Minkowski 3-forms, Flux String Vacua, Axion Stability and Naturalness

We discuss the role of Minkowski 3-forms in flux string vacua. In these vacua all internal closed string fluxes are in one to one correspondence with quantized Minkowski 4-forms. By performing a dimensional reduction of the $D=10$ Type II supergravity actions we find that the 4-forms act as auxiliary fields of the Kahler and complex structure moduli in the effective action. We show that all the RR and NS axion dependence of the flux scalar potential appears through the said 4-forms. Gauge invariance of these forms then severely restricts the structure of the axion scalar potentials. Combined with duality symmetries it suggests that all perturbative corrections to the leading axion scalar potential $V_0$ should appear as an expansion in powers of $V_0$ itself. These facts could have an important effect e.g. on the inflaton models based on F-term axion monodromy. We also suggest that the involved multi-branched structure of string vacua provides for a new way to maintain interacting scalar masses stable against perturbative corrections.


Introduction
Consistency of Poincare invariant field theory implies that the possibilities for Lorentz structure of massless fields is quite limited. Fermions must have spin 1/2 or 3/2, whereas bosons should have a Lorentz structure of any of the kinds C 0 , C µ , C µν or g µν , with g the graviton and the C-fields being antisymmetric. This list should be extended to include 3-index antisymmetric tensors C µνρ . At first sight this extra possibility looks irrelevant, since a Minkowski 3-form has no propagating degrees of freedom.
However the presence of such fields may lead to important physical implications. A very recent example of this fact is discussed in [1], in which it is shown that the ultraviolet behaviour of pure gravity amplitudes changes if 3-form contributions are included in loops, in spite of not having propagating degrees of freedom. More well known is the fact that the corresponding field strength F µνρσ may be non-vanishing and permeate space-time giving rise to a constant contribution to the cosmological constant, and hence to new (quantized) degrees of freedom. Due to this fact Minkowski 4-forms have been considered in the past in trying to address the cosmological constant problem [2][3][4][5][6]. More specifically Brown and Teitelboim [7] considered a background 4-form field strength in space-time, contributing to the vacuum energy. Membranes coupling to the C (3) form can nucleate and give rise to jumps in the c.c. They suggested this contribution adjusts itself dynamically to cancel the rest of the contributions to the c.c. This 4-form is assumed not to couple directly to other fields in the theory.
The main difficulty with this approach is that the 4-form steps required to cancel the c.c. should be extremely tiny and is difficult to construct a working model with the required properties.
Bousso and Polchinski [8] suggested also to consider the contribution of 4-forms to the c.c. within the context of string theory (see also [9]). They argued that in string where F i = ǫ µνρσ F i µνρσ and −V obs denotes the remaining contributions, typically yielding a negative value. In this case the cancellation is not dynamical but is assumed to occur on the basis of anthropic arguments. A difficulty with this proposal so formulated is that within string vacua one cannot separate the issue of the c.c. from that of moduli fixing and one expects the 4-forms to couple to the moduli, making the situation far more complicated. As is well known, soon after a general approach to fix all moduli within Type IIB string theory vacua was proposed [10], in which internal RR and NS fluxes are turned on [11] to fix the complex structure moduli and dilaton in Type IIB orientifolds, with the Kahler moduli assumed to be fixed by non-perturbative effects.
Since then a large amount of effort has been dedicated to the issue of moduli fixing, involving internal fluxes [12,13]. Still the possible role of Minkowski 4-forms has been rarely discussed.
Minkowski 4-forms were discussed in papers by Dvali [14] in which it was shown that the usual strong CP problem and its axion solution may be elegantly described in terms of a composite 3-form, the QCD Chern-Simons term C (3) , with a dynamical 4form proportional to F ∧F . Here the PQ solution to the strong CP problem corresponds to the 3-form becoming massive via a coupling to a 2-form B µν , the latter being the dual of a standard axion.
More recently Kaloper and Sorbo [15,16] showed that 4-forms in field theory provide for a natural definition of quadratic chaotic inflation [17], stable under large field trips of the inflaton. Schematicaly, one starts from an action including not only a quadratic piece for the 4-form but a coupling to an axion-like field φ L = − F 2 4 + µφF 4 + . . . (1.2) with µ some mass parameter. Impossing dC 3 = F 4 through a Lagrange multiplier q and upon using the equations of motion for F 4 , one finds a quadratic scalar potential of the form where q is interpreted as a F 4 vev. Membranes couple to the 3-form C 3 and induce changes ∆q = e, where e is the membrane charge. The interesting point is that this is not a scalar potential but rather a family of potentials or different branches parametrized by the value of q. The family of potentials has a discrete shift symmetry which is spontaneously broken when a minimum φ = −q/µ is chosen. For each local minimum we have a quadratic potential, which can be used e.g. to induce chaotic inflation if φ is identified with the inflaton. The above description in terms of a 4-form is a way of gauging a shift global symmetry for a scalar field without introducing new degrees of freedom. One can formulate the same system by using the dual 2-form B 2 instead of the scalar φ. Here, as in [14] , C 3 gets massive by combining with B 2 , yielding a massive degree of freedom. One then obtains an action of the schematic form [14,18] L = −F 2 4 − µ 2 2 |dB 2 − C 3 | 2 + . . . (1.5) This action is obviously invariant under a gauge transformation which corresponds to the above shift symmetry. This shift symmetry is expected to be broken in a complete theory by non-perturbative effects. However, what makes this elaborated construction of a simple quadratic potential interesting is that the symmetries will protect the potential from perturbative and Planck suppressed corrections.
