The type IIB string axiverse and its low-energy phenomenology

We study closed string axions in type IIB orientifold compactifications. We show that for natural values of the background fluxes the moduli stabilisation mechanism of the LARGE Volume Scenario (LVS) gives rise to an axiverse characterised by the presence of a QCD axion plus many light axion-like particles whose masses are logarithmically hierarchical. We study the phenomenological features of the LVS axiverse, deriving the masses of the axions and their couplings to matter and gauge fields. We also determine when closed string axions can solve the strong CP problem, and analyse the first explicit examples of semi-realistic models with stable moduli and a QCD axion candidate which is not eaten by an anomalous Abelian gauge boson. We discuss the impact of the choice of inflationary scenario on the LVS axiverse, and summarise the astrophysical, cosmological and experimental constraints upon it. Moreover, we show how models can be constructed with additional light axion-like particles that could explain some intriguing astrophysical anomalies, and could be searched for in the next generation of axion helioscopes and light-shining-through-a-wall experiments.


Introduction
The QCD axion 'a' is predicted [1,2] in the most plausible explanation of the strong CP problem, that is the non-observation of a θ-angle term. To accomplish this, it must act as a field-dependent θ-angle term, i.e. it must have a coupling to the topological charge density in QCD: where F b 3,µν is the bth component of the gluon field strength tensor,F b,µν its dual, g 3 is the strong coupling, and f a is the axion decay constant. We take conventions such that the axion is periodic under a → a + 2πf a ; C a3 is then a constant describing the coupling of the axion to QCD, which for a field theory model is equal to the anomaly of the Peccei-Quinn field under QCD. It is usually absorbed into the definition of f a but it shall be more convenient for string models to retain it. Non-perturbative QCD effects at low energies then generate a potential for the QCD axion plus the θ angle, stabilising the physical value at zero, thereby solving the strong CP problem [3], and giving the axion a parametrically small mass m a ∼ m π f π /f a , where m π and f π are the pion mass and decay constant, respectively.
There are beam dump, astrophysical and experimental constraints on the decay constant of the QCD axion and axion-like particles (ALPs) a i via their couplings to photons and electrons: that are summarised in appendix B. There are also searches underway in light-shiningthrough-a-wall (LSW) experiments [4], in searches for solar axions [5] and in direct haloscope searches for axion cold dark matter (CDM) [6]. Direct searches for the QCD axion in beam dumps and the non-observation of its effect on the cooling of stars constrain the QCD axion decay constant to be f a /C a3 10 9 GeV. Furthermore, there are also several tantalising astrophysical hints, which together point to a light ALP with couplings and mass [7][8][9][10][11][12]: f a i C ie ≃ (0.7 ÷ 2.6) × 10 9 GeV , f a i C iγ ∼ 10 8 GeV, m a i 10 −9 ÷ 10 −10 eV , hinting at C iγ /C ie ∼ 10, which we shall review in appendix B. A summary of constraints and hints is given in figure 1. Figure 1: A summary of constraints on and hints for the couplings f ai /C ij of axion-like particles a i to fields j. The green regions from top to bottom correspond respectively to the classic 'axion dark matter window', hints of an axion from white dwarf cooling and transparency of the Universe to very high energy gamma rays. Red regions are excluded, and the orange region would be excluded by red giants but is compatible with the hints from white dwarfs. The blue region would be excluded by dark matter overproduction in the absence of a dilution mechanism or tuning of the misalignment angle. Legend: 'Molecular interferometry' refers to reach of future experiments discussed in [13], see appendix B.2; 'CDM' is constraints on the QCD axion coupling from dark matter overproduction in the standard cosmology, see section B.2; constraints from 'White dwarfs' are discussed in section B.1.1; 'HB/CAST' are constraints from horizontal branch stars and the CAST experiment respectively, see section B.1.1; 'SN1987A' are the bounds from observation of high energy gamma rays from that supernova, see section B.1.2; AGN are the hints from observation of gamma rays from active galactic nuclei, also in section B.1.2.
As a result of these constraints and hints, the QCD axion is necessarily associated with a very high energy scale, and so it is natural to search for it in ultra-violet completions of the Standard Model (SM) such as string theory. Indeed, it has long been known that the low-energy effective field theory (EFT) of string compactifications predicts natural candidates for the QCD axion [14][15][16][17][18], often even an 'axiverse' [19] containing many additional light ALPs whose masses are logarithmically hierarchical.
Despite all of these promising considerations, it has been very hard to construct explicit string theoretic examples with a successful QCD axion candidate, let alone additional light ALPs. In order to understand the origins of these difficulties, we focus on type IIB flux compactifications since this is the theory where moduli stabilisation is best understood. Let us summarise the main features and problems encountered in the study of closed string axions from type IIB compactifications: 1. The low-energy spectrum below the compactification scale generically contains many axion-like particles which are the Kaluza-Klein zero modes of antisymmetric form fields belonging to the massless spectrum of the bosonic type IIB string. Their number is related to the topology of the internal manifold, namely to the number of harmonic forms which can be defined on a Calabi-Yau. Given that this number is generically of the order of hundreds, we expect the low-energy theory to be populated by many axion-like particles. These are all closed string axions which live in the bulk. On top of them there can also be open string axions living on a brane but their presence is more model-dependent, and so we shall focus just on closed string axions.
2. Type IIB string theory compactified on a Calabi-Yau three-fold gives rise to an N = 2 4D EFT which is not chiral. An N = 1 EFT can be obtained via an orientifold projection which projects out many degrees of freedom of the low-energy theory. The kind of axions which are projected out depends on the particular orientifold projection considered, but certainly several axions are removed from the low-energy spectrum.
3. Each axion comes with a real scalar field, the corresponding 'saxion', forming a complex field which is the lowest component of an N = 1 chiral superfield. The real part of this field is the saxion whereas the axion is the imaginary part. The saxions are moduli which parameterise the size of the extra dimensions and their VEV determines crucial features of the EFT like the gauge couplings and the mass spectrum. It is then of primary importance to develop a potential for these moduli also because, if massless, they would mediate long-range unobserved fifth-forces. Moreover the moduli may suffer from the famous cosmological moduli problem (CMP) either decaying after Big-Bang Nucleosynthesis (BBN) has begun or, if stable, overclosing the Universe. Typically this is solved if they have masses larger than O(10) TeV. 4. The mechanism which stabilises the moduli, giving them a mass of the order m mod 10 TeV, might also develop a potential for the axions rendering them too heavy. In fact, the axions enjoy a shift symmetry which forbids the presence of axion-dependent operators but only at the perturbative level since this symmetry is broken by non-perturbative effects. Therefore if the saxions are fixed perturbatively, the axions do not develop any potential, whereas if the moduli are fixed by non-perturbative effects, the axions might obtain a mass of the same order of magnitude. This is indeed the case for KKLT [20] scenarios where the Kähler moduli are fixed by assuming that the superpotential has non-perturbative contributions for each Kähler modulus. All the corresponding axions would then have the same mass as the Kähler moduli, which is generically of the order the gravitino mass, rendering all of them too heavy. 5. In type IIB string compactifications, the visible sector (the MSSM or GUTlike generalisations thereof) is built via space-time filling D7-branes wrapping internal four-cycles whose volumes are measured by the saxions (the so-called Kähler moduli). Each stack of N D7-branes supports a U (N ) = SU (N ) × U (1) field theory where the U (1) factor is generically anomalous. This anomaly is cancelled via the Green-Schwarz mechanism which induces a mass term for the U (1) gauge bosons. The gauge boson masses generated in this way may be as small as an meV [21] for a low string scale, but will often be much heavier than observable physics. This is a standard Stückelberg mass term since the Abelian gauge boson is made massive by eating the axion which is the imaginary part of the Kähler modulus parameterising the size of the wrapped four-cycle. Thus all the axions which are eaten disappear from the low-energy EFT. 6. Even if some axions are not projected out by the orientifold projection, do not become too heavy due to non-perturbative moduli stabilisation and are not eaten by anomalous U (1)s, it is very hard to obtain axion decay constants f a 10 12 GeV, where this bound comes from the requirement in standard thermal cosmology of avoiding the overclosure of the Universe by axion CDM produced by the misalignment mechanism (see appendix B.2 for a brief review). In fact, in many string compactifications the QCD axion and other ALPs generically have a larger decay constant of the order the GUT scale M GUT ∼ 10 16 GeV. 1 This is not necessarily a phenomenological disaster if the axions are diluted by the late outof-equilibrium decay of some heavy moduli, or have their initial misalignment angles tuned small. Nonetheless only ALPs with an intermediate scale decay constant allow a possible observation of axion related effects in astrophysics, cosmology and even in the laboratory.
At present, the best realisation of the axiverse from type IIB string compactifications is based on the generation of a single non-perturbative correction to the superpotential which fixes all the h 1,1 Kähler moduli plus one axion combination, leaving all the other h 1,1 − 1 axions massless at leading order [22]. These axions then develop a potential at subleading order via higher order instanton effects. In order to fix the moduli within the regime of validity of the EFT, the tree-level superpotential W 0 has to be fine-tuned very small, W 0 ≪ 1, of the order the leading non-perturbative correction. This effect can be generated by a Euclidean D3-brane instanton or gaugino condensation on D7-branes wrapping an 'ample divisor' which is given by a combination of all the Kähler moduli with positive coefficients. A microscopic realisation of this interesting scenario based on a Calabi-Yau example with explicit brane set-up and fluxes has not been found yet due to the difficulty to find a rigid ample divisor and to avoid chiral intersections between the instanton and the visible sector.
In this paper we shall present a different realisation of the axiverse from type IIB strings which is valid in the natural regime W 0 ∼ O(1). It is based on a moduli stabilisation mechanism, the so-called LARGE Volume Scenario (LVS), which exploits both perturbative and non-perturbative effects and is characterised by an exponentially large volume of the extra dimensions [23,24].
Explicit LVS examples of globally consistent Calabi-Yau constructions with magnetised D-branes and chirality have been recently derived in [25]. The same paper presents a general discussion of various tensions between brane fluxes and moduli stabilisation (chirality vs non-vanishing non-perturbative effects, cancellation of Freed-Witten anomalies vs generation of more than one non-perturbative contribution, Dterm induced shrinking of the cycles supporting the visible sector) showing how the LARGE Volume Scenario provides a very attractive solution to these problems. Then ref. [25] outlined a general strategy to fix the moduli within the LVS. We shall now show how this moduli fixing mechanism gives rise to an axiverse: • In the LVS, usually all the h − 1,1 moduli are projected out by the orientifold involution so that h + 1,1 = h 1,1 . • The only condition to realise the LVS is the presence of a del Pezzo four-cycle τ dP supporting a single non-perturbative effect whose interplay with the leading order α ′ correction to the EFT naturally yields an exponentially large internal volume V ∼ W 0 e 2πτ dP /n . Notice that the rigidity of a del Pezzo divisor guarantees the generation of this non-perturbative effect, and the fact that the LVS does not require more than one non-perturbative contribution implies that there is no problem with the cancellation of Freed-Witten anomalies. Moreover, a del Pezzo divisor is intrinsically 'local', in the sense that a basis can always be found where this four-cycle enters the volume in a completely diagonal way. Therefore it is easy to decouple this modulus from the one supporting the visible sector, avoiding problems arising from chiral intersections. The final upshot is that this blow-up mode does not appear in the D-terms and the corresponding axion is very heavy, of the order of the gravitino mass.
• The D-terms are the leading order effects in an expansion in inverse powers of the huge internal volume and they might in principle generate a potential for h 1,1 − 1 moduli via the flux induced generation of moduli-dependent Fayet-Iliopoulos (FI) terms. However, if this is the case, the only solution to D = 0 in the absence of SM singlets which can have non-zero VEVs, is the one where all the cycles shrink to zero size. This can be avoided if the FI-terms depend only on d moduli with d < h 1,1 − 1, which can be obtained by an appropriate choice of brane set-up and fluxes. Thus exactly d axions are eaten up, corresponding to the d combinations of Kähler moduli fixed by the D-terms.
• All the remaining n ax = h 1,1 −1−d flat directions have to be fixed perturbatively via α ′ or g s effects. As we have already pointed out, the volume is fixed by the leading order α ′ effects whereas all the other cycles are stabilised by string loops.
• The number of massless axions is n ax which for an arbitrary Calabi-Yau with h 1,1 ∼ O(100) is likely to be very large. Subleading higher-order non-perturbative effects then slightly lift the axion directions giving rise to the 'LVS axiverse'.
As explained in [25], the LARGE Volume Scenario requires n ax ≥ 2, implying that the simplest model involves at least two light axions. If n ax = 1, this remaining flat direction would parameterise the overall volume and be constrained to be fixed at small size by the requirement of obtaining a visible sector gauge coupling g vs of the correct order of magnitude, given that the D-term relations force the volume to scale as the visible sector cycle: V 2/3 ∼ τ vs ∼ g −2 vs . Hence the volume would be rather small in string units and the tree-level superpotential W 0 fine-tuned small otherwise the relation V ∼ W 0 e 2πτ dP /n would yield an exponentially small gauge coupling. However, if n ax ≥ 2, only one of these flat directions is forced to be small to obtain the correct value of g vs while the others can be exponentially large allowing a standard realisation of the LVS for W 0 ∼ O(1).
Clearly, providing a string realisation of the axiverse is not enough if none of these light axions can behave as the QCD axion. This depends mainly on the strength of the corresponding axion coupling to QCD which, in turn, depends on the topology of the Calabi-Yau three-fold and the choice of brane set-up and fluxes. In fact, if the visible sector wraps a small rigid divisor which does not intersect the cycles controlling the size of the large overall volume, the decay constant of the corresponding axion is of the order the string scale f a ∼ M s / √ 4π due to the local nature of this interaction. In all the other cases, this interaction is non-local resulting in axion couplings suppressed by the Planck scale. We shall discuss the consequences of this in section 3.2 where we shall find that, in the cases when non-local axions can play the rôle of the QCD axion, their decay constants are of the order f a ∼ M GUT ∼ 10 16 GeV.
Given that by dimensional reduction the string scale can be expressed as M s = M P / √ 4πV, exponentially large values of the volume can naturally yield a very low string scale. The gravitino mass is instead m 3/2 = g s /2W 0 M P /V , and so TeV-scale soft-terms M soft ∼ m 3/2 ∼ 1 TeV can be obtained for V ∼ 10 14 if W 0 ∼ O(1) and g s ≃ 0.1. Consequently the string scale turns out to be intermediate, M s ∼ 5 · 10 10 GeV, or a few orders of magnitude larger as we moderately tune W 0 smaller, providing an axion decay constant in the classic phenomenological window for the case of a local interaction [15]. In this paper we shall describe how to obtain a QCD axion candidate plus additional ALPs which are not eaten by any anomalous U (1)s and exhibit these interesting phenomenological features.
We stress that LVS models with an intermediate string scale involve light moduli which may suffer from the CMP. For example, the mass of the volume mode V is m V = m 3/2 /V 1/2 ∼ 0.1 MeV. In order to solve the CMP, these light moduli could be diluted either by the entropy released by the decay of some heavy moduli [26] or by a late time inflationary period caused by thermal effects [27]. If a non-local axion with a GUT-scale decay constant plays the rôle of the QCD axion, this dilution mechanism would also dilute this axion, potentially allowing it to evade the present bounds from dark matter overproduction. Consequently, non-local axions might form dark matter and be detectable via future molecular interferometry experiments (see appendix B.2). On the other hand, if the QCD axion is realised as a local axion with an intermediate scale decay constant at the lower end discussed above, f a ∼ M s / √ 4π ∼ 10 10 GeV, this opens up the intriguing possibility to detect it directly in the laboratory in the next generation of LSW experiments. Furthermore, these local axions can provide a nice explanation to several astrophysical puzzles, such as the anomalous transparency of the Universe for TeV photons and the anomalous cooling of white dwarfs; all of these hints and constraints are summarised in appendix B. Moreover, it is worth stressing that in this case axions do not form a sizeable contribution to dark matter, since the latter requires a decay constant in the upper end discussed above, f a ∼ M s / √ 4π ∼ 10 12 GeV, cf. appendix B.2.
We shall illustrate our general claims with four Calabi-Yau examples with stabilised moduli and an explicit choice of brane set-up and fluxes. The main features of each example are: 1. GUT-like model with n ax = 1 non-local axion with f a ∼ M 10D KK /(4π) ≃ 5 · 10 14 GeV that behaves as the QCD axion in the presence of a dilution mechanism.
2. MSSM-like model with n ax = 2 non-local axions, a f with f a f ≃ 10 16 GeV, and The axion a f can be the QCD axion in the presence of a dilution mechanism, whereas a b is a very light and almost decoupled ALP (C bj ≪ 1).
3. Chiral model with n ax = 3 light axions: one local axion a s with f as ∼ M s / √ 4π ≃ 10 10 GeV, and two non-local axions, a f and The local axion a s provides the first explicit example of a closed string QCD axion which is not eaten by any anomalous U (1) and has an intermediate scale decay constant. The two non-local axions are very light and almost decoupled ALPs (C f j , C bj ≪ 1). 4. Chiral model with more than one light local axion with f a ∼ M s / √ 4π ≃ 10 10 GeV.
We stress again that in this analysis we focused only on axions which come from closed strings since only this case is model-independent. In fact, axion-like particles can also be realised as open strings which live on D-branes but this realisation is more model-dependent, since it depends crucially on the particular D-brane construction considered.
This paper is organised as follows. Section 2 presents a simple review of the lowenergy 4D EFT of type IIB string theory compactified on Calabi-Yau orientifolds paying particular attention to axions and their interplay with massive anomalous U (1)s. In section 3 we describe in detail how the LVS moduli stabilisation gives rise to an axiverse giving formulae for the sizes of decay constants, couplings to gauge and matter fields for Swiss-cheese and fibred Calabi-Yau compactifications. The explicit LVS examples mentioned above are analysed in section 4, whereas we present our conclusions in section 5. Due to this success to derive the axiverse from type IIB strings, in appendix A, we work out the low-energy effective action of many ALPs computing in particular their couplings to photons and electrons which are particularly relevant for phenomenology. In appendix B we summarise the results of axion searches in astrophysics, cosmology, and laboratory arguing that various astrophysical hints point to an axion decay constant at the intermediate scale. Finally, in appendix C we compute the axionic couplings to matter at one-loop, and in appendix D we discuss the axion decay constants for general fibred Calabi-Yau manifolds.

