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Warping the Weak

Karta Kooner,1, ∗ Susha Parameswaran,2, † and Ivonne Zavala1, ‡ 1Department of , Swansea University, Singleton Park, Swansea, SA2 8PP, UK 2Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building Liverpool, L69 7ZL, UK The Weak Gravity Conjecture, if valid, rules out simple models of Natural Inflation by restricting their axion decay constant to be sub-Planckian. We revisit stringy attempts to realise Natural Inflation, with a single open string axionic inflaton from a probe D-brane in a warped throat. We show that warped geometries can allow the requisite super-Planckian axion decay constant to be achieved, within the supergravity approximation and consistently with the Weak Gravity Conjecture. Preliminary estimates of the brane backreaction suggest that the probe approximation may be under control. However, there is a tension between large axion decay constant and high string scale, where the requisite high string scale is difficult to achieve in all attempts to realise large field inflation using perturbative string theory. We comment on the Generalized Weak Gravity Conjecture in the of our results. Keywords: String theory, , inflation, D-branes, warped geometries, large axion decay constants.

I. INTRODUCTION to invoke a shift symmetry in the inflaton field, for in- stance by identifying the inflaton with a Goldstone boson, Cosmological inflation stands strong as the leading the axion. The classical example in this vein is Natural mechanism to provide the seeds that gave rise to the large Inflation [5], now tightly constrained by the latest CMB 1 structure we observe today in the . Precision ob- observations . In Natural Inflation, the axion enjoys a servations in the Cosmic Microwave Background provide continuous shift symmetry within the perturbative ap- a window into this very early history of the Universe. proximation. This is broken to a discrete symmetry by The latest results from Planck [1] are in perfect agree- non-perturbative effects, which generate a potential of ment with the simplest inflationary models, driven by the form: the dynamics of a single scalar field rolling down a very  φ flat potential. Current bounds from Planck/BICEP2 on V (φ) = V0 1 ± cos , (3) f the scalar to tensor ratio in the CMB power spectrum are r . 0.12 (95% CL). Any future detection of tensor where f is the axion decay constant. The potential is modes would have the remarkable implications, via the sufficiently flat for slow-roll inflation provided that f  Lyth relation [2–4], that inflation occurred at scales close MP l, and as a consequence the axion can undergo super- to the Planck scale: Planckian field excursions. The current bound on the 1/4 Natural Inflation potential (3), given by PLANCK from 1/4  r  16 > Vinf ≈ × 1.8 × 10 GeV (1) the spectral index, ns, is f/MP l ∼ 6.8 (95% CL) [1]. 0.1 To understand whether or not such ideas are viable and that the inflaton field had super-Planckian excur- requires the embedding of large field models in a theory sions: of . In fact, much interest has recently been generated by the possibility that general features of 1/2 ∆φ >  r  quantum gravity can constrain inflationary models with ∼ 0.25 × . (2) MP l 0.01 observable consequences. The Weak Gravity Conjecture [7] roughly proposes that gravity must be the “weak- “Large field” inflationary models are intriguing not est ” in a quantum theory of gravity, in order to only due to their robust prediction of high scale inflation avoid stable black hole remnants. For example, for a with observable primordial gravitational . They de- four dimensional theory describing gravity and a U(1) pend sensitively on the degrees of freedom comprising the gauge sector with gauge coupling, g, there must exist a ultraviolet completion of gravity. In particular, Planck state with mass, m, which satisfies m . MP l g. More- suppressed corrections to the slow-roll inflaton poten- over, the effective field theory has a new UV cutoff scale, tial typically become large when the inflaton varies over Λ ∼ MP l g. This conjecture was used [7] to rule out Ex- super-Planckian scales. One idea to protect the slow-roll tra Natural Inflation [8], where the inflaton arises from inflaton potential from dangerous quantum corrections is

