Axion Monodromy and the Weak Gravity Conjecture
DDESY-15-242
Axion Monodromyand the Weak Gravity Conjecture
Arthur Hebecker, Fabrizio Rompineve , and Alexander Westphal Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg,Germany DESY, Theory Group, Notkestraße 85, D-22603 Hamburg, Germany
Abstract
Axions with broken discrete shift symmetry (axion monodromy) have recently playeda central role both in the discussion of inflation and the ‘relaxion’ approach to thehierarchy problem. We suggest a very minimalist way to constrain such models by theweak gravity conjecture for domain walls: While the electric side of the conjecture isalways satisfied if the cosine-oscillations of the axion potential are sufficiently small,the magnetic side imposes a cutoff, Λ ∼ mf M pl , independent of the height of these‘wiggles’. We compare our approach with the recent related proposal by Ibanez,Montero, Uranga and Valenzuela. We also discuss the non-trivial question whichversion, if any, of the weak gravity conjecture for domain walls should hold. Inparticular, we show that string compactifications with branes of different dimensionswrapped on different cycles lead to a ‘geometric weak gravity conjecture’ relatingvolumes of cycles, norms of corresponding forms and the volume of the compactspace. Imposing this ‘geometric conjecture’, e.g. on the basis of the more widelyaccepted weak gravity conjecture for particles, provides at least some support for the(electric and magnetic) conjecture for domain walls.December 11, 2015 1 a r X i v : . [ h e p - t h ] J u l Introduction
The fog surrounding Large Field Inflation is rapidly dissolving. On the one hand, if experimentsare going to detect primordial gravitational waves in the very near future, then we will know thattrans-Planckian field displacements are required. On the other hand, ongoing theoretical effortmay rule out large field inflation in effective field theory coupled to gravity before the observationalverdict. In particular, recent constraints focus on models where the inflaton is an axion with asuper-Planckian decay constant.Axions with large periodicities also occur in a different setting: they are a crucial ingredientof the proposed relaxion solution to the hierarchy problem [1] (see also [2–9]).However, trans-Planckian values of decay constants are problematic [10]. Nevertheless, thereexist several proposals to implement large-field axion inflation in effective field theory with sub-Planckian decay constants. They include decay constant alignment (KNP) [13] and N-flation [14](see also [15]). For a biased collection of recent implementations in string theory see [16–25].Another proposal, which is the focus of this paper, is axion monodromy [26] (see also [27–35] forrealisations in string theory).Recently, important quantum gravity arguments have been used to constrain, and in some caseseven rule out, many of these models (see [36–53]). A first criticism to large field displacementsis based on gravitational instantons [42]. A second one is based on the Weak Gravity Conjecture(WGC) [36]. The latter has been successfully used to constrain models of N -flation and decayconstant alignment à la KNP . The implications of the WGC for axion monodromy are less clear(see however [43]). A third criticism is based on entropy arguments ([37], see however [53]).Very recently, developing an idea of [43], the authors of [54] have applied the WGC for domainwalls to axion monodromy, albeit mainly in the different context of the relaxion proposal. Theiranalysis rests on interpreting the monodromy as being due to the gauging of the discrete shiftsymmetry of an axion by a 3-form potential à la Kaloper-Sorbo (KS) [55, 56] (see also [57]). TheWGC for the original 3-form gauge theory says that this system comes with light domain wallswhich, in turn, threaten the slow-roll field evolution in the resulting monodromy model.The WGC is a statement which connects low energy effective field theories and their UV com-pletion (which we assume to be string theory). In the very same spirit, this paper follows twodifferent, but related, directions: the first one is more phenomenological and relevant both toinflation and relaxation, while the second one is more conceptual and deals with string compacti-fications. The link is provided by the WGC.Concerning the first direction, we advocate a different point of view on constraints coming from d membranes in models of axion monodromy (inflation and relaxation). In particular, we take aminimalist effective field theory perspective: a generic realisation of monodromy is characterisedby ‘wiggles’ in the axion potential (see Fig. 1, 2). The latter define a four-form flux and associateddomain walls. Those differ from the membranes analysed in [43, 54], as they do not arise from agauging procedure: they originate purely from the oscillatory axion potential. In particular, theirtension can be made lighter without spoiling slow-roll. In fact, the lower the ‘wiggles’, the easierit is to slow-roll. Therefore the electric WGC does not constrain generic realisations of axionmonodromy (inflation and relaxation). See [11] for very recent work and [12] for a recent discussion of the underlying shift symmetries beyond treelevel.
