Pole N-flation

A second order pole in the scalar kinetic term can lead to a class of inflation models with universal predictions referred to as pole inflation or $\alpha$-attractors. While this kinetic structure is ubiquitous in supergravity effective field theories, realising a consistent UV complete model in e.g. string theory is a non-trivial task. For one, one expects quantum corrections arising in the vicinity of the pole which may spoil the typical attractor dynamics. As a conservative estimate of the range of validity of supergravity models of pole inflation we employ the weak gravity conjecture (WGC). We find that this constrains the accessible part of the inflationary plateau by limiting the decay constant of the axion partner. For the original single complex field models, the WGC does not even allow the inflaton to reach the inflationary plateau region. We analyze if evoking the assistance of $N$ scalar fields from the open string moduli helps addressing these problems. Pole $N$-flation could improve radiative control by reducing the required range of each individual field. However, the WGC bound prohibiting pole inflation for a single such field persists even for a collective motion of $N$ such scalars if we impose the sublattice WGC. Finally, we outline steps towards an embedding of pole N-flation in type IIB string theory on fibred Calabi-Yau manifolds.


Introduction
The paradigm of cosmological inflation seemingly explains the origin of spatial homogeneity and isotropy, as well as the seeding process for cosmic structure formation. However, its physical origin remains unclear. High precision studies of the cosmic microwave background have revealed the primordial curvature perturbation to have extremely simple statistics: gaussian to very high precision and describable by just two numbers, the amplitude and tilt of its power spectrum [1]. It is therefore important to identify what microphysical mechanism could be at the origin of this observation. One possibility is that there is a symmetry present in the underlying theory which ultimately forbids contributions to the inflationary potential capable of giving rise to features in the observed data, i.e. a fundamental reason why the inflation potential must take a simple form. A more recent suggestion is that a simple spectrum could be an emergent property of some type of large N dynamics, be it through a large number of terms contributing to a single scalar field potential [2][3][4] or by the interaction of a very large number of scalar fields [5,6]. A third possibility is that there is some structure in the underlying theory which makes inflation insensitive to a broad array of microphysical details. This would essentially give rise to a universality class, where a diverse range of models result in the same predictions for observable quantities.
A particularly dramatic example of this third possibility is the universality displayed by the class of models termed 'pole inflation' [7,8] (or 'α-attractors' for a special subclass thereof [9,10]). This class of models is defined by the presence of a pole in the kinetic term, such as which renders the dynamics of inflation insensitive to the details of a generic scalar potential, provided this can be expanded around the pole as 1 These models lead to a universal prediction given by n s = 1−2/N e and r = 12α/N 2 e , where N e is the number of e-folds of expansion between the pivot scale leaving the horizon and the end of inflation. These predictions are in remarkable agreement with observations and are fully determined by the order of the pole (which sets the deviation of n s from perfect scale invariance; for general pole of order p, n s = 1 − p/N e -see e.g. [8,11]), its residue (which controls the amplitude of primordial gravity waves), and the value of N e , which depends on details of post-inflationary physics.
The kinetic structure of this class of models is of special interest as they can result from logarithmic Kähler potentials in 4D N = 1 supergravity effective scenarios, which are abundant in the context of string compactifications (a more detailed discussion is given in Sec. 2). Schematically, one can have where the first hermitian function is defined in the 'half plane' T +T > 0, while the domain of the second one is the 'unit disk' ΦΦ < 1. 2 The corresponding component of the Kähler metrics are given by K TT = 3α (T +T ) 2 or with subscripts denoting partial derivatives. Just as eq. (1.1), they have a second-order pole in the real part of T and radial direction of Φ, respectively, such that these fields can act as the inflaton. Interestingly in this context, the presence of the kinetic pole has the geometric interpretation of the existence of a boundary in moduli space. Note that, as both T and Φ are complex, the inflaton always comes together with a partner scalar degree of freedom which we will argue to be axionic. While this class of models arises in 4D N = 1 supergravity, it is important to understand if it corresponds to a low-energy effective description that can consistently be embedded in a quantum theory of gravity, like string theory. Various consistency requirements such as the convergence of the higher instanton corrections, the weak gravity conjecture or the swampland conjectures [13][14][15] place strong bounds on the nature of effective theories that can be embedded in quantum gravity.
A classic example of the importance of this type of bound is provided by natural inflation [16], where a single axion in a shift-symmetric potential is responsible for the cosmological dynamics. In the effective field theory description, the axionic shift-symmetry protects the potential from radiative corrections such that large periodicities, i.e. potentials with a decay constant f M P , can ensure radiative stability even for field displacements larger than M P . However, it is not obvious that such symmetries admit completions in quantum gravity. Specifically, the weak gravity conjecture places a bound on the axion decay constant as f M P . But compatibility with observations suggests the axion in natural inflation to have a decay constant f 5M P .
