Open string multi-branched and Kahler potentials

We consider type II string compactifications on Calabi-Yau orientifolds with fluxes and D-branes, and analyse the F-term scalar potential that simultaneously involves closed and open string modes. In type IIA models with D6-branes such potential can be directly computed by integrating out Minkowski three-forms. The result shows a multi-branched structure along the space of lifted open string moduli, in which discrete shifts in special Lagrangian and Wilson line deformations are compensated by changes in the RR flux quanta. The same sort of discrete shift symmetries are present in the superpotential and constrain the Kahler potential. As for the latter, inclusion of open string moduli breaks the factorisation between complex structure and Kahler moduli spaces. Nevertheless, the 4d Kahler metrics display a set of interesting relations that allow to rederive the scalar potential analytically. Similar results hold for type IIB flux compactifications with D7-brane Wilson lines.


Introduction
A substantial effort in the literature of 4d string compactifications is devoted to construct models with a rich chiral gauge sector and where most neutral scalars are stabilised at a sufficiently high scale [1][2][3][4][5][6][7][8][9][10][11]. Achieving both features simultaneously is a non-trivial task, and probably the most developed models in this sense appear in type II orientifold compactifications with either O3/O7 or O6-planes. In this type II scheme the chiral gauge sector arises from space-time filling D-branes, and moduli are fixed by a potential generated by internal background fluxes and non-perturbative effects [2][3][4]. Remarkably, these two features are usually treated independently, and the presence of the D-branes is ignored when computing the F-term potential that stabilises moduli. 1 1 Except when they source non-perturbative effects or implement a vacuum energy uplifting mechanism. This picture is somewhat reversed for other classes of models at weak coupling, like heterotic/type I or type IIB compactifications with O5-planes. There the techniques to achieve full moduli stabilisation are not so well developed, but on the other hand moduli in the gravity and gauge sector are treated on equal footing. In these setups one can see that, in general, the presence of D-branes/vector bundle sectors modifies the moduli stabilisation potential [12][13][14][15][16]. Of course, all these different constructions are related to each other by dualities like mirror symmetry, which has been used to match superpotentials involving open and closed string modes in type IIA compactifications with D6-branes. However, in the cases that have so far been explored the source of open string moduli superpotential in the type IIA side is exclusively due to worldsheet instantons, which are very suppressed in the large volume limit in which RR flux potentials are valid. This somehow sustains the perception that D6-brane moduli are to be treated separately from those moduli entering the flux superpotential, and that they should only be considered at a second stage, after the effect of fluxes on the closed string sector has been taken into account. More generally, open string moduli involving D-brane Wilson lines are presumed to only develop superpotentials from either worldsheet or D-brane instantons, being insensitive to and negligible for the dynamics of the flux generated superpotential.
Contrary to this expectation, it was pointed out in [17] that D-brane Wilson line moduli can also develop superpotentials at large volume and weak coupling, comparable in magnitude to the flux generated superpotential for closed string moduli. In those circumstances, one should analyse the process of moduli stabilisation by treating open and closed string modes simultaneously, considering a scalar potential that includes both at the same time.
The purpose of this work is to consider type II compactifications where open strings modes (and in particular Wilson lines) enter the potential generated by fluxes on equal footing to the closed string ones. Such a combined description is not only necessary for consistency, but also provides valuable information regarding the N = 1 compactification data. Indeed, global knowledge of the scalar potential and the superpotential provides stringent constraints on the Kähler potential for 4d chiral fields. In our case it will allow us to elucidate how closed and open string moduli are interrelated and the presence of shift symmetries in the latter, typically a relevant aspect in building models of inflation [18][19][20].
In order to implement our approach we will seek for compactifications with two basic features, which allow to reverse engineer the Kähler potential dependence on the open string moduli. First, one should be able to compute the F-term scalar potential without prior knowledge of the Kähler or superpotential. Second, one should be able to involve all the open string moduli in the superpotential, which can also be computed independently.
We find that type IIA compactifications in Calabi-Yau orientifolds with D6-branes are particularly suitable for this purpose, as they fulfil both requirements when we restrict to tree-level potentials at large compactification volumes. Indeed, for type IIA compactifications with background fluxes the tree-level F-term potential for closed strings can be fully computed by integrating out 4d three-forms [21], and along the lines of [22,23] one may generalise this computation to include open string moduli into the potential. Moreover one may always generate an F-term for each D6-brane position and Wilson line moduli, by simply adding an internal worldvolume flux to the D6-brane.
In our analysis we find that a particularly important role is played by a series of discrete symmetries manifest both at the level of the scalar potential and superpotential. we give a simple geometrical interpretation. A similar statement holds at the level of the flux-generated potential, which displays the multi-branched structure discussed in [21] but cations for the open-closed Kähler potential. In Section 6 we use these superpotential and Kähler potential to recover the F-term scalar potential of Section 3 via 4d supergravity.
Many of these results also apply to type IIB compactifications with O3/O7-planes and D7-brane Wilson lines, as we discuss in Section 7. We draw our conclusions and directions for future work in Section 8.

Several technical details have been relegated to the appendices. Appendix A discusses
aspects of the open-closed Kähler metrics and contains the proof of several identities necessary for the computations of Section 6. Appendix B contains a direct derivation of the type IIA flux potential in the democratic formulation of 10d supergravity. Finally, Appendix C illustrates the somewhat abstract definitions used along the main text in the simple case of a T 2 × T 4 /Z 2 orientifold example.

D6-branes in type IIA orientifolds
Let us consider a type IIA orientifold compactification on R 1,3 × M 6 with M 6 a compact Calabi-Yau 3-fold. Following the standard construction in the literature [1,[5][6][7], we take the orientifold action to be given by Ω p (−1) F L R, where Ω p is the worldsheet parity reversal operator, F L is the space-time fermion number for the left-movers, and R is an internal anti-holomorphic involution of the Calabi-Yau. This involution acts on the Kähler 2-form J and the holomorphic 3-form Ω of M as The fixed locus Π O6 of R is given by one or several 3-cycles of M, in which O6-planes are located. In order to cancel the RR charge of such O6-planes one may introduce 4d space-time filling D6-branes wrapping three-cycles Π α of M 6 , 2 such that the orientifold symmetry is preserved. In the absence of NS background fluxes, RR tadpole cancellation requires that the following equation in where Π O6 stands for the O6-plane loci. Finally, for such D6-branes to minimise their energy and preserve the 4d N = 1 supersymmetry of this background they need to wrap special Lagrangian three-cycles. That is they need to satisfy the geometric conditions as well as to have vanishing worldvolume flux F, defined as with σ = l 2 s 2π and the string length given by l s = 2π √ α .