2e then seek for constraints arising from the magnetic WGC. It is generically expected thataxion monodromy models cannot allow for a parametrically large field range when correctly im-plemented in a setup of string compactifications. Our claim is that the magnetic WGC describesprecisely such a limitation to the field range (see also [58] for general constraints in setups ofstring compactifications). It does so by bounding the cutoff of the effective theory of an axionwith periodicity πf and monodromy-induced potential m φ : Λ (cid:46) mf M pl . Our point of view isrelevant not only for models where monodromy is used to realise inflation, but also for relaxationmodels, where the low energy barriers are a fundamental ingredient of the mechanism. Applied toinflation, this condition allows in principle for large field displacements, but forbids models witha too small decay constant. Our constraint may be considered as a positive statement about thefeasibility of Axion Monodromy Inflation. In addition, this drives limits on the amount of resonantnon-Gaussianity [59, 60] from the ‘wiggles’, and on the possibility of slow-roll eternal inflation.We then justify our extension of the magnetic side of the WGC to domain walls. In assumingthat the electric WGC can be extended to any p -dimensional object in d dimensions, we aremotivated purely by the fact that string theory always fulfils the WGC. We argue that, adoptingthis point of view, the extension of the magnetic part is equally well motivated. In fact, we showthat string theory fulfils the WGC precisely by lowering the cutoff of the d description, i.e. theKK scale, rather than by providing objects which are light enough. The electric side is thereforesatisfied as a consequence of the magnetic WGC.The second aim of this paper is to describe some insight concerning extensions of the WGC togeneric p -dimensional object in d -dimensions. This has been already the focus of recent activity[43,48,50,54]. We make progress by showing that, in a setup of string compactifications, the WGCcan be phrased as a purely geometric constraint. In particular, it translates into a requirement onthe size and intersections of the q -cycles wrapped by the p -dimensional objects. Explicitly: V / X | q Σ | V Σ ≥ A d , (1.1)where V X is the volume of the compactification manifold, V Σ is the volume of the q -dimensionalcycle Σ , | q Σ | is the norm of the harmonic form related to Σ using the metric X , and A d is a O (1) number given below. Once assumed for one particular configuration (e.g. one leading to d particles), this constraint is valid for any other s -dimensional object, with s (cid:54) = p , wrapped on thesame cycles of the same CY. Therefore, our strategy shows that string dualities are not neededto generalise the WGC: one can separately constrain theories with different brane configurationscompactified on the same CY. In this sense our approach improves on the existing literature.This paper is structured as follows. Sect. 2 is devoted to phenomenological considerations. Inparticular, in Sect. 2.1 we describe the presence of domain walls in d effective field theory modelsof axion monodromy and deduce the constraints coming from the electric WGC. In Sect. 2.2 weassume the magnetic WGC for domain walls and extract the consequences for Axion MonodromyInflation. In Sect. 2.3 we motivate the extension of the magnetic WGC to domain walls andin Sect. 2.4 we comment on the relation to KS membranes. Sect. 3 is devoted to a geometricinterpretation of the WGC. Finally, we offer our conclusions in Sect. 4.3 Axion monodromy and Domain Walls
In this section we aim at obtaining constraints on models based on axion monodromy (inflationor relaxation). We begin by pointing out the existence of light domain walls in those models.Interestingly, these are different from the membranes inherent to the KS approach to axion mon-odromy. They belong purely to the effective field theory regime and do not descend from a higherdimensional gauge theory. We apply the WGC to these low energy domain walls and then discussthe relation of our result to the recent analysis of [54].
We start by adopting a naive d effective field theory point of view of axion monodromy models.The Lagrangian of such a model is given by: L = ( ∂φ ) − V ( φ/f ) , (2.1)and the inflationary (or relaxion) potential generically consists of a polynomial part and an oscil-latory term, e.g.: V ( φ ) = 12 m φ + α cos (cid:18) φf (cid:19) (2.2)In writing a cosine term in (2.2), we are assuming that the axion φ couples to instantons. Theresults that we will derive in the following subsections rely on the presence of this oscillatoryterm. Let us remark that, in models of cosmological relaxation, such a contribution is crucial.Furthermore, in models of axion monodromy and relaxation one typically has (and for relaxationactually needs) α ≡ α ( φ ) , see e.g. [60–62].The cosine term generates ‘wiggles’ on top of the quadratic potential. For suitable values of m, α and f , namely for α/ ( m f ) > , the potential is characterised by the presence of localminima, see Fig. (1). In this paper we focus precisely on this case. Slow-roll inflation startsat large values of φ , where the quadratic potential is dominant and there are no local minima.Eventually, the field reaches the region where the wells become relevant and minima appear. Wewish to constrain the model with potential (2.2) by focusing on this latter region. ‘Wiggles’ arerelated to the existence of domain walls: once the inflaton (relaxion) gets stuck in one of the cosinewells, there is a nonvanishing probability to tunnel to the next well, which is characterised by asmaller value of the potential. This happens by the nucleation of a cosmic bubble created by aColeman-De Luccia instanton, containing the state of lower energy and its rapid expansion.In order to understand this point, let us adopt a coarse-grained approach: namely, let usconsider a model with V ( φ ) as in Fig. (1) at spatial distances which are larger than the inverse“mass” V (cid:48)(cid:48) ( φ ) − . At these distances, φ is non-dynamical. The “wiggles” are therefore invisible, andwhat remains is a set of points, corresponding to the local minima of the original potential. Thesepoints are naturally labelled by an integer index n . Therefore the energy of the correspondingconfigurations is just: E (cid:39) (1 / m φ min (cid:39) m (2 πf ) n , n ∈ Z . (2.3) Even if axions without coupling to gauge theory or stringy instantons exist, the