1 Although massive modes during inflation can change the classic ∗ [email protected] NI predictions by generating a smaller than unity speed of sound † [email protected] bringing the model back to the allowed parameter region, as ‡ [email protected] shown in [6]. 2 a Wilson line in a five-dimensional U(1) gauge theory decay constant consistently with the Weak Gravity Con- compactified on a circle. jecture. The large decay constants are generated within The Weak Gravity Conjecture might also be general- the perturbative limits of the supergravity approxima- ized to D dimensions, p-form Abelian gauge fields, and tion, and initial estimates support the validity of the their p spacetime dimensional charged objects. Then, probe brane approximation used. Moreover, scalar po- in a four dimensional gravitational theory with a 0- tentials that break the continuous axion shift symmetry form axion, there must exist an instanton with action, to a discrete one are potentially generated by classical or Scl . MP l/f, where f is the axion decay constant. Al- loop effects [8, 40–43]. Therefore, any non-perturbative though this conjecture lacks convincing motivation from instanton effects are by construction exponentially sup- black hole physics, the same phenomenon was observed pressed, and would not rule out single field, large field, in [9] in several diverse string theoretic setups. If valid, it slow-roll inflation. would essentially rule out single field models of inflation Unfortunately, these results do not lead to promising with super-Planckian axion decay constants, as instanton models of large field inflation and observable primordial corrections would always introduce higher harmonics to gravitational waves. This is because in explicit construc- the inflation potential: tions, there is a tension between obtaining large decay constant and high string scale. In fact, as we empha- M n P l inθ size, it is always difficult to obtain a sufficiently high e f e , (4) string scale within the limits of perturbation theory. This effectively limiting the axion field range. Analogous ar- presents an important challenge in building string theo- guments considering gravitational instantons in effective retic models of large field inflation. field theory lead to similar conclusions [10]. Moreover, The paper is organized as follows. In the next section the examples studied in [9] demonstrated the difficulty we introduce inflation from D-branes in warped geome- in obtaining axions with large decay constants within tries and fix our conventions. In Section III we study sce- the limits of perturbative string theory, and raised the narios in which the candidate axionic inflaton is a Wilson question if this is possible at all. line on a wrapped D-brane, and in Section IV we turn to Recent has focused on whether these constraints the T-dual picture of the position modulus of a wrapped from the Generalized Weak Gravity Conjecture on ax- D-brane. The most attractive scenario studied can be ion inflation can be evaded, in particular, by introduc- found at the end of this section. Finally, in Section V ing multiple axion fields [11–19]. There has also been a we discuss our results, and the light they shed on the large amount of work towards developing string theoretic Generalized Weak Gravity Conjecture. models of large field inflation with sub-Planckian axion decay constants. These come under two main classes, II. OPEN STRING INFLATION firstly, stringy mononomial chaotic inflation scenarios where monodromy effects explicitly break the axion shift symmetry [20–28], and secondly, many field models where Our starting point is a generic type IIB string warped multiple axions generate an effective decay constant that compactification from ten to four dimensions with metric 2 is super-Planckian [29–34]. Most constructions have used (in the Einstein frame ): closed string axions in type II string theory. ds2 = h−1/2(r)g dxµdxν + h1/2(r)g dxmdxn , (5) In this letter, we revisit open string, single field mod- µν mn els of axion inflation, in the light of quantum gravity where µ, ν = 0,... 3, m, n = 4,... 