In the absence of D-brane moduli, the 4d effective action for the closed string sector of these constructions has been analysed in great detail, see for instance [22,23,25]. In particular, the moduli space of closed string deformations and its related Kähler potential 2 One may additionally consider D8-branes on coisotropic cycles as in [24], but for simplicity we will restrict our discussion to models where only D6-branes are present.
can be described as follows. On the one hand there are h 1,1 − (M 6 ) complexified Kähler moduli defined as where φ is the 10d dilaton, J is computed in the Einstein frame and l −2 s ω a are harmonic representatives of H 2 − (M 6 , Z) that can be defined as ω a = dImJ c /dImT a . At large volumes compared to the string scale the tree-level Kähler potential for these moduli is given by with K abc = l −6 s M 6 ω a ∧ω b ∧ω c ∈ Z the triple intersection numbers of the compactification manifold. At this level the B-field axions b a ≡ Re T a display a continuous shift symmetry only broken by worldsheet instantons and e −K K is a cubic polynomial on the moduli On the other hand, the moduli space of complex structure deformations is encoded in terms of the harmonic three-form which is even under the orientifold involution. Here C 3 is the three-form RR potential and C ≡ e −φ e 1 2 (K CS −K K ) stands for a compensator term with K CS = −log i l 6 s Ω ∧Ω .
In order to translate this quantity into 4d chiral fields one first describes Ω in terms of a symplectic integer basis (α λ , β λ ) ∈ H 3 (M 6 , Z) as Ω = X λ α λ − F λ β λ [26]. Then one uses the orientifold action to split such a basis into even (α K , β Λ ) ∈ H 3 + (M 6 ) and odd (α Λ , β K ) ∈ H 3 − (M 6 ) three-forms and defines the chiral fields which have a well-defined counterpart in mirror type IIB orientifolds [25]. Finally, one imposes the orientifold constraints to obtain the Kähler potential which should then be translated into the variables n K ≡ Im N K and u Λ ≡ Im U Λ on which it depends. As pointed out in [23] one can always perform a symplectic transformation so that all moduli are of the kind N K . In this case showing such dependence is relatively easy, as we have where Im F KL is a homogeneous function of zero degree on the n K . In general one can show that e −K Q /2 is a homogeneous function of degree two on n K and u Λ , as we discuss in Appendix A.
This effective field theory becomes more involved as soon as we introduce open string degrees of freedom [22,23,27,28]. In particular it was found in [22,23]  and their implications for the scalar potential governing both. As we will see, a key ingredient of our analysis will be the periodic directions that appear in open string moduli space, and the discrete shift symmetries that they correspond to.
Given a particular compactification, one may describe a point in open string moduli space by considering a set of special Lagrangian three-cycles {Π 0 α } that satisfy the RR tadpole condition (2.2) and where D6-branes are wrapped. One may now move in this moduli space by deforming the reference three-cycle Π 0 α to a homotopic special Lagrangian three-cycle Π α . If the deformation is infinitesimal we can describe it in terms of a normal vector X, and define the open string moduli of a D6-brane wrapping such a three-cycle where A is the D6-brane gauge potential along Π 0 3 containing the Wilson lines degrees of freedom, 3 and l −2 s ρ i ∈ H 2 (Π 0 α , Z) is a basis of quantised harmonic two-forms in Π 0 α . 4 Here X = 1 2 l s X j ϕ j is a linear combination of normal vectors to Π 0 α preserving the special 3 The normalisation for Φ is such that the periodicity γ A ∼ γ A + π for D-branes on top of orientifold planes is translated into θ ∼ θ + 1. A similar statement holds for the position moduli. 4 By taking ρ i harmonic we are selecting the lowest of a tower of Kaluza-Klein open string modes, but one may extend this definition to the full tower by taking an appropriate basis of quantised two-forms [22].
For most pourposes we will assume a truncation to the lightest states of each open string KK tower.
Lagrangian condition, that is such that [L X j J] Π 0 α = [L X j Ω] Π 0 α = 0 with L X j the Lie derivative along X j . Following [29] this implies that ι X j J| Π 0 α is proportional to a harmonic one-form on Π 0 α , 5 and therefore i, j = 1, . . . , b 1 (Π 0 α ). Finally we define to make manifest the implicit dependence of Φ on the Kähler moduli.
For finite homotopic deformations Π α = exp X (Π 0 α ) the dependence on the deformation parameters ϕ j should be computed by a normal coordinate expansion [22,23], which adds to the above linear behaviour a higher order dependence on the ϕ's. The open string moduli can then be expressed as where the functions f i α a (ϕ) satisfy the differential equation so that by imposing f i α a (ϕ j = 0) = 0 we recover (2.11) from the leading term in the Taylor expansion f i α a = (η 0 α a ) i j ϕ j α + . . . In general (η α a ) i j and so f i α a may further depend on the Calabi-Yau metric, and therefore on the closed string moduli Im T a , Im N K , Im U Λ .
In simple compactifications like toroidal orientifolds, the η a 's are independent of ϕ, and so the definition (2.11) is exact. Moreover, the normal vectors X j can be chosen such that the η a 's are integer numbers and both θ's and ϕ's are periodic variables of unit period [30,31]. This last statement will remain true for the θ's in general Calabi-Yau compactification while for the f a 's things may become more complicated. The best way to analyse their periodicity is to define the open string moduli in terms of integration chains, along the lines of [22,32,33].
When the three-cycle Π 0 α is homotopically deformed to Π α , a one-cycle γ 0 i in the Poincaré dual class to ρ i will sweep a two-chain Γ i α in M 6 , such that ∂Γ i α = γ i − γ 0 i with γ i the corresponding one-cycle in Π α . One can the define the complexified open string 5 Notice that the same is not true for ι Xj J c | Π 0 α , whose real part could be a non-harmonic one-form. However, one may always choose the profile for A such that A − ι X J c | Π 0 α is harmonic. Alternatively one may take ρ i to be harmonic two-forms, as we do here. coordinates as [32,33] whereF is an extension of the worldvolume field strength F = dA to the two-chain Γ i Using Lefschetz duality one may then rewrite this as as done in [22]. Here C α 4 is the four-chain swept by the three-cycle Π α andρ i the quantised two-form on C α 4 that pulls-back to ρ i on its boundary, whileF is now extended to the whole of C α 4 . By comparing these definitions to (2.13) one obtains that which clearly satisfies (2.14). From this perspective it is easy to understand when the functions f a describe periodic coordinates in the D6-brane moduli space. If a homotopic special Lagrangian deformation is such that C α 4 is a four-cycle in M 6 , the D6-brane system has returned to its original position after performing a loop in its moduli space. This should correspond to a discrete symmetry of the theory, just like shifts of Wilson lines by their period. Notice that if C α 4 is a four-cycle then the f a 's are integer numbers, 6 and so the shifts that are generated by periodic directions in the D6-brane moduli space are with k i α , r i α a ∈ Z. As we will see later on, such discrete shifts are directly related to the discrete gauge symmetries and the multi-branched structure of F-term scalar potentials in type II compactifications with fluxes and D-branes.