presence of gravitationalinstantons (see [42] and references therein) appears unavoidable. V Figure 1: Monodromy potential,as in (2.2). Here α/ ( m f ) (cid:39) . Φ V Figure 2: Monodromy potential,as in (2.2). Here α/ ( m f ) (cid:39) .Such a discrete set of vacua can be described in terms of a four-form field strength F = dA . Indeed, due to gauge symmetry, the theory of a free -form potential in d has no dynamics(as e.g. in [63–66]). The points corresponding to the local minima of the original potential for φ are separated by domain walls. Therefore, the -form lagrangian which provides an effectivedescription of the axion system is: L = 12 e (cid:90) F + (cid:90) DW A , (2.4)where we have included a phenomenological coupling of A to domain walls. It is easy to see that F changes by e across a membrane, such that F / (2 e ) ≡ (1 / e n . By comparison with (2.3),we find: e = 2 πmf .The tension of the domain wall, i.e. the surface tension of the bubble containing the state oflower energy, can be estimated as the product of the characteristic thickness b ∼ ∆ φ/ √ V andheight of the domain wall V ∼ α (see e.g. [67]). One obtains: T DW ∼ √ V ∆ φ ∼ α / f . As wemake the domain walls lighter, i.e. as we lower the value of α , the wiggles become less pronounced,see Fig. (2).In order to ensure that the inflaton (or relaxion) can slowly-roll for a sufficiently large distance,one needs to make sure that the height of the wiggles, i.e. the tension of the domain walls, issmall enough. The crucial point is that lowering the tension of these domain walls goes precisely in the samedirection as required by the WGC. Let us recall that, in its original form [36], the conjectureconcerns d U (1) gauge theories with coupling e and gravity. The electric side of the conjecturerequires that a particle of mass m e exists such that: eM pl /m e (cid:38) . The statement can in principlebe extended to any ( p + 1) -form gauge theory in d dimensions, with p -dimensional electricallycharged objects. The generalisation to domain walls, i.e. p = 2 in d , is actually not straightfor-ward and may present subtleties (see [50, 54]). For the moment, we assume that the conjecture is Jumping ahead, we note that our use of a four-form flux is therefore different from the approach of [55, 57],where F is introduced as the field strength of a -form which gauges the shift symmetry of the d dual of φ . Thiswill be discussed in Sec. 2.4. Nevertheless, the last stages of inflation (or relaxation) may arise as continuous nucleation of cosmic bubbles. T (cid:46) eM pl . (2.5)Applied to inflationary (relaxion) models, this condition leads to T (cid:46) mf M pl . The conjecturerequires a small tension, which is what is needed to have slow-roll inflation (or relaxation).Therefore, we are unable to constrain Inflation/Relaxation models by this logic. In the previous subsection we have seen that the electric side of the WGC, as applied to lightdomain walls, does not constrain models based on axion monodromy. However, there exist twoversions of the conjecture. The aim of this subsection is to show that the magnetic side imposes anon-trivial constraint on the field range in models of Axion Monodromy (inflation or relaxation).We start by providing a statement of the magnetic WGC in the form of a constraint on thecutoff Λ of a gauge theory. To this aim, let us proceed by dimensional analysis. We consider a ( p + 1) -form gauge theory with coupling e p,d in d dimensions with electrically charged Dp -branesand magnetically charged D ( d − ( p + 4)) branes. The magnetic WGC simply states that theminimally charged magnetic brane should not be a black brane. The tension of a black braneis T BHd − ( p +4) ∼ M d − d R p +1 , where R is the Schwarzschild radius of the black brane and M d is thePlanck scale in d dimensions. The tension of a magnetically charged brane can be estimated byintegrating the field strength outside the core, as in the familiar case of the magnetic monopole.In d dimensions and for a p + 1 -form, the coupling has dimensions [ E ] ( p +2) − d/ . Therefore: T d − ( p +4) ∼ Λ p +1 e p,d . (2.6)The magnetic WGC then requires:magnetic WGC: T (cid:46) T BH ⇒ Λ p +1) (cid:46) e p,d M d − d . (2.7)Although this derivation does not go through in d for p = 2 (since we cannot make sense of D ( − branes), we conjecture “by analytical continuation” in ( p, d ) that the constraint applies.We therefore obtain: Λ (cid:46) e / M / P l . (2.8)This is the constraint we were after. We will provide more support for it later on.We now apply (2.8) to axion monodromy models. As we have seen in the previous subsection,the coupling e is related to the axion parameters by: e = 2 πmf . Therefore, we get the condition Λ (cid:46) (2 πmf M pl ) / . The relevant constraint is now obtained by requiring that the Hubble scale is There exists yet another version of the conjecture, demanding that the states satisfying the WGC are withinthe validity range of the effective field theory [48]. In this paper, we do not consider it, since, in string models, thisappears not to hold if one identifies the KK scale with the cutoff. Also, there are further variants of the electricversion (“strong”,“mild”,“lattice”), which we do not discuss. H = ( V / M pl ) / = 1 / √ · ( m/M pl ) φ (cid:46) Λ . This gives an upper boundon the field range: φM pl (cid:46) (cid:18) M pl m (cid:19) / (cid:18) πfM pl (cid:19) / . (2.9)As it stands, the constraint (2.9), although non-trivial, represent only a mild bound on the fieldrange. With m/M pl ∼ − , and πf /M pl (cid:39) one gets φ/M pl (cid:46) , which safely allows largefield inflation. We expect that our dimensional analysis estimate is modified only by O (1) factors(see Sec. 3). However for models with small f the constraint (2.9) may become relevant.It is a generic expectation that, in models of large field inflation, the field range cannot beparametrically large. The discussion of this section confirms this expectation: In the case of axionmonodromy, the magnetic side of the WGC limits the field excursions. However, phenomenologi-cally relevant field ranges are allowed.Let us now very briefly discuss the corresponding constraint for models of cosmological relax-ation [1] based on monodromy [54]. In this case, we take our axion φ to be the relaxion, andcouple it to the Higgs field of the standard model. Therefore, the relaxion potential is: V φ = 12 m φ + α v cos (cid:18) φf (cid:19) + ( − M + gφ ) | h | , (2.10)where M is the cutoff scale and α v ≡ α ( h = v ) . As