Indeed, gauge invariance of F 4 and the shift symmetry of φ force the corrections to appear in powers (F 2 4 /M 4 ) n , with M the ultraviolet cut-off of the theory, rather than arbitrary powers of φ. Thus corrections to an inflationary potential should appear as powers of V 0 /M 4 p , which will be very small for an inflaton potential V 0 < (10 16 GeV ) 4 . This is crucial to get stability of large field inflation in these schemes. This Kaloper-Sorbo Lagrangian is a 4D field theory avatar of a somewhat analogous structure found in the monodromy inflation models of [18][19][20][21]. In those models large field inflation is attained by coupling an axion-like periodic field to an external source of energy, like e.g. a brane tension. Upon a period the field gets a shift in energy, so that the field does not come to the same point but rather perform a large trans-Planckian excursion. In the recent paper of Marchesano, Shiu and Uranga [18] it has been explicitly shown how a structure analogous to that of the KS Lagrangian appears in specific string constructions.
In the present paper we study in a systematic way the role of Minkowski 4-forms in Type II, D = 4, N = 1 orientifold vacua and discuss to what extent the above discussed 4-form avatars do appear in compactified string theory. We also study the connection between the internal RR and NS fluxes abundantly used in moduli fixing and the Minkowski 4-forms. We analyse in more detail the case of Type IIA orientifold N = 1 flux vacua, in which the discussion is more transparent, but also present analogous results for the Type IIB case. In the former case some of the conclusions are as follows • RR and NS closed string fluxes through internal cycles are in one to one correspondence to Minkowski 4-forms. These 4-forms act as auxiliary fields of both Kahler and complex structure moduli as well as for the N = 1 supergravity multiplet.
• The full dependence of the flux scalar potential on RR and NS axions goes always through combinations of Minkowski 4-forms. As a result the scalar potentials of string flux vacua are not any random sugra potential but have a branched structure. The potential has the general form (1.7) Here M a denote collectively both Kahler and complex structure moduli, and ImM a denote the RR and NS axions. The functions Θ i come from the Type IIA Chern-Simons couplings and contain polynomials of the axion fields with coefficients involving linearly the internal fluxes. V local contains the contribution of the D-branes and orientifold planes to the potential, which can be re-expressed in terms of the ReM a upon imposing RR tadpole cancellation. Upon applying the equations of motion for the 4-forms the full scalar potential is obtained, with an axion dependent part which is always positive definite.
• The above scalar potential is in some sense a string multi 4-form and multi-flux generalisation of the Kaloper-Sorbo structure in which the quadratic potential is replaced by more general (up to order six) polynomials. The role of the shift symmetry is played by the duality symmetries of the compactified theories. Under R ↔ 1/R duality symmetries the different Minkowski 4-forms transform into each other. As in the KS field theory model, gauge invariance of the 4-forms combined with the duality symmetries of the compactification constrain the corrections to the potential to come suppressed by powers of V 0 /M 4 p . This shows that flux string vacua is a natural arena to construct large field inflaton models with a stable potential.
The structure in eq.(1.7) resembles the one discussed by Beasley and Witten in the context of M-theory compactified in G 2 manifolds X in the presence of G 4 flux [22].
They found that, although the superpotential W depends explicitly only on the G 4 flux supported on X, it also describes the breaking of SUSY by G 4 flux in Minkowski.
The resulting scalar potential is also branched, in analogy with the Schwinger model in two dimensions [23].
This structure leads to families of scalar potentials parametrized by specific flux choices, some of which are related by orbits of duality transformations. As expected, there can be transitions from one potential to another by membrane nucleation. This has been analysed in a context similar to ours in [24]. The membranes in Type IIA come from D2, D4, D6 and D8-branes wrapping even cycles (for RR 4-forms) and NS5 branes wrapping 3-cycles (for NS 4-forms). Analogous conclusions hold for Type IIA vacua with geometric fluxes. In this case the nucleating membranes will be KK5-branes wrapping 3-cycles. A similar story also applies to N = 1 Type IIB orientifolds with RR and NS fluxes, which we describe more briefly. We also briefly touch upon the issue of non-geometric fluxes. In the Type IIB case the natural objects which appear are complex 4-forms, involving the complex dilaton as well as both RR and NS fluxes in their definition.