Closed string axions from type IIB strings
In this section we will describe how axion-like fields arise in type IIB string theory models and how to determine which of the axion-like fields are eaten by massive U (1) gauge bosons. The low-energy couplings, decay constants, and masses can be obtained by comparing the EFT obtained in this section from dimensional reduction of the IIB string from 10 to 4 dimensions with the general low-energy effective Lagrangian of axion-like particles (A.1) which we shall derive in appendix A.

Effective axionic action
In type IIB flux compactifications on Calabi-Yau orientifolds, the Standard Model or a suitable extension like a Grand Unified Theory (GUT) is localised on a stack of space-time filling D3 or D7 branes. Pseudo-scalar axion-like fields c a and c α arise as Kaluza-Klein zero modes of the Ramond-Ramond (RR) antisymmetric tensor fields C 2 and C 4 [28,29] (for a short review, see the appendices in [21,30]), that is as coefficients of the two-and four-form term, respectively, in their expansion in harmonics of the Calabi-Yau orientifold: The axionic couplings to gauge field strengths ∝ F ∧ F arise then from the Kaluza-Klein reduction of the Chern-Simons term in the action of the D-branes. The effective Lagrangian including U (1) gauge bosons A i and couplings to axions c a and c α can be written as: The quantities in the above expression are defined as follows. M P is the reduced Planck mass (8πG N ) −1/2 ≃ 2.4 · 10 18 GeV, Φ is the axio-dilaton, V is the Einsteinframe volume of the internal space in string units, while q iA and r iA are the couplings of the two-forms ω A and four-formsω A to branes, ignoring any globally-trivial forms, and are given by: where F is the gauge flux on D i which is a generic divisor whose Poincaré dual twoform can be expressed in terms of the basis forms asD Writing the Kähler form as J = t A ω A , we have: s ω A ∧ ω B ∧ ω C are the triple intersection numbers. The above expression makes clear that some axions are eaten by the U (1) gauge symmetries in the model, while the axionic couplings to other gauge groups are inversely proportional to the Planck mass. However, the physical axion decay constants will be determined by the kinetic terms, which -particularly for large volumes -can differ from unity by many orders of magnitude. In this work, we are interested in ALPs whose decay constants may be equal to or less than the string scale, so that their couplings may be equal to or enhanced relative to the QCD axion; an examination of the above action and the expressions for the metrics (2.3) will make clear that the orientifold-odd fields c a will have decay constants equal to or parametrically greater than the string scale (similar to the case for U (1) masses [30]) and so we will neglect them, or focus on models with h 1,1 − = 0. In this case the Kähler moduli are defined as: However, it should be stated that the analysis of the odd two-form fields c a would proceed almost identically to that of the even fields.

Canonical normalisation
From (2.2) it can be seen via topological charge quantisation that the axions have (unconventionally for axions, but conventionally for Kähler moduli) periods equal to integer multiples of M P ; this derives from the quantisation of the Ramond-Ramond forms from which they descend. We diagonalise the axions in two steps: first we make an orthogonal transformation, after which the new fields are still periodic under integer shifts by M P . This leads to kinetic terms: where λ α are the eigenvalues of K αβ . We then canonically normalise by defining c ′ α ≡ 2λ −1/2 α a α ; the new fields are then periodic under shifts by 1 2 λ 1/2 α M P -this allows us to define the decay constants: so that the fields have the conventional periodicity a α = a α + 2πf α . Some of these fields will be eaten by a U (1) gauge boson and thus cannot enter into a term in the superpotential of the form W ⊃ A e −aT where A is a constant (not a function of matter fields) due to gauge invariance -the shift symmetry is gauged. Hence these axions are in some sense orthogonal to their light ALP brothers, and we must first remove them from our consideration. To find the massless axions coupling to a given gauge group, we must find the null vectors to eliminate the dc∧⋆A couplings. Hence the full transformation, leaving only the ungauged axions, is: i.e. the matrix V is the set of null vectors of the mass couplings. When both axions and gauge fields are canonically normalised we have a term L ⊃ Note that for U (1) ⊃ U (N ) we normalise the generators so that tr(T U (1) T U (1) ) = 1, which means that T U (1) = 1/ √ N and tr(T U (1) ) = √ N. There is an alternative notation much used in the literature, involving physical couplings g ji : where τ i is the volume of the four-cycle wrapped by the brane in string units. It is often convenient to make a redefinition of the axions choosing a basis where r iα a α ≡ {â i , 0} depending on whether the cycle is wrapped by a brane or not. Under this transformation, we must not forget to transform the charges appropriately.
With these couplings, we can work out which axions will obtain a mass through any non-perturbative effects -such as coupling to a gaugino condensate or D-brane instanton -since these axions cannot be gauged (for masses that arise through superpotential terms of the form W ⊃ A e −bT where A is independent of matter fields). In determining the light axion and ALPs we must therefore make a further redefinition of the fields to eliminate the heavy massive ones.

Volume scaling of axion decay constants and couplings to gauge bosons
Being the imaginary component of the moduli, whose interaction with matter is gravitational, we expect the physical couplings of axions to matter fields to be suppressed by the Planck scale. This applies for moduli which are able to propagate throughout the Calabi-Yau volume. However, for axion fields that only propagate over so-called 'local' cycles (such as small rigid divisors), i.e. four-cycles which do not intersect any divisor which controls the overall volume, the natural coupling strength is the string scale. Using conventional definitions, the physical coupling is inversely proportional to the axion decay constant f a and proportional to the square of the gauge coupling g 2 ≃ τ −1 a . However, this argument only holds for the scaling with respect to the overall volume, rather than the volumes of small cycles -we expect: As pointed out in section 1, local interactions open up the possibility to obtain axion decay constants which are much lower than the Planck or GUT scale, since their size is controlled by M s instead of M P [15]. Hence if M s ∼ 10 10 GeV, f a turns out to be intermediate in a regime which is particularly interesting for phenomenology. Notice that this value of M s can be naturally obtained for exponentially large values of the overall volume which characterise LVS scenarios. A range of values of f a are then possible depending upon the exact numerical coefficients, the value of the gravitino mass and the tuning of W 0 ; M s ≃ M P m 3/2 W 0 so values of decay constants spanning the classic axion window (10 9 ÷ 10 12 GeV) are possible without fine tuning.
On the other hand, the decay constants of non-local axions may be even smallersince τ a in this case can scale as V 2/3 or even V for anisotropic compactifications. 2 This has important consequences for the amount of dark matter that they can produce, which we shall discuss in section 3.3.
We shall now illustrate these general claims and derive the physical couplings to gauge bosons, C ij , for explicit Swiss-cheese compactifications and in section 2.3.2 for fibred Calabi-Yau examples.

Swiss-cheese Calabi-Yau manifolds
In the simplest Swiss-cheese case, the Calabi-Yau volume takes the form [31]: where τ b is a large cycle controlling the overall size of the internal manifold whereas the small divisor τ s is a local blow-up mode. Writing ǫ 2 ≡ τ s /τ b ≪ 1, and working at leading order in a large volume expansion, we find: The eigenvalues of this are 3 τ −2 b , 3 2 τ −3/2 b τ −1/2 s , and so the decay constants of the relevant axions will be: 12) in agreement with (2.9). In (2.12), M 10D KK denotes the 10D Kaluza-Klein scale: The diagonalising matrix is given by: 2 For anisotropic compactifications we show in appendix D that there is only one axion that has a Planck-sized decay constant -that corresponding to the fibre -with the others being much smaller. and the matrix r iα is the identity. Hence the original axion fields c b and c s can be written in terms of the canonically normalised fields a b and a s as: where we have used V ≃ τ 2/3 b and we have neglected subleading corrections proportional to ǫ. Thus the axionic couplings to gauge bosons scale as: The volume scaling of the leading terms of these couplings are exactly in accordance with (2.9). The constants describing the coupling of the axions to gauge bosons instead become:

Fibred Calabi-Yau manifolds
Now let us consider fibred Calabi-Yau manifolds with a del Pezzo divisor. The volume is given by [32]: where t b is the size of the P 1 base, τ f = t 2 f the volume of the K3 or T 4 fibre and τ s = 3t 2 s parameterises the volume of a small del Pezzo four-cycle. The volume of the divisor including the base of the fibration is instead given by We treat more general fibred examples in appendix D.
In the limit t s τ f ≫ τ 3/2 s , the eigenvalues of the mixing matrix are to leading order 3 V √ τ s , and so the decay constants are: again in agreement with (2.9): the first two axions are non-local. Note that the fibre axion has a potentially large decay constant in the anisotropic limit when τ f ∼ τ s ≪ τ b ; we show in appendix D that this is the only axion that can have a decay constant parametrically above the string scale.
We should, however, distinguish the decay constants from the physical couplings. To determine these we can use the results of [27], by which the original axion fields c b , c f and c s can be written in terms of the canonically normalised fields a b , a f and a s as: The axionic couplings to gauge bosons following from (2.19)-(2.21) scale as: Notice that volume scaling of the leading terms of these couplings are exactly in accordance with (2.9). The constants describing the coupling of the axions to gauge bosons instead become: Let us now see how these quantities behave in the two different cases when the Calabi-Yau has an isotropic or anisotropic shape.
• Isotropic case: in the isotropic limit t b ∼ √ τ f ∼ V 1/3 the decay constants have the same volume scaling as the Swiss-cheese example: 24) and the anomaly coefficients (2.23) reduce to: • Anisotropic case: in the anisotropic limit [21,33], the decay constants become: where f a b is now scaling as the Kaluza-Klein scale of the 6D EFT given by: Moreover, the anomaly coefficients (2.23) take the simplified form: (2.28)