Another set of quantities that will become important in the formulation of the Kähler potential are the functions {g K α i , g α Λ i } on the deformation parameters ϕ's. Such functions provide an alternative parameterisations of the D6-brane moduli space and are defined by the differential equations [32] ∂g K α i In fact they are even integer numbers. However we will also consider the possibility where C α 4 ∪ RC α 4 is a four-cycle without C α 4 being so, which implies odd f a 's. where Here Π α = exp X (Π 0 α ) is again a homotopic special Lagrangian deformation of Π 0 α , and ζ i is a basis of quantised harmonic one-forms on it such that Π 0 α ζ i ∧ ρ j = l 3 s δ j i . Again, these functions can be expressed in terms of chain integrals as [22] 2 l 3 where Σ i α is the three-chain swept by the two-cycle Poincaré dual to ζ i , andζ i is the extension of this one-form to the four-chain C α 4 . More explicitly we have that 3 The scalar potential from Minkowski three-forms The space of background and D-brane deformations described in the last section will be subject to a scalar potential in certain type IIA compactifications. In particular, both Kähler and complex structure moduli will develop an F-term scalar potential when NSNS and RR background fluxes are present [25,34]. In the absence of open string deformations, this potential can been reproduced by applying the usual 4d supergravity expression or by direct dimensional reduction. The latter method involves integrating out the degrees of freedom associated to three-form fields in R 1,3 which give a non-vanishing contribution to the potential [25,34]. In fact, as shown in [21] one may describe the full F-term scalar potential purely in terms of contributions coming from Minkowski threeforms if one performs the dimensional reduction in the democratic formulation of type IIA supergravity. In the following we will adopt this latter approach, as it allows to incorporate the D-brane moduli into the computation and derive a scalar potential for open and closed string modes simultaneously [22,23].
For simplicity let us consider a Calabi-Yau orientifold compactification where only RR background fluxes are present, ignoring for now the presence of localised sources. Working in the democratic formulation we have the RR p-form potentials C p with p = 1, 3, 5, 7, 9 that can be arranged in the polyforms respectively dubbed C and A-basis for the RR potentials in [35]. The corresponding gauge invariant field strengths are given by [35,36] withḠ a formal sum of harmonic (p + 1)-forms of M 6 to be thought as the background value for the internal RR fluxes defined in the A-basis. This basis is particularly adequate to apply Dirac quantisation, since a D(p − 1)-brane will couple to the potential A p via its Chern-Simons action, and so the standard reasoning gives the quantisation condition [37] 1 l p s π p+1 for any cycle π p+1 in the internal space not intersecting a localised source like a background D-brane. In the A-basis a source wrapping a cycle Π a enters the Bianchi identity as with δ(Π a ) the delta-function with support on Π a and indices transverse to it, while F a is the quantised worldvolume flux threading Π a . In the absence of localised sources the rhs of (3.4) vanishes and the A p are globally well-defined, so they do not contribute to the integral in (3.3) which becomes a quantisation condition for the fluxesḠ p+1 . We can then define flux quanta in terms of the integer cohomology of M 6 , namely as whereπ a ∈ H − 2 (M 6 , Z) and π a ∈ H + 4 (M 6 , Z). These definitions need to be generalised if we take into account the effect of localised sources in the above Bianchi identity, as we discuss in the next section. Finally, as we have doubled the p-form degrees of freedom we need to impose the Hodge duality relations G 2n = (−) n 10 G 10−2n . To proceed we define a set of Minkowski four-form field strengths arising from the dimensional reduction of the 10d RR field strengths Here the four-forms (F 0 4 , F a 4 ,F 4 a ,F 4 ) have their indices in R 1,3 and (ω a ,ω a , ω 6 ) are harmonic forms of M 6 . 7 In particular l −2 s ω a are the harmonic representatives of H 2 − (M 6 , Z) defined in the last section and l −4 sω a is a dual basis of harmonic forms in H 4 + (M 6 , Z) in the sense that Finally l −6 s ω 6 = dvol M 6 / √ g M 6 represents the unique harmonic six-form with unit integral over M 6 . Following [21], such Minkowski four-forms enter the 4d effective action as whereV 6 = l −6 s Vol(M 6 ) stands for the covering space compactification volume in the Einstein frame and in string units. In deriving the above expression we have performed the 4d Weyl rescaling g µν → gµν V 6 /2 , we have used that and that In terms of the dimensional reduction of the RR potential to Minkowski three-forms We have that represent the usual metric for harmonic two-forms and its inverse. Finally we have that We may now integrate out the Minkowski four-forms from (3.11), obtaining the following scalar potential which is nothing but the usual type IIA RR flux potential [34] where we have used that K ≡ K abc t a t b t c = 6e 3φ/2V 6 . Alternatively, one may deduce this scalar potential from the dimensional reduction procedure described in Appendix B.

Adding a single D6-brane
This method to obtain the scalar potential has the advantage that it allows to incorporate the open string scalars in a very straightforward way. Indeed, one just needs to add to the 4d action (3.11) the extra four-forms couplings that arise form the dimensional reduction of the D-brane Chern-Simons actions, and then proceed as before. As in [22,23], let us first consider the effect of a single D6-brane wrapping a three-cycle Π α which is a homotopic special Lagrangian deformation from the reference cycle Π 0 α . Furthermore although it may break supersymmetry, we also consider the following worldvolume flux threading the three-cycle where the harmonic two-forms ρ i have been defined in (2.11). The CS action reads dA ∧ e σF (3.18) where µ 6 = 2π/l 7 s . As in the previous section X = 1 2 l s ϕ j X j parametrises the homotopic special Lagrangian deformation Π α = exp X (Π 0 a ), C α 4 is the corresponding four-chain such that ∂C α 4 = Π α −Π 0 α andF is the extension of the field strength made in (2.15). Expanding the RR potentials as in (3.8), we find the following couplings of the D6-brane scalars to the Minkowski four-forms arising from the bulk Here we have defined with f i a as in (2.17) and with the harmonic one-forms ζ i defined as in (2.20). By using that the pull-back and the wedge product commute, one can show that these quantities satisfy which can be used to simplify (3.20).
Incorporating the couplings (3.19) into the 4d effective action and then integrating the Minkowski four-forms will result into performing the following replacements in the expression (3.15) for the scalar potential while leavingρ invariant. Equivalently, one may replace the flux quanta that enter (3.16) by the following quantities while leaving the Romans mass untouched. The scalar potential that we then obtain is where V DBI stands for the total tension of the D6-branes minus that of the O6-planes, with k i such that k i n F i , k i n a i ∈ Z. Consequently the full scalar potential is left invariant by these shifts. Because n F i , n a i ∈ Z, a particular solution is given by taking k i ∈ Z, which corresponds to the first kind of shift in (2.18). Note that such discrete symmetries are nothing but large gauge transformations related to the D6-brane Wilson lines, and therefore discrete gauge symmetries. As such they should be present not only in the scalar potential but also at the level of the superpotential, as we will discuss in the next section.
Similarly, if there are periodic directions in the moduli space of D6-brane positions we may formulate the following discrete shifts where now r i a is such that n F i r i a ∈ Z and n (b i r i c) = s a K abc with s a ∈ Z. These conditions are satisfied whenever the shift correspond to the D6-brane sweeping a four-cycle in M 6 , that is to the second kind of shift in (2.18). Again, because the D6-brane is returning to the same position, V DBI is left invariant under these shifts. 8 In both cases, the presence of these discrete symmetries is directly related to the multi-branched structure of the scalar potential, typical of models with axion-monodromy.