Proposals to evade this contradiction fall into two types: either they realize an axionic approximate shift symmetry via monodromy (by coupling to a 4-form field strength) [17][18][19][20], or they use assistance effects driven by several axions participating during inflation. The second case can arise through the tuned alignment of two axions [21], by arranging a hierarchy of the decay constants [22][23][24], or as a generic assistance effect driven by a large number of axions termed 'N-flation' [25][26][27][28][29]. In the latter, the total inflaton field range ∆ϕ arises through the collective displacements of individual axions, each of them satisfying the constraints of the WGC: ∆φ i ∼ f i M P . In the simplest setup of N-flation with N axions with roughly similar decay constants, it is easy to see that the field displacements are related by ∆ϕ ∼ √ N ∆φ i , such that for large enough N , super-Planckian inflaton displacements seem to be allowed. It turns out that generalizing the WGC bounds to theories with multiple axions is more subtle than simply implementing the bound f M P for each individual axion. Implementing WGC bounds for this case implies using the convex hull condition [30][31][32][33]. Which version of the WGC applies is still an open issue. For rather strong forms such as the lattice WGC [34], one finds a bound on the collective axionic motion forbidding the √ N enhancement with respect to the single field case. The situation for a single field driving pole inflation is morally similar. The universal predictions of this scenario arise when the non-canonical field approaches the kinetic pole, or from a supergravity perspective, the boundary of the moduli space. However, on generic grounds, precisely in this regime we expect numerous quantum corrections to grow large, thus potentially leading to a loss of control of the setup. We can argue both in 4D effective field theory [35] and in string theory [36][37][38] for the appearance of such dangerous terms. In this paper, we explore the possibility of using the collective behaviour of a large number N of moduli-like scalar fields to alleviate some of these problems, in what we call pole N-flation. Specifically, we propose a scenario where the approach to the pole is achieved by the assistance effect of many fields, such that each field is individually further away from the boundary by a factor of √ N . This should suppress some of the generically expected loop contributions which grow large when individual fields approach the boundary.
As a conservative estimate for the domain of validity of the effective description of both pole inflation and pole N-flation, we make use of the WGC constraints. In both cases the moduli of the simplest supergravity models are associated with axionic partners, allowing us to implement bounds on their periodicities as conditions for the consistency with ultra violet physics. We find that imposing WGC bounds on the axionic periodicities directly translates into the impossibility of getting close the boundary, and therefore to a finite inflationary plateau in canonically normalized variables. This has dramatic consequences for the viability of pole inflation in general. While this bound is very stringent for the case of a single superfield, the situation improves considerably in the context of pole N-flation, even when applying the lattice WGC. Unlike in the original N-flation scenario, in this case the collective moduli motion does allow for a field range enhancement as compared to the single field case, enabling a sustained inflationary plateau. The reason for this difference rests on the particular structure and moduli dependence of the axion field space metric in pole N-flation.
The outline of the paper is the following: in §2 we scan the typical kinetic structures derived from string theory and identify the most natural for a scenario with a pole due the collective behaviour of many fields. In §3 we will study a supergravity toy model of pole N-flation, describing the ellipsoid structure of the pole and the subsequent universality behaviour. We use these results in §4 to establish contact with the WGC and the swampland conjectures, and derive a bound on the field range in pole N-flation. With the aim of embedding these ideas in string theory, in §5 we develop an explicit scenario based on type IIB string theory on fibered Calabi-Yau manifolds. We draw our conclusions in §6. Throughout the paper, we will work in reduced Planck mass units (M P = 1).

Kinetic poles in string theory
In order to search for setups with N 1 fields and second order kinetic poles, it is illuminating to analyse the structure of kinetic terms in 4D N = 1 supergravity derived from string theory. String compactifications on Calabi-Yau manifolds generically produce Kähler potentials containing both closed and open string moduli, with the exact number of such moduli given by the underlying geometry and the amount of D-branes. We can classify the possible Kähler potentials as follows: 1. In perturbative string theory, in the large volume and large complex structure limit, there are at most 3 volume and 3 complex structure moduli which describe the total Calabi-Yau manifold. Together with the axio-dilaton, these corresponds to a maximum of 7 chiral fields, which lead to the tree-level Kähler potential with the parameters α i depending on the number of fields (see e.g. [39] in the context of α-attractors).

2.
There are open string moduli describing brane positions (e.g. D3-branes), and/or open string matter fields as well as gauge fields. Their number is usually subject to tadpole bounds and can be as large as O(10 4 ). They appear as a contribution to the volume moduli Kähler with the schematic form where the parameter α depends on the specific configuration of the bulk geometry and the Kähler moduli.
3. Finally, the moduli space of Calabi-Yau manifolds contains singular regions, most easily seen as the conifold points of complex structure moduli space. Near such singularities, the corresponding moduli acquire a Kähler potential of non-polynomial form inside the primary logarithm. For complex structure moduli near a conifold point this generically implies a Kähler potential of the form where z denotes the complex structure moduli parametrizing the vicinity of the conifold singularity, and the u i denote the remaining other complex structure moduli of the Calabi-Yau. The number of such conifold regions in a given Calabi-Yau can be quite large, easily of the order of a few tens.
From this short list we see that a realization of pole inflation involving assistance effects of a large number of fields cannot arise from the large-volume or large-complex-structure type of closed string moduli described in the first class of the list, as their number is intrinsically limited. The study of the third class in the list would require an in-depth analysis of multi-conifold complex structure Kähler potentials and their dependence on non-conifold complex structure moduli. This analysis is beyond the scope of the present paper and we leave it for future work. In this paper we therefore explore the second class, where a large number of open string moduli fields with Kähler potentials of the form eq. (2.2) lead to the pole N-flation scenario.