We also suggest that the above structure of symmetries may provide for a new way to obtain an interacting theory of scalars in which stability against loop corrections may be obtained. This would be a consequence of the multi-branched structure of the axion scalar fields yielding a corrected potential which is itself an expansion in powers of the uncorrected potential. We also speculate about possible applications of this idea.
The structure of this paper is as follows. In the next section we recall a few facts about Minkowski 4-forms in general. In section 3 we study the structure of Minkowski 4-forms in Type IIA orientifolds with RR and NS fluxes. We perform the dimensional reduction starting from the D = 10 Type IIA action and focus on the couplings of the Minkowski 4-forms. We show how they behave as moduli auxiliary fields and how they are invariant under a class of discrete symmetries involving both RR and NS axion shifts as well as internal flux transformations. We also discuss in the toroidal case the action of R ↔ 1/R dualities as well as how the introduction of geometric fluxes modifies the setting. In section 4 we address the case of Type IIB orientifolds and how in this case the RR and NS 4-forms combine to yield complex auxiliary fields, but a structure otherwise analogous to that of the Type IIA case. We also discuss briefly how 4-forms may arise from the open string sector, by dimensionally reducing the duals of the F 2 gauge field strengths, and discuss in some detail the example of reference [21].
In section 5 we present a general discussion of implications of the uncovered symmetry structure for the stability of scalar potentials against perturbatione corrections. We briefly discuss the case of inflation and a possible new way to obtain naturally light interacting scalars. Some conclusions are left for section 6.

Minkowski 3-forms
Before turning to Type II orientifold compactifications, let us recall a few facts about 3-forms (see e.g. [8,9,14,15,25,26]). The bosonic action of a 3-form includes terms Here S bound includes some boundary terms which do not modify the equations of motion and will not play a role in our discussion, so will not be displayed here. On the other hand S mem describes the possible coupling of C 3 to membranes, i.e.
where the membrane charge q has dimensions of mass 2 and D 3 is the membrane world volume. Away from the membranes the equations of motion for C 3 force F 4 to be constant, i.e.
where f is a constant. In the presence of membrane domain walls, the value of this constant varies as ∆f = q as one goes across the wall. As argued e.g. in [8] the value of the 4-form in string theory is quantized in units of the membrane charge, i.e.
In the case of generic string compactifications we will have multiple 4-forms, some coming directly from dimensional reduction and others upon expanding higher order antisymmetric RR or NS tensors in harmonics in the compact directions. In addition, as we will see, unlike the BT or BP scenarios, the 4-forms have couplings to the axions and moduli of the compactification, with a structure for each 4-form with θ a function of the axions and moduli. Upon integration by parts the second piece may be written as This current is conserved, i.e., ∂ ν J νρσ = 0 and the action is invariant under the gauge transformations.
For 4-forms in which θ(φ i ) is just a linear function of a RR or NS axion field, the structure of its contribution to the action is analogous to a Kaloper-Sorbo action. In this case one can dualise the axion into a Minkowski 2-form in the usual way, with Then the φF 4 coupling becomes indicating how through a Higgs mechanism the 3-form gains a gauge invariant mass by swallowing the 2-form. This is the dual of the axion becoming massive in the KS setting, and is what Dvali used for his reinterpretation of the QCD axion physics [14].
The 3-form and 2-form have then gauge transformations This leads to a massive 3-form multiplet, which now contains a massive scalar degree of freedom. This structure of a massive scalar may be connected also with torsion cycles in string compactifications, as emphasized in [18].
Massless 3-forms may be embedded into N = 1 supersymmetric multiplets. They naturally appear as auxiliary fields in non-minimal versions of the N = 1 chiral multiplet [27][28][29][30][31][32][33][34][35][36][37]. And essentially correspond to replacing one or both of the real auxiliary fields of a chiral multiplet by corresponding 4-forms. Similarly, one can formulate non-minimal N = 1 sugra multiplets with one or two real scalar auxiliary fields being replaced by 4-forms. Still these type of multiplets have not been discussed much in the literature. In [33] the SUSY action of a non-minimal chiral multiplet S including one 4-form auxiliary field is discussed in detail. The corresponding superfield may be defined as where V is a real multiplet with the same content as a standard vector multiplet, but with the vector field replaced by ǫ µνρσ C νρσ . The chiral S field has then an expansion with F = ǫ µνρσ F µνρσ and D an auxiliary real scalar. This multiplet contains onshell one complex scalar M and one Weyl fermion λ. It can combine with a linear supermultiplet L, which includes a 2-form antisymmetric field B 2 , to yield a massive 3form multiplet. This is a SUSY generalisation of the Higgs mechanism described around eq.(2.9). In addition these non-minimal chiral super fields S can have superpotential couplings in superspace, i.e.
where W a denotes derivative with respect to S a . Using the equations of motion for C 3 one gets F a = Im(W a ) + f a , with f a a constant. Then the scalar potential has the form [29,33,34] This agrees with the result obtained for standard chiral multiplets with the replacement Let us advance that this multiplet is not enough to describe the structure of 4-forms that we find in Type IIA and IIB orientifolds. In particular we find that for the Kahler(complex structure) moduli in IIA(IIB) orientifolds both auxiliary fields of a chiral multiplet are replaced by 4-forms.