9 and h(r) is the warp constraints and the Weak Gravity Conjecture discussed factor, possibly trivial, depending on a radial-like direc- above. Open string inflatons include Wilson lines on tion in the internal . For example, for an adS5 ×X5 wrapped Dp-branes and the position moduli of moving geometry which describes well a generic warped throat Dp-branes. In fact, these scenarios are T-dual to each generated by branes and fluxes in the mid-throat region, other. By considering D3-branes moving down a long we have: warped throat, Baumann and McAllister pointed out a rigid, sub-Planckian upper bound on the field range 4 4 2 m n 2 2 2 h(r) = L /r and ds6 = gmndx dx =dr + r dΩ5 , [35, 36], and this can indeed be interpreted as a conse- (6) quence of the Weak Gravity Conjecture. However, single 2 i j where L is the adS length scale and dΩ5 =g ˜ijdφ dφ field models have been proposed with moderately super- is the metric on some five-dimensional Einstein-Sasaki Planckian decay constants. Wilson lines on wrapped Dp- space that ensures N = 1 supersymmetry. To construct branes with sub-Planckian decay constants were studied in [37], and with super-Planckian decay constants in [38]. Planckian decay constants from wrapped Dp-branes mov- ing down or around a warped throat were found, respec- 2 Our conventions for going from string to Einstein frame are ϕ0−ϕ tively, in [39] and [40]. E 2 s GMN = e GMN , where ϕ is the dilaton, whose vev hϕi = ϕ We show that warping and wrapped volumes indeed ϕ0 defines the string coupling as gs = e 0 . In these conventions allow for single field models with super-Planckian axion the volumes evaluated in the background are frame independent. 3 a smooth, compact internal space out of the adS throat, III. WILSON LINE INFLATION we take rIR < r < rUV , where the adS region is glued to the tip of the throat at the IR cutoff and to a compact When the (p − 3)-cycle wrapped by the Dp-brane con- Calabi-Yau at the UV cutoff [44]. tains a non-trivial 1-cycle3, parameterized by some coor- The four-dimensional Planck mass after compactifica- dinate φ, the brane can have a Wilson line wrapping its tion takes the form: worldvolume: H w Z eiθ = ei Aφdφ . (12) 2 4πV6 2 w 6 6 p MP l = 2 Ms with V6 ls = d y det gmn h , gs Upon dimensional reduction, the DBI action (10) for (7) the brane in the background (5-6) includes a 4D gauge 2 −2 2 0 where we defined the string scale as ls = Ms = (2π) α , kinetic term for the U(1) worldvolume gauge field: ϕ0 w gs = e is the string coupling and V is the warped vol- 6 Z ume of the six-dimensional internal space in string units. 4 √ 1 µν S = − d x −g 2 Fµν F , (13) For example, assuming most of the volume comes from 4g4 the middle region of an adS throat generated by N D3- branes at its tip, we have: where the four-dimensional effective gauge coupling con- stant, g4, is computed to be:

4 gsN 4 2 (3−p)/4 −1 L = ls (8) g4 = (2π)gsh0 (n Vp−3) . (14) 4V5 Here, h0 is the value of the warp factor evaluated at and the brane position and Vp−3 is the unwarped volume of the (p − 3)-cycle wrapped√ by the brane in units of ls, 1 V lp−3 = R dp−3y det g (notice that V also de- Vwl6 = V L4r2 , (9) p−3 s ab p−3 6 s 2 5 UV pends on the position of the brane, see eq. (6)). Finally n is the wrapping number. R with V5 = dΩ5 the dimensionless volume of the base of At the same , the DBI action leads to a kinetic the cone. term for the Wilson line modulus, θ = 2πAφ, which takes To the above space we add a probe space-filling Dp- the form brane wrapping a (p−3)-cycle in the internal space, with Z 2 4 √ f µ worldvolume coordinates ξA (A, B = 0, . . . , p + 1) and S = − d x −g ∂µθ∂ θ (15) 2 DBI action (in the Einstein frame): where the axion decay constant, f, is computed to be: Z p+1 p (p−7)/4 SDBI = −Tp d ξ − det (γAB + FAB) . (10) −1 2 n gs h0 Vp−3 f = 3 2 (16) (2π) R0 The tension of the brane in the Einstein frame is given Here, 2πR0 is the unwarped length of the 1-cycle with by: 2 2 2 2 2 line element ds = R0 dφ = r0g˜φφdφ . The axion, θ, has a continuous shift symmetry de- −1 −p 0 − (p+1) 2 scending from the higher dimensional gauge symmetry, Tp = µp gs with µp = (2π) (α ) . (11) which may be broken to a discrete one. A slow-roll po- M N tential can thus be generated for the Wilson line axion, Also, γAB = gMN ∂AX ∂BX is the pullback of the ten-dimensional metric onto the brane (M,N = 0,... 