3 The pole N-flation picture

Kinetic structure and universality
We start our analysis by looking at the Kähler potential given by eq. (2.2). Once the Kähler modulus T is stabilized, the relevant dynamics in the EFT is described by where the new coefficients are rescaled by the VEV of T , such that A i = a i / T +T . The corresponding kinetic term is given by which has a pole for R 2 ≡ k A k Φ kΦk = 1, which is the equation of an ellipsoid in field space with N independent radii of length directly related to the brane contributions A i . As can be seen by the form of the denominator, the N fields collectively contribute to reach the boundary without any Φ k reaching the boundary itself. 3 Therefore, when all fields equally contribute to inflation, the displacement of each individual field will be reduced by a factor of √ N . This property will protect the model against radiative corrections which grow as the fields individually approach the pole.
To further understand how the presence of these fields affects the model and its dynamics, we make the following change of variables where Ω i (ψ β ) is the spherical angular element such that i Ω 2 i (ψ β ) = 1. Here the index β = 1, · · · , N − 1. As will be discussed in §4, the angles θ i can be associated with axions and will play a crucial role in our understanding of the UV consistency of the setup.
In these variables the line element becomes where ∂ β ≡ ∂/∂ψ β and In this form, the field-space metric has useful features. First, it is independent of the axionic variables θ i . It is also diagonal in the variables R and ψ β . The mixed terms associated to ∂R∂ψ β vanish due to i Ω i ∂ β Ω i = 1/2 ∂ β i Ω 2 i = 0, and so do the terms ∂ψ β ∂ψ γ for β = γ due to trigonometric relation i ∂ β Ω i ∂ γ Ω i = 0, proved in Appendix A.
One can easily identify R as the variable with a kinetic pole of second order. Upon canonically normalizing the kinetic term, one has such that the boundary at R → 1 is equivalent to ϕ → ∞. Writing the system in this canonical variable, just like in the single-field pole inflation case, makes evident how the model is stable with respect to considerable deformations of the inflaton scalar potential. The potential can be generated by means of several mechanisms: via an inflaton-dependent superpotential (with stabilizer superfield [9,41] or without it [10]), by Kähler [42,43], loop [35][36][37][38] or higher-derivative [44] corrections. A generic expansion looks like with constant coefficients b ij,p . We therefore see that in the vicinity of the pole the scalar potential decomposes into an exponential fall-off from a de Sitter plateau as with V 0 and V 1 functions of the angular variables as dictated by eq. (3.7). It is interesting to note that the residue of the pole does not depend on either A i or Ω i , leading the exponential plateau to have a universal nature. The slope of the exponential fall-off is therefore not affected by the particular radial direction in field space. The amplitude of the plateau is exclusively determined by the angular and axionic dThe WGC therefore constraints the maximum displacement in the radial directions (ψ β , θ i ). This effect can be observed in fig. 1. If inflation occurs purely in the R direction, the observable predictions of this model, regardless of the inclusion of multiple fields, retain the universality properties extensively discussed in the literature (see e.g. [7][8][9]41]) for the single field case, where N e is the number of e-folds of expansion between the pivot scale leaving the horizon and the end of inflation. However, the angular and axionic fields might play a role in the inflationary dynamics leading to multifield effects that can modify the predictions. To assess this, we need to study the hierarchies in the mass spectrum.

Scaling of mass spectrum
The hierarchies in the mass spectrum are intimately related to the eigenvalues of the field space metric given by eq. (3.4) and eq. (3.5). Using the canonically normalised field ϕ we define the parameter as a measure of proximity to the moduli boundary: We can then see that the line element scales with the proximity to the boundary as where we have neglected higher order contributions in . In general, it is not straightforward to compute the eigenvalues of this kinetic metric but we can consider specific field configurations which simplify the situation. We are Figure 1. Potential for the N = 2 case plotted in the polar coordinates {ϕ, ψ}, with angles θ 1 and θ 2 minimized. The top-left plot shows the potential eq. (3.7) up to terms with p = 1, in the case of equal A i . The top-right plot shows the same potential but for A 1 = 1 and A 2 = 0.2. The bottomleft plot shows the potential with terms up to p = 4, in the case of equal A i . The bottom-right plot shows the same potential but for A 1 = 1 and A 2 = 0.2. We can see that the elliptical structure of the model makes valleys in the potential bundle-up. While different radial directions have plateaus with different amplitudes, the exponential fall-off has the same signature for all initial conditions, or in other words, values of the angular coordinate ψ β . The oscillating plateau of the circular case were already noted for a real 2-disk α-attractor (without supergravity) in [45].
particularly interested in the regime where all the fields Φ i contribute equally when approaching the boundary, i.e. when all the branes are equally displaced from the origin and the inflationary dynamics is determined by their collective motion. In this maximally multifield case, this corresponds to the choice Ω 2 i = 1/N for any i. In this case, following Appendix A, the metric element for the angular coordinate ψ β is which ranges from 3α/2 to 3α/N . Also in this regime, close to the boundary, the metric for the axionic fields θ i eq. (3.5) can be written as where J ij is the all-ones matrix (a square matrix with all entries equal to 1). The eigenvalues of this metric are easy to compute: the matrix J ij has rank one, with one single non-zero eigenvalue equal to Tr(J ij ) = N . The identity matrix is invariant under any transformation and therefore G ij has all but one eigenvalue equal to v. We denote the corresponding eigenvectors of the N −1 equal eigenvalues, Θ a , with a = 1, . . . , N −1. The last eigenvalue, corresponding to what we define as the ϑ direction, is (3.14) At the point in field space where the metric has these eigenvalues, we can locally canonically normalize the angular fields by defininĝ and write the potential near the pole as Owing to the local canonical normalization of the kinetic terms, the mass spectrum therefore scales as for a generic scalar potential where usually O(V 0 ) ≈ O(V 1 ). The mass scaling of the N − 1 elliptical angular fieldsψ β ranges from V 0 to N V 0 , i.e. from the same scaling as the light radial field ϕ to the scaling of the heavier axionic fieldsΘ a . We should therefore allow for the possibility that some of these fields might contribute to the inflationary dynamics. These effects might lead to multifield deviations from the simplest predictions given by eq. (3.9). The N − 1 axionsΘ a , in the large N limit, correspond to a heavier sector which we do not expect to contribute significantly to the dynamics. The singleθ direction, in the limit of ϕ 1, is exponentially lighter than all other sectors and becomes a true spectator; in this deep plateau limit the dynamics ofθ is frozen in deep slow-roll and will resemble the case of the single angular field of ref. [46].