discussed in [1], the following constraintsapply to this class of models: ∆ φ (cid:38) M /g to scan the entire range of values of the higgs mass. (2.11) H (cid:46) α / v to form the low energy barriers. (2.12) H > M /M pl for the energy density to be inflaton dominated. (2.13)Furthermore, the slow roll of φ ends when the slopes of the perturbative and non-perturbativepotential terms are equal, i.e.: m φ ∼ α v /f . This should happen at a generic point in the rangeof φ . Hence, from (2.11), φ ∼ M /g and thus: m M g ∼ α v f . (2.14)We can now find the consequences of the magnetic WGC for this class of models. We apply (2.8)and require, as in the inflationary case, H (cid:46) Λ (cid:46) (2 πmf ) / M / pl . We express f in terms of α v by means of (2.14). By also imposing (2.13), we are able to constrain the cutoff M as follows: M (cid:46) (cid:18) πgm (cid:19) / α / v M / pl . (2.15)A similar constraint was given in [54] (we review this approach in Sec. 2.4), where a more detaileddiscussion can also be found. Even if α v is as low as f π m π this constraint is not fatal.Before moving on to the next subsection, we would like to remark on a well-known problemof all the axion models, which also affects our setup. In these models there are always instantonsassociated to the slowly-rolling axion. If they all contribute to the axion potential, there is no flat7irection on which to inflate (relax). It is a non-trivial task to suppress the higher order instantons(our ‘wiggles’), and strategies to do so and evade the WGC have been an important focus of recentwork (see [45] for a proposal which realises a loophole of the WGC [41, 43]).Let us now motivate, as promised, our extension of the magnetic WGC to generic p -dimensionalobjects. It has been suggested in [54] that there is no magnetic side of the conjecture for domain walls, astatement which conflicts with our previous discussion. Here we would therefore like to motivateour use of the magnetic WGC. From now on, we work in units where M pl ≡ .From the point of view of string theory, there are two possible ways of satisfying the elec-tric WGC. On the one hand, string compactifications may provide light objects whose tensionand coupling satisfy the inequality T (cid:46) e . However, Dp -branes in D are extremal, i.e. theymarginally fulfil the WGC. Under compactifications, the resulting objects are not guaranteed tobe extremal, unless SUSY is preserved. Therefore, it is not clear whether objects arising fromstring compactifications could violate the WGC.On the other hand, there exists another mechanism by which the conjecture can be satisfiedin string compactifications: It is the presence of a maximal scale up to which a 4d effective fieldtheory description is valid. In many cases such a cutoff is set by the KK scale M KK ∼ /R ,where R is the typical length scale of the compactification manifold. Above M KK , one has towork with the full 10D theory. In particular, if the tension of the objects descending from stringtheory is larger than M KK , then these objects simply do not exist in the low energy effective fieldtheory. Therefore, by lowering the KK scale, one can ensure that the WGC is not violated, bysimply removing the dangerous objects from the spectrum of the low energy theory. A low cutoffis precisely what is required by the magnetic side of the WGC for a weakly coupled theory.Explicitly, consider a q -dimensional object descending upon compactification from a p -dimensionalbrane in 10D. The ratio between its tension and the appropriate power of the KK scale is givenby: τ q /M q +1 KK ∼ M p +1 s R p +1 /g s . We are assuming that we are in a controlled regime, i.e. either g s < or R > or both. Therefore as R increases the corresponding object simply disappearsfrom the 4d theory.The bottomline of this discussion is that, in many cases, string theory satisfies the WGC byimposing a low cutoff to the 4d effective field theory, not by providing objects which are lightenough. In other words, setups of string compactifications satisfy the magnetic side of the WGCand, as a consequence, the electric side as well.This is the reason why we think that the magnetic constraint is the more fundamental conjec-ture among the two version of the WGC. Therefore, we assume that the magnetic WGC is validfor any p-form, and in particular for domain walls.Recently, the electric WGC has been applied to another class of membranes in the context ofrealisations of axion monodromy models à la Kaloper-Sorbo (KS) (see [55] for the KS proposal, [54]for the recent developments) . In the next subsection we describe the relation of this work to ourfindings. 8 .4 Relation to domain walls à la Kaloper-Sorbo We begin by reviewing the strategy of [43, 54] to constrain nucleation rates in models based onaxion monodromy. In this subsection, we follow the notation of [54], which differs from the oneused in the previous subsections.The KS proposal [55] to implement monodromy models in a d setup is to introduce a 3-formgauge potential A and to couple the corresponding 4-form field strength to the axion: L = −
12 ( ∂ µ φ ) − | F | + gφF , (2.16)where | F p | ≡ p ! F µ ...µ p F µ ...µ p . Notice that this setup is different from the dual picture that wehave described in Sec. (2.1). We used just one scalar field theory with a discretuum of vacua, whichcorresponds to a gauge theory with the same discretuum of vacua. By contrast, the lagrangianin (2.16) consists of a scalar field theory (first term) and a gauge theory (second term), eachwith its own set of vacua. The third term couples these two theories. The potential A couplesto fundamental d domain walls via S ∼ q (cid:82) − branes A . The field strength F varies across themembranes and is quantised in units of the membrane charge, i.e. F = nq ( (cid:63) . A shift in thevalue of F is a part of the residual gauge symmetry of the KS lagrangian. Under this symmetry,also the scalar field shifts: φ → φ + 2 πf, nq → ( n − k ) q, n, k ∈ Z , (2.17)with the consistency condition πf g = kq , and f being the axion periodicity. Due to this residualgauge symmetry, we are left with only one set of vacua, labeled by one integer.The quadratic potential for φ arises from integrating out the field strength F : V KS = 12 ( nq + gφ ) . (2.18)The crucial point is that each value of n corresponds to a different branch of the potential.The gauge symmetry (2.17) provides a way to identify these branches. In this sense, crossing amembrane corresponds to an alternative way to move one step down