Type IIB Flux Vacua and Tadpole Cancellation

We consider flux vacua for type IIB orientifold compactifications and study their interplay with the tadpole‐cancellation condition. As a concrete example we focus on T6/Z2×Z2 , for which we find that solutions to the F‐term equations at weak coupling, large complex structure and large volume require large flux contributions. Such contributions are however strongly disfavored by the tadpole‐cancellation condition. We furthermore find that solutions which stabilize moduli in this perturbatively‐controlled regime are only a very small fraction of all solutions, and that the space of solutions is not homogenous but shows characteristic void structures and vacua concentrated on submanifolds.


Introduction
String theory is argued to be a consistent theory of quantum gravity including gauge interactions. It is defined in ten space-time dimensions, and in order to make a connection to four-dimensional physics six spatial dimensions have to be compactified. Such compactifications have to satisfy a number of consistency conditions, for instance, in the presence of D-branes and closed-string fluxes the tadpole-cancellation condition and the Freed-Witten anomaly-cancellation condition have to be satisfied. (For a review of the former see [1] and for the latter see [2].) These conditions relate the closed-and open-string sectors to each other and put strong restrictions on the allowed background configurations. More concretely, 1) compactifications of string theory to four dimensions are typically performed on Calabi-Yau three-folds. The resulting effective theory contains a number of massless scalar fields corresponding to deformations of the background, which should be absent due to experimental constraints. A way to achieve this for type II theories is to deform the background geometry

DOI: 10.1002/prop.201900065
by Neveu-Schwarz-Neveu-Schwarz (NS-NS) and Ramond-Ramond (R-R) fluxes, which generate a potential in the four-dimensional theory and provide mass-terms for the moduli fields. This procedure is called moduli stabilization, and for early work in the type IIB context see for instance [3][4][5] and for work in the type IIA context see [6][7][8]. However, especially in the type IIB setting some of the moduli cannot be stabilized by (geometric) fluxes. One therefore includes non-perturbative effects which lead to the KKLT [9] and large-volume [10] scenarios. Moduli stabilization often results in anti-de-Sitter or Minkowski vacua, while it is difficult to obtain de-Sitter solutions. [11] 2) A gauge-theory sector for type II theories can be engineered using D-branes. D-branes filling four-dimensional space-time and wrapping submanifolds in the compact space have a gauge theory localized on their world-volume. Chiral matter can be localized at the intersection loci of different D-branes in the compact space, and in this way four-dimensional gauge theories can be constructed in a geometric way (for a review see for instance [1] ). However, when introducing D-branes one typically has to perform an orientifold projection of the theory. The fixed-loci of this projection correspond to orientifold planes which generically have negative mass and negative charge.
Moduli stabilization and the construction of a gauge-theory sector are two important aspects of connecting string theory to realistic four-dimensional physics. These tasks are often approached independently, however, as emphasized for instance in [12], there is a complicated interplay between them. This interplay can prevent moduli from being stabilized or can modify the stabilization procedure. Following this line of thought, the purpose of this paper is to study how parts 1) and 2) are connected via the tadpolecancellation condition (schematically) fluxes = D-branes + O-planes. (1.1) We approach this question by analyzing how properties of the space of flux vacua depends on the contribution of fluxes to the left-hand side of (1.1). We perform our analysis for the type IIB orientifold of T 6 /Z 2 × Z 2 , and consider R-R three-form flux, geometric NS-NS H-flux as well as non-geometric NS-NS Q-flux. The geometric fluxes generically stabilize the complex-structure moduli and the axio-dilaton, while the non-geometric fluxes allow for stabilization of the Kähler moduli. We then determine distributions for how the values of the stabilized moduli depend www.advancedsciencenews.com www.fp-journal.org on the tadpole contribution of the fluxes. Note that distributions of flux vacua have been discussed extensively in the literature before. For instance, for type IIB compactifications various aspects have been studied in [13][14][15][16][17][18][19][20][21][22][23][24] and for type IIA related work can be found in [8,25]. In the context of M-theory similar questions have been discussed in [26], and for F-theory see [27]. Recently also topological data analysis has been used to investigate properties of flux vacua in [28,29]. The main results of our analysis can be summarized as follows: r We observe that the space of flux vacua is not homogenous but shows characteristic structures such as circular voids. [16] We show furthermore that solutions can be accumulated on submanifolds in the space of solutions.
r We find that flux configurations which stabilize moduli in a weak-coupling, large complex-structure and/or large-volume regime are only a very small fraction of all possible configurations. The number of reliable flux vacua is therefore much smaller than naively expected.
r In order to stabilize moduli in a perturbatively-controlled regime at weak coupling, large complex structure and large volume, the flux contribution to the left-hand side of tadpolecancellation condition (1.1) has to be larger than a certain threshold. The more reliable these vacua are required to be, the larger this threshold has to be. However, the contribution of D-branes and orientifold planes to the right-hand side of (1.1) is typically small. It is therefore difficult to perform moduli stabilization in a perturbatively-controlled regime and to satisfy the tadpole-cancellation condition.
Our findings for the structure of the space of flux vacua agree with for instance [16,29] for the axio-dilaton, but we extend their analysis by including the complex-structure moduli. Our observation concerning the difficulty of obtaining reliable flux vacua is consistent with for instance, [30] who find that type IIB solutions at weak string-coupling are rare. Similarly, in [31] the authors argue that in order to avoid a certain runaway behaviour large fluxes have to be considered, also in [32] large fluxes are needed to obtain reliable solutions, and related difficulties are encountered in [33].
This work is organized as follows: in section 2 we review type IIB orientifold compactifications with geometric and non-geometric fluxes, we discuss the corresponding tadpolecancellation conditions, we specialize to the example of the T 6 /Z 2 × Z 2 orientifold and determine the relevant dualities. In section 3 we study moduli stabilization for the axio-dilaton, in section 4 we discuss the combined moduli stabilization of the axio-dilaton and the complex-structure moduli, and in section 5 we stabilize all of the closed-string moduli at tree-level. At the end of sections 3, 4 and 5 we have included brief summaries for each section, which may help the reader to get an overview of the main results. Our conclusions can be found in section 6.

Flux Compactifications
In this paper we are interested in compactifications of type IIB string theory on Calabi-Yau orientifolds with geometric and non-geometric fluxes. In order to fix our notation, we start in sections 2.1 and 2.2 by briefly reviewing orientifold compactifications and tadpole-cancellation conditions for general Calabi-Yau three-folds. In section 2.3 we specialize to the example of T 6 /Z 2 × Z 2 , and in section 2.4 we discuss duality transformations for this background.