The kinetic scaling of the mass spectrum of the theory for a configuration where all fields contribute equally to inflation is illustrated on the left-hand side of fig. 2. On the right-hand side of fig. 2 we show the kinetic scaling of the mass spectrum when couplings to quantum gravity are included, as discussed in the following section. The couplings to quantum gravity limit the proximity to the boundary to be 1/ √ N ; the right-hand side plot shows the mass spectrum for the maximum value of = 1/ √ N . In this case, the angular fieldθ does not behave as a spectator and instead has a mass of order the Hubble scale. Figure 2. Mass spectrum of pole N-flation. The left-hand side shows the mass hierarchies of eq. (3.17) in the deep plateau limit, ϕ 1, and large N limit. The right-hand side shows the mass hierarchies when the proximity to the moduli boundary is constrained by weak gravity arguments. As discussed in §4, imposing consistent couplings to weak gravity limits 1/ √ N ; in this regime the fieldθ has generically a mass of order H.

The fate of inflationary plateaus -weak gravity strikes back
Effective field theories arising from a theory of quantum gravity are constrained by consistency conditions such as the Weak Gravity Conjecture (WGC) [14] or the Swampland Distance Conjecture (SDC) [15,[47][48][49]. The SDC states that whenever one moves an infinite distance in moduli space, an infinite tower of states becomes massless causing the break of the effective description. While we will make the connection between the SDC and α-attractor models later in this section, for now we focus on enforcing consistency arguments coming from the weak gravity conjecture.
The WGC arose as a proposal to argue that black holes should always be able to evaporate via Hawking radiation, such that the final state of any charged black hole would be able to decay and leave no remnant. For this to happen it is necessary that any theory of quantum gravity has at least a fundamental object fulfilling certain condition on its charge to mass ratio, such that the decay process is possible for any black hole. A number of attempts were made to further constrain the particular object fulfilling the WGC condition. These have given rise to the multiple current versions of the WGC. It is still unknown which version of the conjecture (if any) is the right one and it is not our intention to provide any new insight in this direction, so we just refer the interested reader to [31][32][33][34] for extensive discussion. Our purpose, instead, will be to apply the WGC constraints to the α-attractor models as a consistency requirement to allow for a string theory embedding of these effective field theories.
Since string theory contains fundamental charged objects of different dimension, first note that the above argument can be extended to other black objects of different dimensionality. This implies one is not completely free concerning the assignment of charges and tensions of the fundamental objects of any theory. For our purposes we will be interested in how the WGC applies to instantons. In this case, the WGC sets the bound where S E is the euclidean action of the instanton, f the decay constant of the axion coupled to it and n the instanton number/charge. This bound will be our starting point to set the limitations of α-attractors. To do so while working in a controlled effective field theory, we want to guarantee a controlled instanton expansion, which in turn implies S E > 1. Furthermore, string theory includes fundamental objects with all possible charges (see [75] for some important progress in this direction). In particular, this means that there will always exist an instanton with n = 1. Recalling that we are taking M P = 1, this implies that the above WGC bound, when applied to the single complex field case (single axion) simplifies to in agreement with evidence from string theory compactifications [13].
For effective field theories including many axions, the above argument needs to be extended. In the black hole picture this corresponds to a theory with multiple U (1) gauge factors, first studied in ref. [30]. The key factor in this case is that black holes can be charged under more than a single U (1) factor at the same time, and thus the WCG constraints cannot be implemented by considering each of these U (1) factors separately. Therefore, one needs to consider the lattice of charge-to-mass ratios of the theory and the single U (1) bound extends to the so called convex hull condition on this lattice. This condition was first translated to the axion language in ref. [31]. Again, there exist several versions of the WGC on this multiple axion case. The one we apply to α-attractor models is related to the lattice WGC [34]: starting with a set of axions coupled to instantons of charge n = 1, we consider these instantons to be the ones relevant for the convex hull condition. If the axions have initial periodicity θ i ∼ θ i + 2π, then, we can draw the corresponding convex hull by computing the eigenvalues of the axion field space metric (the squares of the decay constants after performing a change of basis in the axionic sector). In this case, the vertices of the polygon lie on top of the axis at a value given by the inverse of each eigenvalue. The convex hull condition states that this polygon needs to contain the unit ball for the effective field theory to satisfy the WGC.
Since we are setting bounds on axion decay constants, we next argue that the fields θ i in §3 are indeed axions. It is nowadays standard in supergravity to denominate axions those fields that do not appear in the Kähler potential and whose potential is periodic. In order to apply WGC arguments, these conditions are necessary but not sufficient: one needs to argue that the axion potential can only be generated by instantons. In order to do so, recall that the string theory picture corresponding to our inflationary setup corresponds to the motion of D3-branes, whose position we parametrized by the fields Φ i . It is known that in standard type IIB compactificationsà la GKP [76,77], the radial part |Φ i | of the position moduli is massless at the perturbative level unless supersymmetry is broken, while their axion phases θ i remain flat directions always in perturbation theory. So, it is necessary to also include non-perturbative objects on the compactification in order to stabilize the θ i .