4-forms in Type IIA orientifolds
We turn now to describe how 4-forms appear in Type IIA orientifold compactifications down to four dimensions. The compactification of ten-dimensional massive Type IIA string theory on a Calabi-Yau threefold in the presence of background fluxes has been thoroughly studied in e.g. [38][39][40][41][42]. Here we perform the same compactification but keeping trace of all the Minkowski 4-forms which appear upon dimensionally reducing the 10d RR and NSNS fields. This leads to a new formulation of the scalar potential in terms of Minkowski 4-forms as in eq.(1.7) and the intriguing result that the full dependence of the flux scalar potential on RR and NS axions comes only through couplings to the said 4-forms.

4-forms , RR and NS fluxes in IIA orientifolds
Let us consider Type IIA string theory compactified on a Calabi-Yau threefold Y in the presence of O6 planes. The massless ten-dimensional bosonic content of the closed string spectrum contains the metric, the dilaton and the antisymmetric two-form B 2 from the NS-NS sector and the p-form fields C p from the RR sector. We will work in the democratic formulation [43] in which all the p-form fields C p with p = 1, 3, 5, 7 are present, so we will have to impose the Hodge duality relations at the level of the equations of motion in order to avoid overcounting of the physical degrees of freedom. The gauge invariant field strengths are defined as [40,43] where H 3 = dB 2 , F p = dC p−1 and F is a formal sum over all the RR fluxes F p . The background field strength G 0 may be regarded as the mass parameter (also known as Romans mass) of massive Type IIA supergravity, G 0 = −m. The massless 4d fields (before introducing the fluxes) are in one-to-one correspondence with the harmonic forms of the internal manifold Y, so the multiplicity is counted by the dimension of the cohomology groups H (p,q) (Y ). To implement the orientifold projection we split the harmonic forms into forms with even or odd parity under the orientifold projection.
The elements of the cohomology basis satisfy the following relations, where ω,ω, α denote a 2-form, 4-form and 3-form respectively. Notice that since the volume form is odd under the orientifold projection and the Hodge star involves contraction with the volume form, the dual form of an odd 2-form ω a is actually an even 4-formω a . Therefore ω a ∈ H The metric, the dilaton, C 3 and C 7 are even under the orientifold projection while B 2 , C 1 and C 5 are odd.
We are interested in the presence of Minkowski 3-form fields in the fluxed induced scalar potential. In addition to the universal RR 3-form C 3 one can also get 3-forms by dimensionally reducing higher RR and NSNS fields, C 5 , C 7 , C 9 and H 7 , and considering three of the indices in Minkowski space. By allowing also for the presence of internal fluxes, the RR field strengths can be expanded as The parameters e 0 , e i , q i , m refer to internal RR fluxes on Y and we get 2h Similarly the NS H 3 background is intrinsically odd under the orientifold projection so it can be expanded as while the dual H 7 can be expanded in terms of even 3-forms obtaining h + 2,1 + 1 additional Minkowski 4-forms H I 4 coming from the NSNS sector. Moreover, the fields B 2 and C 3 can be expanded as where b i and c I 3 are 4d scalars and correspond to the axionic part of the complex supergravity fields T, S, U as follows, The Hodge dualities of eqs.(3.1) relate the Minkowski 4-forms and the internal magnetic fluxes as we proceed to explain in the following. Separating each field strength into Minkowski and internal parts and using eq.(3.6), the duality relations given by (3.1) imply 4k ω i ∧ * ω j is the metric in the Kahler moduli space, k is the volume and k ijk the topological triple intersection number.
The type IIA ten dimensional supergravity action can be divided into three terms, where the RR and NSNS actions are given by and S loc refers to the contribution from localized sources like D6-branes and O6-planes.
Let us start analyzing the part of the action involving the RR fields. By using the duality relations (3.1), the kinetic terms for the RR fields can be written as Plugging eqs.(3.6)-(3.9) into the above RR action and integrating over the internal dimensions we get the following effective scalar potential in four dimensions This scalar potential can be rewritten by using eq.(3.13) in the general form depending polinomially on the axionic fields and the internal fluxes. Analogously, the kinetic term for the NSNS field leads to the following contribution, where c IJ = β I ∧ * β J is the metric on the complex structure moduli space. By Hodge duality the Minkowski 4-form background is related to the NS internal flux by * H I The contribution from the localized sources can be written as [40] where T a is the tension of the object and Σ the worldvolume. Assuming that tadpole cancellation is satisfied, this contribution can be related to the fluxes and the real part of the moduli so that [40] with s, u i , v i the real parts of the S, U i , T i moduli respectively. which is independent of the configuration of localized sources as long as they preserve N = 1 supersymmetry.