Axionic couplings to matter fields
After having computed the axionic couplings to gauge fields, let us now move to the computation of the axionic couplings to matter fields. Written in terms of twocomponent Weyl spinors, these couplings take the form We discuss the low energy phenomenology of these couplings in appendix A, where we also explain that we shall neglect 'Yukawa' couplings of the form a i ψψ -since they arise from non-generic and model dependent non-perturbative terms.
In string theory, closed string axions couple generically to matter fields from higher-order Kähler potential terms, and can thus be derived from the effective supergravity action. In a globally supersymmetric theory, the axion couplings arise from the iθσ µ θ∂ µ T α component of the superfield T α = τ α + i c α + ...: If we write for simplicity a diagonal matter Kähler metric as ∂ ΨΨ K ≡K Ψ (the generalisation to non-diagonal cases being obvious) then the moduli dependence of this gives us the couplings: So in the physical basis (when we canonically normalise the matter fields) the coupling of axions to fermions is ∂ Tα lnK Φ . Given that the above expression has to be gauge invariant, the axions can only appear in the combination ∂ µ c α + M P π q α A µ , and so we have additional anomalous gauge couplings. However, in a local supergravity theory, there is an additional contribution: and so when we canonically normalise the matter fields we obtain physical couplings: Note that these are different to the coupling to the real parts of the moduli, which couple to the fields via the kinetic term only. The best conjecture for the Kähler metric of chiral matter supported on curves at the intersection of cycles a and b is [34,35]: (2.34) Thus: For matter supported all on one cycle we haveK .
We then translate these into ALP-matter couplings (to axions a i ): matter at a singularity (2.36) It is thus necessary to consider the canonical normalisation of the axions. A local axion will thus couple typically asX in the first two cases, and non-local ones will couple much weaker than Planck strength (or not at all). Note that we expect these couplings to violate parity -the parity violating couplings are proportional tô X i ψ L −X i ψ R , which are generically non-zero since the left-and right-handed fields will be localised on different curves or branes -but as discussed earlier these should not have severe consequences. We can parameterise the coupling as: . An important special case is the couplings of the 'large' cycle axion. When the SM is realised on a local cycle, C bi ≃ O(1) and only the universal term contributes; the couplings are then highly suppressed, withX 3 for a Swiss-cheese compactification. The coupling to matter fields will then be dominated by the effect generated at one loop via the photon coupling; most important among these for phenomenology is the coupling to electrons. We shall give a full discussion of the oneloop effects in appendix C, but here present a useful approximation to the one-loop corrections to the above couplings: while there is an additional correction from pion loops to the QCD axion to electron coupling, unchanged in SUSY theories, equal to

The axiverse in the LARGE Volume Scenario
In this section we will focus on the type IIB LARGE Volume Scenario and show how this moduli stabilisation mechanism gives rise to an axiverse with many light axions. We shall then discuss the phenomenology and cosmology of local and non-local axions in different scenarios where the Calabi-Yau has a isotropic or anisotropic shape and the SM is supported on a local or non-local four-cycle.

The axiverse and moduli stabilisation
As we have already pointed out in section 1, there are two regimes for the type IIB axiverse depending on the value of the VEV of the tree-level superpotential W 0 .
Case W 0 ≪ 1: the axiverse from an ample divisor In the case where W 0 ≪ 1, the axiverse can be realised by an E3-instanton or gaugino condensation on D7-branes wrapping an ample divisor D am whose volume is given by τ am [22]. If D i is a basis of H 4 (X, Z) where X is a generic Calabi-Yau three-fold, then: is 'ample' if and only if λ i > 0 ∀ i = 1, ..., h 1,1 . The non-perturbative effects supported by this divisor yield a superpotential which looks like [36]: where a = 2π/n with n = 1 for an E3-instanton and n = N for gaugino condensation on N D7-branes. By fine-tuning W 0 ∼ A e − a Tam ≪ 1, this single non-perturbative effect can generate a minimum for all the Kähler moduli within the regime of validity of the EFT. However it can lift only one axion corresponding to the imaginary part of the ample divisor modulus: c am = Im(T am ). All the remaining h 1,1 − 1 axions are massless at leading order and develop a potential only via tiny higher order instanton effects of the form [22]: where T i is a combination of moduli orthogonal to T am ∀ i = 1, ..., h 1,1 − 1. Two main concerns regarding the microscopic realisation of this scenario are related to the difficulty to find an ample divisor which is rigid (and so definitely receives nonperturbative effects), and the possibility to choose gauge fluxes that avoid chiral intersections between the instanton and the visible sector.
In the natural regime where W 0 ∼ O(1), the LVS gives rise to an axiverse since only one Kähler modulus, a del Pezzo divisor T dP , is fixed by non-perturbative effects whereas all the other moduli (if not fixed by D-terms) are stabilised perturbatively by α ′ or g s effects. Hence only one axion, c dP = Im(T dP ), becomes heavy whereas all the other ones (except those eaten up by anomalous U (1)s) remain light and develop a potential via subleading higher order instanton effects. Before providing the details of this mechanism, let us summarise three issues that have to be addressed when combining moduli stabilisation with the presence of brane fluxes within the same compactification [25,32]:

Tension between chirality and non-vanishing non-perturbative effects
Chiral modes at the intersection of D-branes are induced by a non-zero worldvolume flux F = 0. In the presence of chiral intersections between the visible sector and a cycle T np supporting non-perturbative effects of the form: the prefactor A depends on visible sector modes φ i : A = A(φ i ). However, in order to prevent the breaking of the visible sector gauge group at a high scale, φ i = 0, and in turn A = 0, implying the vanishing of W np . This observation sets the following constraint on the flux choice: there has to be no chiral intersection between the visible sector and the cycle supporting non-perturbative effects. A straightforward solution to this problem is obtained by placing the non-perturbative effects on a del Pezzo divisor D dP since, due to the local nature of this blow-up mode, it is always possible to find a basis of H 4 (X, Z) such that D dP is an element of the basis that intersects no other element. Moreover, due to the rigidity of this four-cycle, the generation of the non-perturbative effect is definitely under control.

2.
Tension between the cancellation of Freed-Witten (FW) anomalies and the generation of more than one non-perturbative effect The cancellation of worldsheet FW anomalies requires to turn on a half-integer flux on any divisor D ns which is 'non-spin', i.e. its first Chern class c 1 (D ns ) = −D ns is odd: In order to obtain a non-zero contribution to the superpotential, the total flux F = F − B on an E3-instanton or gaugino condensation stack wrapping an invariant four-cycle D a , has to vanish: F a = 0. However, in the case of a nonspin divisor, F a = 1 2D a = 0, forcing to choose B = F a in order to obtain F a = 0. Notice, however, that once B is fixed in this way, generically it cannot be used anymore to cancel the FW flux on a second non-spin divisor D b . Thus the total flux on D b is non-zero, F a = 1 2 D b −D a = 0, preventing the generation of a second non-perturbative contribution to W . This tension is clearly absent if the Kähler moduli are frozen by using only one non-perturbative effect. If this is supported by a del Pezzo divisor, then there is also no problem with chirality and the rigidity of this divisor guarantees the generation of W np .

D-term induced shrinking of the cycles supporting the visible sector
The world-volume flux turned on to produce chirality generates also modulidependent Fayet-Iliopoulos (FI) terms: resulting in a total D-term potential which looks like: For vanishing VEVs of charged open string fields, φ j = 0, the supersymmetric D-term conditions give rise to a system of a homogeneous linear equations ξ a = 0 ⇔ q aj t j = 0 which might in general depend on all the h 1,1 Kähler moduli. Notice that in order to avoid unwanted matter in the adjoint representation, the visible sector is generically built by wrapping D7-branes around rigid divisors. If this divisor is a del Pezzo T dP , then one equation reduces just to t dP = 0, forcing the shrinking of the four-cycle supporting the visible sector. Hence the EFT is driven to a regime where the supergravity approximation is not under good control. For this reason, the visible sector should be built on rigid but not del Pezzo divisors. On the other hand, as we have stressed above, a del Pezzo divisor should support the non-perturbative effects. Thus this four-cycle does not enter in the D-term equations which turn out to depend in general on h 1,1 − 1 unknowns. In order to prevent the shrinking of some four-cycles, the brane configuration and the fluxes have to be chosen such that the rank d of the matrix of the system is d < h 1,1 −1, implying that the D-term conditions stabilise d combinations of Kähler moduli leaving n ax = h 1, Notice that the combinations of two formsc j , dual to the axions c j , which couple to the anomalous U (1)s are also given by q ajc j . Therefore the d combinations of axions which are eaten up are the same as the d combinations of Kähler moduli which are fixed by the FI-terms.
As explained in [25], the type IIB LARGE Volume Scenario provides a promising way to address these issues, allowing the construction of explicit Calabi-Yau examples with magnetised D-branes and stabilised moduli which are chiral and globally consistent. The scalar potential receives several contributions which have a different scaling with the exponentially large overall volume V: Let us briefly describe each of these contributions: . Given that it is the leading order effect in an expansion in inverse powers of V, it is minimised supersymmetrically at first approximation by imposing V D = 0. This condition fixes d combinations of Kähler moduli with d < h 1,1 − 1. Therefore d axions are eaten up. Notice that if n ax = h 1,1 − 1 − d = 1, the D-term equations force all the moduli to be of the same size: τ i = τ * ∀i = 1, ..., h 1,1 − 1. In turn, τ * is required to be small in order to obtain a visible sector gauge coupling g −2 vs ∼ τ * of the observed order of magnitude. This observation then prevents the volume to be exponentially large. This is because, requiring that there is a standard model which does not wrap at least one "small" diagonal del-Pezzo divisor, if d = h 1,1 − 2 then the large cycle will be fixed relative to the small cycles. This is not the case if n ax ≥ 2, and so we conclude that the simplest realisation of a standard LVS requires at least n ax = 2, and so h 1,1 = 3 + d, giving h 1,1 = 4 for d = 1. 3 Notice that this bound applies only for models with the visible sector in the geometric regime. For example, in the case of a canonical strong Swisscheese geometry, D-terms force the visible sector to the quiver locus, leaving just one light axion (the volume axion). The visible sector could be kept in the geometric regime by giving a VEV to a open string singlet, resulting again in at least two light axions: the volume axion and the phase of the open string mode.
• V tree F is the tree-level F-term scalar potential which also scales as V tree but it is zero due to the 'no-scale structure'.
and it is the non-perturbative scalar potential arising from corrections to the superpotential W coming from an E3-instanton or gaugino condensation on a D7-stack wrapping a single del Pezzo divisor T dP : This effect fixes T dP at a small size giving it a mass of the order the gravitino mass. Hence the corresponding axion becomes very heavy [37]: • V p F is the perturbative potential which is derived from the α ′ and g s corrections to the Kähler potential K. This potential lifts all the n ax flat directions left over by the D-terms and the non-perturbative effects, leaving all the corresponding axions massless. This is the origin of the 'LVS axiverse'. The α ′ correction to K is of the form [38]: It gives rise to a scalar potential which scales as V α ′ F ∼ O(V −3 ) and its interplay with W np , for natural values W 0 ∼ O(1), yields a supersymmetry-breaking AdS minimum at exponentially large volume [23]: Consequently, the volume mode obtains a mass of the order [37]: and so it is much lighter than the gravitino. This can cause a CMP if M soft ∼ m 3/2 [37] whereas there is no problem in sequestered models where M soft ∼ m 3/2 /V ≪ m 3/2 [35]. Notice that this vacuum can be uplifted to a dS solution within a manifestly supersymmetric EFT by using the positive contribution coming from E(-1)/D3 non-perturbative effects at a singularity [39]. On the other hand, g s corrections to K are subleading with respect to the α ′ -corrections due to the 'extended no-scale structure' which forces their contribution to V to scale as V gs F ∼ O(V −(3+p) ) with p > 0 [40]. Hence the remaining n ax −1 flat direction can be lifted at subleading order either by open string loops (if the corresponding cycle intersects a cycle wrapped by a brane or the cycle itself is wrapped by a brane with vanishing flux so that the axion is not eaten up) [24] or by closed string loops [41]. 4 Tiny subdominant higher-order non-perturbative effects similar to the ones in (3.3) then slightly lift the n ax axion directions via the generation of the following potential: Notice, however, that the axions can become massive also by non-perturbative corrections to the Kähler potential of the form K np ≃ V −1 i e −n i a i T i which are not required to be invariant under the orientifold involution [15]. These effects generate a potential for the axions which looks like: (3.14) In [15] it was determined that the size of the potential generated by the above effects could only dominate over the QCD axion potential for a single instanton superpotential contribution but the presence of such an instanton is incompatible with chirality.
On the other hand, the mass generated by a single Kähler potential contribution would be: where we used the fact that the relevant four-cycle volume for the QCD axion is τ a = α −1 QCD . This gives m a O(10 −11 ) eV for m 3/2 ∼ 100 TeV and the suppression would be even greater for higher order effects. Hence we expect ALP masses to be typically smaller than those of the QCD axion.
For non-local axions, given that they correspond in general to non-rigid cycles, single non-perturbative effects are likely to be absent due to their zero-mode structure. However, even if they are present, they are suppressed with respect to perturbative corrections by a huge exponential factor. Two typical examples of non-local cycles whose non-perturbative effects would be minuscule are the volume mode of Swisscheese manifolds for V ≃ 10 4 this factor is less than 10 −600 -and the fibre modulus T f of fibred Calabi-Yau constructions which might receive only poly-instanton contributions of the [33,42]. On the other hand, non-local axions may obtain small masses via a QCD-like effect in the hidden sector, as we shall discuss in section 3.2; these are of the order m non−local ∼ Λ 2 /M P , where Λ is the relevant strong coupling scale. These last two mass contributions are very model dependent.
In summary, generic non-local axions are effectively massless (although in certain cases some may obtain non-negligible masses, still typically smaller than local axion masses) while local axions obtain masses of the form: m local ∼ e −nπτ local × M P Superpotential terms or QCD-like masses m 3/2 Kähler potential terms (3.16) Therefore the axions acquire a mass spectrum which is logarithmically hierarchical giving rise to the 'LVS axiverse'. As pointed out above, the LVS requires n ax ≥ 2, implying that the simplest realisation of the LVS axiverse involves at least two light axions with a QCD axion candidate plus one ALP. More generic models for h 1,1 very large will include an arbitrarily large number of light axions.

The QCD axion in the LVS axiverse
In order to have a phenomenologically viable axiverse, it is crucial to identify potential QCD axion candidates among all the light axions. A very important observation related to this task is the following. If we restore the QCD vacuum angle θ, we have: where the coefficient C a3 gives the effective QCD anomaly. Since QCD is periodic under shifts of θ by 2π, we have the standard axion situation only if C a3 ≃ O (1). However, given that a only varies through [0, 2πf a ), if C a3 ≪ 1, like in some cases described in sections 2.3.1 and 2.
This implies that axions with a volume suppressed anomaly coefficient C a3 cannot solve the strong CP problem, and so they cannot be the QCD axion. Let us investigate which axions can play the rôle of the QCD axion in the case when the SM is on a local or a non-local cycle.