The inclusion of Euclidean D3-branes will indeed generate a potential and stabilize these moduli. In the 4D supergravity language these instantons generate a non-perturbative superpotential [78,79] W

3)
T J being the superfield describing the size of Σ J where the ED3-brane is wrapped and F a holomorphic function of the D3-brane moduli as well as other geometric moduli z I , such as the complex structure ones. Note here that holomorphy of the superpotential will ensure the potential to be compatible with the axion periodicity arising from our choice of polar coordinates θ i ∼ θ i + 2π. Using this fact, we conclude that the potential of the θ i is indeed generated by instantons and compatible with the usual identification coming from each axion living on a S 1 . 4 As a final step before applying WGC constraints, note that it is always possible to rescale the axionsθ i = f i θ i , such that if the periodicity of θ i is given by θ i ∼ θ i + 2π, then forθ i it isθ i ∼θ i + 2πf i . Therefore the decay constant changes with rescaling, and so does the axion kinetic term. In the single field case, the relevant scale for WGC arguments is the one giving rise to a canonical kinetic term for the axion, or equivalently, the square root of the prefactor in the kinetic term for an axion with periodicity θ i ∼ θ i + 2π. In the multiple axion case, we need to apply the same criterion to each axion in a base where there is no kinetic mixing. Due to our initial coordinate choice where θ i ∼ θ i + 2π for all axions, this task can easily be carried out: we just need to compute the eigenvalues of the field space metric. In fact, after a change of coordinates the kinetic Lagrangian will be − i f 2 i 2 ∂ µ θ i ∂ µ θ i which is canonically normalized by the changeθ i = f i θ i , such that f i are the decay constants of interest for WGC arguments.
We are now ready to start studying the consequences for the single (complex) field case. In this case, the Kähler metric is where ΦΦ = φ 2 , and leads to an axionic partner of φ with a decay constant (4.5) As the field φ approaches the moduli boundary, the decay constant of the axionic partner diverges; this behaviour has been noticed in ref. [82]. The WGC therefore constraints the maximum displacement in the radial direction. To understand how stringent this constraint is, it is interesting to compare the WGC bound f 2 1 with the slow-roll condition SR < 1.
Assuming inflation occurs primarily in the radial direction and following eq. (3.4), we have that in the slow-roll limit (4.6) 4 We note that the 4D low-energy effective description of instantons from string theory reduces in many cases to the Giddings-Strominger type gravitational instantons [80,81]. Expanding the axion on a ninstanton background provides the usual axion-instanton term enforcing θi ∼ θi + 2π for the path integral to be well defined.
Taking the potential to be expanded as in eq. (3.7) and dominated by O(φ 2 ) terms, inflation occurs when which comparing with eq. (4.5) is equivalent to f 2 O(1). We conclude that sufficient inflation requires starting much closer to the moduli boundary than permitted by WGC considerations. This observation is independent of the value of α. The simplest supergravity α-attractor model, when embedded and coupled to quantum gravity, is therefore in direct conflict with consistency requirements coming from the weak gravity conjecture.
We now apply a similar argument to the N -field case studied above and see how the collective assistance of multiple fields alleviates these constraints. We saw in §3 that, taking polar coordinates for the complex field, the field-space metric on the axionic sector is not diagonal. In order to apply the WGC bound we choose a configuration where the inflationary dynamics is carried out by the collective motion of all D3-branes, i.e. when all fields contribute equally and all angular functions Ω i have the same value Ω 2 i = 1/N . Diagonalizing the field space metric in this configuration gives rise to the axionic fields Θ a , a = 1 . . . N − 1, and ϑ with diagonal kinetic terms The decay constants are given by eqs. (3.13) and (3.14) 5 and near the pole read Again, we see that for the limit → 0 (or ϕ → ∞) the decay constants diverge, violating the constraints from the WGC. Nevertheless, as we now show, it is possible to have configurations with rather small and yet compatible with the convex hull condition. We now study the limiting case in this regime. The largest decay constant near the boundary is f ϑ , so requiring it to be sub-Planckian provides a bound 3α 2N . (4.10) The minimum value of is the closest point to the boundary allowed by the WGC. This corresponds to the extremal case f ϑ 1, that is when one axis of the convex hull approaches very closely the unit ball (see the right diagram of fig. 3). As the other set of decay constants satisfies the relation f 2 a 2 f 2 ϑ 2 , they have smaller values and the remaining N − 1 axes can be long enough for the polygon to contain the unit ball. This is definitely the 5 Let us remark that, in α-attractor models, inflation does not involve an active axion or linear combination of axions. Therefore, the only input needed from the axionic sector of the theory are the periodicities of the canonically normalized axions. This is unlike models where inflation occurs in the axionic sector, such as N -flation [26], where it is necessary to have information about the instanton numbers on each cycle in order to compute the potential and the fundamental field-space domain of the axions; see e.g. [53,83] for a discussion of models of this type. case in the large-N limit, which allows for parametrically small . Of course, cases where f ϑ < 1 simply correspond to a bigger volume of the polygon and are again compatible with the condition imposed by the WGC. However, in this circumstance, the available range for inflation is reduced as the distance from the boundary (parametrized by ) is bigger.
