free 21 yield a run away behavior of M 3/2 with n F − n B < 0.
However, setups that lead to exponentially suppressed or positive values of V 1-loop may be of particular interest. Indeed, for n F − n B = 0, it is conceivable that the suppressed terms at one loop combine with higher loop corrections to stabilize the dilaton and M 3/2 in a perturbative regime. In that case, the resulting cosmological constant should be small, and the issue raised in Ref. [64] avoided. Moreover, cases where n F − n B > 0 may shed light on the existence or non-existence of de Sitter vacua after stabilization of the string coupling and supersymmetry breaking scale.
To be specific, the existence of 2 brane configurations without tachyons at one loop 21 and satisfying classically a Bose/Fermi degeneracy at the massless level were shown to exist in Ref. [32]. Moreover, 4 more tachyon free setups with n F −n B > 0 were also found. In all these instances, reaching these conclusions was possible thanks to the absence of moduli in the ND+DN sectors, and thanks to anomaly-induced supersymmetric masses for all blowing-up modes of T 4 /Z 2 . Furthermore, 2 extra brane configurations were presented [32], one with n F − n B = 0 and the other with n F − n B > 0, which we analyze further in the present section. Indeed, it was established that they both yield nonnegative squared masses at one loop for all moduli in the NN, DD and untwisted closed-string sectors, and that they possess moduli fields in the ND+DN sectors. Given the result of the previous section, we are going to see that the latter are non-tachyonic.

NN, DD and closed-string sector moduli masses at one loop
Before describing the two brane configurations of interest, let us review the stability conditions established in Ref. [32] for all moduli fields that are not in the ND+DN sector.

Moduli in the NN and DD open-string sectors:
The If perturbatively all choices of coefficients d ii and n ii are allowed, it turns out that they must satisfy constraints, whose origin is six-dimensional, in order to guaranty the consistency of the model at the non-perturbative level [5]. Before stating these conditions, we need to Notice that (R,R) characterizes disconnected components of the moduli space. Indeed, when the model is decompactified to six dimensions, R is the number of rigid pairs of D5-branes in T 4 /Z 2 , whileR counts the pairs of D5-branes T-dual to the D9-branes that are rigid inT 4 /Z 2 . There is no gauge-theory phase transition in six dimensions that can describe a variation of (R,R). The components that are fully consistent non-perturbatively have R andR equal to 0, 8 or 16 [5]. 24 Moreover, when R (orR) is 8, the rigid pairs of D5-branes (D5-branes T-dual to the D9-branes) must be located on the 8 fixed points of one 22 We will see in a second step that some of the 16 + 16 positions inT 2 of the D3-branes T-dual to the D5or D9-branes have a tree-level mass proportional to the open-string coupling. 23 This number is not reached when the Wilson-line backgrounds on the worldvolumes of the D5-branes lead to some D i = 0 modulo 4 with some D ii = 2 modulo 4. Physically, this corresponds to "eating" moduli fields when the gauge group is Higgsed in its Coulomb branch. 24 In four dimensions, this results in constraints on i d ii and i n ii .
In the language of D3-branes, the mass-squared terms of the positions around the fixed points ii have been derived in [32]  Bose-Fermi degeneracy at the massless level, the tadpole of M 3/2 vanishes and the latter is an extra flat direction. In supergravity language, the spontaneous supersymmetry breaking scale is known as the "no-scale modulus," which parametrizes a flat direction of the classical potential in Minkowski space [65]. Hence, in the particular string setups satisfying n F − n B = 0, the no-scale structures valid in the classical backgrounds are preserved at one loop, up to exponentially suppressed terms. For this reason, they are designated as "super no-scale models" [36,37,39].
As explained in Ref. [32], the heterotic/type I duality can be used to show that the dependency of the one-loop effective potential on the RR two-form components C I J and C IJ appears only in the exponentially suppressed terms (even when the D3-branes are located in the bulk of the internal space). Hence, the expression (1.1) of critical points of V 1-loop remains valid when C I J and C IJ are switched on. In other words, these moduli parametrize flat directions (up to the suppressed terms).