9). θ, by fluxes, brane backreaction, warping, loop contri- Finally, F = B + 2πα0F , with B the pull- butions and/or other effects, including non-perturbative AB AB AB AB ones [8, 37, 49, 50]. Large field excursions for θ are en- back of the NSNS 2-form onto the brane, and FAB the field strength associated to the worldvolume gauge coded in the field’s decay constant by f/MP l > 1, with: field. We will choose static coordinates for the brane, so f 2 g h(p−7)/4l2 n V ξA = (xµ, ya) with ya the (p − 3) internal coordinates s 0 s p−3 2 = 4 2 w . (17) along the brane. We will comment on the validity of the MP l 2(2π) R0 V6 probe approximation below. Notice that for r0 > L, the warp factor can enhance Possible open string inflaton fields in the above system f/MP l, although the final behaviour of the latter also include Wilson line moduli associated with the worldvol- depends on the wrapped volumes. ume gauge field [37, 38] and the moduli describing the position of the Dp-brane in the compact space [45–47]. We will consider single field, large field inflation and the Weak Gravity Conjecture in these setups, which are re- 3 Note that this does not require the full six-dimensional internal lated to each other by T-duality. space to have a 1-cycle, see [48] for examples. 4

As a first example, consider a Dp-brane in an un- Considering an adS5 × X5 throat (6) as a prototype, warped, possibly anisotropic toroidal orientifold com- the warped length of a string stretching from a D5-brane pactification, with j directions of size R, and 6 − j direc- at r0 near the top of the throat all the way down to the tions of size L. Taking the brane to wrap a (p − 3)-cycle bottom of the throat is: with j directions of size R (so j ≥ j ) and p − 3 − j b b b Z r0   1/4 r0 directions of size L (so 6 − j ≥ p − 3 − jb), and the Wil- ` ∼ h dr = L ln . (23) Φ r son line wrapping one of the R-directions (so jb ≥ 1), we rIR IR have: Taking rIR ∼ ls and r0 ∼ rUV we then obtain from eq. f 2 n g  l 2+j−jb l 9−p−j+jb (21): = s s s . M 2 2(2π)4(2π)9−p R L P l (2π)2 r2/l2 r2 V (18) 1 < 0 s 0 5 , (24) 4 ng (ln(r /l ))2 l2 V A large f/MP l would require a substring scale cycle s 0 s s 2 R < l and/or L < l . In this limit the perturbative de- s s where again recall that V goes as r2/l2. Note in partic- scription breaks down, and the T-dual description should 2 0 s ular that the Weak Gravity Conjecture does not put an be used. upper limit on r , so both the Weak Gravity Conjecture Next, consider a wrapped D5-brane in an adS × X 0 5 5 and large axion decay constant could indeed be satisfied warped throat. The axion decay constant is then: with a brane at the top of a long throat. 2 6 2 2 f gs nV2 ls r0 ls In the above analysis, we have neglected the backreac- 2 = 4 6 2 2 . (19) tion of the wrapped D5-brane onto the background geom- M (2π) V5 L r R P l UV 0 etry. In fact, a D5-brane would alter the warp factor and 2 2 Since V2 goes as r0/ls, the possibility now arises that internal geometry (c.f. (6)), and introduce a non-trivial f/MP l > 1 can be achieved with a brane at the top of a profile for the dilaton. However, if its contribution to long throat, r0 ∼ rUV and rUV  L. the Einstein equations is much smaller that that of the Now let us check whether the conditions for large ax- stack of N D3-branes sourcing the warped throat, then ion decay constant can be fulfilled consistently with the we can safely consider the D5-brane as a probe. The lo- Weak Gravity Conjecture. Indeed, probe D-branes can cal contribution from a Dp-brane to the traced Einstein’s be used to test the consistency of a string theory con- equation goes as [39, 44]: figuration by asking whether their worldvolume gauge m µloc (9−p) theories and charged satisfy the Weak Gravity Tm − Tµ = (7 − p)Tp∆ (Σp−3) (25) Conjecture, wherever they are placed. The Weak Grav- (9−p) (9−p) p ity Conjecture implies a new UV cutoff, Λ = MP l g4, where ∆ (Σp−3) = δ (Σp−3)/ det g9−p is the which in the present case is: covariant delta function on the wrapped (p − 3)-cycle, Σ , with transverse volume R pdet g . The condi- (2π)2 2 Vw p−3 9−p Λ2 = 6 . (20) tion that the backreaction of the wrapped D5-brane is 2 (p−3)/4 gs l n Vp−3 s h0 negligible can then be written as: Matter fields charged under the U(1) arise from strings n T ∆(4)(Σ ) 5 2  1 . (26) stretching between the Wilson line Dp-brane and a sep- 2N T ∆(6)(Σ ) arated parallel Dp-brane. Largest masses arise when 3 0 2 the branes are very distant, with m ∼ Ms `Φ and `Φ This condition ensures that the probe approximation is the brane separation. So the Weak Gravity Conjecture, good when the probe is close to the N source D3-branes. 