Orientifold Compactifications
Type IIB orientifold compactifications on Calabi-Yau three-folds give rise to a N = 1 supergravity theory in four dimensions. This theory can be characterized in terms of a superpotential, Kähler potential and D-term potential, which we determine in the following.

Calabi-Yau Orientifolds
We begin with type IIB string theory on R 3,1 × X , where X denotes a Calabi-Yau three-fold. The latter comes with a holomorphic three-form ∈ H 3,0 (X ) and a real Kähler form J ∈ H 1,1 (X ), and to perform an orientifold projection we impose a holomorphic involution σ on X which acts on and J as The involution leaves the non-compact four-dimensional part invariant, and hence the fixed loci of σ correspond to orientifold three-and seven planes. For the full orientifold projection σ is combined with P (−1) FL , where P denotes the world-sheet parity operator and F L is the left-moving fermion number. The latter two act on the bosonic fields in the following way However, more general cases can be considered as well. For the third de-Rham cohomology group H 3 − (X ) we choose a symplectic basis as follows with all other pairings vanishing. For the even Dolbeault cohomology groups we introduce bases where |g X | denotes the determinant of the metric for X . These bases can be chosen such that they satisfy a condition analogous to (2.4), in particular, we can require www.advancedsciencenews.com www.fp-journal.org

Moduli
When compactifying string theory from ten to four dimensions, the deformations of the six-dimensional background become dynamical fields in the four-dimensional theory. These moduli fields are contained in the multiforms [35] where the sum over all R-R potentials C = p C p has been separated into a flux contribution and a moduli contribution as C = C flux + C mod . Complex scalar fields τ and T A can be determined by expanding + c into its zero-and four-form components as follows where T 0 ≡ τ is called the axio-dilaton and T A contain the Kähler moduli of X . In general this expression also contains two-forms anti-invariant under σ , which are however vanishing due to our choice (2.3). The complex-structure moduli U i with i = 1, . . . , h 2,1 − are contained in the holomorphic three-form .

Fluxes
We furthermore consider non-vanishing fluxes for the internal space. These are the R-R three-form flux F = d C flux | 3 as well as geometric and non-geometric fluxes in the NS-NS sector. The latter are the three-form flux H, the geometric flux f , and the nongeometric Q-and R-fluxes. These fluxes can be interpreted as operators acting on the differential forms as and can be conveniently summarized using a generalized derivative operator [36] 10) The precise action of the fluxes on the cohomology will be specified below. We furthermore summarize the action of the combined world-sheet parity and left-moving fermion number on the fluxes as follows [36,37] For our assumption (2.3) this implies that f and R are vanishing. We also note that the R-R and NS-NS three-form fluxes have to satisfy quantization conditions of the form (see e.g. [38]) where ∈ H 3 (X , Z) is an arbitrary three-cycle on the Calabi-Yau three-fold X . For orbifolds and orientifolds this condition can be modified, and we come back to this point on page 16 below. Furthermore, as will be explained in section 2.4, the NS-NS fluxes are related among each other through T-duality transformations, and hence also the geometric f -and the non-geometric Q-and R-fluxes should be appropriately quantized.

Supergravity Data
When compactifying type IIB string theory on orientifolds of Calabi-Yau three-folds, the resulting four-dimensional effective theory can be described in terms of N = 1 supergravity. [34] In particular, the Kähler potential takes the following form whereV denotes the volume of the Calabi-Yau manifold in Einstein frame. The superpotential is generated by the fluxes and can be expressed using the Mukai pairing ·, · of the multiforms (2.7) and the generalized derivative (2.10) in the following way [23,36,39] (2.14) In general, the fluxes (2.9) also generate a D-term potential which can be expressed using the three-form part of D(Im + ) [37] (see also [40]). However, due to (2.11) the latter belongs to the σ -even third cohomology and vanishes when taking into account our requirements (2.3). In our setting therefore no D-term potential is generated.

Bianchi Identities and Tadpole-Cancellation Conditions
Finally, the R-R and NS-NS fluxes have to satisfy a number of Bianchi identities. These can be expressed using the generalized derivative D as follows where NS-NS sources stand for NS5-branes, Kaluza-Klein monopoles or non-geometric 5 2 2 -branes (see for instance [41] for a review and collection of references). However, in this work we assume these to be absent and therefore require D 2 = 0. The R-R sources stand for orientifold planes and D-branes, and the second condition in (2.15) is also known as the tadpole cancellation condition. We discuss this condition in more detail in the following section.

Tadpole-Cancellation Condition
The tadpole-cancellation condition is an important consistency conditions for type I string theories. It links the closed-string to the open-string sector and puts strong constraints on the allowed D-brane configurations (for a review see for instance [1] ). From a conformal-field-theory point of view the tadpole-cancellation condition ensures the absence of UV divergencies in one-loop amplitudes (see e.g. [42,43] for textbook reviews) and therefore plays an important role for string theory being a consistent theory of gravity. From an effective-field-theory point of view, the tadpole-cancellation condition is the integrated version of the equation of motion for the R-R potentials and ensures the absence of certain anomalies in type II orientifold compactifications via the generalized Green-Schwarz mechanism. [44] The tadpolecancellation condition is thus an important consistency condition for string compactifications.

Explicit Expressions
We now formulate the tadpole-cancellation condition for the setting of the previous section. The contribution of the R-R-sources can be described using the charges [45,46]  The open-string gauge flux on the D-branes F appears in the Chern character, the tangential and normal part of the curvature two-form R appear in theÂ-genus and the Hirzebruch polynomial L, and Q p = −2 p−4 denotes the charge of an orientifold p-plane. For more details we refer for instance to section 8.6 in [41]. Denoting the orientifold image of a D p-brane with a prime, the Bianchi identity for the R-R fluxes then reads where the sum is over all D-branes and orientifold planes present in the background. The Freed-Witten anomaly-cancellation condition [2] for D-branes takes the general form [41,47] DQ D p = 0, where we included the corresponding expression for an orientifold p-plane. Equation (2.17) can therefore be interpreted as a relation in D-cohomology.
For the setting discussed in this paper, the orientifold projection satisfies (2.1) and therefore leads to spacetime filling O3and O7-planes. Taking into account (2.3) and that F in (2.17) is a three-form flux, we find the following explicit expressions Here, N D3 and N O3 denote the number of D3-branes and O3planes, N D7a denotes the number of coincident D7-branes in a stack a, [ D7a ] = n A a ω A is the Poincaré dual of the cycle D7a expanded in the basis (2.5), F a is the (quantized) two-form gauge flux on the stack of D7-branes a in the fundamental representation, and χ ( ) denotes the Euler number of the cycle . The D-brane sums are over all D7-branes, and due to h 1,1 − = 0 the orientifold images give a factor of two. For details on the derivation of these expressions see for instance. [48] Orientifold Contributions Let us now discuss the contribution of orientifold planes to the right-hand sides in (2.19). Typically orientifold planes give a positive contribution while D-branes give a negative contribution. For some classes of models the orientifold-contributions can be estimated as follows: r For orientifolds of T 6 /Z M or T 6 /Z M × Z N the numbers of O3and O7-planes have been computed for instance in [49] for some examples. Here, the authors find that N O3 , N O7 60 and the contribution of the Euler numbers to (2.19) are vanishing. The contribution of orientifold planes to the right-hand sides of (2.19) is therefore typically positive and of order O(10).
r For del-Pezzo surfaces the possible orientifold projections have been classified in [50]. The number of orientifold threeand seven-planes are of order O (10), and in some examples the Euler numbers of the four-cycles are of order O(100). Also here, the contribution of orientifold planes to the right-hand side of (2.19) is positive and of order O(10).
r In F-theory the geometry of Calabi-Yau four-folds Y encodes the geometry of D7-branes and orientifold planes in Calabi-Yau three-folds X . If a lift from type IIB orientifolds to F-theory is possible, one finds that