Combining all the pieces and using (3.13) we get the following scalar potential which in terms of the moduli and the internal fluxes becomes as has been previously obtained in the literature [39]. This potential can also be recovered from the standard Cremmer et al. supergravity description in terms of the N = 1 4d effective Kahler potential and superpotential, see [38].
We would like to recall that the full axionic part of the scalar potential can be written in terms of the above couplings to Minkowski 3-form fields and it is always positive definite. This is one of the main results of the paper.
It is worth mentioning a subtlety regarding the process of integrating out the 3-form fields. By looking at (3.18) the equation of motion for the 3-form field implies where c is a constant and ρ the function depending on the axionic moduli defined in The underlying well-defined structure of the scalar potential in terms of the 4-forms is also remarkable. In this description it is clear that the solution of minimum energy will correspond to have all 4-forms vanishing, and can be obtained by solving eqs. (3.13) and (3.21) in which the left side of each equation is equal to zero. We recover then the AdS minima with b i = q i /m previously studied in detail in [41,42,44]. This suggests that moduli fixing might be more intuitive in terms of 4-forms.
Note that there are in total 2h 11 − 4-forms, denoted above as F i 4 and F a 4 , which act as auxiliary fields for the h 11 − Kahler moduli of the compactification. This means that the SUSY multiplets associated to the Kahler moduli should contain two 4-forms acting as auxiliary fields. On the other hand there are h 3 + 4-forms H I 4 associated to the h 3 + complex structure fields. In this case the associated SUSY multiplets would only include one 4-form auxiliary field, like the multiplets discussed in [33]. In addition there are two 4-forms F 0 4 , F m 4 which seem to be associated to the N = 1 supergravity complex scalar auxiliary field. In this connection the relation imposed by the equations of motion between the 4-forms and the moduli of the compactification is interesting.
By looking at eqs. (3.13), the Minkowski 4-forms satisfy where W is the N = 1 type IIA RR superpotential given by It would be interesting to understand if this structure is consequence of the possible identification of 4-form fields as auxiliary fields of the moduli/gravity multiplets. More generally, it would seem that non-minimal N = 1 supergravity formulations, with auxiliary field scalars replaced by Minkowsk 4-forms, as in refs. [27][28][29][30][31][32][33], could be the appropriate formulation to describe the multi-branched nature of string flux vacua.

Symmetries
The above effective action features remarkable shift and duality symmetries which play an important role in constraining the structure of the scalar potential. In particular the latter is invariant under discrete group transformations acting both on the moduli and the internal fluxes. They correspond to shifts on the axionic components of the Kahler and complex structure moduli combined with the corresponding changes on the internal fluxes. In particular, a shift on the Kahler axion given by The metric is given in general by and transforms under the two T-dualities as The RR part of the scalar potential is invariant under this T-duality if the functions defined in (3.19) are also interchanged Therefore T-duality seems to exchange Minkowski 4-forms with each other. Recall that each 4-form comes from dimensionally reducing the field strength of the different higher dimensional RR fields. Then it can be checked that the result matches with the known transformation rules for the RR fields under T-duality, propagates along the T-dual direction (3.45) propagates along the T-dual direction (3.46) Finally, if the internal manifold is T 6 we can perform a T-dual transformation along all the internal dimensions, obtaining and the potential is invariant if consistent with the transformation rules for the RR fields. Note that the fact that T-dualities relate the different 4-forms make that e.g. only the full V RR combination, involving all 4-forms, will be invariant under dualities and shift symmetries, we will come back to this issue in section 5. Let us conclude by mentioning that non-vanishing values for the 4-forms will generically break SUSY (since they are auxiliary fields).
However the discrete symmetries will remain unbroken, since the 4-forms are invariant under them.

4-forms and geometric fluxes in toroidal Type IIA orientifolds
It is known that beyond standard RR and NS other, less well studied NS fluxes may be present. These include the geometric fluxes in toroidal models that appear in the context of Scherk-Schwarz reductions. In this section we will just explore whether the addition of these fluxes change in any important way the above discussion. We will take an effective viewpoint on geometric fluxes and focus on the essential results.
See [40,42] and references therein for a more thorough discussion of geometric fluxes.
We are interested to see how the presence of geometric fluxes change the 4-forms described in eqs. (3.13). Geometric fluxes are easiest described on a factorized 6-torus ⊗ 3 i=1 T 2 i with O6-planes wrapping 3-cycles. In addition we assume there is a Z 2 × Z 2 orbifold twist so that only diagonal moduli survive projection. In this case we are left with 3 Kahler moduli and 4 complex structure moduli (including the complex dilaton).