Eq. (4.10) also shows that, when many fields contribute to the dynamics, the constraint on the approach to the field-space boundary is alleviated. Nonetheless, the amount of D3branes that we can include on a compactification is finite, and therefore this constraint indicates we cannot get arbitrarily close to the boundary. The maximum displacement for the canonically normalized radial field ϕ, in the large-N limit and for α = O(1), is enhanced as Unlike the single field case, the collective assistance in pole N-flation makes it possible to sustain slow-roll inflation at the plateau while fulfilling the WGC bounds. Therefore, we argue that models with a large number N of fields are better candidates when trying to embed pole inflation or α-attractors in string theory. We would like to contrast this with the case of axion N-flation. As previously discussed in the literature (see e.g. [32]) and shown in the left diagram of fig. 3, large enhancements in axion N-flation are not compatible with the lattice WGC: the larger the individual decay constants f i ≡ f are, the smaller the polygon is, and it is not possible to fulfill the convex hull condition. On the other hand, if one starts with small decay constants (f ∼ 1/ √ N ), the polygon is large and a √ N enhancement is available as compared to the possible individual displacements, but cancellation of factors of √ N in this case does not allow for super-Planckian displacements in field space. Meanwhile, in pole N-flation we showed that it is possible to obtain an enhancement while fulfilling the convex hull condition. This is possible when f ϑ 1 for N 1, since it implies that the other axions related to the inflationary process Θ a have small decay constants, fulfilling the convex hull condition. The key difference is that in the latter case we are inflating with a collective modulus ϕ which is not an axion, and we are limiting the inflaton displacement by the dependence of the axion decay constants on this modulus. This crucial difference with axion N-flation allows us to buy finite field displacement in the modulus direction at finite large N , while maintaining the convex hull condition. As a final remark, note that on the approach to the boundary (when → 0) not only the decay constants grow exponentially according to eq. (4.10), but also the masses of the axions become exponentially small as can be seen in eq. (3.17). This behaviour arises from the existence of the second order pole in the field-space metric, which requires an infinite geodesic displacement in moduli space in order to reach the boundary. So this behaviour at → 0 not only violates the WGC at finite N , but also corroborates the recent arguments in favor of the SDC provided in ref. [84,85] in that reaching the pole would require N → ∞ complex fields whose axions become light in that limit.
As a consequence of these observations, we conjecture that the truly infinite inflationary plateaus of pure supergravity α-attractor models cannot exist once embedded in string theory.

Towards pole N-flation in type IIB string theory
As we have seen in the previous sections, the Kähler potential (2.2) becomes singular exactly at the same point where the kinetic Lagrangian develops a pole. This fact poses strict limitations on the possibility of realizing the pole inflation scenario within a supergravity framework. Indeed, the F-term scalar potential V = e K (. . .) will in general not be regular at this point in field space and the inflationary plateau will be easily spoiled. To avoid this situation, one can tune the superpotential such as to cancel the pole induced by the exponential pre-factor in V , but this appears to be a quite non-generic and model-dependent situation. But even granted this possibility, we will interpret the field approaching the pole as a shrinking volume of extra dimensions in string theory. If this volume is the total volume of the extra dimensions, sending this to zero will send perturbative corrections soaring in magnitude and thus compromising control.
A rather more appealing alternative is to find a class of models where the form of K has a regular behaviour while still inducing a pole in the corresponding kinetic structure. Interestingly, stabilizing the overall volume of fibred Calabi-Yau (CY) geometries [86][87][88] using the Large Volume Scenario (LVS) mechanism [89] provides a large class of string models with a Kähler potential with the desired properties.
In the following, we will review the main characteristics of this framework and show how to embed the pole N-flation picture therein. We will also discuss moduli stabilization of this setup, pointing out its limitations given the current status of knowledge on quantum corrections. Finally, we will provide an analysis of the model's dynamics and cosmological predictions.

Pole N-flation from fibred Calabi-Yau manifolds
A large fraction of CY manifolds are K3-fibred. This means that the positive part of the CY volume takes the form in terms of the 2-cycle volumes v i , and κ 122 is the intersection number between the 2-cycles on the given Calabi-Yau manifold. The 4-cycle volumes τ i are related to the 2-cycles by allowing us to write the volume of a K3-fibred CY in terms of the 4-cycle volumes as The corresponding Kähler potential then takes the form where we have introduced the volume moduli T i , which are related to the 4-cycle volumes by means of 2τ j = T j +T j while their axionic partners are 2c j = (T j −T j )/i. Now assume the CY to possess a warped near-conifold region. Assume further that the 4-cycle Σ 4 2 with volume τ 2 reaches somewhat into the warped region. This is not particularly restrictive, as we can stabilize part of the complex structure moduli using flux near conifold points for a large fraction of all K3-fibred CYs. Finally, place a number N of D3-branes at the IR end of the warped region.
The Kähler potential for models in this class will look like where we define as before with Φ i parametrizing the positions of the D3-branes. The O7-orientifolding enforces the relation between 2-cycle volumes, 4-cycle volumes and D3-brane coordinates such that [77,90] The corresponding expression for the CY volume now reads An alternative construction might instead shift τ 1 by the D3-brane Kähler potential R 2 /2. In this case, the CY volume would become V ∼ τ 2 τ 1 − R 2 /2.