D-Brane Contributions
We furthermore note that the tadpole-cancellation conditions (2.19) are the integrated versions of the R-R Bianchi identities (2.15). The former are therefore less restrictive than the latter, but for a proper string-theory solution also the Bianchi identities with localized sources have to be solved. When placing Dbranes directly on top of orientifold planes solutions may be constructed more easily, but in general this a difficult task (see for instance [54] ). However, we can make the following general argument: because D-branes have a non-vanishing mass, their probe approximation breaks down when too many D-branes are placed into a compact space (away from the orientifold planes). In this case the back-reaction of D-branes on the geometry has to be taken into account, and an extreme case for this mechanism is the formation of black holes. It would be desirable to make this more precise, but we can argue that for ignoring back-reaction effects the contribution of D-branes to the right-hand sides in (2.19) should not be arbitrarily large.

Flux Contributions
Turning now to the flux contribution on the left-hand sides in (2.19), we note that for vanishing Q-flux the H ∧ F -term typically has to be positive in order to obtain physically-relevant solutions. Since the right-hand sides are bounded from above by the orientifold contributions, the flux contributions should not be larger than O (10) to O(10 5 ). In the presence of non-geometric Q-flux the left-hand sides in (2.19) can be negative -but since also the D-brane contributions are bounded, again the flux contributions should not be too large. This is an important point for our approach in this paper, which we summarize as In order to solve the tadpole-cancellation condition (2.19) and ignore the back-reaction of D-branes, the contribution of fluxes to the left-hand sides in (2.19) should not be too large. Depending on the setting, known bounds are of orders O (10) to O(10 5 ).

T 6 /Z 2 × Z 2 Orientifold
Let us now turn to a specific example for a compactification space. We consider the orbifold T 6 /Z 2 × Z 2 which provides a simple example of a Calabi-Yau three-fold with only few moduli. For our purposes it is sufficient to stay in the orbifold limit and not blow-up the fixed-point singularities, that is we ignore the twisted sectors. This model has been extensively studied in the literature, and we refer for instance to [55][56][57][58] for more details in the present context.
For this model the contribution of orientifold planes to the tadpole-cancellation condition (2.19) only allows for a small num-ber of different flux choices. In order to be able to study general properties of the space of solutions, in the following we therefore ignore the precise form of the tadpole cancellation condition and allow for arbitrarily-large values of H ∧ F and Q • F . We do however keep in mind that these tadpole contributions are bounded by the D-brane and orientifold contributions.

Compactification Space
We start from the following six-dimensional orbifold construction which has the properties of a Calabi-Yau three-fold On each of the two-tori we introduce complex coordinates as where x i and y i denote real coordinates with identifications x i ∼ x i + 1 and y i ∼ y i + 1, U i denote the complex structures on each of the T 2 , and no summation is performed in (2.23). The orbifold action is given by where and are the two generators of the orbifold group Z 2 × Z 2 . In addition, we perform the following orientifold projection σ :

Cohomology
Next, we turn to the cohomology of (2.22). We note that there are no one-or five-forms invariant under the orbifold action (2.24), and that the invariant three-forms are given by the following combinations Choosing the orientation of the six-dimensional space (2.22) such that we have d x 1 ∧ d x 2 ∧ d x 3 ∧ dy 1 ∧ dy 2 ∧ dy 3 = 1, the three-forms in (2.26) satisfy the intersection relation (2.4). We can furthermore define a holomorphic three-form Turning to the orientifold action (2.25), we see that all three-forms (2.26) are odd under σ and therefore h 2,1 − = 3 and h 2,1 + = 0. We also note that is odd under the orientifold action as required by (2.1).
For the even cohomology we observe that the zero-and sixform cohomologies are even under the orbifold action (2.24). For the second cohomology we find the following invariant (1,1)forms with no summation over A, and we define invariant (2,2)-forms as Note that these satisfy the relations shown in (2.6). For the orbifold (2.22) we can now define a real Kähler form in the following way where the t A are the (real) Kähler moduli. The forms (2.29) and (2.30) are all even under the orientifold projection (2.25) and therefore h 1,1 + = 3 and h 1,1 − = 0. We also note that J is even under σ , in agreement with (2.1).

Moduli
With the explicit expressions for the even cohomologies discussed above, we can now determine the moduli fields contained in + c via Equation (2.8). For the R-R zero-and four-form potentials (purely in the internal space) we use the following conventions and evaluating (2.8) in the present situation leads to with the Einstein-frame Kähler moduli defined as t We also note that the R-R two-form potential C 2 is odd under the combined world-sheet parity and left-moving fermion number (cf. (2.2)) and should therefore be expanded in the σ -odd (1,1)-cohomology, which however vanishes. Finally, the complex-structure moduli U i are contained in as can be seen from (2.28).

Fluxes
Let us now turn to the fluxes. Using the basis of three-forms (2.26), the NS-NS and R-R three-form fluxes can be expanded in the following way where I = 0, . . . , 3. The expansion coefficients f I , f I , h I , h I are quantized due to the flux quantization conditions for F and H shown in (2.12). For the remaining fluxes in the NS-NS sector we note that due to (2.3) and (2.11), the geometric f -and the nongeometric R-flux vanish. The Q-flux is in general non-vanishing, and we specify it by its action on the third and fourth cohomology as Here we have again A = 1, 2, 3 and the flux quanta are integers. In order to shorten the notation for our subsequent discussion, we combine the H-flux with the Q-flux by defining Let us briefly discuss a subtlety concerning the flux quantization condition (2.12). It was first pointed out in [59] that on orbifold (or orientifold) spaces besides bulk cycles inherited from the covering space, twisted cycles of shorter length can exist. This implies that the quantization condition of the fluxes shown above is slightly modified. For the present example of the type IIB T 6 /Z 2 × Z 2 orientifold this observation has been mentioned in [23,60] and has been analyzed in detail for instance in [61,62]. More concretely, for Z 2 × Z 2 orbifold actions with and without discrete torsion (see [63]) one finds that fluxes on generic bulk cycles have to satisfy As mentioned at the beginning of this subsection, in this paper we ignore the twisted sector which effectively implies that we consider models without discrete torsion. [61] Fluxes will therefore be quantized in multiples of eight. In the literature similar orientifolds have been studied, [13,16,20,23,60] although with slightly different quantization conditions.