2 2 2 m < MP l g4 requires: We should exercise some caution as the probe becomes very distant from the source branes, at the top of the (2π)2 l2 2 Vw 1 < s 6 . (21) throat. 2 (p−3)/4 gs ` n Vp−3 Φ h0 For example, for the D5-brane in the adS√ 5 × X5 back- ground, we can take the parameters L ∼ 3 ls, gs ∼ 0.3, The Weak Gravity Conjecture (21) then imposes an up- r ∼ r ∼ 500 l , R = r g˜1/2 ∼ l , n ∼ 100, per limit on f/M (see eq. (17)): 0 UV s 0 0 φφ s P l 2 2 3 V2ls ∼ πr0 and V5 ∼ π . To compute the backreaction,√ f 2 h−1 l4 we estimate transverse volume elements as det g ∼ 0 s √ 4 2 < 2 2 2 . (22) 3 3/2 5 MP l (2π) `ΦR0 h0r0 sin θ1 and det g6 ∼ h0 r0 sin θ1 sin θ2, for some internal coordinate angles θ1, θ2. Thus, we can consis- −5 tently achieve f ∼ 4.2 MP l, Ms ∼ 1.4 × 10 MP l and Λ ∼ 4.5 × 10−2M . Note that we are safely within the 4 Vanishing cycles do occur in more general string geometries, for P l example when blowing-up a singularity. The vanishing cycles limit of perturbative supergravity, but it would be impor- in the conifold were used in [51] to achieve small axion decay tant to study in detail if the probe approximation remains constants from string theory, and in [52] to achieve string axion good when the D5-brane is at the top of the long throat. N-flation models with near Planck scale axion decay constants. In this sense, the scenario presented at the end of Section 5

IV will be more reliable. As we will also see there, it is The new UV cutoff implied by the Weak Gravity Con- also conceivable that more complicated warped geome- jecture is the same as for the Wilson line scenario, see tries than the adS mid-throat geometry may allow one (20). Charged matter may again arise due to strings to achieve even higher f/MP l. stretching to parallel D-branes that are separated by a To summarise, it is plausible that a Wilson line on a distance `Φ along the throat or around the θ-direction. wrapped D5-brane can give rise to an axion with large The Weak Gravity Conjecture, m < g4MP l constraint is axion decay constant consistently with the Weak Gravity the same as before, eq. (21), and implies the following Conjecture. This can be achieved by ensuring a warped upper bound on the axion decay constant,5 (28): throat geometry with a 1-cycle inside the 2-cycle that is f 2 g wrapped by the D5-brane in a region with small warp fac- 2 θθ 2 < (2π) 2 . (29) 1/2 MP l `Φ tor, h0  1, and thus requires a highly anisotropic long warped throat. A concrete realisation of the proposal We can now consider these constraints in some explicit would require a throat geometry with these properties. constructions. Take first a possibly anisotropic unwarped It would likely not, however, correspond to a stringy re- toroidal orientifold compactification, with j directions of alisation of the Extra-Natural Inflation proposal [8], as large size R and 6−j directions of small size L. Consider the string scale would probably turn out to be too low. a Dp-brane wrapping jb large directions and p − 3 − jb Indeed, note that taking r0 ∼ rUV large to achieve large small directions, cycling a large direction (so jb ≤ j − 1). f/MP l drives the string scale to smaller values. We will The position modulus has axion decay constant given by: comment further on this below. First we turn to the 2  2+jb−j  7−p T-dual description of a D-brane Wilson line, which is a f n gs 1 R ls 2 = 9−p . (30) D-brane position field. In this case, more explicit, reli- MP l 2 (2π) L L able constructions are possible. So a large decay constant could be achieved with a large angular direction R  L for j − jb = 1. We can test the consistency of the required parameters by using the Weak IV. BRANE