in the potential. This isdifferent from rolling over or tunneling through a “wiggle” of Fig. (1). The KS membranes canpotentially spoil the slow-roll behavior allowed by small “instanton-induced wiggles”. As usual, theprobability for such tunneling events is described in terms of a nucleation rate for the correspondingbubbles.Since this probability is exponentially suppressed, one might wonder whether this effect rep-resents a concern for Axion Inflation. The nucleation rate Γ is given by e − B , where B ∼ T /H inthe relaxion regime (see [68]).In [43], the authors show that a strong suppression of the nucleation rate requires a violationof the WGC. More recently, in [54], the authors follow the same direction to constrain models ofrelaxion monodromy. In this case, the WGC requires T (cid:46) πf g . By requiring B > , the authorsobtain a constraint which is similar to (2.15). In particular, the parametric dependence on α v isthe same. Furthermore, the authors of [54] obtain a stronger constraint by requiring that B > N ,where N is the number of e-folds. This requirement arises from demanding that there are nodomain walls in the part of the universe created during the above N e-folds. Such a constraint9annot be obtained by using our low energy wiggles, because the latter arise only much later,when inflation is in its last stages.Applied to inflationary models, T (cid:46) eM pl and T (cid:29) H lead to the same constraint that wehave found in (2.9). However, we have obtained it by using a different, arguably simpler, effectivefield theory point of view, based on the magnetic, rather than the electric WGC. Notice also thatthe objects that we have described in Sec. (2.1) can be naturally lighter than the KS membranes.We have seen that the Kaloper-Sorbo procedure consists in gauging an axionic theory by a -form potential. The original theory of a free -form potential has domain walls to which theWGC can be applied. However, gauging corresponds to a discontinuous, qualitative change of themodel. It is hence not clear whether the relevant parameters, i.e. the tension and the coupling, andtherefore the consequences of the WGC, remain unchanged. In particular, it may be hard tocheck the changes of the coupling and tension of the domain walls, since the dualisation proceduredescribed in [57] does not always lead to an explicit determination of the F lagrangian. Therefore,it is desirable to work with constraints which do not appeal to the situation before gauging.Crucially, after the gauging both the fundamental KS domain walls and the ‘wiggle-induced’effective domain walls are present.We are then left with two possibilities: The first is that the electric WGC has to be separatelysatisfied by both the KS and by the effective domain walls described in this paper. In this case theconstraint given in [54] and based on the electric WGC for the (heavier) KS domain walls applies.The same constraint arises as a consequence of the magnetic WGC. Everything is consistent andthe present paper provides an alternative derivation of the same constraint.The other possibility is that the electric WGC needs to be satisfied only by the lightest domainwalls. These are the effective domain walls, but due to their lightness no interesting constraintarises. The heavier KS domain walls provide no further constraints. Thus, the magnetic WGCprovides the only useful constraint, as described in this paper.Our conclusion is that in both cases the field range is constrained according to (2.9). In thefirst case the latter comes from the electric side applied to KS membranes, as explained in [43, 54],and from the magnetic side applied to “wiggles” membranes. In the second point of view, whichwe adopt in this paper, no UV information on the origin of the gauge theory is required and (2.9)follows only from the magnetic WGC. In this section we want to address the extension of the WGC to domain walls. We will do so inthe framework of 10D string theory compactified on a CY manifold.
In [50], the authors provide the following statement of the WGC for any p -form in d dimensions: In particular, the conceivable limiting procedure of taking the gauge coupling to zero and hence going fromthe gauged to the ungauged case is forbidden by the WGC itself. Also, in string constructions gauging oftencorresponds to (necessarily discrete) changes in the flux configuration or even in the geometry of the compactspace. α p ( d − p − d − (cid:21) T p ≤ e p : d q M d − d . (3.1)In the absence of a dilaton background, the inequality is degenerate for p = 0 (axions) and p = d − (strings). Moreover, for p = d − , i.e. for domain walls, the inequality cannot be satisfied, asalready pointed out in [50]. Therefore, one may worry that there is no statement of the electricWGC for domain walls.An idea to extend the WGC to generic p -dimensional objects, as noticed in [54], is to use stringdualities. This follows very closely the strategy of [43], where the conjecture is extended to axionsand instantons. In that case, the authors consider type IIB on a CY 3-fold with D branes andtheir associated C gauge potential. Wrapping the branes on -cycles and compactifying to 4d,one obtains a theory of C axions and D instantons. This type IIB theory is then T-dualised totype IIA with D branes and their associated C potential. Since this theory is strongly coupled,one actually uses the M -theory picture, introducing a further compact dimension. Again, bywrapping the branes around -cycles and compactifying, one obtains a d theory with a U (1) gauge field and M particles. This is the original content of the WGC, which can therefore beapplied to this particular 5D setting. Finally, one can translate the constraints obtained on theparticles/vector fields side to the axion/instanton side, by using the T-duality relations betweenIIA and IIB couplings and mass scales.In [54], the authors propose to implement the very same idea to constrain domain walls.Starting with a 10D theory with p = d − objects, they propose to T-dualise twice along directionstransverse to the branes, so that the dual theory is of the same type but with p = d − branes.One can then apply (3.1) to the latter setup, then translate the constraints to the domain wallsside.We agree with the authors of [54] that the apparent problems of the WGC for domain wallsdisappear when considering them in a string theory setup. Notice that the dualisation procedureworks for any p -dimensional object in dimensions reduced to a q dimensional object in d