Bianchi Identities
Turning to the Bianchi identities (2.15), we recall from (2.5) that the collective basis for the even cohomology is denoted by ω A and from (2.34)  of the Bianchi identities we then introduce the following general notation where A, B = 0, . . . , 3. Note that these expressions can be combined into an anti-symmetric five-by-five matrix of the form The right-hand side of the Bianchi identities (2.15) correspond to NS-NS and R-R sources, and schematically we have the relations where in particular Q A for A = 0, . . . , 3 are the contributions to the R-R tadpole cancellation conditions (2.19). As mentioned above, in this paper we do not consider NS5-branes or nongeometric 5 2 2 -branes which leads to the requirement Q AB = 0 for A, B = 0, . . . , 3.

Supergravity Data
Let us finally determine the Kähler and superpotential for the T 6 /Z 2 × Z 2 orientifold compactification. Evaluating (2.13) we find for the Kähler potential (2.41) up to an irrelevant constant term. Turning to the superpotential (2.14), the expansions of the fluxes in (2.34) and (2.35) give rise to where a summation over A = 0, . . . , 3 and i = 1, 2, 3 is understood. For ease of notation we also defined the symmetric symbol σ i j k which has the only non-vanishing components The scalar F-term potential is determined in terms of the Kähler K and superpotential W according to where φ α collectively labels the complex scalar fields of the theory. The Kähler metric is computed from the Kähler potential as G αβ = ∂ α ∂ β K, and the covariant derivative reads We also note that due to our assumption h 2,1 + = 0 shown in (2.3), no D-term potential is generated by the fluxes.

Dualities
We now want to discuss dualities for the orientifold of T 6 /Z 2 × Z 2 introduced in the previous section. We are interested in transformations which leave the physical properties of a system invariant but which are not necessarily symmetries of the action. In particular, we note that an extremum of the F-term potential (2.44) is reached for vanishing F-terms (2.46) and in our subsequent analysis we are interested in duality transformations which map solutions of (2.46) to new solutions.

Overall Sign-Change
Let us start by noting that the F-term potential (2.44) as well as the F-term Equations (2.46) are invariant under changing the sign of all fluxes [23] f I , This Z 2 transformation maps W → −W, which indeed leaves the scalar potential (2.44), the Equations (2.46) and the tadpole contributions (2.38) invariant.
Next, we consider the group of large diffeomorphisms for each of the two-tori in (2.22). [23] For a single T 2 this group is SL(2, Z), which is generated by T -and S-transformations of the form  67,1900065 www.advancedsciencenews.com www.fp-journal.org where (g I , g I ) stands collectively for ( f I , f I ) and (q I A , q I A ), and σ i j k was defined in (2.43). Under S-transformations of the complex structure moduli, the fluxes transform as follows and similarly for U 2 and U 3 . Note that for the fluxes this is not a Z 2 but a Z 4 action, which is however reduced to Z 2 using (2.47). We also note that for a simultaneous S-transformation of all three complex-structure moduli, the transformation reads Furthermore, all Bianchi identities and tadpole contributions Q A and Q AB are invariant under these transformations.

T-Duality
We now turn to T-duality transformations. It is well-known that performing an odd number of T-dualities for type IIB string theory results in the type IIA theory and vice versa, and applying two or six T-dualities to type IIB string theory with O3-/O7-planes results in type IIB with O5-/O9-planes. For T-duality to map the present setting of type IIB with O3-/O7-planes to itself, we therefore have to perform four collective T-duality transformations.
Let us now consider more closely the T 6 /Z 2 × Z 2 orientifold with O3-/O7-planes. Using the Buscher rules, [64,65] a collective Tduality transformation [66] say along the first and second T 2 results in the following transformation of the moduli T-duality along z 1 , z 2 : (2.52) In (2.52) we have only shown how the moduli transform, but also the fluxes transform in a non-trivial way under T-duality. However, (2.52) contains an S-transform of the complex-structure moduli U 1 and U 2 . To better show the underlying structure, let us undo the U i transformation in (2.52) using (2.50). We then obtain the transformation which by a slight abuse of notation we will refer to as T-duality in the following. Similar transformations are obtained for T-duality along the second & third and first & third two-torus. 2 We furthermore observe that the F-term Equations (2.46) are invariant under a permutation of the Kähler moduli T A and fluxes (q I A , q IA ). This is just a re-labelling of indices and corresponds to the permutation group S 3 . Using now the T-duality action (2.53) together with the permutation of Kähler moduli, we see that S 3 is enhanced to S 4 acting on T A = (T 0 , T A ) and fluxes (q I A , q I A ). Indeed, for S A B ∈ S 4 the superpotential (2.42) is invariant under 54) and the flux contribution to the Bianchi identities shown in (2.38) transform as We emphasize that four collective T-duality transformations for type II orientifold compactification are permutations of moduli and fluxes. They do not correspond to transformations which invert T A .

S-Duality
We finally consider the SL(2, Z) duality of type IIB string theory. For vanishing Q-flux its action on the axio-dilaton and the F -and H-flux takes the following form where the parameter b A has to be quantized as b A ∈ Z. This corresponds to a gauge transformation of the R-R zero-and four-form potentials.
r For an S-transformation τ → −1/τ in the presence of nongeometric fluxes, the F-term Equations (2.46) are in general not invariant. To restore the SL(2, Z) duality the authors of [68] and T A , respectively. Under a single T-duality the R-R potentials transform as C p → C p±1 , [67] where the upper/lower sign is for a transformation transversal/longitudinal to C p . For a collective T-duality along four directions we therefore map C 0 → C 4 and some components of C 4 → C 0 . This agrees with (2.53).

Moduli Stabilization I
As a first example for moduli stabilization on the T 6 /Z 2 × Z 2 orientifold we consider a specific choice of fluxes which stabilizes the axio-dilaton τ , fixes the complex-structure moduli U i to a symmetric point but leaves the Kähler moduli T A unstabilized. This setting has been studied for instance in [13,16,20], and here we use it as a toy model for the more involved settings in the subsequent sections.