POSITION INFLATION Gravity Conjecture. To this end we introduce another probe Dp-brane, separated by a distance `Φ ∼ πR around Once again consider a space-filling Dp-brane wrapping a large angular direction. The Weak Gravity Conjecture a (p − 3)-cycle, and now assume that the position of the (21) then requires: brane in one of the compact angular dimensions corre- sponds to an inflaton field. Such scenarios have been 8  L 2+jb−j  L 7−p 1 < (2π)9−p (31) studied, for example, in [25, 39, 40, 53, 54]. Dimensional n gs R ls reduction of the DBI action (10) gives, in exactly the same way as for the Wilson line scenario, a U(1) gauge and so imposes an upper bound on the decay constant: group in the four-dimensional effective field theory, see f 2 < 4 , (32) (13), (14). It also leads to a kinetic term for the angular M 2 position modulus, θ, of the form (15). The axion decay P l constant, f, is now given by: which is at most f ∼ 2 MP l. For example, taking a D3- 6 brane, with gs ∼ 0.3, L ∼ ls, R ∼ 1.5×10 ls, j = 1, jb = (2π) g−1 0 and n = 1, we obtain f ∼ 1.9 M , M ∼ 2.8×10−7M f 2 = s h(p−3)/4g n V . (27) P l s P l 4 0 θθ p−3 and Λ = 1.4 × M . Note that to achieve f ∼ 2 M , the ls P l P l string scale is very low. A large axion decay constant requires f/MP l > 1 with: Let us next consider a Dp-brane moving in a warped

2 throat. The prototypical example is the Klebanov- f 1 (p−3)/4 n Vp−3 Strassler warped throat, produced by placing N D3- 2 = gsh0 gθθ 2 w . (28) MP l 2 lsV6 branes at the tip of the deformed conifold. Away from the conical deformation, in the mid-throat region, the Now, taking the brane at the tip of the throat, r  L, 0 base of the conifold can be taken to be T 1,1, and the 10D we see that the warp factor enhances f/M , although P l metric takes the form (5), with h = L4/r4 and again the wrapped volumes will also affect the overall behaviour. 2 2 1 2 2 2 2 ds = dr + r dθ + sin θ1dφ The axion, θ, corresponds to a periodic direction and 6 6 1 1 1 so inherits a discrete shift symmetry. Again, the inflaton + r2 dθ2 + sin2 θ dφ2 potential may be induced by fluxes, brane backreaction, 6 2 2 2 warping, loop contributions and/or other effects, includ- 1 2 + r2 dψ2 + cos θ dφ + cos θ dφ  . (33) ing subleading non-perturbative effects. For instance, 9 1 1 2 2 [40] found a potential of the Natural Inflation form (3), by considering the experienced by a D-brane mov- ing in the geometry of the warped resolved conifold glued 5 We thank Zac Kenton and Steve Thomas for pointing out an to a compact Calabi-Yau. error in the previous version of this equation. 6

Allowing for a Zk orbifolding in the ψ direction, the vol- In fact, one can immediately infer from Eqs (28), (34), 1,1 16 3 2 2 4π 2 4 4 ume of T is given by V5 = 27k π and the adS radius (9), gθθ = r0/6, V2ls = 3 r0 and h0 = L /r0, that the of curvature (8) is: axion decay constant is given by:

27k f 2 3 nk g l2r2 L4 = g Nl4 . (34) s s 0 3 s s 2 = 2 2 2 (41) 64π MP l 8π L rUV Now consider a D5-brane wrapping a 2-cycle in the throat 6 As r0 . rUV and L > ls, in the end warping and wrapped [39]: volumes conspire so that only the orbifold and wrap- r = r , ψ = ψ , θ = −θ , φ = −φ (35) ping numbers can help to increase f. With large wrap- 0 0 1 2 1 2 ping numbers and orbifold numbers, moderate9 super- Planckian decay constants are attainable, arguably at the and moving in an angular direction, say, θ2. The condition that the backreaction of the probe D5- limits of the supergravity approximation and consistently brane can be consistently neglected, eq. (26), evaluates with the Weak Gravity√ Conjecture. For example, taking in the present case to78: gs ∼ 0.3, L ∼ 3 ls, rUV ∼ r0 ∼ 30 ls, n ∼ 30 and −3 k ∼ 100, we obtain f ∼ 3.4 MP l, Ms ∼ 3.1 × 10 MP l n L2 and Λ = 7.1 × 10−2M . Beware again, however, that sin θ  1 . (36) P l 2 1 the backreaction of the D5-brane may actually be larger 12N ls than the estimate given in (36), as the brane is at the top The parameters are also constrained by the Weak of the throat, far from the source N D3-branes. Also, the Gravity Conjecture. To see how, we introduce another string scale against results to be quite low (c.f. eq. (1)). probe Dp-brane, firstly separated from r0 up the throat