SM on a local cycle
In this case the visible sector is localised on a small rigid divisor which in sections 2.3.1 and 2.3.2 we have denoted as τ s . The local axions a s are natural QCD axion candidates since they have C ss ≃ O (1) and an intermediate scale decay constant of the order the string scale f as ≃ M s / √ 4π ∼ 10 10 GeV for V ∼ 10 14 which gives rise to TeV-scale SUSY for W 0 = 1, or f as somewhat larger for smaller W 0 . This has already been pointed out in [15] but without providing an explicit chiral example where the local axion is not eaten up by an anomalous U (1). In sections 4.3 and 4.4 we shall present two explicit examples where the SM is on a local cycle and the QCD axion is given by a local axion which is not eaten up by any anomalous U (1) and does not receive large mass contributions from single instanton effects. This is a crucial issue since non-local axions, like the volume axion a b of section 2.3.1, cannot be the QCD axion when the standard model is built on a local cycle. In fact, consider the case where the local axion c s coupling to QCD is eaten by an anomalous U (1), and the only other axion that can couple is the non-local one c b . The theory can be written as: which can be diagonalised via: giving: We can then make a chiral rotation of the fermions charged under the massive U (1) to eliminate the axion a s at low energies, leaving only 1 Notice that the axion a s , which is eaten up, corresponds to the axion c s , which is gauged, only at leading order since the kinetic terms induce also a subleading mixing with c b . In fact, the canonical normalisation (3.19) implies: (3.20) It is this subleading mixing that induces the coupling of a b to QCD. However, as can be seen in (2.25), the non-local axion a b has C bs ≃ O V −3/2 ≪ 1, and so it cannot play the rôle of the QCD axion. Moreover, there is necessarily an anomalous global U (1) symmetry left over. When this remaining global symmetry is spontaneously broken, an additional axion will be introduced, and only one linear combination of this new axion and a b is made massive by QCD. If, on the other hand, the strongly coupled gauge group was not QCD, and the global U (1) was not spontaneously broken (but merely anomalous -and hence broken to a discrete subgroup by the strong coupling effect) then a b could obtain a mass, of order where Λ is the strong coupling scale. The dark matter density produced through misalignment would be set by sending f → M P , Θ i → Θ i /V 2/3 (where Θ i is the initial misalignment angle) in the usual axion calculation, and would therefore be comfortably tiny. This is at first rather puzzling, since the initial Lagrangian is independent of the expectation value of c b , and a b ∝ c b . However, we note that a shift of a b induces a shift in c s -and this is where the mass generation arises. That both axions become massive appears because we have two different mass terms -and a mass mixing term (the combination eaten by the gauge field is a s , which depends on both c s and c b ).
On the other hand, in the presence of a non-perturbative superpotential of the form: the local axion c s is made more massive than the QCD axion, while the non-local axion c b develops a non-perturbative potential only at subleading order and its coupling to QCD is extremely small. In fact, the Lagrangian: would be diagonalised by a transformation which looks like: giving: and so the coupling of a b to F ∧ F is suppressed by e −a b τ b ≪ 1 (less than 10 −600 for V ≃ 10 4 ) for τ b ≃ V 2/3 ≫ 1. This implies that the anomaly coefficient C bs in this case would scale as C bs ∼ e −a b τ b V −2/3 ≪ 1, and so it is in practice zero for all intents and purposes. Such a light non-local axion would effectively produce no dark matter. Thus we have shown that non-local axions cannot behave as the QCD axion if the SM is on a local cycle. There is however an exception to this conclusion. If the extra dimensions are anisotropic, as shown in (2.28), the fibre axion a f has C f s ≃ O (1), and so it can be the QCD axion in the presence of a dilution mechanism as its decay constant is of order f a f ≃ M P /τ s ∼ M GUT ∼ 10 16 GeV.

SM on a non-local cycle
In this case the SM is localised on a divisor which controls the overall size of the Calabi-Yau, and so non-local axions can play the rôle of the QCD axion. In fact, considering the Swiss-cheese example of section 2.3.1 or isotropic fibred manifolds of section 2.3.2, the visible sector is now supported on τ b . As can be seen from (2.25) and (2.28), now C bb ≃ O (1), and so the non-local axion a b can now be the QCD axion. However, given that the visible sector gauge coupling scales as has to be set of the order V ∼ 10 4 , while W 0 has to be fine-tuned small to obtain TeV-scale soft terms. In turn, the axion decay constant f a b ∼ M P /V 2/3 ∼ M GUT ∼ 5 · 10 15 GeV, and so a dilution mechanism is needed. We shall present an explicit example of this situation in section 4.1.
Alternatively, if the Calabi-Yau has an anisotropic shape, the SM can be localised on the fibre divisor τ f and the volume can be set exponentially large V ∼ 10 14 without any need to tune W 0 since the visible sector gauge coupling does not depend on the overall volume g −2 vs ≃ τ f ∼ τ s . Moreover, as shown in (2.28), the anomaly coefficient of the fibre axion a f is of order unity, C f f ≃ O (1), allowing a f to play the rôle of the QCD axion provided that it can be diluted by some mechanism as f a f ∼ M P /τ s ∼ M GUT ∼ 10 16 GeV. We shall present an explicit example of this situation in section 4.2.

Cosmology of the LVS axiverse
The cosmology of the LVS axiverse takes a very different form depending on whether the value of the Hubble scale during inflation H inf is larger or smaller then the gravitino mass m 3/2 . The standard case in string cosmology is H inf ≤ m 3/2 , creating a well-known tension between inflation and supersymmetry breaking since the requirement of generating enough density perturbations sets, in general, H inf ∼ M GUT , forcing m 3/2 to be well above the TeV scale [43]. Given that the moduli tend to develop a mass of the order m mod ∼ m 3/2 , the cosmological moduli problem and its solution are closely related to the above mentioned tension between inflation and TeV-scale supersymmetry. The best available solutions to this problem are: 1. The value of m 3/2 after inflation is very different from the value of m 3/2 during inflation, so that the relation H inf ≤ m 3/2 does not hold today where we can instead have m today 3/2 < H inf allowing TeV-scale SUSY for V ∼ 10 14 . This is achieved if the volume mode is the inflaton [44]. However this model is very finetuned and reheating is a big problem since the inflaton at the end of inflation becomes extremely light.
2. The value of m 3/2 after inflation is the same as the value of m 3/2 during inflation but one has a sequestered scenario where the soft terms are suppressed with respect to m 3/2 by some power of the volume [35]. Hence M soft can be of order TeV even if m 3/2 is very large. The most promising sequestered models have M soft ∼ m 2 3/2 /M P ∼ M P /V 2 , and so TeV-scale SUSY requires V ∼ 10 7 instead of the 'traditional' V ∼ 10 14 .
Let us now analyse the two different cases separately.

Case
In this case, as we have pointed out above, we can set V ∼ 10 14 , so to obtain M soft ∼ m 3/2 ∼ O (TeV), 5 and the string scale turns out to be intermediate GeV. This is the case considered in subsection 3.2.1, and found in sections 4.3 and 4.4 where the SM is on a local cycle and the QCD axion is given by a local axion. We stress again that in the literature there is no explicit inflationary model with a successful reheating process that realises this scenario [45]. We shall however assume that this is possible and proceed to analyse the cosmological implications.
1. The complex structure moduli and the dilaton do not play any significant cosmological rôle since they are trapped at their minima, i.e. their initial displacement is tiny [46]. 4. The del Pezzo modulus τ dP and the corresponding axion a dP fixed by nonperturbative effects, obtain a large mass of the order m τ dP ∼ m a dP = m 3/2 ln V ≃ 35 TeV. However this modulus suffers from the CMP since it has a very small coupling with the SM, of the order 1/(M P √ V) [45], reflecting the fact that this cycle blows up a point which is geometrically separated from the point blown up by the SM cycle. Thus τ dP and a dP would decay after BBN at a temperature of the order: Notice that τ dP and a dP would have a much stronger coupling, of the order 1/M s , to hidden sector degrees of freedom living on this del Pezzo divisor, and so the CMP would be solved if they could decay to hidden particles. However, if the hidden sector consists of an E3-instanton or a pure N = 1 SYM theory that develops a mass gap, no light hidden degree of freedom is present.
5. Some of the local Kähler moduli supporting the SM are fixed by the D-terms which induce masses of the order M s , and so these moduli disappear from the EFT. The directions left massless by the D-terms (the ones corresponding to the uneaten axions) are instead fixed by string loops and obtains a mass of the order m τ SM ≃ α SM m 3/2 ≃ 40 GeV for α SM ≃ 1/25. This scaling can be understood from the fact that the g s potential is of the order V gs (τ SM ) ≃ 1/(V 3 √ τ SM ) [25,40], while the mass can be estimated as m 2 These moduli couple locally as 1/M s but, as shown in [27], they decay before dominating the energy density. Hence their decay does not reheat the Universe and cannot dilute any unwanted relic. Therefore the best way to solve the CMP seems to be the presence of a late time period of thermal inflation [47]. Different considerations apply to the case considered in subsection 3.2.2 and found in section 4.1 where the SM is supported on the non-local cycle τ b controlling the overall volume of the Calabi-Yau. As we have already stressed, the volume cannot be set very large, otherwise the visible sector gauge coupling would be too small, whereas W 0 has to be fine tuned small in order to obtain TeV-scale soft terms. For V ≃ 10 4 , g s ≃ 0.1 and W 0 ≃ 1.5 · 10 −10 , we would obtain M soft ∼ m 3/2 ≃ 3 TeV. The string scale turns out to be quite high: M s ∼ 5 · 10 15 GeV. Let us present a brief analysis of the cosmology of this scenario: 1. The non-local Kähler modulus τ b is light and suffers from the CMP given that its mass is of the order m τ b = m 3/2 / √ V ≃ 30 GeV.
2. The non-local axion a b associated with the large cycle can play the rôle of the QCD axion since C bb ≃ O (1). Its decay constant is however rather high f a b ∼ M P /(4πV 2/3 ) ∼ 5 · 10 14 GeV, and so it would naively lead to dark matter overproduction (without tuning the initial misalignment angle, or, as is in fact the case, entropy dilution).
3. The del Pezzo modulus τ dP and the corresponding axion a dP fixed by nonperturbative effects obtain a larger mass of the order m τ dP ∼ m a dP = m 3/2 ln V ≃ 30 TeV. Hence, given that this modulus has an ordinary Planck-strength coupling to the SM, it does not suffer from the CMP since it would decay at a temperature of the order: which is above the BBN temperature but below the freeze-out temperature of thermal dark matter. Thus, dark matter will be produced again non-thermally by the decay of τ dP which will also dilute the modulus τ b suffering from the CMP and the QCD axion a b . Contrary to the previous case, a b would constitute a relevant component of dark matter, and so the constraints from isocurvature fluctuations would apply to this model. Conventional haloscope searches exploiting microwave cavities, however, would not be sensitive to a b dark matter due to its tiny mass and extremely small coupling. Molecular interferometry may be a future option to search for it if one allows for a bit of fine-tuning in W 0 such that f a b ≃ 10 16 GeV.
We finally briefly analyse the model of section 4.2, where the Calabi-Yau has an anisotropic shape and the SM is localised on the fibre divisor τ f . In this case the visible sector gauge coupling does not depend on the overall volume g −2 vs ≃ τ f ∼ τ s , and so we can set V ≃ 10 14 The del Pezzo modulus and its axion have a mass of the order m τ dP ∼ m a dP ≃ 35 TeV and couple to the SM as 1/M P . Therefore they would decay at T τ dP ∼ T a dP ≃ 5 MeV, producing non-thermal dark matter and diluting the abundance of both the light moduli and the QCD axion.
It is worth pointing out that the cosmology of an axiverse with a high string scale was also discussed in [22,26]. As we have seen, our scenarios have some similarities and crucial differences to that case.