Setting
We start by specifying the superpotential (2.42). We consider a configuration with vanishing non-geometric fluxes (2.35), which implies that the Kähler moduli T A do not appear in the potential and hence are not stabilized. The remaining R-R and NS-NS fluxes (2.34) are chosen as follows whereh 0 andh 0 should not be zero simultaneously. Since the superpotential W is independent of the Kähler moduli the F-term Equations (2.46) take the simple form Ignoring unphysical values for U i and τ with negative or vanishing imaginary part, we obtain the following solution to (3.3) The fluxes in (3.2) are not arbitrary but are subject to the Bianchi identities (2.38). Since all non-geometric Q-fluxes vanish, the only nontrivial condition is the D3-tadpole contribution where the requirement of Q 0 being positive is related to having Im τ > 0. Note that due to the quantization condition for the fluxes in (3.2), Q 0 is a multiple of 768.

Finite Number of Solutions for Fixed Q 0
Next, we briefly review the arguments of [16,20] showing that the number of physically-distinct solutions (3.4) is finite for finite Q 0 . We restrict the values of the axio-dilaton τ to the fundamental domain of the corresponding SL(2, Z) duality (2.56) For ease of notation we express the axio-dilaton in terms of its real and imaginary part as τ = τ 1 + iτ 2 , and for the solution (3.4) to the F-term equations we have For a fixed positive value of Q 0 the imaginary part of τ is bounded from above as τ 2 ≤ Q 0 768 , sinceh 0 andh 0 are integer multiples of eight which cannot be zero simultaneously. We now argue along the following lines: r The tadpole contribution Q 0 is invariant under the SL(2, Z)

transformations (2.56). Using then a T -transformations acting on the axio-dilaton as
This T -transformation is a duality transformation, and therefore we have the equivalencẽ r Next, using an S-transformation τ → −1/τ (possibly together with additional T -transformations) we can bring τ into the fundamental domain F τ . In F τ a lower bound for the imaginary part τ 2 is obtained by considering τ 1 = − 1 2 for which τ 2 ≥ √ 3/2. Using (3.7) we then find which leaves only finitely many possibilities for the integers h 0 ,h 0 . Together with (3.9), this implies also a finite number of choices forf 0 .
r The remaining fluxf 0 is now determined via the tadpole contribution Q 0 shown in (3.5).
In summary, for a fixed positive value of the D3-tadpole contribution Q 0 , the F-term Equations (3.3) have only a finite number of of physically-distinct solutions for τ .

Space of Solutions
In [16,20] it was shown that the solutions (3.4) mapped to the fundamental domain for τ are not distributed homogeneously. In particular, the space of solutions contains voids with large degeneracies in their centers. In this section we review these findings and provide some new results on the dependence of these distributions on the D3-tadpole contribution Q 0 . Our data has been obtained using a computer algorithm to generate all physicallydistinct flux vacua for a given upper bound on the D3-brane tadpole contribution.

Distribution of Solutions
As we argued above, for a fixed value of Q 0 , the number of physically inequivalent solutions for the axio-dilaton τ is finite. Using the SL(2, Z) duality (2.56) we can map these solutions to the fundamental domain (3.6), and we have shown the corresponding space of solutions in Figure 1  r For Figure 1 we have included all flux configurations for which the tadpole contribution satisfies 0 < Q 0 768 ≤ 300, and in order to have a symmetric plot we have added points on the boundary of the fundamental domain at τ 1 = + 1 2 . We see that the space of solutions for (3.4) is bounded as τ 2 ≤ 300, and that solutions are located on lines with fixed τ 1 .
r In Figure 2 we show a zoom of Figure 1 for a small range of τ 2 . Here we see a characteristic structure of voids [16] with accumulated points in their centers. The large voids are encircled by smaller ones, and we note that the higher density of points near |τ | 2 = 1 is not a physical property as we have not taken into account the metric on moduli space.
Let us next note that the moduli space of the axio-dilaton τ is hyperbolic. Indeed from the Kähler potential (2.41) we can derive the corresponding Kähler metric with components Fortschr. Phys. 2019, 67, 1900065 www.advancedsciencenews.com

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A convenient way to visualize this hyperbolic space is by mapping the Poincaré half-plane to the Poincaré disk via the conformal transformation (3.12) The space of solutions for the axio-dilaton mapped to the Poincaré disk is then shown in Figure 3, which is the mapping of Figure 1 under (3.12).
r In Figure 3 the characteristic structure of voids is visible. In this plot effects of the moduli-space metric are incorporated.

Analysis of Voids
The number of physically-distinct solutions for the axio-dilaton is finite for fixed tadpole-contribution Q 0 . The number of solutions N with Q 0 ≤ Q 0 max can be determined numerically, which leads to the following scaling behaviour for large Q 0 max . We now want to study how the voids change depending on N, or, equivalently, depending on Q 0 max . In particular, we are interested how the size of the voids depends on Q 0 max . Qualitatively, this behaviour is illustrated in Figure 4: r In Figure 4 the space of solutions for the axio-dilaton around the point τ = 2i is shown. The blue points correspond to solutions which satisfy Q 0 768 ≤ 300, and the red points correspond to solutions with Q 0 768 ≤ 3000. For larger Q 0 max the void around τ = 2i therefore becomes smaller, and finer void structures appear. These results are in agreement with the topological data analysis in [29].
Let us denote the origin of a void by τ void , and define its size by the distance to the nearest solution τ sol (not located at τ void ). The geodesic distance d is measured using the metric (3.11) on the axio-dilaton moduli space, for which we have (3.14) As we can see for instance from Figure 3, in the proper distance the voids can be approximated by a circle whose radius we define as The scaling behaviour of R void with Q 0 max has been obtained for instance in [16,20] as R 2 void ∼ 1/Q 0 max , and below we have determined the prefactors for some families of voids numerically. For voids located in the fundamental domain on the Poincaré plane we have the following relation between the radius of the void R void , the tadpole contribution Q 0 max and the number of solutions located at the center of the void n void (3.17)

Solutions at Small Coupling
The imaginary part of the axio-dilaton τ is bounded from above by the D3-tadpole contribution Q 0 , which via (2.33) implies a restriction on the string coupling g s as Recall that in our conventions Q 0 is a multiple of 768. In the following we determine the number of physically-distinct solutions N c which satisfy τ 2 ≥ c for some cutoff c > 0 so that we have Note that in order to ignore string-loop corrections and corrections from world-sheet instantons, we need to stabilize the axiodilaton at small g s . This implies that Q 0 and c should be sufficiently large. Using then the exact data for the space of solutions, we can obtain fits for N c for values of Q 0 max of the order  Figure 5. We see that for a particular c in Fortschr. Phys. 2019, 67, 1900065 www.advancedsciencenews.com www.fp-journal.org  g s ≤ 1/c, the D3-tadpole contributions Q 0 768 has to be larger than some threshold. Furthermore, above this threshold the number of solutions is not large but only O(10).