The LVS scheme of volume stabilization can now proceed if we assume the total CY to have a third pure blow-up Kähler modulus τ 3 , such that the CY volume becomes We therefore include the leading order type IIB α -correction into the Kähler potential and τ 3 acquires an ED3 instanton contribution in the superpotential, in addition to the constant piece from 3-form fluxes, such that This setup will stabilize the modulus τ 3 and the whole leading-order volume combination In order to reproduce the pole N-flation dynamics, schematically encoded by eq. (3.1), we would like to stabilize the modulus τ 2 separately. For this purpose, we first observe that the scales of LVS stabilization operate at O(V −3 ). This rules out the possibility of stabilizing τ 2 supersymmetricallyà la KKLT, by adding a non-perturbative effect to W . The resulting potential terms from the KKLT mechanism would indeed appear at O(V −2 ) and eventually spoil the LVS mechanism.
Hence, we need to stabilize τ 2 perturbatively, presumably using an interplay of string loop corrections and higher-order F -term contributions to the scalar potential, which operate starting at O(V −10/3 ). However, in the known simple cases, where we can compute some of the string loop corrections to K and the F 4 -terms in the scalar potential [44], these depend on the 2-cycle volumes v i [35][36][37][38]. Therefore, looking at expressions (5.7), these corrections do not affect τ 2 individually but rather τ 1 and the whole combination τ 2 −R 2 /2.
At this point, we content ourselves with merely pointing out as a challenge the need to explicate a perturbative stabilization mechanism which will stabilize τ 2 just by itself. From now on, we will simply assume that such stabilization for τ 2 exists.
As a final remark, we wish to point out that we could have instead looked at the case where the whole combinationτ 2 ≡ τ 2 − R 2 /2 is given a potential and is stabilized by string loop corrections such as those discussed above. For those models, one can show that the structure of the kinetic terms, in terms of τ 1 ,τ 2 , R and the angular variables ψ α , θ i , reduces to L kin. = − 1 If the potential only has contributions of the type discussed above, this setup resembles precisely the original fibre inflation setup [86] (see also [87]) in terms of the effective half-plane variables τ 1 ,τ 2 [91] except for the extra 2N massless spectator fields: 2N − 1 angular fields ψ α and θ i and one field direction given by a linear combination of R and τ 2 orthogonal toτ 2 . If in general these 2N fields are also given a potential, we expect a rich mass spectrum and possible multifield phenomenology in analogy with §3. In this work we do not study this type of model, focusing instead on the stabilized τ 2 case.

Dynamics of fibred pole N-flation and universal predictions
The effective Kähler potential of fibred pole N-flation reads once we assume stabilization of τ 2 . Note that the last contribution is identical to eq. (3.1), with 3α = 2 and up to a multiplicative factor in R. Therefore in the following analysis we can employ the results derived in §3.
The kinetic Lagrangian of the dynamical degrees of freedom is given by Applying the LVS procedure for volume stabilization forces 2τ 1 = T 1 +T 1 to be a function of R, such as with V 0 being the stabilized volume. This implies an additional contribution to the total kinetic term of R of the form Therefore, after volume stabilization, the field-space metric for the radial direction is determined by eq. (3.4) together with the contribution of the D3-branes from eq. (5.16) (with the R properly rescaled): In order to absorb the τ 2 dependence, we defineR ≡ R/ 2 τ 2 such that the kinetic term becomes This allows us to define the canonically normalized field ϕ corresponding to the radial field R as We see thatR → 1 corresponds to ϕ → ∞, which we can use to express ϕ in terms of This expression is precisely eq. (3.10) for α = 2. Using analogous arguments to those in §3, we make an expansion of the scalar potential as eq. (3.7). This generic structure of the scalar potential of Φ i as a power law series around its minimum often arises for open string moduli in setups with controlled moduli stabilization and supersymmetry breaking. For example, refs. [92,93] argue explicitly that mobile D3-branes at the IR end of the warped throat of a KKLT or LVS compactification acquire a scalar potential of the general form of eq. (3.7). This results into a computable spectrum of discrete values of p ≥ 1 while the coefficients a i,p , b ij,p are tunable Wilson coefficients except the one arising from the conformal curvature coupling of the D3-brane moduli. Finally, in analogy with §3, if the motion is purely radial, we see that for an arbitrary number N of open string moduli Φ i driving exponential plateau inflation, we arrive at as universal observable predictions. Similar to the simplest fibred inflation models, if the shift by the D3-brane Kähler potential was made on τ 1 rather than τ 2 , the effective α = 1/2. The predictions for inflation happening along the radial direction would therefore be These predictions can be altered if the angular directions are active during inflation and truly multifield dynamics takes place (see e.g. [94]).

Weak gravity conjecture implications for fibred models
Similarly to what discussed in §4, we now explore the constraints that the weak gravity conjecture imposes on fibre inflation models. In the original models with Kähler potential eq. (5.4), such as ref. [86], one of the half-plane Kähler moduli variables acquires the second order pole. There the kinetic terms of the two Kähler moduli of the bulk fibred CY manifold and their C 4 -axion partners read where T j = τ j + ic j denote the N = 1 chiral Kähler moduli multiplets. LVS stabilization imposes the constraint τ 1 = V 2 /τ 2 2 , and string loop or higher-superspace-derivative corrections decide if τ 1 = exp (κρ) or τ 1 = exp(−κρ) provides the canonically normalized inflaton, while providing a scalar potential with a minimum at τ 1 , τ 2 . Depending on whether the inflationary plateau appears at large or small τ 1 (which give α = 2 and α = 1/2, respectively, see [91]), one of the two axion decay constants diverges in the parametric limit Demanding f M P again limits the field range on the plateau, dictating ρ 12M P and ρ 4M P , respectively, using typical values for the volumes from ref. [86]. We conclude that the second branch of fibre inflation with α = 1/2 is in tension with the WGC.