The multi-branched structure of the closed string sector of this potential was analysed in [21]. In this sense, the presence of open string modes in the potential and the discrete symmetries (3.28) and (3.29) describe how open string modes are related to these branches of the scalar potential, as already pointed out in [38] for the case of the Wilson line.

Adding all the D6-branes
Let us now consider a full compactification with several D6-branes, wrapping the special Lagrangian three-cycles Π α and their orientifold images. Recall that a consistent configuration must satisfy the RR tadpole condition (2.2), which is equivalent to the existence of a four-chain Σ 4 ⊂ M 6 that connects all the D6-branes and O6-planes. Physically, one can interpret this four-chain as follows. If we wrap a D6-brane on it, we will construct a domain wall in 4d connecting two different vacua: one of them with all the D6-branes on top of the O6-planes and the other one with the D6-branes wrapping the three-cycles Π α and their orientifold images. Now, considering such a global configuration allows to take into account terms of the Chern-Simons action which we implicitly neglected in the single D6-brane case, when computing the couplings (3.19). These terms are those without derivatives, namely where in the first equality we have used that A 7 and F vanish on top of the O6-planes and then applied Stokes' theorem on the reference four-chain Σ 0 4 , defined such that ∂Σ 0 . For this one needs to extend of the worldvolume flux F from the boundaries to the four-chain Σ 0 4 connecting them, extension which we dubbedF . In terms of the D6-brane domain wall described above,F would be the actual worldvolume flux of the D6-brane along Σ 0 4 .
Setting the Wilson lines on all the D6-branes to zero, the rhs of this equation gives 4d couplings of the form (3.19), which will eventually translate into shifts of the flux quanta.
In particular we have the shifts Let us now add the contribution of each D6-brane due to turning on the Wilson lines and deforming its embedding away from the reference cycles Π 0 α . One then obtains where the index α runs over each independent brane but not their orientifold images.
Finally, since such contributions can be described in terms of four-chains C α 4 such that and define the dressed fluxes in (3.27) in terms of it where we have absorbed the Wilson line dependence in the definition ofF . Notice that this reproduces and extends the result in [22], in the sense that it gives an expression for the scalar potential globally valid in the open string moduli space. In this last expression the discrete shift symmetries leaving invariant the potential are particularly transparent.
Such discrete symmetries will be useful to determine the superpotential that corresponds to this F-term scalar potential, as we discuss next.

Open-closed superpotential and axion monodromy
In the absence of D-branes, the superpotential generated by type IIA RR fluxes is [39,40] Indeed, one can check that plugging this superpotential and the Kähler potential (2.6) in the standard 4d supergravity formula (3.16) is recovered as an F-term scalar potential [22].
Adding D6-branes degrees of freedom should modify the superpotential as where W D6 contains open and closed string moduli. By the discussion of the last section we should impose that the full superpotential is invariant under the discrete gauge symmetries related to integer shifts of the Wilson lines and completing loops in the moduli space of special Lagrangians. If we just consider the effect of a single D6-brane on Π α , such symmetries amount to (3.28) and (3.29), fixing the new superpotential piece to where we have used (2.13) and defined W 0 D6 as the superpotential at Π α = Π 0 α . This superpotential contains a bilinear term of the form n i a Φ i T a whose microscopic origin was described in [17] and its applications to large field inflation in [38]. As pointed out in [17] one can derive such a bilinear superpotential from the general expression [41,42] with C α 4 a four-chain such that ∂C α 4 = Π α − Π 0 α . Taking a homotopic deformation and applying the definition (2.16) one sees that the superpotential (4.3) is also recovered.
To consider the full set of D6-branes in the compactification let us follow [17] and set where Σ 4 is the four-chain connecting all the D6-branes to the O6-planes, as described below (2.2). Now, by splitting this four-chain as Σ 4 = Σ 0 4 + α C α 4 ∪ RC α 4 we obtain the following generalisation of (4.3) where Σ 0 4 the four-chain connecting the reference three-cycles wrapped by the D6-branes, and as usual α only runs over half of the D6-branes, excluding orientifold images. Notice that this superpotential also satisfies the appropriate invariance under discrete shifts.
Indeed, either taking the definition of dressed fluxes in (3.32) or (3.33) one can show that which is the statement made in (3.27) but now at the level of the superpotential. Therefore we find that the dependence of the superpotential on the open string deformations enters uniquely through the dressed fluxes {ẽ 0 ,ẽ a ,m a }. The latter are to be thought as the gauge invariant quantities including flux quanta and axions that typically appear in models with F-term axion monodromy [43]. Therefore, following the philosophy in [44][45][46][47], one expects that in the 4d scalar potential the dependence on the open string axions also only appears This result is clearly true for the scalar potential (3.26) computed at tree-level, but the claim is that it should also hold after all kind of UV corrections have been taken into account, which is particularly important in order to build models of large field inflation.
This dependence on {ẽ 0 ,ẽ a ,m a } is also directly related to the multi-branched, domainwall connected structure of the scalar potential. From the viewpoint of Wilson line axion dependence this structure was partly discussed in [38], which considered the case with n F i = 0. In that case the discrete symmetry with Λ 4 the appropriate choice of four-cycle in M 6 . 9 If a D6-brane closes a loop in moduli space by sweeping a non-trivial four-cycle Λ 4 α , then there will be a non-trivial change in 9 In technical terms, we are computingm a by integratingω a over a cycle in the relative homology group the integral over Σ 4 , which can be compensated by replacing equivalently by shifting m a by integers. In the 4d effective theory, the latter corresponds to crossing a domain-wall made up of a D6-brane wrapping Λ 4 α . Finally, when we switch on n F i the same will apply, but now such a domain wall D6-brane is magnetised internally and it will have induced D4-brane charge, that shift the e a quanta as well.
As a final remark 10 let us point out that the open-closed superpotential can be described in the compact form with G defined in (3.2). In the absence of D6-branes this is obvious, since then dA is exact and does not contribute to the integral, so we recover (4.1) directly from the definitions (3.5). When we include D6-branes the flux polyform dA + G is still quantised, but it is not closed as it satisfies the Bianchi identity (3.4). 11 In particular we have that which has a globally well-defined solution for the two-form flux dA 1 +Ḡ 2 due to the tadpole condition (2.2). The corresponding contribution to the superpotential reads In the first equality we have used the fact that the two-form dA 1 +Ḡ 2 is quantised as in (3.3), and that J 2 c is closed to convert the integral over M 6 into an integral over the four-chain Σ 4 , following [53] and also [17,28]. As before, this chain is such that , and so it is determined only up to a closed four-cycle Λ 4 , which can be understood as the contribution to the superpotential coming from W K . In the second equality we have simply used the expression (4.8) for the dressed two-form fluxm a .
Also, in eq.(4.11) a piece of the open-closed superpotential is computed by integrating J 2 c over a relative homology cycle. This formulation for the open-closed superpotential is analogous to the one for type IIB compactifications with D5-branes, see e.g. [48][49][50][51]. 10 See also the discussion in section 3.3 of [52]. 11 Or in other words dA + G does no longer belong to the standard cohomology H * (M 6 , Z) but it does belong to an element of the relative cohomology group H * (M 6 , Π D6 , Z 6 ). See [28] for other applications of such relative (co)homology groups to type IIA compactifications with D6-branes.