Summary
Let us summarize the results and observations of this section for moduli stabilization of the axio-dilaton with the choice of fluxes given in Equation (3.4): r As already known before, for a fixed D3-brane tadpole contribution Q 0 , the number of physically-distinct solutions to the F-term equations for the axio-dilaton τ is finite due to the corresponding SL(2, Z) duality. [16,20] r The solutions for the axio-dilaton in the fundamental domain are not distributed homogeneously, but show characteristic void structures as illustrated in Figures 2 and 3. r With increasing upper bound Q 0 max on the tadpole contribution, the area of these voids shrinks and the number of solutions located at the center n void increases as shown in (3.16). The precise behaviour for the radius R void and n void is proportional to a constant depending on the location of the void, however, the ratio n void /2π R 2 void is universal.
r The string coupling is bounded from below by the tadpole contribution as 768 Q 0 ≤ g s . In order to ignore string corrections and trust the solutions (3.4), we have to demand g s 1 which implies Q 0 768 1. This is in contrast to our discussion of the tadpole-cancellation condition on page 12 which requires Q 0 768 to be small, and illustrates the difficulty of obtaining reliable solutions to the F-term equations.
r We have furthermore analyzed the number of physicallydistinct solutions satisfying g s ≤ 1/c. Requiring a small string coupling of for instance g s ≤ 1/10, we find that only about 10% of the solutions satisfy this condition. If we require g s to be smaller, then the corresponding fraction of solutions is smaller.

Moduli Stabilization II
In this section we extend our previous discussion by including the complex-structure moduli U i . We choose flux configurations which stabilize the axio-dilaton and fix the complexstructure moduli at an isotropic minimum with U 1 = U 2 = U 3 . Such vacua have previously been studied for instance in [20,60].

Setting
We start again by specifying the superpotential (2.42) and set to zero the non-geometric fluxes (2.35) and for the R-R and remaining NS-NS fluxes (2.34) we choose the following restricted setting Since the superpotential is independent of the Kähler moduli T A , the F-term Equations (2.46) simplify as in (3.3) and we obtain Due to the isotropic choice of fluxes in (4.2), the complexstructure moduli U i are stabilized such that 4) and the F-term Equations (4.3) reduce to www.advancedsciencenews.com www.fp-journal.org The R-R and NS-NS fluxes in (4.2) are furthermore subject to the Bianchi identities (2.38), and due to the vanishing Q-fluxes the only nontrivial relation is again given by the D3-brane tadpole contribution Note that due to the quantization condition for the fluxes, the tadpole contribution Q 0 is an integer multiple of 64. However, as it has been explained in footnote 10 of [60], in order to obtain physically-viable solutions Q 0 receives an additional factor of three. The tadpole contribution is therefore always a multiple of 192, which is also what we see explicitly in our data.

Finite Number of Solutions for Fixed Q 0
The two equations for the complex-structure modulus shown in (4.5) define an overdetermined cubic system for U, which in general does not allow for a solution in closed form. Since the coefficients in (4.5) are real, one can bring these equations into the form where u 0 , u 1 ,ū 1 denote the solutions. Physically-acceptable solutions have to satisfy Im U > 0, and therefore the F-term equations (4.5) have at most one solution for U of interest to us. The Equation (4.6) can be solved for the axio-dilaton as which however depends on U. More details on these solutions can be found in appendix A, where we follow the discussion of [20,60]. As reviewed in section 2.4, in the absence of nongeometric Q-fluxes the axio-dilaton and the complex-structure moduli enjoy SL(2, Z) dualities. These can be used to bring τ and U into their fundamental domains where we again split τ and U into their real and imaginary parts as τ = τ 1 + i τ 2 and U = U 1 + i U 2 . We furthermore note that the two SL(2, Z) dualities leave the D3-tadpole contribution Q 0 invariant. Now, as shown by [20,60] and reviewed in appendix A, the dualities can be used to show that the number of physicallydistinct vacua in the fundamental domain is finite for fixed Q 0 . In the following we explore how the properties of the space of solutions for τ and U depend on Q 0 .

Space of Solutions
In this section we study the space of solutions to the F-term Equations (2.46) for the combined axio-dilaton and complex-structuremodulus system. Since for the axio-dilaton system we found two-dimensional circular voids in the two-dimensional moduli space, it is natural to expect four-dimensional spherical voids in the four-dimensional moduli space. However, we can not confirm this expectation. Our data has again been obtained using a computer algorithm, which generated all physically-distinct flux vacua for a given upper bound on the D3-brane tadpole contribution Q 0 .

Distribution of Solutions
In [20,60] (as well as in appendix A) it is shown that for fixed Q 0 the number of physically-distinct solutions is finite. We have determined all solutions for the setting described in section 4.1 numerically, and have visualized them in the following figures.
r In Figure 6 we have shown the solutions for the fluxes of the form (4.2) projected onto the τ and onto the U-plane. [20] All solutions satisfy the bound on the tadpole contribution Q 0 192 ≤ 1000, and in order to have a symmetric plot we included points on the boundary of the fundamental domains. These plots are similar to the one in Figure 1. When comparing Figures 6a and 6b, we note that for the same Q 0 the maximum values for τ 2 and U 2 differ significantly. Furthermore, we note that the number of different values for τ 1 is much larger than for U 1 .
r In Figure 7 we show sections through the four-dimensional space of solutions for τ 2 ≤ 2, characterized by different values of the complex-structure modulus. All solutions satisfy Q 0 192 ≤ 1000, and these plots show void structures similar as in Figure 2. We note however that although the location of the voids stays the same when going from U = i to U = 2i and similarly from U = √ 2i to U = 2 √ 2i, the density of solutions decreases. This appears to be a general feature which we observe in the data.
r In Figure 8 we have shown three-dimensional sections of the four-dimensional space of solutions for U 1 = 0. All solutions have been mapped to the fundamental domain. Figures 8a and  8b show two different points of view, which illustrate that the three-dimensional section of the space of solutions is not homogenous. Solutions are accumulated on planes for particular values of U 2 , while the space between these planes is only sparsely populated. This is in agreement with our observations in Figure 7, which also show that the density of solutions varies. The lines in Figure 8a and 8b connect voids for different values of U 2 and are described by the following equations for t ∈ R + orange l 1 (τ 1 , τ 2 , U 1 , U 2 ) = (0, 1 + t, 0, 1 + 1 1 t), red l 2 (τ 1 , τ 2 , U 1 , U 2 ) = (0, 2 + t, 0, 1 + 1 2 t), purple l 3 (τ 1 , τ 2 , U 1 , U 2 ) = (0, 3 + t, 0, 1 + 1 3 t), green l 4 (τ 1 , τ 2 , U 1 , U 2 ) = (− 1 2 , 3 2 + t, 0, 1 + 2 3 t). (4.11) Fortschr. Phys. 2019, 67, 1900065 www.advancedsciencenews.com www.fp-journal.org r In Figure 9 we have shown the same three-dimensional section of the space of solutions as in Figure 8. The point of view in Figure 9a is along the line l 1 (orange) of (4.11) and the point of view in Figure 9b is along the line l 2 (red). In these threedimensional sections of the four-dimensional space of solutions we therefore have a cylindrical void centered around the lines in (4.11).