We now turn to the model involving N open string moduli described by the Kähler potential eq. (5.5). In this case the kinetic Lagrangian looks like where L kin. ((∂ψ β ) 2 , (∂θ i ) 2 ) is equivalent to the angular piece of eq. (3.4) upon the transformation R 2 → R 2 /2 τ 2 and 3α = 2. In this case the approach to the pole corresponds, by volume stabilization, to τ 1 → ∞ and can be parameterized by a small parameter The full kinetic terms for the axion sector have the structure where G ij is defined in eq. (3.5). If again we focus on the case where all D-branes contribute equally to the approach to the pole, G ij contains terms ∼ Ω 2 i = 1/N and Ω 2 i Hence, at parametrically large-N the dominant axion kinetic terms are (∂c 1 ) 2 and (∂c 2 ) 2 as they are N -independent, and their axion decay constants are The decay constant for c 1 goes to zero on the approach to the boundary, whereas f c 2 diverges for → 0. Therefore, the decay constant for c 2 provides an N -independent limit on the approach to the pole, i.e. a limit on the field range of the canonically normalized radial coordinate ϕ of O(10 M P ) for such models of pole N-flation embedded into fibred type IIB Calabi-Yau compactifications.

Conclusions
Pole inflation/α-attractors is an intriguing class of models that suggests that the observed primordial power spectrum may be a universal consequence of a pole in the field space metric. That is to say, regardless of a broad range of microphysical considerations, ultimately observables are determined by just a few key parameters characterising the pole. This property is two-sided. On one hand, such a mechanism seriously limits the potential for learning about fundamental physics from cosmology, given there are many fundamental parameters one simply cannot hope to infer from cosmological data. On the other hand, such robust predictions provide an especially appealing target for future observational surveys and in principle would enable a small number of exceptionally sharp statements about the underlying theory. For example, the model predicts that primordial gravitational waves may be detectable. In the context of this model, such a detection would imply the existence of a hyperbolic moduli space [12,95], which in turn may be viewed as indirect evidence for extra dimensions. 6 To make such statements, however, it is crucial to understand the robustness of the mechanism both from a phenomenological viewpoint and from the perspective of its possible embedding in string theory or another theory of quantum gravity. Considerable progress in this direction has been done by showing that the so-called 'fibre inflation' model [86][87][88] is a string realization of α-attractors with α = 1/2 , 2 [91]. Furthermore, investigations on the effects of string moduli backreaction [96] and Kähler corrections [42] have given strong evidences of the special resilience of this attractor mechanism.
In this paper, we have taken a step forward towards a consistent realization of the pole inflation dynamics in string theory, by exploring the possibility of assistance of many fields in the inflaton sector. The proposed pole N-flation model consists of several open string moduli, such as D3-branes, whose collective motion reduces the distance each brane should traverse in order to yield the inflationary attractor phase. Allowing each individual brane to be sufficiently far from the moduli boundary improves the radiative stability of this model.
In §4, we focus on the limitations that UV physics imposes on the effective description of pole inflation when this is embedded into supergravity as a low-energy limit of string theory. We find the existence of axionic partners with decay constants which explicitly depend on the distance to the boundary. This fact has direct consequences for inflation. The bounds which the weak gravity conjecture (WGC) imposes on the periodicity of the axions (f M P ) automatically result in a net constraint on the available length of the exponentially flat plateau typical of pole inflation. We show that in the original single superfield pole-inflation, with a single brane, the inflaton is not even allowed to reach the plateau region of the scalar potential. When inflation is driven by the assistance of N branes, however, these constraints are relaxed -we find that the upper bound on the canonical radial field range set by the WGC scales like ∆ϕ ∼ log N . Note that even if the plateau region of the potential is now available for slow-roll inflation, it cannot be infinite for finite N . This means that the effective description is only valid at finite distances from the pole before it breaks down, as one would also expect from the swampland distance conjecture. We interpret these findings as an important WGC bound on the range of validity of the effective field theory of this cosmological scenario.
The universality of the pole inflation/α-attractor mechanism also emerges in our analysis. Despite the presence of N fields, the form of the exponential plateau remains unaltered from the single field case. This implies that when inflation occurs along the collective radial direction, we recover the single field predictions. This may be contrasted with other many-field inflationary constructions, where the predictions at large N are typically distinct from the single field limit [5,25,[97][98][99][100][101][102] (however, see Ref. [28] for a counter example). That said, a full analysis of the large-N dynamics of this model remains to be explored, as a subset of the angular field directions may also be sufficiently light to play a role in the inflationary dynamics. This may give rise to richer phenomenology through multifield effects which have the capacity to modify the original predictions of the model. While studying the complete dynamics will be a computationally heavy task, the necessary tools have recently been made publicly available [6,[103][104][105]; we leave this for future work.
Regarding the implementation of pole N-flation in type IIB string theory, while we have made first steps in §5 by embedding the model in fibred geometries, developing a consistent program for moduli stabilization within this scenario remains an important step to be addressed. We see this as an exciting avenue to be explored.
This relation, together with eq. (A.5), implies that in the configuration of interest