dimensions. Indeed the moduli of the theory, i.e. the compactification radius and the stringcoupling, disappear from the charge-to-tension ratio on both sides of the duality. Were this notthe case, we would not be able to extract a sensible constraint on the objects in the d theory.This property suggests that the WGC in 10D string theory can be phrased as a constraint onsome geometrical data of the particular compactification manifold, independently of the specific p -dimensional object. Once the geometry of the compactification manifold is constrained, one canextract the consequences for any other q -dimensional object in the theory. This is the novel pointof this section. Our focus in this section is the electric statement.Our approach implies that there is no need of T-dualising in order to extend the WGC toobjects other than d particles. In the next subsection, we will verify this statement focusingon the case of domain walls. Let us therefore outline the strategy to extend the conjecture toany p -dimensional object, without using dualities. One starts with a type IIB setting with Dp branes wrapped on p -cycles of the internal manifold X . Upon compactification, this leads to a4d theory of particles and gauge fields. One then applies the standard WGC to this setting: theresult is a constraint on the metric on the space of p -cycles in X . For example, in [43] the authorsobtain a constraint on K ab ∼ (cid:82) w a ∧ (cid:63)w b , where w a is a basis of H ( X, Z ) . Once this constraint isobtained, it is valid for any brane setup on the same CY. One can then consider Dq branes, with11 (cid:54) = q wrapped on the same p cycles and obtain inequalities for the tension and couplings of the d theory derived by compactification on X . Following our discussion, we now perform an explicit computation to prove our claim. We firstfocus on obtaining particles in d = 4 . As a starting configuration we choose type IIB with D3branes compactified on a CY 3-fold X . Other choices are equally valid. We work in the conventionsof [69]. The relevant 10d action reads: S ⊃ κ (cid:90) M (cid:20) g s R (cid:63) − F ∧ (cid:63)F (cid:21) + µ (cid:90) D C (3.2)where κ = (1 / π ) α (cid:48) and µ = 2 π (4 π α (cid:48) ) − . Now let us perform dimensional reduction, bywrapping the D on 3-cycles of X . We focus on the gauge kinetic term. We consider a symplecticbasis w i = ( α a , β b ) of H ( X, Z ) , i.e. s.t.: (cid:90) X α a ∧ β b = δ ba , (3.3)and the other intersection numbers vanish. By Poincaré duality, one can define the integralcharges: q ki = (cid:90) Σ k w i = (cid:90) X w i ∧ w k , (3.4)where Σ k is a -cycle in X and w k is its dual form. By (3.3), these charges are either vanishingor unit.The 4d action is obtained by expanding the five-form flux and the four-form potential in termsof the symplectic basis of H ( X, Z ) : F = N (cid:88) i =1 F i ( x ) ∧ w i ( y ) , C = N (cid:88) i =1 A i ( x ) ∧ w i ( y ) , (3.5)then integrating over X . Here N is the number of cycles of X . D -branes wrapping a -cycle Σ generate particles in the d theory. For the moment being, let us focus on one such cycle. We willlater extend our results to particles descending from different cycles. Let us introduce the metricon the space of -forms: K ij ≡ (cid:90) X w i ∧ (cid:63)w j . (3.6)Before moving to the d theory, an important remark is in order. This concerns the self-dualityof F , i.e. (cid:63)F = F . This constraint cannot be implemented at the level of the d action (3.2).Therefore one actually starts with a more general theory where F (cid:54) = (cid:63)F . Nevertheless, thekinetic term in (3.2) is normalised with a prefactor / instead of / , as will be appropriate afterself-duality is imposed. To obtain consistent d equations of motion, the coupling in (3.2) shouldactually read: S ⊃ µ (cid:90) D C + µ (cid:90) ˜ D ˜ C , (3.7)12here at the moment different branes source the dual potentials (see also [70], footnote n. 6).Self-duality can then be consistently imposed at the level of the d equations of motion, derivedfrom this action. This goes together with identifying D and ˜ D .We now consider the d theory descending from dimensional reduction. In d there are certainconstraints on the field strengths F i and ˜ F i = (cid:63)F i descending from self-duality of F in d . Forthe sake of our analysis, we first focus on the set of unconstrained F i , exactly as we we did with F in d . The d action then reads: S ⊃ M pl · (cid:90) M g s V X K ij F i ∧ (cid:63)F j + q Σ i µ (cid:90) − brane A i . (3.8)Here M pl = V X /κ g s is the d Planck mass. The equations of motion arising from (3.8) read d (cid:63) F i K ij = 2 V X M pl g s µ q Σ i dj . (3.9)From the latter, it is clear that only a certain linear combination of gauge fields is sourced by theparticle with charge q Σ i . To make this visible in the 4d action, we define the field A and its fieldstrength F = dA by A i ≡ A K ij q Σ j . (3.10)In terms of A and F the 4d action reads S ⊃ M pl · | q Σ | g s V X (cid:90) M F ∧ (cid:63)F + | q Σ | µ (cid:90) − brane A , (3.11)where | q Σ | ≡ K ij q Σ i q Σ j . Now we consider the realistic setup where the dual field strength ˜ F andits associated ˜ A are also included. Therefore we add to (3.8) the action: ˜ S ⊃ M pl · (cid:90) M g s V X K ij ˜ F i ∧ (cid:63) ˜ F j + ˜ q Σ i µ (cid:90) (cid:94) − brane ˜ A i (3.12)where ˜ A and ˜ q Σ i are analogous to A i and q i Σ . The coupling term can be obtained from dimen-sionally reducing (3.7). Finally we relate F i and ˜ F i by dimensionally reducing d self-dualityof F . In particular, d self-duality implies F J = (cid:63) F K H JK , where the matrix H JK is definedby (cid:63) w K = H JK w J . In imposing this constraint, we also identify the branes sourcing A i and ˜ A i .Thus, adding (3.11) and (3.12) corresponds to a doubling of the action (3.11). Therefore, the finaltheory which we will constrain via the WGC has action S ⊃ M pl · | q Σ | g s V X (cid:90) M F ∧ (cid:63)F + | q Σ | µ (cid:90) − brane A . (3.13)In order to extract the d gauge coupling, we normalise the gauge potential. Finally, we obtain: S ⊃ e (cid:90) M F ∧ (cid:63)F + (cid:90) − brane A , (3.14)13here we have kept the same notation for the normalised fields and the d gauge coupling isdefined as: e = 2 V X µ | q Σ | M pl g s . (3.15)The result of this procedure is therefore a d theory of a U (1) gauge field with coupling (3.15). Theparticle descending from the D brane wrapped on Σ has mass M Σ = ( T /g s ) (cid:82) Σ (cid:63) = ( T /g s ) V Σ ,and T = µ .We are now ready to apply the WGC to the d theory defined by (3.14) with particles of mass M Σ : eM pl M Σ ≥ √ ⇒ V / X | q Σ | V Σ ≥ . (3.16)Before moving to the case of domain walls, let us pause to extract the full meaning of (3.16).The WGC for particles arising from a string compactification translates into a purely geometricconstraint on the size and intersections of the cycles of the manifold, in this case -cycles. Crucially,all couplings and d scales have disappeared from the final statement. Despite the presence ofvolume factors, the charge-to-mass ratio is independent on any rescaling of the d metric ˜ g mn .This statement is actually true for any p -cycle: indeed the metric K ij on the dual space of p -forms contains (3 − p ) powers of the d metric, so the numerator scales as ˜ g p/ mn , but so does thedenominator.The conclusion is as follows: the procedure that we have followed works for any p -dimensionalobject and associated field strength defined on a chosen manifold X and dimensionally reduced toa q dimensional object in d . In particular, (3.16) is a constraint on the -cycles of X . As such,it can be applied applied to any other 4d object descending from any p -brane on X wrapped onthe same -cycles.We are particularly interested in constraining d domain walls. In order to apply our previousresult, we study the case in which the membranes arise from compactifications of type IIB stringtheory with D branes wrapped on -cycles. The action is obtained by simply replacing the D branes with D branes in (3.2): S ⊃ κ (cid:90) M (cid:20) − F ∧ (cid:63)F (cid:21) + µ (cid:90) D C + S DBI , (3.17)with µ = µ / (2 πα (cid:48) ) . Dimensional reduction to d goes as in the previous case, therefore we donot repeat the computation. The d action reads: S ⊃ e DW (cid:90) M F ∧ (cid:63)F + (cid:90) D A (3.18)with: e DW = 2 V X µ | q Σ | M pl g s . (3.19)The tension of the d domain wall is: T DW = T /g s V Σ . The charge-to-tension ratio is: e DW M pl T DW = (2 V X ) / | q Σ | V Σ . (3.20)14s expected, (3.20) is the same as (3.16). Therefore the WGC constraint on particles translatesinto the following inequality for the charge-to-tension ratio of domain walls:WGC: e DW M pl T DW ≥ √ . (3.21)This is the result we were after, namely a WGC for domain walls.One can give a general inequality for a ( q + 1) -dimensional object in d dimensions descendingfrom a s -brane wrapped on a ( s − q ) cycle of a CY X , by relating its charge-to-mass ratio tothat of particle descending from a p -brane wrapped on the same ( s − q ) cycle. For consistency s − q = p . The WGC then states that the charge-to-tension ratio of the D ( q ) -brane must satisfythe condition: e p M pl T p ≥ (cid:114) d − d − . (3.22)Finally, let us generalise our results to the case of N cycles Σ k , k = 1 , . . . , N . Correspondingly, wehave a set of charge vectors q Σ k . These vectors belong to R N equipped with metric K ij definedas in (3.6). With the same notation as above, consider Dp -branes wrapped on p -cycles of a CYmanifold. These lead to particles in d dimensions with mass M k . The Convex Hull Condition(CHC) for the p -cycles reads: The convex hull spanned by the vectors z k ≡ V / X q Σ k V Σ k , must contain the ball of radius r = (cid:113) d − d − . Now consider a q -brane in d dimensions obtained by wrapping a D ( s ) -brane on p -cycles of thesame CY. The tension and the charge vectors of the ( q +1) -dimensional objects are respectively T kq and e q q Σ k , where e q is the prefactor in (1 / /e q ) (cid:82) K ij F iq +2 ∧ (cid:63)F jq +2 + q Σ i (cid:82) Σ k A iq +1 in the effectivetheory. Assuming the CHC for particles, we obtain the following statement for the q -branes: The convex hull spanned by the vectors Z k ≡ e q q Σ k M d T kq must contain the ball of radius r q = (cid:113) d − d − . It is important to remark that (3.21) has been obtained without using any string duality: theWGC for particles imposes a constraint on the geometry of CY three cycles. This constraint,applied to objects derived from any p -brane in the 10d setup, translates to a corresponding WGCfor these particular objects. This line of reasoning can be applied also to the case of axions andinstantons. In that case one starts from a Dp brane wrapped on p cycles, then considers D ( p − branes wrapped on the same cycles. Obviously this requires a change in the theory, e.g. from typeIIB to type IIA/M-theory on the same CY. However, the constraints obtained in the IIB settingare still just geometric constraints on p -cycles of the CY, therefore there is no need of performinga duality between the two theories. It is sufficient to consider a type IIA/M-theory setup withthe appropriate branes, and impose on this setup the previously determined geometric constraint.It would be interesting to think about manifolds with backreaction and fluxes. In this case, thetransition from IIA and IIB (or other setups) would not be so straightforward.15 Conclusions
In this paper we have investigated two different aspects of the Weak Gravity Conjecture. Firstly,we have discussed its consequences for models based on Axion Monodromy (Inflation and Relax-ation). Secondly, we have provided a geometric interpretation of the conjecture in the frameworkof string compactification. We now provide a detailed summary of our results.In the first part of this paper, we have adopted an effective field theory point of view. Namely,given a certain scalar potential, we have tried to constrain its use in models of monodromy inflation.In particular, inflaton (relaxion) potentials in models of Axion Monodromy are characterised bythe presence of ‘wiggles’ on top of a polynomial potential. The resulting local minima imply theexistence of d domain walls. This is more evident by using an effective description in terms of afour-form flux, whose value changes across these membranes.We assumed that the WGC can be extended to domain walls. In our setup, its electric versiongives an upper bound on the tension of the d membranes. Crucially, this condition agrees withwhat is required to realise slow-roll: as the tension decreases, the height of the ‘wiggles’ decreasesand slow roll can be seen as a continuous nucleation of cosmic bubbles. Therefore, we concludethat, in this logic, the electric WGC does not constrain models of axion monodromy (Inflationand Relaxation).For this reason, we focused on the constraints imposed by the magnetic side of the WGC,which we stated as an upper bound on the cutoff of a generic ( p + 1) -form