Solutions at Small Coupling and Large Complex Structure
We now consider the number N of physically-distinct solutions for the combined axio-dilaton and complex-structure moduli system defined in section 4.1. This number is finite for fixed D3tadpole contribution Q 0 , and since we have the numerical data we can determine this number explicitly. For large Q 0 the dependence takes the form (4.12) We next note that in the fundamental domains, the imaginary parts of the axio-dilaton and complex-structure moduli satisfy a lower bound similarly as in the previous example. An upper bound can be obtained from the numerical data, which can be expressed as 3 3 More precisely, with x = Q 0 192 the bound on U 2 can be expressed as Note that in our conventions the tadpole contribution Q 0 is a multiple of 192. However, as we have seen in (4.9), the solution for the axio-dilaton depends on the complex-structure modulus. Although this dependence is difficult to analyze analytically, the numerical data gives the following bound on the solutions This bound is stronger than in (4.13), and it implies that for fixed Q 0 the imaginary parts of τ and U cannot be made simultaneously large. In particular, in order to have solutions at small coupling g s = 1 τ 2 1 and large complex structure U 2 1, the tadpole contribution has to be sufficiently large. Let us make this more precise and determine numerically the number of solutions N c with Q 0 ≤ Q 0 max for which In the limit of large These approximations do not describe the data very well, but are sufficient for our purposes here. In particular, we see that at Fortschr. Phys. 2019, 67, 1900065 www.advancedsciencenews.com www.fp-journal.org leading order N c depends quadratically on Q 0 max and that the ratios N c /N are rather small. Thus, only a small percentage of the solutions to the F-term equations are in a perturbativelycontrolled regime. More interesting is the limit of small Q 0 192 , which we have illustrated in Figure 10. We see that for c = 2 (blue) there are solutions starting at Q 0 192 = 16. For c = 5 (orange) we find solutions starting at Q 0 192 = 100, and for c = 10 (green) solutions can be obtained starting at Q 0 192 = 400. The main conclusion we want to draw from this analysis is that for solutions at weak coupling g s 1 and large complex structure U 2 1, the D3-tadpole contribution Q 0 192 has to be large. As discussed on page 12, this is in tension with the tadpole cancellation condition.

Summary
Let us summarize the results obtained in this section for the space of solutions of the combined axio-dilaton and complexstructure-moduli system with fluxes characterized by the setting described in section 4.1: r As known before, for a fixed D3-brane tadpole contribution Q 0 the number of physically-distinct solutions to the F-term Equations (2.46) for the axio-dilaton and complex-structure moduli is finite. This is again due to the SL(2, Z) dualities for the axiodilaton and the complex-structure moduli.  of solutions. As shown in Figures 9, the solutions are accumulated on submanifolds in the four-dimensional space with few points in between. We also find void structures in the space of solutions, which are however not spherical but take a cylindrical form in three-dimensional sections (cf. Figures 9).
r The imaginary parts of the axio-dilaton and the complexstructure moduli τ 2 and U 2 are bounded from above and below as shown in Equation (4.13). In our data we find however the stronger bound on the product τ 2 U 2 ≤ 3 4 Q 0 192 , which implies that in the weak-coupling and large-complex-structure regime the tadpole contribution Q 0 192 has to be large. This is again in contrast to our arguments regarding the tadpole-cancellation condition on page 12 which requires Q 0 192 to be small, and illus-trates the tension between the closed-and open-string sectors for obtaining reliable solutions.
r We have furthermore shown that the fraction of reliable flux solutions within all solutions for fixed tadpole Q 0 192 is only of orders O(10 −3 ), which is a reduction as compared to the setting of section 3.

Moduli Stabilization III
We now generalize the setting from section 4 by including non-geometric Q-fluxes. The fluxes are restricted such that the complex-structure and Kähler moduli are fixed isotropically as T 1 = T 2 = T 3 := T and U 1 = U 2 = U 3 := U, which reduces the system to the three complex moduli fields τ , U and T . Such vacua have previously been analyzed for instance in [23].

Setting
We specify the superpotential (2.42) by imposing the following restrictions on the R-R and NS-NS fluxes (2.34) and (2.35) which leaves four independent R-R F -flux components, four independent H-flux components and six independent Q-flux components. As discussed around Equation (2.37), these fluxes are subject to the quantization conditions Together with the Kähler potential (2.41), the F-term Equations (2.46) can then be determined explicitly. Since in the present situation the superpotential W depends on the Kähler moduli T A , the condition W = 0 is in general not obtained and the resulting system of equations is more involved. However, provided that solutions to the F-term equations with non-vanishing imaginary parts exist, then for the fluxes (5.1) the moduli are stabilized isotropically A necessary condition to achieve this stabilization is that q 1 1 = q 1 1 and q 11 =q 11 . The system of seven F-term equations for τ , U i , T A then reduces to the following three equations The R-R and NS-NS fluxes are furthermore subject to the Bianchi identities (2.38), and using the restrictions (5.1) we find for the tadpole contributions , Finally, as we discussed in section 2.4, the present setting is duality invariant under SL(2, Z) transformations of the complexstructure modulus U whereas the SL(2, Z) duality (2.56) of the axio-dilaton is broken to constant shifts (2.57) due to the nonvanishing Q-flux. Furthermore, T-duality (2.54) is in general broken because of the isotropic choice of fluxes in (5.1).