Case
In this case, inflation can be reconciled with TeV-scale SUSY if the SM is sequestered from the SUSY-breaking sector. This can be achieved if the SM is built via fractional D3-branes at a singularity obtained by shrinking a blow-up divisor. The most promising scenario involves the following hierarchy M soft ≃ m 3/2 /V which leads to M soft ≃ 2 TeV for m 3/2 ≃ 10 10 GeV with g s = 0.1, W 0 = 1 and V ≃ 10 7 . This is also the value of the volume which gives the right amount of density perturbations in the inflationary model of [50] where the inflaton is a blow-up mode fixed by non-perturbative effects. Therefore, contrary to the case H inf > m 3/2 , we now have a more robust construction with an explicit realisation of inflation and reheating [45]. Notice that the string scale turns out to be relatively high: M s = M P / √ 4πV ≃ 10 14 GeV. Let us now present an outline analysis of the cosmological implications of this scenario.
1. The del Pezzo modulus τ dP , and the corresponding axion a dP fixed by nonperturbative effects, play the rôle of the inflaton and are very light during inflation. After the end of inflation, when the classical condensate oscillates coherently around the minimum, they obtain a large mass of the order m τ dP ∼ m a dP = m 3/2 ln V ≃ 5 · 10 11 GeV. Given that these moduli couple to the SM as 1/(M P √ V), their perturbative decay gives rise to reheating at a temperature: Notice that τ dP and a dP have a much stronger coupling, of the order 1/M s , to hidden sector degrees of freedom living on this del Pezzo divisor, and so they would dump all their energy to hidden, instead of visible, particles. However, this problem is absent if the hidden sector consists of an E3-instanton or a pure N = 1 SYM theory that develops a mass gap since no light hidden degree of freedom would be present [45].
2. The non-local Kähler moduli are all very heavy in this case and do not suffer from the CMP. In fact, the mass of the volume and fibre moduli are m τ b = m 3/2 / √ V ≃ 5 · 10 6 GeV and m τ f = m 3/2 /V 2/3 ≃ 500 TeV. Given that they both couple to the SM with Planck-strength they would decay respectively at: and : Since T τ f is above T BBN ≃ O(1) MeV but below T freeze−out ∼ O(10) GeV, nonthermal dark matter will be produced by the decay of τ f which will also dilute the abundance of any axion-like particle. 6 3. If the Calabi-Yau is isotropic, the non-local axions associated with the base and the fibre divisors cannot be the QCD axion and behave as very light and almost decoupled ALPs with a decay constant of the order the 10D KK scale, GeV. Notice that these nonlocal ALPs do not create problems with dark matter overproduction since they are very light and their decay constants are very low. On the other hand, if the Calabi-Yau is anisotropic, the fibre axion a f can be the QCD axion since C f f ≃ O (1), but its decay constant is very high: f a f ≃ 10 16 GeV. However this is not a problem since a f will be diluted by the decay of the non-local Kähler moduli. The axion of the base divisor, a b , is instead again a very light and decoupled ALP.
4. In the case of fractional D3-branes at del Pezzo singularities, all the local closed string axions are eaten up, and so cannot be the QCD axion. Correspondingly, all the local Kähler moduli are fixed by D-terms, and so obtain masses of the order M s and disappear from the EFT. In fact, each dP n has n two-cycles which can be trivial or non-trivial within the Calabi-Yau plus one two-cycle corresponding to the canonical class which is always also a two-cycle of the Calabi-Yau dual to the del Pezzo divisor. Moreover each dP n has exactly two anomalous U (1)s which become massive by eating the axions given by the reduction of C 2 on the canonical two-cycle and C 4 on the del Pezzo divisor [51]. Given that these are the only two local cycles, all the local closed string axions are eaten up. In fact, if some of the n two-cycles are non-trivial within the Calabi-Yau, they would correspond to non-local cycles, and so to non-local axions. Two possible way-outs could be the following: • Consider more complicated singularities whose resolution involves more than two local cycles. In this case, some local closed string axions might be left over, and they could play the rôle of the QCD axion. The corresponding decay constant would be rather high, f as ≃ M s / √ 4π ≃ 5 · 10 13 GeV, but the abundance of the QCD axion would be diluted by the decay of the non-local Kähler moduli.
• Focus on the phase of an open string axion φ to realise the QCD axion in a more model-dependent way. In this case, one should check that the D-terms, which give a VEV to the radial part of φ that sets the axion decay constant, do not resolve the singularity obtained by setting the Fayet-Iliopoulos term ξ to zero. In fact, the D-term potential used to fix the blow-up mode τ blow at the singularity looks like: Hence, if |φ| = 0, V D = 0 for ξ = 0 implying τ blow = 0. However, if φ is a SM singlet which can acquire a non-zero VEV, V D = 0 would set |φ| = √ ξ ≃ √ τ blow M s , creating a tension between τ blow = 0 and |φ| = 0. However τ blow could still have a non-zero VEV, as long as it is below the string scale, since we would still be at the quiver locus if τ blow = V −2α with α > 0. If this kind of VEV were induced by the F-terms, the VEV of |φ| would become |φ| ≃ M s /V α . Given that this VEV also sets the axion decay constant f a , the volume suppression factor might bring f a at the intermediate scale. At that point, the QCD axion would not form a relevant component of dark matter due to the dilution induced by the decay of the non-local Kähler moduli, and the constraints from isocurvature fluctuations would not apply to this case.

Explicit LVS models
Not all the axions of the examples studied in section 2.3 will survive in the EFT since some of them will become heavy due to non-perturbative effects whereas others will be eaten up by anomalous U (1)s. According to the general discussion of the axiverse and moduli stabilisation performed in section 3.1, only n ax = h 1,1 − d − 1 axions remain light. Moreover, the LVS requires n ax ≥ 2 guaranteeing the presence of a QCD axion candidate and at least one ALP. However, the couplings of the n ax light axions depends crucially on the Calabi-Yau topology and the particular choice of brane set-up and fluxes. In particular, it is important to determine under what circumstances one of these n ax light axions can have an intermediate scale decay constant. This axion should be associated to a local blow-up mode but if we consider the simplest case of a GUT model with SU (5) that lies on a diagonal del Pezzo four-cycle τ dP that does not intersect any other divisor, this axion is definitely eaten up. In fact, even if the D-term induced shrinking of τ dP is avoided by giving a non-zero VEV to a SM singlet, since SU (5) descends from a U (5) gauge group, we must make the U (1) ⊂ U (5) massive via a diagonal flux upon the stack, which gauges the axion corresponding to τ dP .
The way to circumvent the above issue is to introduce additional local cycles that intersect the GUT/MSSM stack. This, then, introduces the additional problem of how to stabilise the corresponding moduli. In [15], this was performed by an instanton superpotential that included two blow-up moduli, and then their relative sizes were fixed by loop corrections, leaving a massless axion. However, there may be difficulties with that approach due to chirality, since it required an instanton coupling to the SM cycle. Moreover, the massless axion left over is likely to be eaten up as in the example above with a single del Pezzo divisor.
We shall instead advocate a different strategy, where D-term constraints are used to fix the relative sizes of local cycles, and then the overall size of these is fixed by loop corrections following the general strategy outlined in [25]. In this way, we are sure that the massless axion left over is definitely not eaten up by any anomalous U (1), and so it can play the rôle of QCD axion with an intermediate scale decay constant.
On top of these local axions, as we discussed in previous sections, the LVS ensures the presence of at least one additional non-local axion associated to the overall volume. This axion is also a QCD axion candidate, albeit with a GUT-scale decay constant. Due to the large size of this decay constant (and also of the decay constants of other possible non-local ALPs), severe constraints from overproduction of dark matter have to be considered with care (for a complete set of constraints upon such ALPs see [6]). However, as explained in section 3.3, these constraints can be avoided in the presence of a late out-of-equilibrium decay of heavy moduli which dilute the non-local axions.
In the following sections, we shall discuss several explicit examples with different topologies and choices of brane set-up and fluxes which lead to different phenomenological features: • We shall first focus on a fibred example with h 1,1 = 4 taken from [25].
-In section 4.1 we shall analyse an isotropic GUT-like model with d = 2 D-term constraints leading to n ax = 1 light axion with f a ≃ M 10D KK /(4π) ≃ 5 · 10 14 GeV which is associated to the overall volume. As we have seen in section 3.3, this axion can be the QCD axion in the presence of a dilution mechanism (or tuning of the initial misalignment angle).
-On the other hand, in section 4.2, we shall study an anisotropic SU (3) × SU (2) model with d = 1 D-term equation leading to n ax = 2 light axions: the fibre axion a f with f a f ≃ 10 16 GeV, and the base axion a b with f a b ≃ M 6D KK /(4π) ≃ 5 TeV. As explained in section 3.3, a f can be the QCD axion in the presence of a dilution mechanism, whereas a b is a very light and almost decoupled ALP.
• In section 4.3 we shall then consider another fibred Calabi-Yau example with h 1,1 = 5 taken from [52], which allows a chiral model with n ax = 3 light axions.

A GUT-like model with a QCD axion with large decay constant
Here we shall study the model of section 4 of [25] and derive the low-energy properties of the closed string axions. This type IIB model involves intersecting magnetised D7branes, tadpole and Freed-Witten anomaly cancellation and all the geometric moduli stabilised in a way compatible with the presence of chirality. The internal space is an orientifold of a K3 fibred Calabi-Yau three-fold with h 1,1 + = 4 and h 1,1 − = 0 realised as a hypersurface embedded in a toric variety. The relevant divisors for model building are the K3 fibre D 1 , two intersecting rigid four-cycles D 4 and D 5 and a del Pezzo divisor D 7 . Expanding the Kähler form J in this basis as: and so the overall volume reads: The volume of the basis divisors, which are the real part of the Kähler moduli T j = τ j + i c j , are given by: The P 1 base of the fibration is given by the intersection of the two rigid cycles D 4 and D 5 : and so the expression for the volume (4.3) can also be rewritten as: The GUT-like model is realised by wrapping N a = 5 D7-branes around D 4 and N b = 2 D7-branes around D 5 (this number has to be even in order to cancel the K-theory charges). Given that each divisor is transversally invariant under the orientifold, we obtain symplectic gauge groups Sp(2N a ) and Sp(2N b ) which are broken down to unitary groups by switching on appropriate magnetic fluxes F a and F b on the worldvolume of the D7-branes, so that Sp(10) → SU (5) × U (1) and Sp(4) → U (1) × U (1). As explained in the previous sections, the anomalous U (1)s become massive via the Stückelberg mechanism by eating up a closed string axion. The U (1) charges of the Kähler moduli are given by: and so the combinations of two-formsc i dual to the axions c i which are eaten are: Therefore the visible sector gauge group at low-energy is SU (5) × U (1) where the massless U (1) factor turns out to be very weakly coupled, and so it behaves as a dark force [21]. The gauge fluxes F a and F b induce also moduli-dependent Fayet-Iliopoulos (FI) terms which fix two Kähler moduli at: Notice that the combinations of axions that are eaten up are the same as the combinations of Kähler moduli which are fixed by the FI-terms: Given that the two moduli fixed by the D-terms obtain an O(M s ) mass, the EFT can be studied just in terms of the two remaining Kähler moduli τ 1 and τ 7 . Thus we can simply integrate out the fixed moduli and work with an 'effective' volume form to determine the masses of the light axions. This is particularly pertinent given that D-term corrections to the potential typically dominate over all others. The D-term stabilisation (4.8) gives a volume of the form: where α = 86/(57 √ 19) and γ = 1/(3α). The form of the volume (4.10) is very similar to (2.10), and so the canonical normalisation takes the same form as the simplest Swiss-cheese example of section 2.3.1: In order to stabilise τ 1 and τ 7 , the authors of [25] introduced also N gc = 4 D7-branes around the del Pezzo divisor D 7 which give rise to a pure N = 1 Sp(8) hidden sector that undergoes gaugino condensation. The interplay between this non-perturbative effect and the leading order α ′ correction to the Kähler potential fixes both τ 1 and τ 7 following the standard LVS procedure. The minimum is at: For values of W 0 of order unity, the volume, and so τ 1 , would be exponentially large resulting in an exponentially small value of the visible sector gauge coupling since from (4.8) we realise that α −1 GUT = τ 4 is proportional to τ 1 . Thus this phenomenological requirement tells us that we need to fine tune W 0 ≪ 1. This tuning is actually also needed to obtain TeV-scale SUSY since, for V ∼ O(10 4 ), the gravitino mass (and the soft terms generated via gravity mediation) turns out to be of the order the TeV scale only if W 0 ∼ O(10 −10 ). Moreover the non-perturbative effects on D 7 generate a potential for the axion c 7 , and so the canonically normalised axion a 7 obtains a mass of the order m 3/2 . This axion is definitely too heavy for playing the rôle of the QCD axion, and so we shall focus on the remaining axion a 1 which remains massless since the corresponding saxion, i.e. τ 1 , is fixed purely perturbatively via α ′ corrections.
The couplings of a 1 to the gauge bosons living on D 4 and D 5 scale as: (1) ). (4.14) The volume scaling of these couplings can again be understood in terms of the locality argument of (2.9). In fact, the two rigid divisors D 4 and D 5 are non-local effects since they control the size of the overall volume by defining the P 1 base of the fibration: t base = Vol (D 4 ∩ D 5 ). Hence we correctly found that the massless axion couples to these gauge bosons with Planck strength.
The axion decay constant instead scale as the 10D KK scale: while the anomaly coefficient is of order unity, C 11 ≃ O(1), and so a 1 can play the rôle of the QCD axion. As explained in section 3.3, the abundance of the axion a 1 will be diluted by the late out-of-equilibrium decay of τ 7 and a 7 .

An SU (3)×SU (2) model with a QCD axion with large decay constant plus a light ALP
Here we study the second example of [25] which has two massless axions. This is another globally consistent chiral model with magnetised D7-branes and moduli stabilisation within type IIB Calabi-Yau compactifications. The internal manifold is the same described in section 4.1. However this time the gauge group is not GUT-like since the brane fluxes are chosen in such a way to give rise to just one D-term condition, so that the volume of the Calabi-Yau is no longer proportional to an inverse power of the visible sector gauge coupling. In turn, W 0 can be set of order unity and the volume can be exponentially large lowering the string scale to the intermediate value M s ≃ 5 · 10 10 GeV and obtaining TeV-scale supersymmetry without any fine-tuning. Thus there is no space anymore for GUT theories but only for an MSSMlike construction. This is achieved by wrapping N a = 3 D7-branes around D 4 and N K3 = 1 D7-branes around the K3 fibre D 1 . The corresponding gauge groups are Sp (6), which is broken down to SU (3) × U (1) by switching on appropriate magnetic flux F a on D 4 , and Sp(2) ∼ = SU (2). Notice that the total gauge flux on D 1 is zero. The anomalous U (1) becomes massive via the Stückelberg mechanism by eating up a closed string axion. The U (1) charges of the Kähler moduli are given by: and so the combination of two-forms (dual to the axions) which is eaten up is: Therefore the visible sector gauge group at low-energy is SU (3) × SU (2). The gauge flux F a induces also a moduli-dependent FI term which fixes one Kähler modulus at: and so the combination of axions eaten is: Given that the modulus fixed by the D-term has an O(M s ) mass, the EFT can be studied just in terms of the three remaining Kähler moduli τ 1 , τ 5 and τ 7 . The D-term stabilisation (4.16) gives a volume of the form: where the variables τ 1 and τ 5 have been traded for τ s and τ b defined as: where the label s stays for 'small' and the label b for 'big'. This redefinition is suggested by the D-term condition (4.16) which implies that the combination (τ 1 − τ 5 ) has to be fixed small in order to obtain a visible sector gauge coupling α −1 vis = τ 4 which is not too small The form of the volume (4.18) is very similar to (2.18) once we identify the base t b with τ b / √ τ s and the fibre τ f with τ s . Given that we are interested in the regime where τ s is much smaller than τ b , we shall consider the anisotropic limit where the size of the base √ t b is much bigger than the size of the fibre τ In order to stabilise V ≃ √ τ s τ b and τ 7 , the authors of [25] introduced also N gc = 3 D7-branes around the del Pezzo divisor D 7 which give rise to a pure N = 1 Sp (6) hidden sector that undergoes gaugino condensation. The interplay between this nonperturbative effect and the leading order α ′ correction to the Kähler potential fixes both the combination V ≃ √ τ s τ b and τ 7 following the standard LVS procedure. The minimum is at: Moreover the non-perturbative effects on D 7 generate a potential for the axion c 7 , and so the canonically normalised axion a 7 obtains a mass of the order m 3/2 . This axion is definitely too heavy for playing the rôle of the QCD axion, and so we shall focus on the remaining massless axions a s and a f . These axions remain massless since the corresponding saxions are fixed purely perturbatively via α ′ and string loop corrections to the Kähler potential. The couplings of a s and a f to the gauge bosons living on D 4 and D 1 scale as: (2) ). (4.24) The volume scaling of these couplings can again be understood in terms the locality argument of (2.9). In fact, both the K3 fibre τ 1 and the rigid divisor D 4 are non-local effects since they control the size of the overall volume (recall that D 4 controls the size of the P 1 base of the fibration). Hence we correctly found that the massless axions couple to these gauge bosons with Planck strength or weaker. In the anisotropic limit τ b ≫ τ s ∼ τ 7 ≃ O(10), the axion decay constants become: while the anomaly coefficients look like (see (2.28)): As described in section 3.3, a s can be the QCD axion with an O(M GUT ) decay constant. This value is not in the allowed window but it might still be viable in the presence of the late out-of-equilibrium decay of the heavy moduli τ 7 and a 7 which would dilute the axion a s [22]. The other axion a b is instead a light ALP with a very low decay constant which does not create any problem with dark matter overproduction. We finally mention that the SU (2) gauge group is a hyperweak interaction since it is realised via D7-branes wrapping an exponentially large four-cycle.

A chiral model with a QCD axion with intermediate decay constant plus two light ALPs
In this section we shall present an explicit type IIB model with the presence of a closed string modulus which is a perfect QCD axion candidate since it is not eaten up by any anomalous U (1), does not develop any potential by non-perturbative effects in g s and has a decay constant at the intermediate scale.
The model is very similar to the one used in [52] to obtain a modulated reheating scenario with the production of detectable non-Gaussianities of local form. It is based on an orientifold of a Calabi-Yau three-fold with a K3 or a T 4 fibration structure and h 1,1 + = 5 and h 1,1 − = 0. Again, this internal space can be realised as a hypersurface embedded in a toric variety [32]. The relevant divisors for model building are the K3 or T 4 fibre D 1 , the divisor D 2 controlling the P 1 base of the fibration, a del Pezzo four-cycle D 3 and two rigid divisors D 4 and D 5 which intersect each other but have no intersections with the other four-cycles. The Kähler form J can be expanded in this basis as: where we took a minus sign for the rigid cycles in order to have positive dual twocycles. All the non-vanishing intersection numbers are positive and are: k 122 , k 333 , k 444 , k 445 , k 455 , k 555 , (4.27) with the relation k 2 455 = k 445 k 555 . Thus the overall volume reads: where α, γ 3 , γ 4 , γ 5 and x are O(1) constants which depend on the intersection numbers (in particular x = k 445 /k 455 ). In [52], the authors consider a particular set-up suitable to build a phenomenologically viable model of modulated reheating. The visible sector is built via two stacks of intersecting D7-branes wrapping two different combinations of the rigid divisors D 4 and D 5 . Moreover an E3-instanton wrapped around D 5 gives rise to poly-instanton corrections to the superpotential since D 5 is assumed to be a rigid four-cycle with Wilson lines, i.e. h 2,0 (D 5 ) = 0 while h 1,0 (D 5 ) = 1 [42]. The gauge fluxes are chosen in such a way to yield chirality at the intersection between the two visible sector stacks but with the generation of just one FI-term. Furthermore, no chiral matter at the intersection of the visible sector with the E3-instanton is induced. This set-up allows to keep the field τ 5 light since it develops a potential only via tiny poly-instanton effects. This is a crucial condition to obtain a viable modulated reheating scenario. More precisely, three effects develop a potential for τ 4 , τ 5 and the absolute value of a open string field |φ| which is a visible sector singlet: 1. D-terms fix |φ| , 2. String loop corrections to the Kähler potential fix the combination (τ 4 − x τ 5 ) , 3. Poly-instanton corrections to the superpotential fix τ 5 .
Moreover, as in standard LVS, τ 3 is fixed non-perturbatively, the volume by the interplay of non-perturbative effects on D 3 and α ′ corrections to the Kähler potential, while the fibre modulus τ 1 is stabilised by string loops. Due to the locality argument of (2.9) only the axions of τ 4 , τ 5 and φ (the corresponding phase) couple to the visible sector as 1/M s , and so can be good QCD axion candidates if the string scale is intermediate. However, the axion of τ 5 cannot be the QCD axion since it develops a non-perturbative superpotential. On the other hand, a combination of the axion of τ 4 and the phase of φ will be eaten up by the anomalous U (1) while the other can play the rôle of the QCD axion. If one is not interested in keeping τ 5 very light, then one could drop the assumption that D 5 has Wilson lines resulting in the absence of poly-instanton corrections. Then also this divisor would be frozen by string loop effects, and the corresponding axion would also remain massless and have the right coupling to QCD.
In the analysis of [52], the open string field φ plays a crucial rôle because in its absence it is impossible to obtain a light modulating field. In fact, a combination of τ 4 and τ 5 would be fixed by D-terms while another would be stabilised by string loops which beat the tiny poly-instanton effects and generate a mass which is not below the Hubble constant (a crucial condition to find a viable modulated reheating). However, given that we are not interested in modulated reheating, we shall explore exactly this simpler possibility since it involves no open string fields, and so we consider it to be more model-independent.
In our set-up the visible sector is realised by two stacks N a and N b of intersecting D7-branes wrapped around the divisors D 4 and D 5 : D a = D 4 and D b = D 5 . We also turn on the following world-volume fluxes: where the half-integer contributions come from the cancellation of the Freed-Witten anomalies 7 . The U (1)-charges of the Kähler moduli induced by these fluxes can be written one in terms of the other as: The moduli-dependent FI-terms take the form: The chiral intersections between the stacks of D7-branes on D 4 and D 5 depend on the gauge fluxes in the following way: Setting g 4 = −k 555 /(2k 455 ) and g 5 = 0, we obtain q b 4 = q b 5 = 0 which implies ξ b = 0, and so we are left over with just one FI-term. Consequently, just one closed string axion will be eaten up. Moreover, we obtain I ab = q a 5 which implies q a 5 = 0 in order to have chiral matter. From (4.30) we notice that we can choose the fluxes in such a way to have q a 5 = 0 and q a 4 = 0 at the same time, so that the FI-term (4.31) simplifies to 8 : where we introduced the combinationτ 4 defined asτ 4 ≡ τ 4 − x τ 5 . In the absence of open string singlets which can develop a non-zero VEV, the vanishing of the D-terms implies ξ a = 0, and so one combination of τ 4 and τ 5 is fixed at: (4.34) This combination can be fixed within the regime of validity of the EFT due to the intersection between D 4 and D 5 . In fact, in the absence of this intersection for x = 0, (4.34) would imply the shrinking of the rigid divisor D 5 (see also [25] for related issues). Moreover, (4.34) implies that the combination of axions eaten is: Given that the modulus fixed by the D-term receives an O(M s ) mass, the EFT can be studied in terms of the four remaining Kähler moduli τ 1 , τ 2 , τ 3 andτ 4 . Hence, after the D-term stabilisation (4.34), the volume (4.28) simplifies to: The form of the volume (4.35) is very similar to (2.18) once we identify the base t b with τ 2 / √ τ 1 and the fibre τ f with τ 1 (setting α = 1 without loss of generality). Hence the canonical normalisation, in the limit V ≃ √ τ 1 τ 2 ≫ τ 3 ∼τ 4 , takes the same form as the fibred example of section 2.3.2: Notice that the mixing terms between the axions a 3 and a 4 associated to the blow-up modes τ 3 andτ 4 are very suppressed due to the fact that these two rigid divisors do not intersect each other, and so they correspond to the resolution of singularities which are localised in different regions of the Calabi-Yau background [45]. The overall volume and τ 3 are fixed by the leading order F-term scalar potential following the standard LVS procedure which requires non-perturbative effects on D 3 and the leading order α ′ correction to the Kähler potential. The minimum is at: The non-perturbative effects on D 3 generate a potential for the axion c 3 , and so the canonically normalised axion a 3 obtains a mass of the order m 3/2 . This axion is definitely too heavy for playing the rôle of the QCD axion, and so we shall focus on the remaining massless axions a 1 , a 2 and a 4 . These axions remain massless since the corresponding saxions, i.e. V, τ 1 andτ 4 , are fixed purely perturbatively via α ′ and string loop corrections to the Kähler potential [24,25]. As explained in [24], in order to fix τ 1 via g s effects, two stacks of D7-branes have to be wrapped around D 1 and D 2 , whereas the string loops which stabiliseτ 4 are sourced by the visible sector stacks on D 4 and D 5 . In fact, following the systematic analysis of string loop corrections performed in [40], the leading order behaviour of the effective scalar potential generated by g s effects is: where µ 1 and µ 2 are positive constants which depend on the complex structure moduli, while µ 3 is a positive constant that depends on the VEV of stabilised moduli like τ 3 once we include the intersection of the branes wrapping D 4 and D 5 with the O7-planes and additional Whitney-type branes for tadpole cancellation (see [25]). For natural O(1) values of the coefficients of the string loop corrections, the potential (4.41) can develop a minimum at:τ This minimum is indeed located at smallτ 4 since for example for µ 1 = µ 2 = 1 and µ 3 = 10, we obtainτ 4 = 25.
The couplings of a 1 , a 2 and a 4 to the gauge bosons living on D 1 , D 2 , D 4 and D 5 scale as (recalling that τ 4 =τ 4 + x τ 5 and τ 5 = λτ 4 ⇔ τ 4 = (1 + x λ)τ 4 ): 9 We stress that we managed to obtain a closed string axion, a 4 , which is not eaten up by any anomalous U (1) and couples to the visible sector on D 4 and D 5 as 1/M s opposed to the standard 1/M P gravitational couplings. The axion decay constants instead become: whereas the anomaly coefficients look like: The field a 4 is a perfect QCD axion candidate since its decay constant is intermediate and C 4i ≃ O(1) ∀i = 4, 5. Therefore we have managed to build an explicit example where a closed string axion with these features [15] survives in the EFT since it is not eaten up by any anomalous U (1). As pointed out in section 3.3, this axion will not form a relevant component of dark matter since it will be diluted by the decay ofτ 4 . Thus the constraints from isocurvature fluctuations do not apply to this case. Moreover, there are two further ALPs, a 1 and a 2 , whose behaviour is different in the case of a isotropic or anisotropic Calabi-Yau.
• Isotropic case: in the isotropic limit τ 1 ∼ τ 2 ∼ V 2/3 , the decay constants of a 1 and a 2 scale as the 10D KK scale: 44) and the anomaly coefficients simplify to: The two non-local axions a 1 and a 2 correspond to two light and almost decoupled ALPs that do not create any problem with dark matter overproduction. Moreover the gauge theories on τ 1 and τ 2 are hyperweak.
• Anisotropic case: in the anisotropic limit τ 2 ≫ τ 1 ∼τ 4 , the decay constants of a 1 and a 2 become: while the anomaly coefficients take the simplified form: Now the anomaly coefficient C 1i is of order unity, and so also a f can play the rôle of the QCD axion but with a GUT-scale decay constant. However, there are no problems with dark matter overproduction since, as explained in section 3.3, the abundance of the axion a 1 will be diluted by the decay ofτ 4 . Notice that in this case τ 1 ∼τ 4 , and so the visible sector could also be placed on the fibre divisor. However, the gauge theory on the base divisor τ 2 is still a hyperweak interaction. Finally the non-local axion a 2 is a light (almost decoupled) ALP with a very low decay constant of the order the TeV-scale.

A chiral model with a QCD axion and many ALPs with intermediate decay constants
The previous example has shown how it is possible to obtain models with a QCD axion in the axion window. However, it also shows how there can be additional ALPs coupling to the visible sector with a very similar decay constant. For this, we simply need additional local cycles intersecting the visible branes, but without additional D-term conditions. The same method of moduli stabilisation will apply: for n intersecting local cycles, d of them are fixed by D-terms (and so d axions are eaten up) while the remaining n − d moduli are frozen by loop corrections, leading to n − d light axions with intermediate scale decay constant. We stress that the number of D-term conditions can be reduced either by not wrapping some of the intersecting cycles or by setting some world-volume fluxes to zero. To be concrete, consider the blow-up of a Z 7 singularity which has no non-compact exceptional divisors. Thus it will not be altered by the space in which it is embedded, and so it can be studied locally. This is not necessary: a much larger class of examples could be found, for example, via the blow-ups of del-Pezzo singularities, but then we would need to take into account the global embedding; we leave the construction of these rich and fully realistic examples to future work. We denote the overall volume V 0 ≃ V (which could be a Swiss-cheese or a fibred Calabi-Yau) with a small cycle τ s stabilised at τ s ∼ ln V by a gaugino condensate wrapping D s . The volume form is then: 46) and the Kähler cone is given by: We wrap a 'visible' stack on D a = D 1 , and another on D b = D 2 , turning a flux of F a = 3D 3 + 1 2D 1 , leading to I ab = 3 and no other chiral intersections. The D-term generated stabilises τ 3 in terms of τ 1 and τ 2 at: 47) or equivalently: Hence we can write the volume form effectively as: V 0 is stabilised in the usual LVS way; string loops then stabilise the remaining moduli τ 1 and τ 2 via terms in the potential of the form (4.41) (the full loop potential is rather complicated even in this simple case so we neglect to present it here).
The axions corresponding to moduli τ 1 and τ 2 are thus left massless at this level, and couple with string strength to branes a and b, since the effective mixing matrix is (restricting indices to i, j = 1...3, since the mixing with τ s and τ b is negligible): This is non-degenerate with full mixing between the axions. For τ 1 = τ 2 = τ * , the D-term relation (4.47) gives also τ 3 = τ * and so the mixing matrix simplifies to: Therefore the gauge couplings of groups a and b are equal, g 2 a = g 2 b ≡ g 2 ∼ τ −1 * . Taking for example τ * = 10, the axionic couplings turn out to be: giving a 'QCD axion' a QCD = (0.86 a 1 −0.5 a 2 ) together with an ALP a ALP = (0.5 a 1 + 0.86 a 2 ) and: Hence if we interpret the cycle 'b' as a U (1) stack, the ALP will couple with approximately equal strength to the photon as the QCD axion does. We believe this is a generic prediction of this class of models: it is highly non-generic to have an ALP that couples parametrically more strongly to the photon than the QCD axion does.
In order to achieve this, it would be necessary for there to be a local stack which carries a U (1) which contributes to the hypercharge, but which does not intersect the QCD stack, and has no mass mixing (through D-terms) with the axion of the QCD stack. This implies a construction with at least four stacks of branes; it would be an interesting scenario to realise.