gauge theory (in thespirit of [36]). We then applied the condition to inflationary models, i.e. we required H (cid:28) Λ .This gives a non-trivial constraint on the field range: φ (cid:46) m − / f / M / pl . The latter howeverallows for large field displacements, but forbids models with a small decay constant.We then discussed our extension of the magnetic WGC. We argued that string theory lowers theKK scale to fulfil the WGC for objects which descend from compactifications of string theory with Dp -branes, rather than making them light enough. As a consequence, heavy “stringy” objects,which could potentially violate the WGC are confined above the cutoff M KK . Therefore theydo not exist from an effective field theory point of view. Of course, low energy light objectsare allowed, as is the case for our domain walls. Consequently, the electric side is automaticallysatisfied. We suggest that the magnetic WGC should be seen as the fundamental constraint amongthe different versions of the WGC.Recently, the electric WGC has been applied to membranes arising from the realisation of AxionMonodromy á la Kaloper-Sorbo (KS), in the context of new realisations of relaxion models [54].When the tension of these membranes decreases, the probability of tunneling to another branch ofthe potential increases. Such a transition can spoil slow-roll, as it corresponds to discrete “jumps”in the axion trajectory. The requirement that the tunneling rate is suppressed parametricallyleads to the same constraint on the field range that we obtained by studying the domain wallsarising from ‘wiggles’ in the axion potential.However, KS membranes are different from the low energy domain walls described in thispaper. This may have implications for the various constraints.There are, in fact, two possibilities. On the one hand, one could impose the WGC separatelyon the two classes of membranes. In this case, the constraints given in [54] for relaxion modelsapply and can be extended to inflationary models. The magnetic WGC applied to the low energydomain walls gives the same constraint. 16n the other hand, it is possible that only the lightest domain walls have to satisfy the WGC.In this case, the electric WGC applied to the low energy domain walls does not give any constraint.By contrast, the magnetic side gives a bound on the field range, hence playing a central role. Asdiscussed in this paper, there are reasons related to the KS gauging which make this secondpossibility relevant.In the second part of this paper, we worked in the framework of string compactifications. Westarted with 10D type IIB with D branes and compactified to d by wrapping the branes around -cycles of a CY manifold. Therefore, we obtained particles and gauge fields in d . We appliedthe original WGC to this setup. Very interestingly, the final constraint does not depend on thecouplings and moduli of the 10D setup. The electric WGC translates into a purely geometricconstraint on the size and intersection of the -cycles of the CY. The same happens for any p -dimensional object wrapped on the same -cycles. Therefore, by constraining the geometry ofthose cycles through the D /particles case, we obtain a WGC for any p − -dimensional object in d arising from compactification of type IIB with Dp -branes wrapped on -cycles. In particular,by taking p = 5 we obtain the WGC for d domain walls. Crucially, we do so without the use ofstring dualities.The same procedure applies to any p -dimensional object wrapped on some q -cycle of a CY, toobtain a p − q -dimensional object in d . Therefore, our approach provides a simple strategy toextend the electric WGC to any q -dimensional object, without the use of string dualities.Let us close our discussion with observing two further consequences implied by the con-straint on the tension of the low-energy domain walls from the WGC. We note firstly, thatwe get a fundamental upper bound on the size of resonant oscillating non-Gaussianity inducedby the ‘wiggles’ in the scalar potential. Following the analyses of [59, 60], the magnitude f res.NL of this type of non-Gaussianity with an oscillating shape in k -space is approximately given by f res.NL ∼ bM pl / ( f φ ) / . Here, b = α/ ( m f φ ) denotes the ‘monotonicity’ parameter of the scalarpotential with ‘wiggles’ ( b < corresponds to V (cid:48) > for φ > ). We can rewrite this as b = αf / ( m f φ ) ∼ T DW / ( m f φ ) < m f M pl / ( m f φ ) = M pl / ( f φ ) where the inequalityarises from the WGC T DW < eM pl = mf M pl . Hence, we get a bound f res.NL (cid:46) M pl / ( f φ ) / ,to be evaluated at φ = φ ∼ M pl for the observable CMB scales. The bound thus fi-nally reads f res.NL (cid:46) × − ( M pl /f ) / . The typical range for the axion decay constant is − M pl (cid:46) f (cid:46) . M pl (see e.g. [60]). Consequently, for f (cid:38) × − M pl this fundamentalupper bound on f res.NL becomes stronger, f res.NL (cid:46) O (1) for f (cid:38) × − M pl , than the currentobservational bounds [71].Secondly, we observe that in a quadratic potential the boundary to slow-roll eternal inflation(defined as the value of φ = φ (cid:63) where (cid:15) ∼ V ) φ (cid:63) ∼ M / pl m − / can be higher than our magneticWGC field range bound φ < m − / f / M / pl for values of f (cid:46) − M pl , because COBE normal-ization of the CMB fluctuations fixes m ∼ − M pl . Intriguingly, recent analyses such as [62, 72](see also e.g. [73] for earlier work on the WMAP 9-year data) of the PLANCK data searchingfor oscillating contributions to the CMB power spectrum and the 3-point-function hint with thehighest significance at very-high-frequency oscillating patterns with f ∼ − M pl . If this werecorroborated in the future, then jointly with the magnetic WGC this would rule out slow-rolleternal inflation in quadratic axion monodromy inflation potentials in the past of our part of theuniverse.We leave the generalization of both of these observations to more general axion monodromy17otentials V ∼ φ p with ‘wiggles’ as an interesting problem for the future. Acknowledgements
We thank J. Jäckel, P. Mangat, E. Palti and L. Witkowski for useful discussions. This work waspartly supported by the DFG Transregional Collaborative Research Centre TRR 33 “The DarkUniverse". F.R. is partially supported by the DAAD and the DFG Graduiertenkolleg GRK 1940“Physics Beyond the Standard Model”. The work of A.W. is supported by the ERC ConsolidatorGrant STRINGFLATION under the HORIZON 2020 contract no. 647995.
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