In fact, this discussion provides a new interpretation ofm a . Indeed, let us use Hodge decomposition to split the two-form dA 1 +Ḡ 2 such that G 2 is a purely harmonic two-from, while dA 1 is a sum of an exact and co-exact two-forms. Then we necessarily have that (4.12) That is, the gauge invariant quantitym a is nothing but a harmonic component of the two-form flux, computed including the D6-brane backreaction. Notice that due to such backreactionm a does not need to be quantised, since dA 1 also contributes to integrals over two-cycles π 2 ∈ M 6 via its co-exact component.

Holomorphic variables and the Kähler potential
As discussed in [22,23], by dimensionally reducing the 10d type IIA supergravity and D6brane actions one finds that the open string moduli and RR bulk axions mix kinematically.
In terms of the 4d N = 1 effective field theory this is interpreted as a redefinition of the  [58,[65][66][67] and these are difficult to compute in general [68].
In the following we would like to apply an alternative approach, based on the discrete symmetries discussed in the previous sections. In particular we will implement in our setup the reasoning of [62], in which the definition of holomorphic variables was related to their transformation under discrete shifts of the open string fields.
To proceed we need to understand how to dualise correctly in 4d the two forms dual to the axions Re N K . To do so we can add to the 4d effective action the following Lagrange where ρ K are the two-forms dual to the 4d axions Re N K , arising from the RR fiveform potential as C 5 = ρ K ∧ β K . Such two-forms couple to the D6-brane moduli via its Chern-Simons action and in particular through the term from where we obtain the four-dimensional coupling As a result, we have that the bulk axions transform as follows under discrete shifts of the open string moduli where we have defined which for a four-cycle Λ α 4 are integer numbers. It is easy to see that these transformations are not holomorphic. However, as in [62] one may redefine the complex structure moduli N K to new variables that transform holomorphically under the above shifts. Indeed, let us define where H K α a are real functions of the three-cycle position moduli defined by The functionals H K α a are similar to those defined by Hitchin in [32] in order to describe the metric on the moduli space of special Lagrangian submanifolds, 12 and were used in [22] 12 To connect with [32] notice that (5.8 to propose a Kähler potential including D6-brane moduli. From the definitions of section 2 one can see that along a periodic direction of the D6-brane position the function g K α i must be of the form where we have defined ∆f i α a and ∆g K α i as in (5.6). Here M K α i is a periodic function of period one in s, with mean m K α i and such that M K α i (0) = 0. We then have that where P K α a (s + 1) = P K α a (s). Hence along this periodic direction H K α a shifts as and therefore the redefined variables shift as giving the desired holomorphic behaviour.
The same reasoning can be applied to the complex structure moduli U Λ , obtaining the redefined variables that show the appropriate holomorphic behaviour.
Performing this change of variables into the Kähler potential will implement its de- pendence on the open string fields. More precisely we have that the piece (2.6) remains invariant, while (2.9) should be rewritten in terms of the new holomorphic variables. That is, we should again consider K Q = −2log(G Q (n K , u Λ )) (5.15) but express n K = Im N K and u Λ = Im U Λ in terms of the new holomorphic variables, namely with n K = Im N K , u Λ = Im U Λ and t a = Im T a . Although our definition of holomorphic variables is different from the proposal in [22], it embeds the Hitchin functionals into the Kähler potential in a similar fashion, reproducing the same Kähler metrics for the open string fields. Finally, one can check that in the toroidal case this redefinition reduces to and similarly for u Λ , in agreement with standard result in type IIB toroidal orientifolds [54,56,58,62,63].
Notice that our reasoning partially relies on the existence of periodic directions in the moduli space of D6-branes, and such may not exist for the position moduli in generic compactifications. In these cases the definition of the functions H could be different, as they are not constrained by periodic position shifts. Nevertheless, as discussed in Appendix A, in order to reproduce the appropriate metrics for the chiral fields one needs a redefinition of the form (5.16) with the H's defined as above. Hence, at the level of approximation that we are working, the redefinition of the complex structure moduli seems sufficiently constrained.
One important consequence of these results is that, due to the above redefinitions, the piece of the Kähler potential (5.15) also depends on the Kähler moduli, and therefore the complex structure and Kähler moduli spaces no longer factorise. In principle this greatly complicates the computation of quantities in the 4d effective field theory, like for instance the F-term scalar potential. One can nevertheless see that, despite this complication, the redefinition of the holomorphic variables implies several non-trivial identities for the Kähler metrics which will be crucial for the computations of the next section.
For instance, recall that in (5.15), G Q is a homogeneous function of degree two on the variables n K = Im N K and u Λ = Im U Λ . As shown in Appendix A, these two variables are in turn homogeneous functions of degree one in {ψ β } = {t a , n K , u Λ , φ i }, where for simplicity we have absorbed the D6-brane index α into the index i in φ i = Im Φ i α . As a result, G Q is a homogeneous function of degree two on the variables {ψ β }, which in turn implies that the full Kähler potential K = K K + K Q satisfies as well as where in both cases the sum is taken over all fields ψ α .
In addition, from the same definition of the Kähler potential and some simplifying assumptions several important relations for the elements of the inverse Kähler metric follow. First we have Second it happens that K ab = (∂ a ∂bK K ) −1 (5.21) or in other words the inverse Kähler metric for the Kähler moduli is exactly the same as in the absence of open string degrees of freedom. Finally for the specific definition of H taken above we have that where G ij D6 is the inverse of the natural metric for one-forms in the three-cycle Π α In fact, we have that (5.22) fixes H up to a linear function on the φ i 's, a freedom that can be used to redefine the reference cycles Π 0 α . We refer the reader to Appendix A for further details on all these identities. 6 The scalar potential from 4d supergravity In this section we combine all the results from the previous sections together. In particular we will show that if we take the superpotential of section 4 and the Kähler potential of section 5, we derive the scalar potential of section 3 from the usual F-term 4d supergravity where α runs over all the fields {Ψ α } = {T a , N K , U Λ , Φ i } the Kähler potential depends on. Now using the relations (5.18) and (5.19) we can rewrite things as and then analyse this expression term by term.
First we have that where we have defined and have merged the contribution of W 0 D6 in (4.6) and the closed string piece of the superpotential W K Second we have that Hence, summing the two results we find Finally, from the relation (5.20) we can express the remaining term in (6.2) as where here a, b run over the Kähler moduli and i, j over the open string moduli that appear in the superpotential. This implies that Using (3.23) we can simplify the first term in the second line of (6.7) to as well as the terms appearing in the third and fourth lines of (6.9), that now read where we have expressed everything in terms of the dressed fluxesm a = m a + q a . Adding these last two terms we obtain K ab K ad K bc + K d K c m d +mb d m c +mb c = 16e 3φV 2 6 g dc m d +mb d m c +mb c (6.12) where we have used (5.21) and more precisely (A.36).