Explaining astrophysical anomalies with a stringy ALP
Since the above construction is built from intersecting branes in the geometric regime, the coupling of the local axion to matter fields should be given by the first expression in (2.36). Hence we expect the ALP coupling to electrons to be suppressed only by a factor of the gauge coupling squared relative to the photon coupling. This is perfectly consistent with the astrophysical hints for an ALP, -an anomalous energy loss of white dwarfs (see section B.1.1 in the appendix) and an anomalous transparency of the Universe for TeV photons (see section B.1.2) -which would be explained by: For models in this class -i.e. built from local constructions of typical size τ * -we find generically: where N iγ is a purely numerical factor coming from the canonical normalisation of the ALP. For m 3/2 = 10 TeV, g s ≃ 0.1 and W 0 ∼ 10 we can easily arrive at the desired figure for N iγ ∼ 10.
The ALP mass could be generated, for example, by a single Kähler potential instanton, which for m ALP ∼ 10 −10 eV would require: Note that there is then a small amount of tension with the ratio C iγ /C ie , but this is the least well predicted quantity. Hence it is possible to explain these astrophysical hints for an ALP via a natural intermediate string-scale scenario with stable moduli. Note that such an ALP would constitute only a tiny (O(10 −9 )) fraction of dark matter, and would satisfy all constraints -interestingly being just one or two orders of magnitude in mass heavier than the black hole super-radiance constraint [19,53] (see section B.2) and within the reach of future experiments (see figure 1 and appendix B).

Conclusions
String phenomenology holds the promise of an axiverse -the QCD axion plus a (possibly large) number of further ultralight axion-like particles, possibly populating each decade of mass down to the Hubble scale, 10 −33 eV. However, although a plenitude of axions is a generic prediction of string theory, there may be few or no light axions remaining once constraints such as tadpole cancellation and moduli stabilisation are taken into account (for example in the KKLT scenario each axion would be made massive by an exponential term in the superpotential) or, particularly for D-brane models, when considering which axions are eaten in making U (1) symmetries massive. This last issue is particularly important, for example, in GUT models where the diagonal U (1) ⊂ U (5) is made massive in this way; we have shown that in such cases (and more generally for any model where the diagonal U (1) ⊂ U (3) is made massive like this) if the Standard Model is built on a local cycle, the non-local axions cannot solve the strong CP problem, and particular care must be taken to ensure that there are additional light local axions that can do the job.
We have shown that the promise of an axiverse is fulfilled in the LARGE Volume Scenario of IIB string compactifications. In fact, the least fine-tuned (i.e. W 0 ∼ O(1)) models of globally consistent constructions with magnetised D-branes and chirality have at least two light axions. In general, the closed string axions separate into local and non-local classes, where the non-local axions may have small decay constants but also small couplings and vanishingly small masses, and so are unlikely to overproduce dark matter (unless the Standard Model also wraps a non-local cycle but in this case the axions would be diluted by the decay of heavy Kähler moduli).
We have also determined for the LVS axiverse the couplings of axions to matter fields, both at tree level and one loop. The results at one loop generalise the existing non-supersymmetric formulae to supersymmetric models. However, these rely upon the conjectured form of the Kähler metric for matter fields; it would be interesting to calculate these couplings in specific examples.
Moreover, we have shown how to build models that exhibit a QCD axion with a decay constant f a ∼ M P / 4π √ V ∼ 10 10 ÷ 10 12 GeV, and an additional axion-like particle (or even more than one) having the same decay constant and coupling to the photon 10  The particular model of inflation is also relevant for the cosmological constraints. In the LARGE Volume Scenario there are well-defined models of inflation [54], and so these questions can be readily addressed.
If the Hubble scale during inflation is larger than the gravitino mass, the latter can be of the order the TeV scale giving rise to an intermediate string scale which allows couplings for the axions compatible with the astrophysical hints. However, reheating at the end of inflation is problematic (since in this case the inflaton is the lightest modulus, i.e. the volume mode) and it is an interesting question as to how this can be achieved.
Alternatively, if the Hubble scale during inflation is lower than the gravitino mass, the latter has to be much above the TeV scale. This singles out sequestered models where the soft terms can be around the TeV scale even if the gravitino mass is at the intermediate scale [35]. The string scale turns out to be around the GUT-scale, allowing again for good QCD axion candidates. Even in this case there can be heavy particles that decay and dilute other relics, but now, due to a higher string scale, the axion will be a component of non-thermal dark matter. Contrary to the situation with H inf > m 3/2 , inflation and reheating are better understood, and so we can have more control over all the cosmological evolution of the axions and the moduli from the early Universe till today.
Since the phenomenology of axions is so rich, with astrophysical constraints and hints, and many experiments searching for them, it is highly worthwhile to investigate what values of the couplings can be predicted from string theory. We have compiled a comprehensive analysis of these in the LVS context where both local and global questions can be asked about the models in a consistent way. We hope that this will be of use to experimentalists and model builders alike.

A Phenomenology of theories with many axions A.1 Effective Lagrangian at the weak scale
For the low energy phenomenology of axions, their couplings to gauge bosons other than gluons, specifically to photons, and to SM fermions, specifically to light quarks and electrons, are of uttermost importance. It is known that for the QCD axion a great deal about its low energy couplings can be learned from effective field theory techniques, which allow to take electro-weak symmetry breaking and QCD effects into account. In this spirit, generalising the considerations in [55] to the case of several axion-like fields, a i , we write down the most general effective Lagrangian at the weak scale, to first order in the fields a i , which consists of anomalous couplings to gauge bosons and derivative couplings to matter: where θ 0 is an initial theta-angle, F W and F Y are the SU (2) and hypercharge field strengths, respectively, g 2 and g Y the corresponding gauge couplings, M is the mass matrix of the fermions, written as Weyl fermions ψ L and ψ R . Note that in general the couplings to fermions need not conserve parity (see [56] for a related discussion on this issue). Furthermore, the Yukawa couplings on the last line are the leading terms in the expansion of Yukawa couplings of the form: since we expect the full theory to respect the axionic shift symmetries, only broken by non-perturbative effects. Those contributed other than from QCD are encapsulated in the potential V (a i ), which we expect to give a mass to all ALPs at some scaleexcept for those that are gauged, and obtain their mass through being eaten (these shall be discussed in section 3.2). When an ALP couples to a gaugino condensate or D-brane instanton, we expect the mass to be large, and we can simply exclude it from the analysis to follow. These are leading contributions in the superpotential. On the other hand, for ALPs that do not obtain masses in that way, the masses induced are many orders of magnitude smaller, coming from multi-instanton effects or Kähler potential corrections, and have been argued to be many orders of magnitude below the QCD axion mass [15]. Therefore in the following we shall neglect ALP masses and comment on their inclusion at the end.
In the text of the paper have derived the predictions for the dimensionless couplings C i3 , C iW , C iY , X i ψ L , X i ψ R , and g i , and the decay constants f a i from particular string compactifications. Here, in the next subsection, we will further scale the theory down through electro-weak symmetry breaking, heavy quark thresholds and QCD chiral symmetry breaking to arrive at the low energy Lagrangian involving only the QCD axion, ALPs, photons and electrons, which are then be compared to astrophysical and cosmological observations, as well as to laboratory experiments, in section B.

A.2 Effective Lagrangian below chiral symmetry breaking
After electro-weak symmetry breaking, we diagonalise the fermion mass matrix, transforming to the physical basis with real masses m Ψ , noting that this can lead to flavour-changing axionic couplings) and write Ψ = ψ L ψ R : The mass of the QCD axion is generated by the coupling to QCD, but is complicated by the presence of the light quarks -if there were none then its mass would be that the η ′ acquires. One can however use chiral perturbation theory to calculate it: with chiral rotations of the quarks one may eliminate the F 3F3 term and consider the effective pion Lagrangian. In the approximation m s ≫ m u , m d , one can consider only the π 0 and its mixing with the axions (one combination of η and η ′ is made massive by the strange quark mass, the other by the anomaly). Making the rotations Ψ → exp(−iα Ψ γ 5 )Ψ, with: will eliminate the F 3F3 term and couple the axions to the quarks.
We are now ready to define the QCD axion a, with decay constant f a and setting C a3 = 1, cf. eq. (1.1), and the remaining ALP fields a ′ i , in terms of linear combinations of the original axion-like fields a i : resulting in the effective Lagrangian: The kinetic mixing term on the last line between the QCD axion, ALPs and pion can be ignored provided thatX u −X d ≪ f a i /f π ; this is because the non-unitary transformation that diagonalises the kinetic terms can be chosen so that the pion merely rescales ); these lead to subleading (and therefore negligible, for the expected sizes of f a i compared to f π ) corrections to the physical masses and couplings. In fact, the all-orders diagonalisation is performed by: and thus there are no new couplings induced between the ALPs and the photon at any order.

A.2.1 Coupling to photons
Neglecting the mixing terms χ i , the mass of the pion, the mass of the QCD axion and its coupling to photons are then found as usual by choosing the coefficients c u = 1 1+z , c d = 1 − c u , where z ≡ m u /m d , to eliminate the mass mixing term above; this gives: m a = m π fπ fa √ z 1+z and the QCD axion two-photon coupling: while the couplings of the ALPs a ′ i to photons are just: The upshot is that the QCD axion has a 'universal' coupling to photons ∝ ∆C aγ , but the ALPs do not; their coupling to photons and electrons comes entirely from the high-energy theory. Up to this stage, we did not consider mass terms for ALPs. These will have effects on the 'universal' couplings to photons. In fact, we shall treat any ALP heavier than the pion as being already integrated out and ignore it for these purposes, and for ALPs of comparable mass to the QCD axion or heavier the analysis is necessarily involved. However, in the case that additional ALPs are lighter than the QCD axion, we can neglect the kinetic mixing terms as before, and, introducing ALP masses m 2 i , the QCD axion is approximately shifted by: and so the ALPs a j acquire universal couplings, too, which are, however, suppressed in comparison to the universal coupling of the axion, by the ratio m 2 i /m 2 a : Conversely, since the ALPs are shifted by a similar amount, we expect the QCD axion to photon coupling to be shifted by an amount: This is relevant in the case that an ALP couples much more strongly than the QCD axion to the photon and has some coupling to QCD.

A.2.2 Couplings to electrons
Finally, we consider the coupling to electrons which is also crucial for low energy phenomenology. Other than the direct coupling, there will also be one generated through the two-photon coupling at one loop. In supersymmetric theories, the twophoton coupling is accompanied by a coupling to gauginos: and so we have a quantitatively different result. We also have a qualitatively different result, in that we find parity violating vector-like terms arising from the differences in squark masses. We can parameterise the coupling as: whereX A,V j e ≡ 1 2 (X j e R ±X j e L ) and ∆ QCD [C A ae ] = 3α 2 4π ∆C aγ ln(Λ QCD /m a ) is a correction from pion loops to the QCD axion to electron coupling, unchanged in SUSY theories. We shall calculate the full contribution ∆ iγγ C A,V ie in section C, but we note that to a rough approximation we can take: where M soft is the scale of superpartner masses, and Λ the cutoff of the theory, of the order of the string scale.
B Axion searches in astrophysics, cosmology, and laboratory The couplings of QCD axions and ALPs to photons and electrons give rise to a number of effects which can be searched for in astrophysics, cosmology, and in laboratory experiments. In the remainder of this section, we will summarise the present knowledge about the dimensionless couplings and the decay constants. Some astrophysical observations seem to point to a preferred scale of order ∼ 10 9 GeV.

B.1 Axions from astrophysical sources B.1.1 Stellar emission
Light and weakly interacting particles such as the QCD axion or ALPs are produced in hot astrophysical plasmas via their coupling to photons or via their coupling to electrons or nucleons. One of the strictest bounds on the coupling of axions to photons 11 arise from either the non-observation of an anomalous energy loss of Horizontal Branch (HB) stars due to axion emission [57] or (slightly stronger) from the non-observation of solar axions in the CAST experiment [58]: This translates into a bound This is usually used to provide a lower bound for the decay constant of the QCD axion; due to the 'universal' contribution to its two-photon coupling, for a generic field-theoretical model, C aγ ∼ 1. On the other hand, there are some models with C aγ ≪ 1 where the axion decay constant can be lowered. The CAST bounds (B.1) and 11 For a recent update on the bounds on this coupling, see [6].
(B.2) can be improved in the next generation of helioscopes to g iγ ∼ 10 −12 GeV −1 → f a i /C iγ ∼ 10 9 GeV [5]. Turning now to the axial-vector couplings to electrons, let us first note that, in many tree-level interactions, an axial-vector vertex is equivalent to a scalar one: where we have used the equations of motion. Of course, this only applies for tree-level processes involving the fermions as on-shell external states. Excessive axion emission from red giants places the following constraint on axion couplings to electrons [57]: A slightly better limit arises from pulsationally unstable white dwarfs (ZZ Ceti stars), where the period increase provides a measure of the cooling speed. From the measured period increase of the star G117-B15A, it was found [59][60][61][62]: Hence, for C ie 10 −2 C iγ , the limits from electron couplings are more constraining than the direct two-photon coupling. Noting that the 'universal' coupling of the QCD axion to electrons is suppressed by O(10 −4 ), we see that the direct couplings could be important, as we stressed in their discussion in the context of the LARGE Volume Scenario.
However -and intriguingly -reanalysis of the same white dwarf with improvements in the accuracy and precision of the pulsational data identifies a non-standard energy loss [12,63,64]. Improvements in the luminosity distribution of white dwarf stars were also found to require additional energy loss, an absence thereof predicting an excess of white dwarfs at luminosities log L/L ⊙ ≃ −2. The observed excess in cooling speed is consistent with the existence of an axion with mass m i keV and a coupling to electrons of strength: which is compatible, within the uncertainties, with the limits from red giants. On parity violating couplings to the electron, there is a limit [57,65] on a scalar Yukawa coupling g V Y ie eea i of the order: On the other hand, we will generically find in string models parity-violating vectorlike couplings Here however the integration by parts yields a coupling to the divergence of the electron number current; hence we expect the simplest amplitudes involving these couplings to vanish, and the constraints upon them to be very weak. The limits on couplings to nucleons C aN are most strongly constrained by (lack of) energy loss from supernova SN1987A, and applies for the range 0.002 eV C aN m a 2 eV [57]: 3 × 10 9 GeV f a /C aN . (B.9)