In this setting there are 12 geometric fluxes ω M N K that are convienently put in a 3-vector a i and a 3 × 3-matrix b ij , see [42,45] for notation.
Geometric fluxes can be used to convert a p-form into a (p+1)-form via: In particular we find: From an effective viewpoint, geometric fluxes change the field strengths of B, C 3 and C 5 as follows [40]: Putting these field strengths in the in the IIA action and integrating over the internal dimensions as before we find an extra coupling in the NS sector, 55) and two in the RR sector, In this way the 4-forms get modified as The intersection numbers k ijk are equal to one if all the indices are different and 0 otherwise, since we have a toroidal compactification space. It can be shown that the scalar potential that is obtained from these 4-forms and eq.(3.24) can also be obtained from the superpotential given in [40,42].
One interesting question is how the discrete symmetries are modified in the presence of geometric fluxes. One finds that the 4-forms are still invariant under shifts of the axion in the Kahler moduli and complex structure moduli in combination with the shifts of the previous section. All in all, the general structure for 4-forms we described above remains in the pressence of geometric fluxes, as expected.

4-forms in Type IIB orientifolds
We turn now to the case of Type IIB D = 4 orientifolds. We concentrate on 4-forms coming from the closed string sector but we also briefly mention an example of 4-form arising from the open string sector.

4-forms and the IIB flux induced scalar potential
Compared to Type IIA the structure in IIB [46,47] is in principle slightly simpler because the CS couplings are simpler. Only the NS H 3 and RR F 3 tensors have a role in the context of CY N = 1 orientifolds. It is convenient to define the complex 3-form where S is the complex dilaton, S = 1/g s + ic 0 . The relevant piece in our discussion are the kinetic terms of the 2-forms, which in this complex notation may be written as As we did in the Type IIA case we can now expand G 7 in terms of internal harmonics with coefficients given by Minkowski 4-forms. We will consider here only IASD G 3 fluxes of class (3, 0) and (2, 1), which can induce SUSY-breaking. The contribution from ISD G 3 fluxes does not depend on the moduli and it is proportional to the topological number giving the flux contribution to the D3 RR charge [11,48], so it appears (combined with the contribution from localised sources) in the tadpole cancellation conditions. Then the relevant expansion is given by where Ω is the holomorphic (3, 0) form, and the χ a form a basis of the h 21 3-forms in the CY X. Here G 0 4 ans G a 4 are complex Minkowski 4-forms which may be written in terms of NS and RR pieces F 4 , H 4 as The basis of (2, 1) forms may be expressed in terms of the holomorphic 3-form Ω and the complex structure Kahler potential K as [49] χ a = ∂Ω ∂U a + K Ua Ω . From the term G 3 ∧Ḡ 7 we then get the kinetic terms for the Minkowski 3-forms and some couplings to the dilaton and complex structure moduli. For G a 4 the coupling is given by where W GV W is the Gukov-Vafa-Witten superpotential and D a are the Kahler covariant derivatives with respect to the complex structure fields U a . For the remaining 4-form G 0 4 one gets the coupling The kinetic terms of the 7-form yield the quadratic pieces where G ab is the metric of the complex structure fields and where K c.s. (U a ) is the Kahler potential of the complex structure moduli. Collecting all the pieces, the ten dimensional action (4.2) reduces to the following four dimensional effective lagrangian in terms of the Minkowski 4-forms, By inserting eqs. (4.11) in the Lagrangian (4.10) we get the following scalar potential where we have used that K S = −log(S + S * ) and redefined g 0,a ≡ (f 4 − iSh 4 ) 0,a . If the shifts vanish, we recover the standard formula for the N = 1 supergravity scalar potential. Note that, due to the no-scale structure (there is no dependence of the superpotential on the Kahler moduli), after using the equations of motion for the 4- Before closing this section let us make a few remarks about non-geometric fluxes [44,50] in toroidal Type IIB orientifolds, see e.g.( [45]) for a brief description of these fluxes.
Non-geometric fluxes are still poorly understood although their existence is implied by T-dualities. They are known to induce additional terms in the superpotential. Type IIB orientifolds allow only for so called Q-type non-geometric fluxes (we use notation in [44])). The fluxes have index structure Q N P M with antisymmetric upper indices and they are odd under the O(3) orientifold involution. In IIB there are no geometric ω M N P nor non-geometric R M N P fluxes which are even. The effect of the Q-fluxes on the Gukov-Vafa superpotential is captured by the replacement where the 4-form with T i the three diagonal Kahler moduli and Going back now to the 4-forms in IIB, eq.(4.7) gets modified as (4.16) The right-hand side is nothing but the Kahler derivative (with respect to the complex structure) of the extension of the GVW super potential to include non-geometric fluxes.