The remaining terms can be arranged as follows. One may first rewrite the first two lines of the rhs of (6.9) as where we have again used (3.23), the definition ofm a and thatẽ a = e a − n a i θ i − n F i f i a . Then, by using that K ab K a = 4 3 Kt b we can add the second line of this equation to the last term in (6.7) and obtain K 2 m 2 = 4e 3φV 2 6 m 2 (6.14) Summing all these contributions we find the following F-term scalar potential where we have defined Hence, we do indeed recover an F-term scalar potential which is the sum of two terms.
The first one is the potential V RR+CS computed in section 3 which has the form of the usual type IIA scalar potential generated by RR fluxes but with those replaced with the dressed fluxes {ẽ 0 ,ẽ a ,m a , m} that contain the open string moduli dependence. The second piece (6.16) should then correspond to the DBI contribution to the F-term scalar potential, which is non-trivial when σF − J c does not vanish over Π α . Whenever such source of supersymmetry breaking is small in string units, the corresponding excess of energy is given by [38] where G ij D6 is the inverse of (5.23). We then see that the choice of Kähler metric taken in the previous section also reproduces the expected contribution from the DBI action to the F-term potential.

Type IIB models with D7-brane Wilson lines
In this section we translate our results obtained in type IIA compactifications with O6planes to type IIB compactifications with O7/O3-planes. We consider compactifications of type IIB string theory on R 1,3 × M 6 where M 6 is taken to be a compact Calabi- where α (p,q) stands for the Hodge type of the differential form α.
The 4d effective action for the closed string sector contains the axio-dilaton τ = C 0 + ie −φ , Kähler moduli and complex structure moduli, with a factorised moduli space, see [69]. The Kähler moduli are obtained from the dimensional reduction of the Kähler form J, of the B-field and of the RR forms C 2 and C 4 as With these definitions at hand we may define the following 4d chiral multiplets where we defined the triple intersection numbers The Kähler potential at large volume for these moduli is then written as an implicit function of the chiral multiplets as For an explicit expression it is necessary to invert the relation between the volume of 2-cycles v α and the chiral coordinates T α . While this is hard to do in general, one can see that e −K K is a homogeneous function of degree three on the moduli. The quantities (X I , F I ) are called periods of the holomorphic 3-form Ω and depend on the complex structure of M 6 . In particular it is possible to prove that the F I are functions of the X I implying that the latter may be chosen as good projective coordinates on the complex structure moduli space. Fixing X 0 = 1 we will identify the remaining X I with the complex structure moduli with I = 1, . . . , h . Additionally it is possible to prove that the F I = ∂ I F where the function F , usually called the prepotential, is homogeneous of degree 2 in the projective coordinates X I . The Kähler potential for the complex structure moduli may be written using the periods Finally we need to consider the axio-dilaton whose Kähler potential is simply The introduction of an open string sector will add moduli in the 4d effective field theory.
The case of D7-branes was analysed in [59] where it was found that for D7-branes the moduli fall in two separate classes, brane position moduli and Wilson line moduli. In [59] it was found that brane position moduli, counted by the cohomology group H (2,0) (S a ), give a redefinition of the axio-dilaton and therefore modify the Kähler potential ( of H 1 (S a , Z) which allows us to express the differential forms γ i as 13 with f a ij (z) holomorphic in the complex structure moduli and (Im f a ) ij is the inverse of its imaginary part. Setting ξ a i = η i + f a ij (z)θ j we obtain that the dimensional reduction ansatz for the gauge field (7.12) becomes (7.14) implying that the fundamental periods for the axionic fields (η i , θ i ) is η i ∼ η i + 1 and θ i ∼ θ i + 1, independent on the complex structure moduli. As already anticipated the presence of Wilson line moduli will give a redefinition of the Kähler moduli of the Calabi-Yau threefold. The resulting redefinition is the followinĝ Sa ω α ∧α j ∧β i . To motivate this redefinition we can follow a similar logic as the one followed in section 5. Following [59] we add the following Lagrange multiplier term in the effective action where ρ α are duals of the RR axions C α in 4d coming from the dimensional reduction of C 4 along 2-forms ω α . These two forms have additional couplings to the Wilson line moduli of the D7-branes coming from the term (7.17) in the Chern-Simons action of the D7-brane. Performing dimensional reduction of this term we obtain which closely resembles the result obtained in [62]. Following the discussion there, it is possible to see that the variable (7.15) shifts holomorphically under discrete shifts of the Wilson line moduli.

Superpotential and discrete symmetries
When turning on RR fluxes the superpotential obtained in the 4d effective field theory has the simple form l s W F = X I e I + F I m I , (7.19) where the closed string flux F 3 has been expanded as l −2 s F 3 = m I α I + e I β I . (7.20) Introduction of D7-branes will add some terms in the superpotential. Specifically the D7-brane superpotential is [42,59] where, similarly to the case considered in type IIA, Σ 5 is the five-chain connecting all D7-branes to the O7-planes, and F an extension of the worldvolume flux in it. Splitting the five-chain as Σ 5 = Σ 0 5 + a C a 5 + RC a 5 we obtain the following D7-brane superpotential where the index a runs over the D7-branes but not their orientifold images. In the following Choosing the configuration of the D7-branes such that on the reference 4-cycles S 0 a Wilson lines are turned off it is possible to drop the last two terms in (7.23). At this point we may use (7.14) to write the superpotential in terms of the Wilson line scalars as By inspection of the D7-brane superpotential we observe that the combined flux and brane superpotential may be nicely written in terms of dressed fluxes l s W IIB = l s W F + l s W D7 = X Iẽ I + F Im I , (7.26) where the redefined flux quanta arẽ This demonstrates that the superpotential is invariant under the discrete shifts in a quite analogous fashion to their type IIA counterparts (3.28) and (3.29).
While this description has the advantage of making manifest the discrete symmetries of the superpotential, it is not obvious that the superpotential is a holomorphic function of the Wilson line moduli. To write the brane superpotential as a holomorphic function of the Wilson line moduli it is necessary to impose the following condition 14 which fixes the function f a ij (z) for those Wilson lines that appear in the superpotential.

Scalar potential
Knowing the form of the superpotential and the holomorphic variables we may perform a similar computation to the one in section section 6 to derive the F-term scalar potential.
For simplicity we will work in the large complex structure limit where the prepotential takes the form where from now on I = 1, . . . , h (2,1) − (M 6 ) and moreover we will write the complex structure moduli in terms of their real and imaginary parts as z I = X I /X 0 = u I + iw I . In addition, we will absorb the D7-brane index a into the Wilson line index i. We start by noting 14 Note that the condition (7.29) is necessary to have Sa Ω ∧ γ i = 0 thus ensuring that the differential forms γ i are of Hodge type (1,0) even when Ω| S4 is non-trivial. that in this limit the relations (5.18), (5.19) continue to hold if we assume that f a ij is a linear function on the complex structure moduli, which we shall assume henceforth. 15 The relation (5.20) is then replaced by Using these properties it is possible to see that where the index A runs over all fields Ψ A . Since the remaining of the computation is very similar to the one considered in section 6 we will omit most details. We find that where analogously to the type IIA case we have defined where Summing up all contributions we obtain the final form for the scalar potential As shown in Appendix A these properties rely solely on e −K being a homogeneous function in either the real or imaginary part of the moduli, which is true whenever Im f a ij is a homogeneous function of degree one on the w's. This can be easily achieved by assuming that f a ij is linear, and in the particular case of large complex structure by setting p a = h a = 0 and (c a ) i 0 = (d A ) 0 i = 0, which we assume below. It would be interesting to explore more general cases. This seems compatible with the results of [72] where it was found that the function f a ij is linear in the large complex structure limit.
where the modified flux quanta are given by (7.27) and we defined Similarly to the type IIA case, one could have arrived to the piece within brackets in  for models of large field inflation involving D-brane moduli, as will be discussed in [76].