B.1.2 Conversion of astrophysical photon fluxes
Due to the F emFem ∝ E · B coupling, photons can oscillate into axions in largescale magnetic fields. These oscillations can lead to observable effects in astrophysical photon spectra, but with high sensitivity only for very small masses. Correspondingly, these observations typically constrain only ALPs, but not the QCD axion. The absence of a γ-ray burst in coincidence with SN 1987A neutrinos provides, for m i 10 −9 eV, the most restrictive limit on the two-photon coupling of ALPs [66,67]: an order of magnitude better than the one inferred from the lifetime of HB stars and from CAST. This constraint challenges the ALP interpretation of recent measurements of γ-ray spectra from distant active galactic nuclei (AGN), extending to energies TeV -a puzzling observation, since the γ-ray absorption due to e + /e − pair production off the extragalactic background light is expected to cut-off the spectra at very high energies 12 . Intriguingly, apart from conventional explanations, such a high transparency of the Universe may be explained by photon ↔ ALP oscillations: the conversion of gamma rays into ALPs in the magnetic fields around the AGNs or in the intergalactic medium, followed by their unimpeded travel towards our galaxy and the consequent reconversion into photons in the galactic/intergalactic magnetic fields, requiring a very light, m i 10 −9 eV, ALP coupling to two photons with strength [7][8][9]: in the same ball-park as the constraint from the non-observation of γ-rays from SN 1987A. 12 For a recent compilation of the available data and comparison with the expectation, see [68].
A similar effect can occur when gamma ray-ALP conversions are induced by intergalactic magnetic fields B, which have a strength of 0.1 ÷ 1 nG. In this case, steps in the power spectrum can be observed at a critical energy [69]: It has been reported in [10] that such steps are indeed seen at a critical energy of ∼ 100 GeV, with the spectrum consistent with g iγ ∼ 10 −11 GeV −1 , giving: m ALP ∼ 10 −9 ÷ 10 −10 eV. (B.13) Very recently, it was pointed out [11] that a similar value of the photon coupling is required for a possible explanation of the rapidly varying very high energy (E > 50 GeV) emission from the flat spectrum radio quasar PKS 1222+216, which represents a challenge for standard blazar scenarios: in the latter one is forced to invoke the existence of a very compact (r ∼ 10 14 cm) emitting region at a large distance (R > 10 18 cm) from the jet base, in order to avoid absorption of gamma rays in the dense ultraviolet radiation field of the broad line region (BLR). In ref. [11] it was shown that one can also use a standard blazar model for PKS 1222+216 where gamma rays are produced close to the central engine, if one assumes that inside the source photons can oscillate into ALPs, with a similar coupling corresponding to f a i /C iγ ∼ 10 8 GeV.
Remarkably, the values of f a i /C ie and f a i /C iγ , inferred from very different astronomical observations, turn out to be in the same region 13 . However, there is one crucial difference: while the WD value (B.6) can be realised both with the QCD axion or with a generic ALP, the anomalous transparency value (B.11) can not be realised with the QCD axion, since the mass of the latter, m a ∼ m π f π /f a ∼ 10 meV 10 9 GeV/f a , is much higher than the phenomenologically required values, which are in the sub-neV range.

B.1.3 Black hole super-radiance
Any axion that has a Compton wavelength comparable to the radius of black hole event horizons can cause the spin-down of the black hole. Since black holes with masses between 2M ⊙ and 10 10 M ⊙ have been observed, this corresponds to masses of the order 10 −10 to 10 −21 eV. The more precise upper bound, independent of the cosmological abundance of the axions, is then [19,53]: (B.14) Interestingly, this is just below the range required for the astrophysical hints.

B.2 Cosmic axions B.2.1 Dark matter
The QCD axion [72][73][74] and more generally ALPs [6,19,22,75,76] may constitute populations of cold dark matter (CDM), produced non-thermally via the misalignment mechanism. The latter relies on assuming that fields in the early Universe have a random initial state (arising from quantum fluctuations during inflation) which is fixed by the expansion of the Universe; fields with mass m i evolve on timescales t ∼ m −1 i . After such a timescale, the fields respond by attempting to minimise their potential, and consequently oscillate around the minimum. If there is no significant damping via decays, these oscillations can behave as a cold dark matter fluid since their energy density is diluted by the expansion of the Universe as ρ i ∝ a −3 , where a is the Universe scale factor. Without fine-tuning of the initial conditions and without further dilution by e.g. late entropy production, the expected cosmic mass fraction in QCD axion CDM is then [77]: where Θ a is the initial misalignment angle, while the one in ALP CDM is [6]: where m i is the mass of the ALP.
Galactic halo QCD axions, as well as ALPs -if they couple to photons, C iγ = 0can be directly searched for via their resonant conversion into a quasi-monochromatic microwave signal with frequency ν = m a /(2π) = 0.24 GHz × (m a /µeV), in a high-Q electromagnetic cavity permeated by a strong static magnetic field [78]. A number of experiments of this type have already been done [79][80][81][82][83] and further improvements are underway [84][85][86]. These haloscope searches reach currently sensitivities of order in the (1 ÷ 10) µeV mass range.
Recently, [13] reported the possibility of probing for QCD axion dark matter for decay constants between 10 16 and 10 19 GeV via molecular interferometry experiments. While it would not apply to ALPs -relying on the coupling to QCD -it would be a very interesting project to probe such weak couplings.

B.2.2 Isocurvature and tensor modes
If the axions constitute a significant proportion of dark matter, then they produce isocurvature and tensor fluctuations, which may be used to place bounds on the inflationary scale. These have been discussed for the case of many axions, in, for example [19]. The proportion of isocurvature fluctuations |S 2 | to the total A S ≡ |S 2 (k 0 )| + |R 2 (k 0 )| ≃ |R(k 0 )| 2 (where R are the adiabatic fluctuations and k 0 is the pivot scale) is constrained by [87] to be less than 0.077; we have: is the standard deviation of axion angle fluctuations and Ω m is total matter fraction of the critical energy density. Assuming that these fluctuations are dominated by one axion with Θ i ≫ σ Θ i (i.e. no tuning of the angle) we have For the single-ALP model to explain the astrophysical anomalies (section B), this gives H inf 10 4 GeV. Since local axions will dominate over non-local ones in terms of energy density (since they couple much more strongly to the non-perturbative effects) we can use f a i ∼ M P / √ V. Assuming that the dark matter is entirely one axion gives us the restriction H inf < m 3/2 ∼ M P V for V 10 9 , which restricts the possibility of entropy injection by large numbers of moduli for large volumes. Thus, if we insist upon axion dark matter and entropy injection, we are restricted to GUTscale strings; but on the other hand, intermediate-scale strings produce axions only within the cosmologically allowed window, eliminating the need for entropy injection.
Tensor fluctuations of axions are typically negligible compared to those induced by metric fluctuations; the constraint upon the ratio of tensor to scalar fluctuations is r ≡ A T /A S < 0.36, where A T = which is typically much weaker than the isocurvature bound.

B.2.3 Rotation of CMB polarisation
For very light axions/ALPs that begin to oscillate between the CMB era and today (corresponding to masses between 4 × 10 −28 and 10 −33 eV), that in addition have a significant coupling to photons, polarisation of the CMB may occur [19]. The limit is: This is only potentially constraining for local axions, since non-local ones will have a parametrically suppressed C iγ . With a combination of different cosmological observations, including galaxy redshift and weak lensing surveys, it should be possible to detect a fraction of ultralight axions to dark matter of a few percent [88].

B.3 Laboratory searches for axions
Currently, pure laboratory searches for very light (mass < eV) axions can not compete with the constraints from astrophysics and cosmology. The best sensitivity is obtained in light-shining-through-a-wall experiments, which are searching for γ → a i → γ conversions in an external magnetic field and thus probing the axion to two-photon coupling. Recently, a number of such experiments were performed; among them, ALPS was the most sensitive one, improving the limit to [89]: g iγ < 6.5 × 10 −8 GeV −1 → f a i C iγ 1.7 × 10 5 GeV, (B. 22) for masses below m i ≪ meV. The next generation of these experiments is designed to reach a sensitivity of g iγ ∼ 10 −11 GeV −1 → f a i /C iγ ∼ 10 8 GeV [4], thus starting to probe the region of interest for the possible explanation of the anomalous features in AGN spectra reported above.
C Axionic couplings to matter at one-loop In addition to the tree level axionic couplings computed in section 2.4, there will also be corrections to the Kähler metric at one loop. These have not yet been calculated for all cases of interest: the result is only known for the coupling of non-local axions to matter at a singularity [90,91] and to adjoint matter [92]. It would be very interesting to calculate the corrections involving local axions (i.e. blow-up moduli) for matter at a singularity along the lines of [93] but we leave this for future work. Here we shall analyse the contributions from field theory, and how they relate to the existing and future results. The couplings to matter fields at one loop are generated via the coupling to photons or other gauge groups. This is to be distinguished from renormalisation of the axionphoton coupling, since we do not perform a chiral rotation -which would introduce axion couplings to mass terms. In non-supersymmetric theories, the correction from a loop involving photons and electrons is ∆ iγγ [C A,V ie ] = 3α 2 4π 2 C iγ ln fa i me -which depends upon the axion decay constant as a cutoff in the loop, or can be interpreted as a renormalisation of the axion-matter coupling. However, in supersymmetric theories, the two-photon coupling is related to a coupling to gauginos: and so we also have loops involving gauginos and selectrons which will change the result. The cutoff-dependent contributions arise even before supersymmetry is broken, and are related to the corrections to the Kähler metric at one loop, whereas there are also more complicated finite contributions that arise when supersymmetry is broken. Let us recall that the Kähler potential at one loop is given by [94]: whereK is the Kähler metric,K −1/2 is the matrix that diagonalisesK:K −1/2 TKK −1/2 = δī j ,ŵ ≡ ∂ 2 ij W and: This is a global supersymmetry calculation, so we must be careful how we interpret the result. Considering a model where f gh = δ gh g 2 g (1 + g ig T i ) for superfields T i ⊃ ic i containing the axions c i , we can compute the gauge corrections to the Kähler potential involving quadratic terms in matter fields Φ i at first order in T (θ, θ): where we have inserted a mass m V for the gauge boson as an infra-red regulator. This is necessary in a theory without a superpotential and thus massless particles, but when the matter fields obtain masses these should appear instead. We thus find the physical coupling: The first term in square brackets is generated by loops involving the tree-level coupling and thus we expect should be corrected to ln e − K 0 2K in a supergravity calculation, whereas the second term is the one truly generated by the axionic coupling. The above is related to the dependence on the moduli of the anomalous dimension of the fields. If we add a superpotential W ⊃ Y Φ 1 Φ 2 Φ 3 , we have: These two terms are apparently quite different. However, we list it for completeness to compare with the results for the behaviour of models of branes at orbifold singularities, where the anomalous dimension vanishes. There the gauge coupling is given by the dilaton, we haveK = e K 0 3 and we should take Y = √ 2 g e K 0 2 with g 2 = S +S −1 .
Hence the dilaton-axion coupling cancels between the two terms. However, it does indicate the very interesting result that at one loop the corrections to the Kähler potential involving twisted (blow-up) moduli should not only not vanish but involve a large logarithmic factor. It would be interesting to confirm this with a calculation along the lines of [93] where we could learn about the cutoff scale Λ introduced in string theory but we leave this to future work.
Since the corrections at one loop due to the ∂ T i lnK term should be smaller than the tree-level term, we can take the dominant loop contribution to be: in the physical basis where the fields ψ j are canonically normalised. Note that this is for a left-handed fermion, so to obtain the electron couplings we should add together the couplings for left-and right-handed components. If we take the approximation that the electron couples only non-chirally to the photon, then no parity-violating vector-like coupling is generated. However, when we take the chiral nature of the electro-weak interactions into account we obtain the approximation (A.13).
Once the supersymmetry breaking terms are included, there are also finite pieces that should in principle be included, even if they are subdominant to the above. Then the effect of electro-weak symmetry breaking must be taken into account. This leads to non-diagonal gaugino and gauge boson couplings, and the final result is rather complicated. We simply note for reference that the contribution to the coupling of a Weyl fermion (i.e. the left-or right-handed component of a matter fermion) will be derived from coupling terms of the form: where g 1 , g 2 ,g 12 involve gauge couplings and mixing matrices of the gaugino mass eigenstates. The result is: Here we have given expressions in terms of gauge boson masses m g , m h , fermion masses m ψ , sfermion masses M Φ and gaugino masses M 1 , M 2 . In the case that m g = m h = m V , g 1 = g 2 = g g = g h , the leading cutoff-dependent part recovers the diagonal supersymmetric result above. Note that in general we have parity-violating terms.

D Axion decay constants for general fibred Calabi-Yau manifolds
The analysis of the axion decay constants performed in section 2.3.2 is simple since the volume form can be written explicitly in terms of the τ moduli. However, when we have a more complicated model, this may not be the case: we may only have the expression in terms of the Kähler parameters t α and the τ α as quadratic expressions of these. In this case, we must use these to examine the metrics K 0 and K −1 0 . An interesting example of this is a general K3 or T 4 fibration in the anisotropic limit (which may in principle involve many moduli), where some t α ∼ xt 0 α become large; we can expand all of the two-cycles, four-cycles and the volume as: and thus: The matrix C AB diagonalises both of these, and we can determine it order by order by considering the two matrices separately. Since the leading matrix in the expansion of (K 0 ) AB is of order one and rank one (it is proportional to t 0 A t 0 B ) clearly there is exactly one axion that has a decay constant of the order of the Planck mass. This is the axion corresponding to the fibre: it is proportional to t 0 A a A so the associated modulus is proportional to t 0 A τ A ; since t 0 A τ 0 A = 0 this must be a small cycle, and the coupling to matter is therefore Planck suppressed.
There are also several axions with string scale decay constants, but on the other hand, by examining (K 0 ) AB we see that there are axions that have decay constants of order M P V −1 ! These would be very interesting phenomenologically if they coupled strongly to small cycles; however, (2.9) tells us that they must correspond to cycles with very small gauge couplings, and since in this limit there are only cycles τ 0 A ∼ V or τ A ∼ 1 the former case must apply: hence they have Planck-suppressed couplings to gauge groups. They are given by the eigenvectors c i B a B of the matrix −V 0 K 0 AB +τ 0 A τ 0 B ; note that K AB has signature (+, −, −, −, ...) and K 0 AB is the matrix of intersections that have large sizes. Hence there may be more than one large cycle: in this case, they must all be fixed relative to each other via D-terms to give an effective canonical fibration volume form for moduli stabilisation [21].