So it seems that also in the presence of this class of non-geometric fluxes the structure of the Minkowski 4-forms acting as auxiliary fields in the effective action persists. In the open string sector of Type II string theory, Minkowski 3-forms may arise from the dual magnetic potentials of the worldvolume gauge fields of the D-branes.

Minkowski 4-forms and open string moduli
In particular, for a D7-brane the magnetic gauge potential is a 5-form A 5 , whose field strength can be expanded as where ω 2 is a (2,0)-form associated to the position modulus Φ of the D7. This field can be expanded as Φ = φω 2 where φ is a 4d complex scalar. Notice that unlike the 4-forms coming from the closed string sector, now F 4 is a complex Minkowski 4-form. We are going to focus on the Abelian case, but a priori it could be generalised to non-Abelian gauge groups.
Consider ISD G 3 bulk fluxes inducing a B-field on the brane given by [21,[51][52][53][54][55]] where we have denoted the non-supersymmetric ISD (0,3)-flux as G ≡ G123 and the supersymmetric (2,1)-flux as S ≡ ǫ 3jk G 3jk (see [55] for notation). The relevant part of the DBI action to leading order in α ′ , ie. in the Yang-Mills approximation, is given by [21,51,55] S DBI = µ 7 σ where σ = 2πα ′ . Plugging the decomposition (4.17) into the above Lagrangian and performing dimensional reduction we obtain leading to the following effective four dimensional Lagrangian Here ρ = S 4 ω 2 ∧ * 4ω2 and we have used that * 4 ω 2 = −ω 2 . Upon integrating out the 3-form field we get where f is an integration constant which can be identified with the magnetic flux presence being important to keep the potential under control in large field inflationary models. The idea again is that the underlying shift symmetry and the gauge invariance of the 3-form field protects the potential from dangerous higher order corrections, as we will discuss in section 5. Once a specific branch is chosen, ie. the flux background is fixed, we can inflate with the position modulus inducing the monodromy and allowing for large field excursions. Let us finaly note that we recover only half of the complete scalar potential because we are missing the Chern-Simons part of the action, which because of supersymmetry will give the same contribution as in eq.(4.23) (see [56] for a similar computation of the flux-induced scalar potential obtained from the Chern-Simons action of a D5-brane, in which the authors also keep explicetely the presence of the 4-forms coming from the RR fields).
In addition to the above quadratic piece, the position modulus can also have couplings with other matter fields. Yukawa couplings with the D7 Wilson lines A I can be obtained by considering the non-abelian part of F 2 in eq. (4.20) and can also be written in terms of the Minkowski 3-form field.
One can think of exploring a similar structure within Type IIA orientifolds. The magnetic gauge field is a 4-form A 4 , which has to be expanded in a basis of 1-forms on the D6-brane 3-cycles in order to get a Minkowski 3-form field. This can be done e.g. in toroidal models, recovering the T-dual picture of the D7-brane models discussed above, and also CY's with appropriate 3-cycle topology. One can expand also the magnetic gauge field in a basis of torsion cycles obtaining the models of Massive Wilson lines discussed in [18]. We leave the study of this IIA case for future work.
where r, u, s i , t a are integers and the c's are coefficients depending on the non-axionic components of the Kahler and complex structure moduli. F 0 , F i , F a , F m are the 4-forms discussed in section 3 (contracted with the Levi-Civita tensor), and V non−axionic collects pieces of the potential which will not depend on the axion fields, like the local contributions in V local discussed in section 3. Note that for a single factor, e.g t 1 = 0, g 11 = 1 with the rest of the integers vanishing, one gets the familiar Kaloper-Sorbo structure of the type and a discrete symmetry q 1 → q 1 − m, b 1 → b 1 + 1. As claimed in [15], due to this symmetry and the gauge invariance of axions, in the sense that they may have masses and Yukawa couplings. They could be called multi-branched axions since they feature a discrete shift symmetry as long as internal fluxes are also shifted. This is the branched structure which has appeared in the past in the context of F-term monodromy inflation [18]. Moreover, the quadratic potential of Kaloper-Sorbo is replaced by more general polynomials up to order six.
In fact the situation in string theory is often much simpler than what eq. (3.24) so that actually the corrections to the RR potential will appear as an expansion in powers of V RR itself. More generally, the full duality group of a specific Type II orientifold would often force the corrections to the original potential to be an expansion in powers of the leading potential potential V 0 , i.e.one expects in these cases where V 0 is the tree level, leading order in α ′ , flux potential.