There are a number of directions in which our analysis can be generalised and that would be interesting to explore in the future. For instance, in our type IIA analysis we have only considered the scalar potential and superpotential generated by RR fluxes. While this is sufficient for the scope of this work, it would be important to also include NS fluxes in order to incorporate open string moduli in type IIA models of moduli stabilisation [77][78][79]. In particular it would be interesting to analyse the interplay of two different effects of such NS fluxes: how they modify the closed string scalar potential and how they generate a discretum of D-brane positions [80]. In a similar spirit, it would be interesting to generalise our type IIA analysis to non-Kähler flux compactifications. Since in these more general backgrounds the D6-brane deformations are also determined by the number non-trivial one-cycles of the wrapped three-cycle [81,82], one may again consider applying Hitchin's functionals to describe the open string Kähler metrics. Furthermore, it would be interesting to analyse the different corrections that will modify the scalar potential analysed here. There would include threshold corrections to the Kähler potential [58,[65][66][67][68], worldsheet instanton corrections to the type IIA superpotential and D-brane instanton corrections. In particular, it would be nice to incorporate the latter directly into the three-form derivation of the scalar potential of section 3, following the recent proposal in [83].
Finally, while our analysis has been restricted to non-chiral D-brane fields it would be interesting to analyse the consequences of our results for more realistic 4d models in which chiral matter arises from D-branes intersections. In this respect notice that throughout our discussion a key role has been played by the discrete shifts in open string moduli space, and in particular those which leave invariant the open-closed superpotential.
Remarkably, the same kind of shifts are the ones generating discrete flavour symmetries in simple orientifold models [84]. Therefore it would be interesting to see if there are semi-realistic string models in which the structure of the moduli stabilisation potential is directly related to the flavour structure of the D-brane sector.

A Details on the Kähler metrics
The Kähler potential that describes type IIA orientifold compactifications is given by with the definitions made in section 2. Due to the orientifold geometry the holomorphic three-form of the Calabi-Yau M 6 takes the form where each of the coefficients are functions of the real parameters n K and u Λ that define the complex structure of M 6 . Following [26] one can apply the equality in the above expression for CΩ to show that which means that G Q is a homogeneous function of degree two on the variables n K , u Λ In addition, it is easy to see that G K is homogeneous of degree three on the variables t a = ImT a . Therefore the real function G K G 2 Q that appears in (A.1) is homogeneous of degree seven on the variables {t a , n K , u Λ }.
As discussed in section 5, in order to introduce the open string moduli Φ i into the Kähler potential one needs to express n K and u Λ in terms of new variables that depend on the open string position moduli. Recall that such moduli are described in terms of the where in the last equality we have used the definition (2.17) and that ω a = d Im J c /dt a .
If we see φ i a as functions of t a and ϕ j α , the above relation means that φ i a is homogeneous function of degree one on the Kähler moduli t a , or in other words that we have the following scaling behaviour Finally, as pointed out in the main text f i α a may also depend on the complex structure moduli n K and u Λ . Now because the harmonic two-forms ω a are invariant under an overall rescaling of the holomorphic three-form Ω, they can only depends on quotients of their periods, and so the same should be true for f i α a . Therefore these functions should also be invariant under the rescaling n K → λ n K and u Λ → λ u Λ (A.11) with λ independent from λ in (A.9). Finally, a similar reasoning can be applied to the functions g K α i and g α Λ i , defined in (2.22). Indeed, from such a chain integral expression one can argue that these functions should also be invariant under (A.9) and (A.11) separately, and in particular homogeneous functions of zero degree on the variables {t a , n K , u Λ , φ j α }. These observations are relevant for the redefinition of holomorphic variables proposed in section 5, which imply that we must perform the following replacement in G Q with n K = Im N K , u Λ = Im U Λ the imaginary parts of the new holomorphic variables.
Here H K α a and H α Λ a must satisfy (5.8) and (5.14), or equivalently so we can see them as functions of the variables {t a , n K , u Λ , φ j α }. Since g K α i and g α Λ i are homogeneous of zero degree on these variables, the same can be assumed for H K α a , H α Λ a . Finally, by recursively performing the replacement (A.12) we can see H K α a , H α Λ a as homogeneous functions of zero degree on {t a , n K , u Λ , φ k α }. We then see that the rhs of (A.12) are homogeneous functions of degree one on the real fields {t a , n K , u Λ , φ k α } on which the Kähler potential depends. As a consequence, when we perform such replacements we obtain that G Q remains a homogeneous function of degree two in the new fields, namely Finally, we have that is homogeneous of degree seven on the whole set of fields ψ α ≡ {t a , n K , u Λ , φ k }, where for simplicity we have absorbed the D6-brane index α into the index i.
From this simple observation several useful relation can be derived [85]. For instance with Ψ a any of the complex fields of the compactification. To see this we first we rewrite this relation as which is easier to check. Then we compute and then use the relations that arise from the homogeneity of G to prove (A.17). Moreover, using the first identity in (A.20) again, one can show that the no-scale relation follows automatically. Finally, one can use these relations to get a simple expression for the inverse Kähler metric where G αβ is the inverse of ∂ α ∂ β G. For a recent general discussion on no-scale Kähler potentials based on homogeneous functions an their generalisation see [86].

Relations for the inverse metric
We would now like to discuss several identities relating the inverse Kähler metric components which are important for the computations of section 6, and see how they may arise from the above Kähler potential. For simplicity we will work in the symplectic basis defined above eq.(2.10) so that G Q is a homogeneous function of degree two purely on the variables n K , which in turn depend on the holomorphic fields N K , T a and Φ i α and their conjugates through (A.12). As this dependence can in general be quite involved, in here we will make the simplifying assumption that the functions f i α a and g K α i only depend on ϕ j α , which will allow us to carry the computations analytically. 16 Because ϕ j α ≡ ϕ j α (φ i α , t a ), we may then see f i α a and g K α i as functions of φ i α and t a , but nevertheless such that df i α a /dt b = dg K α i /dt b = 0. Given the definition (5.8) the same applies to H K α a , and so we have that which will be used later.