Configurations with non-tachyonic ND+DN-sector moduli
A complete computer scan of the 32 + 32 D3-brane distributions on O3-planes allowed in the non-perturbatively consistent components of the moduli space was performed in Ref. [32].
Six tachyon-free configurations with n F −n B ≥ 0 were found, while two more deserved further investigation due to the existence of moduli fields in their ND+DN sectors. Let us analyze them.  (5) when the Wilson-line background on T 2 is switched on. The anomaly-free gauge group in four dimensions is thus   As follows from the mass-term coefficients (7.5), all D3-brane postions along T 4 /Z 2 and T 4 /Z 2 turn out to be rigid or massive. By taking into account the Green-Schwarz mechanism which stabilizes automatically 14 linear combinations of continuous locations alongT 2 , the mass-matrix of the 18 remaining positions can be found and leads to the conclusion that they are all massless except one which is massive. As explained in the previous subsection, all untwisted closed-string moduli are flat directions, including M 3/2 thanks to the vanishing of n F − n B . The moduli in the ND+DN sectors are realized as strings "stretched" between the stack of 2 D3-branes T-dual to D9-branes located on the left side of Fig. 5a and any of the four stacks of 8 D3-branes T-dual to D5-branes. Indeed, they share the same coordinates inT 2 . In all four cases, we have in the notations of Eqs. (6.51) and (6.52, Finally, the counting of the classically massless degrees of freedom can be done as in the brane configuration of Fig. 5a and leads to n F − n B = 32.

Conclusions
In this work, we have calculated the quadratic mass terms of the moduli fields arising in the ND+DN sector of the type IIB orientifold model compactified on T 2 × T 4 /Z 2 , when N = 2 supersymmetry is spontaneously broken via the Scherk-Schwarz mechanism implemented along one direction of T 2 . Assuming the string coupling is weak, this is done at one loop by computing the two-point functions of "boundary-changing vertex operators" inserted on the boundaries of the annulus and Möbius strip. The main difficulties of the derivation to which we have paid particular attention are the following: • Using the stress-tensor method, the correlators of ground-state boundary-changing fields and spin fields are found up to "integration constants," which are functions of the Teichmüller parameters of the double-cover tori. This leads to an ambiguity in the full amplitude of interest, which is lifted by taking the limit where the insertion points of the vertices coalesce. Indeed, the expression reduces in this case to contributions of the partition function that arise from states with specific Chan-Paton indices only.
This very fact makes this step of the computation more involved than its counterpart for closed-string amplitudes of twisted-sector states for which this identification is made with the entire partition function.
• The two-point function can be split into two parts referred to as "external" and "internal." The former, which is dressed by a kinematic factor, can be used to derive the oneloop correction to the Kähler metric and involves only correlation functions of "groundstate boundary-changing fields." By contrast, the internal part which captures the mass correction also requires correlators of "excited boundary-changing fields". These extra ingredients contain two contributions: 27 One arises from periodicity properties of the orbifold-background coordinates, and the other from pure local monodromy effects.
Although the latter have often been neglected in favor of the former in the literature, both turn out to be of equal order of magnitude, as we have shown explicitly.
The squared masses of all moduli fields have been derived at one loop and up to contributions that are suppressed 28 when M 3/2 is lower than the other scales of the background.
When the results are strictly positive, the corresponding scalars can be stabilized dynamically during the cosmological evolution of the universe [38,[66][67][68][69][70][71]  to exponentially suppressed corrections) seems to be a more severe difficulty. However, heterotic/type I duality can be used to show that non-perturbative contributions of D1-branes, 27 Denoted as (1) and (2) in the correlators τ u τ u qu = τ u τ u qu . 28 They can be exponentially or power-like suppressed.
which are captured by a one-loop computation of the effective potential on the heterotic side, can stabilize some of these moduli [38,[72][73][74]. 29 However, for large Scherk-Schwarz direction i.e. small M 3/2 ≡ M s √ G 55 /2 compared to M s , this mechanism is ineffective for the components (G + C) 5J and (G + C) 5J (which include the degree of freedom of M 3/2 itself) for which extra physics should be invoked to yield their stabilization. that gives the predecessor in this list, In this derivation, we have used the fact that the matrix W j is diagonal and that its components αα for α ∈ H ii are e 4iπ a i · a j . Our assumption on the convergence of ε being consistent, we have shown the existence of solutions U 1 with the above behavior.