Infinite Number of Solutions for Fixed Q A
In contrast to the settings of sections 3 and 4, for non-vanishing Q-flux the number of solutions for fixed tadpole contributions Q A is infinite. [23] This can be illustrated with the following example from [23]: the D7-brane tadpole contribution is fixed as Q 1 = 0 and the fluxes are chosen as follows where m, n ∈ 8Z and b ∈ 8Z is restricted such that f 0 ∈ 8Z. To obtain non-trivial solutions we require m, n, b = 0, and the above choice of fluxes always satisfies the Bianchi identities (5.5). A solution to the equations of motion (5.4a) is given by where the sign takes the same value for all three moduli. In order for the imaginary parts to be positive we require that this sign is chosen appropriately and that Q 0 > 0 and b n > 0. Note that (5.7) describes an infinite set of vacua since m, n are not bounded, which is in contrast to the situations studied in sections 3 and 4. However, in order to trust these solutions we have to require that Im τ, Im U, Im T > 1 which translates into the conditions For a fixed Q 0 there is only a finite number of choices for (m, n, b) which satisfy (5.8), and therefore the number of reliable solutions for the particular flux choice (5.6) is finite for fixed Q 0 . We remark however that duality transformations can change the form of (5.6), and therefore a similar analysis has to be performed for the transformed flux choices. We do not know whether this leads to a finite number of reliable flux solutions.

Space of Solutions
Since the number of physically-distinct solutions for fixed tadpole contributions Q A is in general infinite, for the present setting we cannot construct a complete data set of flux vacua for fixed tadpole contribution. However, we can generate flux vacua using Monte-Carlo sampling.

Data Set
Our data set of flux vacua for the setting described in section 5.1 has been obtained in the following way: r We restrict the contributions to the D3-and D7-brane tadpoles Q 0 and Q 1 shown in Equations (5.5) as Note that due to the flux-quantization condition (5.2) the Q A are always a multiple of 64, and that Q 0 and Q 1 can be negative while still leading to positive imaginary parts for τ , U, T . r For these flux contributions all moduli are fixed, however, not all of these extrema are stable. The number of vacua with all moduli fixed and without tachyonic or flat directions is 2.9 · 10 6 .

Solutions at Small Coupling, Large Complex Structure and Large Volume
Since we do not have a complete set of solutions for fixed tadpole contributions Q 0 and Q 1 , the same analysis as for the previous cases cannot be performed. However, for our data set we have determined the number of solutions N c for which τ 2 = Im τ , U 2 = Im U and T 2 = Im T satisfy For sufficiently large c, this corresponds to the weak-coupling, large-complex-structure and large-volume regime. We furthermore denote by |Q A /64| min the lowest value of the expression |Q A /64| = (Q 0 ) 2 + 3(Q 1 ) 2 /64 in the set of vacua determined by (5.11 In Table 1 we have collected some concrete examples for fully stabilized vacua with imaginary parts greater than one.

Distribution of Solutions in the Q A -Plane
For moduli stabilization of the axio-dilaton and complexstructure moduli studied in sections 3 and 4 we observed that the tadpole-contribution Q 0 has to be positive in order to obtain physical solutions with positive imaginary parts τ 2 and U 2 . However, when including non-geometric fluxes we see that positive as well as negative values of Q 0 and Q 1 can results in positive imaginary parts τ 2 , U 2 and T 2 .
Having a large data set available, we have analyzed the distributions of vacua over Q 0 and Q 1 . For the set of stable vacua without flat or tachyonic directions we obtain stable Q 0 Q 1 fraction of all vacua ≤ 0 ≤ 0 0.518765 ≤ 0 > 0 0.155262 >0 ≤ 0 0.325822 >0 > 0 0.000151 (5.14) It is somewhat surprising that the fraction of vacua with tadpole contributions Q 0 > 0 and Q 1 > 0 is suppressed by three orders of magnitude compared to having at least one Q A negative, but we have no explanation for that. For our data set of solutions which include potentially tachyonic directions no such difference is found all Q 0 Q 1 fraction of all vacua ≤ 0 ≤ 0 0.360664 ≤ 0 > 0 0.256953 >0 ≤ 0 0.254567 >0 > 0 0.127816 (5.15) We also note that both data sets do not contain any solution with Q 0 = Q 1 = 0.

Distribution of Solutions in Moduli Space
We have also analyzed the distribution of solutions to the F-term Equations (2.46) within the moduli space. Since the density of solutions is very small, we were not able to identify any patterns or structures.

Summary
We briefly summarize the main results obtained in this section for the combined moduli stabilization of the axio-dilaton, complex-structure moduli and Kähler moduli by the fluxes shown in Equation (5.1): r For fixed D3-and D7-brane tadpole contributions Q 0 and Q 1 the number of physically-distinct vacua is in general infinite. [23] We therefore were not able to generate a complete data set but used Monte-Carlo methods to randomly generate 1.3 · 10 7 solutions to the F-term equations which fix all moduli.
r We have shown that reliable solutions at weak string coupling, large complex structure and large volume are only a small fraction of all vacua. For instance, stable solutions with τ 2 , U 2 , T 2 ≥ 5 make only a fraction of 3.4 · 10 −7 of all solutions.
Requiring the solutions to be more reliable requires the tadpole contributions Q A to be larger, which is in tension with the tadpole-cancellation condition as discussed on page 12.
r In Table 1 we have shown some concrete examples of stable vacua with axio-dilaton, complex-structure moduli and Kähler moduli fixed at imaginary parts greater than one.
r Finally, we have pointed out that stable vacua with all tadpole contributions Q 0 and Q 1 positive are statistically disfavored. We do not have an explanation for this observation.

Discussion
In this work we have studied moduli stabilization with R-R and NS-NS fluxes in type IIB string theory for the example of the T 6 /Z 2 × Z 2 orientifold. We have analyzed the interplay between moduli stabilization and tadpole cancellation, in particular, we have shown how properties of the vacua depend on the flux contribution to the tadpole-cancellation condition.

Summary of Results
More concretely, the axio-dilaton and complex-structure moduli are fixed by geometric fluxes while the Kähler moduli are fixed at tree-level using non-geometric Q-flux. In section 3 we have focussed on the axio-dilaton only and mainly ignored the complex-structure and Kähler moduli. In section 4 we included the complex-structure moduli, and in section 5 we studied moduli stabilization for all closed-string moduli. We analyzed the space of solutions to the F-term equations for these settings and found that it is not homogenous: r For the axio-dilaton the space of solutions contains characteristic void structures (see Figure 2). [16,20] The radius of these voids depends on the flux contribution Q 0 to the tadpole-cancellation condition, and for larger Q 0 the radii become smaller. When including the complex-structure moduli, we observe that vacua are accumulated on submanifolds within the space of solutions (see Figure 8). On these planes we again find void structures, which are connected by lines between different planes. We therefore find cylindrical voids in (threedimensional sections of) this four-dimensional space of solutions.
Furthermore, in section 2.2 we have argued that the flux contribution to the tadpole-cancellation condition cannot be arbitrarily large. In particular, for many known examples this contribution