From the assumption that H K α i does not depend on the complex structure moduli one 16 This assumption is valid in simple cases like toroidal compactifications, and it should be a good approximation in the large volume and complex structure regions of the Calabi-Yau moduli space. Indeed, in general, f i α a and g K α i are homogeneous functions of zero degree on {t a , n K , u Λ , φ j α }, invariant under (A.9) and (A.11). As such they depend on ϕ j α and also on quotients of Kähler moduli t b /t a and complex structure moduli n K /n J , u Λ /u Σ . The dependence on these quotients is very mild for large values of these bulk fields, that is in the regions of large volume and complex structure, where the bulk harmonic forms ω a , β K do not vary significantly with respect to variations of the closed string moduli.
can see that the Kähler metric can be written in the form 24) where we have defined the matrices and From this expression we find that the inverse metric is given by where From here we deduce the relation and so the relation (5.20) can be rephrased as 33) or equivalently where as before t a = Im T a and φ i = Im Φ i . Using that ∂ φ j n K = − 1 2 g K i and the assumption dg K i /dt a = 0 we recover the desired identity. From these expressions one also obtains that where in the last equality we have used (A.23). Hence the inverse Kähler metric for the Kähler moduli is exactly the same as in the absence of open string degrees of freedom.
More precisely we have that where we have defined K as in (6.4) and K ab is the inverse of K ab in there.
where D i is given by Now in order to match the DBI potential from section 6 we need that 39) or equivalently [29]. Comparing with (A.38) this implies that in agreement with (2.19). In fact, notice that we will also reproduce the same set of results if in (A.13) we shift g K j by a constant, which means that the t a H K a are only determined up to a linear function on φ j .

B The closed string scalar potential
The literature already offers several approaches to obtain the scalar potential for the Kähler moduli in presence of background RR fluxes through dimensional reduction of (massive) type IIA supergravity, both excluding [21,25,34] and including [22,23] couplings to D-branes. In the absence of D6-branes, the scalar potential arises from the kinetic terms of the RR field strengths upon dimensional reduction of the standard formulation of (massive) type II supergravity. In the presence of D6-branes, the road to follow for the dimensional reduction passes through the democratic formulation of massive type IIA supergravity, where aside from the RR-potentials C 1 and C 3 and Romans mass parameter m also the dual RR-potentials C 5 , C 7 and C 9 are taken into consideration and the Hodge duality relations are usually imposed by hand. In this formulation the scalar potential for the Kähler moduli emerges upon dualisation of the four-dimensional four-forms associated to the dimensional reduction of the RR-field strengths in favour of the RR-flux quanta.
This appendix offers a first principle approach to deduce a four-dimensional mother action allowing for the dualisation of the four-forms also in the presence of D6-branes, while taking into account the considerations about Dirac quantisation around equation (3.3).
As advertised in [35,36,87]  16 yields the scalar potential from the RR fluxes as in equations (3.15) and (3.16). Note that a mother action is not unique and a (classical) theory expressed in a particular set of degrees of freedom can arise from two different mother action; the only requisite however is that mother actions reproduce identical equations of motion (and Bianchi identities).
Such non-factorisable three-cycles can be related [24] to coisotropic D8-branes through T-dualities, and share with the latter the property that the lagrangian condition J| Πc = 0 is violated for anisotropic untwisted Kähler moduli, i.e. T 2 = T 3 . This potential violation is the source for the bilinear coupling in the open-closed D6-brane superpotential in this toy model. Since the non-factorisable three-cycles π 1 ⊗ π 3 ⊗ π 4 and π 1 ⊗ π 5 ⊗ π 6 are Rodd under the orientifold projection, they do not yield non-vanishing RR tadpoles. Each fractional three-cycle is frozen at four separate fixed points (ij) on T 4 /Z 2 , such that the D6-brane position moduli along T 4 /Z 2 are projected out. These fractional three-cycles thus only retain the position-moduli Φ 1 α=a,b,c along the first two-torus, as indicated in figure 1 by virtue of arrows. Furthermore, the exceptional part for each fractional threecycle forms a linear combination of four R-odd exceptional basis cycles ε ij , such that the sum of a fractional three-cycle with its orientifold image only wraps bulk three-cycle generators. This consideration implies that the twisted RR tadpoles automatically vanish and drastically simplifies the bulk RR tadpole cancelation conditions (2.2), The SO(2N a ) enhancement of the a-stack gauge group follows when the a-stack lies on top of the ΩR-plane along T 2 (1) , while the U Sp(2N b )-enhancement of the b-stack gauge group occurs for a b-stack on top of the ΩRZ 2 -plane along T 2 (1) . In case the cycles are pulled away from the O6-planes along T 2 (1) , the gauge group supported by the three fractional three-cycles corresponds to U (16 − N ) a × U (16 − N ) b × U (N ) c . Given that all three fractional cycles are parallel to the ΩRZ 2 -plane along the first two-torus T 2 (1) , the Ktheory constraints are trivially satisfied. For this consistent D6-brane configuration, we can now infer the structure of the Kähler potential and superpotential. We therefore introduce local coordinates z i = x i + i y i on each two-torus T 2 (i) with periodicity x i ∼ x i + 1 and y i ∼ y i + 1 along the basis one-cycles. In first instance, we compute the quantities (Q K α ) ij defined in (2.20) by identifying the harmonic one-form l −1 s ζ 1 = dx 1 ∈ H 1 (Π 0 α , Z) compatible with the D-brane normal deformation X = 1 2 l s ∂ y 1 parallel to π 2 for all fractional three-cycles. An explicit computation for the three D6-brane stacks then shows: (Q 0 a ) 11 = 0, (Q 1 a ) 11 = −1, (Q 0 b ) 11 = 1, (Q 1 b ) 11 = 0, (Q 0 c ) 11 = 1, (Q 1 c ) 11 = −1. (C.7) The components (Q K=2,3 α ) 11 vanish as the interior products of β K=2,3 with respect to X 1 vanish, implying that only the complex structure moduli N 0 and N 1 will be redefined by the open string moduli Φ 1 α . The rigidity of the fractional three-cycles along T 4 /Z 2 also implies that components (Q K α ) ij with i, j ∈ {2, 3} vanish. Secondly, we have to compute the quantities (η 0 α a ) i j as defined in (2.12). To this end, we introduce the harmonic twoform l −2 s ρ 1 α ∈ H 2 (Π 0 α , Z) for each cycle α ∈ {a, b, c}, such that it forms the Poincaré dual to ζ 1 on Π 0 α . A straightforward computation then reveals: (η 0 a 1 ) 1 1 = −1, while all other components vanish. Hence, the full Kähler potential for the bulk moduli is given by: where we used the redefinition (5.17) for the complex structure moduli N 0 and N 1 . And lastly, in order to determine the open-closed superpotential W D6 in (4.6) for this D6-brane configuration we have to calculate the geometric quantities n α ai defined through: Keeping in mind the special Lagrangian condition for the fractional three-cycles enables us to pull-back the two-form ω α with respect to Π α and to obtain the geometric quantities (C.13) The contributions to the redefined flux quantam a can be easily computed using equation (3.23) and the quantities n α ai and (η 0 α a ) i j calculated above. The DBI-part follows from expression (6.16) and the open-closed superpotential (C.12): (C.14) The inverse metrics G 11 D6α on the open string moduli spaces have to be determined for each D6-brane Π α separately and depend implicitly on the Kähler modulus T 1 and the complex structure moduli N 0 and N 1 .