In the applications of this setting to inflation, the higher order corrections in V 0 , although under control, may be non-negligible in the particular case of large field inflation. In fact in that case they may lead to a flattening effect [57] so that, although for small field the uncorrected V 0 gives an appropriate description, the asymptotic behaviour of the potential for large field gets flattened. This has been observed in the context of certain monodromy inflation models. It may also appear as an effect of the interaction of the inflaton with heavy modes. This flattening effect is important, since e.g. it makes standard quadratic chaotic inflation become e.g. linear and be consistent with Planck/BICEP limits. Note however that this flattening effect does not modify the value of masses at the origin.
In models of inflation, in which the inflaton appears in the Dirac-Born-Infeld action of a D-brane, the α ′ corrections to the scalar potential do appear as a series expansion in the leading scalar potential V 0 , see e.g. [18,19,21,58], in agreement with (5.4). It arises upon expansion of the square root in the DBI action or, in the case in which the scalar is an open string mode, due to a non-canonical redefinition of the scalar kinetic term. An example of this effect is discussed in [21] in which the inflaton (which in this case is identified with a MSSM Higgs) has an action with terms of the general form where in this case V 0 is just quadratic and ξ is a constant factor proportional to α ′ .
After setting the kinetic term in canonical form, α ′ corrections to the potential appear as a power series in V 0 , giving rise to a linear behaviour for large φ. In this case φ Let us also note in closing that kinetic term redefinitions like that appearing in eq.(5.5) also appear in computing the higher derivative corrections to general N = 1 supergravity Lagrangians, see [59,60] and references therein.

Multi-branched axions, scalar stability and naturalness
There are essentially a two well stablished ideas in order to make stable scalar masses against loop corrections and get naturally light scalars, i.e. naturalness. One of them A very important difference with the axion or, in general, Goldstone boson symmetry is that the symmetry is consistent with interactions. Consider as an explicit model that provided by the RR potential V RR of Type IIA string compactifications in section 3. The potential presents polynomial interactions of the scalars b i , which are still consistent with shifts b i → b i + n i as long as the e 0 , e i , q a parameters are transformed as in eq.(3.34). There is in fact a landscape of potentials corresponding to different choices for these parameters, and the symmetries relate different potentials and vacua. In view of these symmetries and the landscape structure, the perturbative corrections to this scalar potential should appear as a power series in the tree potential V RR . One expects for the corrected potential to a be a function V = f (V RR ) of the classical potential.
Assume that the uncorrected potential V 0 RR (φ i ) has a minimum at some values φ i = φ 0 i , Then at that minimum so that the corrected potential has also in general a extremum there. For the masses at that point one then has and for ∂f ∂V RR > 0 the corrected potential has also a minimum there.
The obvious observation is that both mass matrices are proportional and hence, for an analytic function f , if the uncorrected potential has a massless scalar, the corrected potential will also have a massles scalar. There exists a sort of scalar chirality for the scalar potential if the structure (5.4) is true. In particular it would seem that modifications like e.g. a quadratic loop divergence for the masses of scalars should be forbidden, since they would violate explicitly this property. One way to understand this is to think that, from the effective field theory point of view, upon renormalisation we should have to use a regulator which is consistent with 4-form gauge invariance and the shift and duality symmetries, and that such a regulator should not allow for a quadratic cut-off. While working on a given branch, a perturbative calculation would yield e.g. a quadratic scalar divergence for interacting scalars like these. However that would violate the branched structure of the theory. Once taking the latter into account such divergences should be forbidden by the symmetries.
It is tempting to speculate about the application of this stability property to the SM hierarchy problem and the Higgs. Although the Higgs field is not directly an axion, there are string constructions in which the Higgs degrees of freedom may be identified with complex Wilson lines (see e.g. [21] and references there in) or their T-dual, D-brane position moduli. In such a case the scalar Higgs may posses a multi-branch structure and one could conceive such an scenario. The symmetries would forbid quadratic divergences for the Higgs mass. Still the scalar potentials considered in this paper contain only moduli and no gauge interactions with charged fields. Furthermore, in order to be useful, one should be able obtain a SUSY breaking scale much larger than the discrete symmetry breaking scale, so that it is the latter which is mantling scalars light rather than SUSY. In this respect one should play with the different flux degrees of freedom and compact volumes, which requires a complete scheme of moduli fix ing in De Sitter. It would be interesting to see whether models with these characteristics and the property (5.4) can be built. We leave the study of that possibility to future research.

Conclussions and outlook
In this paper we have studied the role of be of interest for large field models of inflation in the string theory context, particularly in models of F-term monodromy [18], which directly contain this Kaloper-Sorbo structure. The symmetries will protect the corresponding axion/inflaton when featuring trans-Planckian trips. Furthermore one expects the loop corrections to preserve also the structure in (5.4). In particular we have argued that quadratic divergences are not expected to appear for these multi-branched axions even though they can have non-trivial couplings and masses and no supersymmetry. This would provide for a new mechanism to maintain interacting scalars stable against quadratic loop corrections.
We hope to report on possible applications of these ideas in future work.