More complicated warped geometries may allow for to rUV , and secondly separated around the throat along more reliable and even larger axion decay constants. For θ2 at r = r0. We estimate the mass of the corresponding example, a D5-brane moving in the warped resolved coni- 2 charged matter, m = Ms `Φ, as (c.f. (23)): fold was used in [40] to obtain a super-Planckian decay r constant, and proposed as a model for Natural Infla- m = M 2L log UV , (37) s r tion. The 10D metric for the warped resolved conifold 0 is [55, 56]: πL m = M 2 √ , (38) s 6 2 −1/2 2 1/2 2 ds10 = h(r, θ2) ds4 + h(r, θ2) ds6 , (42) where we have taken into account that the lengths of the with stretched strings, `Φ, are warped. The Weak Gravity Conjecture (21) thus implies that (use eqs. (9), (35), 2 −1 2 1 2 2 2 2 2 4π 2 4 4 ds6 = κ (r)dr + r dθ1 + sin θ1dφ1 V2ls = 3 r0 and h0 = L /r0): 6 1 2 2 2 2 2 32π2 1 r2 + r + 6u dθ2 + sin θ2dφ2 1 < UV , (39) 6 3 nk g l2 1 2 s s + κ(r)r2 dψ2 + cos θ dφ + cos θ dφ  ,(43) 9 1 1 2 2 where notice that this ensures that the wrapping number, n, and orbifold number, k, are not too large. The Weak where Gravity Conjecture thus gives the following upper bound r2 + 9u2 on f: κ(r) = (44) r2 + 6u2 2 2 f 4r0 and u is the resolution parameter. The warp factor 2 < 2 . (40) MP l L h(r, θ2) is given by: Comparing this with result for the torus (32), we see that ∞ 4 X warping has relaxed the upper bound on the axion decay h(r, θ2) = L (2l + 1)Hl(r)Pl(cos θ2) , (45) constant, and larger super-Planckian values may now be l=0 possible within the supergravity approximation. where  2  2 Cβ 9u Hl(r) = 2 2+2β 2F1 β, 1 + β, 1 + 2β; − 2 (46) 6 9u r r Note that for p = 3, the warp factor cancels out in f/MP l, eq. (28), and cannot be used to make f/MP l large. 7 The volume orthogonal to the D5-brane embedded into space- time with the static gauge is obtained by setting θ1 = φ1 = 0. 8 9 Note that this condition is easier to achieve than the one re- Larger values for f/MP l are possible compared to [39] by consid- quired in [39], as in the latter the backreaction is enhanced by ering the brane moving in an angular direction rather than the the brane’s velocity. radial direction. 7

and reliability of the probe approximation. See also [40] for a more systematic check. 2β 2 r (3u) Γ(1 + β) 3 Also, the Weak Gravity Conjecture (21) now implies Cβ = and β = 1 + l(l + 1) . Γ(1 + 2β) 2 that: (47) 128(171 − 10 log 10)π2 1 r2 The warped geometry is sourced by N D3-branes placed 1 < 0 . (54) 6561 g nk l2 at the tip of the throat, at r = 0 and θ2 = 0. Very s s close to the branes, at θ2 = 0 and small r, the geometry where we used that the masses of warped strings stretch- 5 4 4 corresponds to adS5 × S with h = L /r and L given in ing between probe D-branes separated up the throat and (34). Further away from the branes, but still at small r < around the throat are, respectively (c.f. (37), (38)): u, the warp factor is h = L4/(ru)2. Taking the smeared Z rUV limit for the N source D3-branes, the warp factor can be 2 1/4 −1/2 m = Ms drh κ , (55) approximated by the l = 0 mode as [55, 56]: r0 u1/2 2L4 2L4  9u2  m = M 2πL . (56) h(r) = − log 1 + . (48) s 1/2 9r2u2 81u4 r2 r0 Notice again that the Weak Gravity Conjecture (54) en- Using this approximation to compute the warped volume, sures that the wrapping number, orbifold number and one obtains string couplings are not too large, as expected. It leads 3 to a simple upper bound on the axion decay constant (see w 64π 4 2 V = s L r (49) eq. (52)): 6 81k UV f 2 4 r u with < 0 . (57) M 2 L2 1 r2   9u2  P l s =2 + UV 1 − log 1 + 2 2 It is now possible to achieve large f/M within the 9 u rUV P l limits of perturbative control, small brane backreaction 1 r4  9u2  − UV log 1 + . (50) and consistently with the Weak Gravity Conjecture. For 81 u4 r2 √ UV example, taking gs ∼ 0.3, L ∼ 3 ls, u ∼ rUV ∼ 65 ls, The string scale is then given by: r0 ∼ 2ls, n ∼ 10 and k ∼ 10, we find f ∼ 6.8 MP l, −4 −2 Ms ∼ 2.1×10 MP l and Λ = 1.2×10 MP l, consistently 81k g2 l6 with (53) and (54). M 2 = M 2 s s . (51) s P l 4 4 2 Although we have seen that warping and non-trivial in- 4(2π) s L rUV ternal geometries allow one to consistently obtain axions In the near-tip region where the warp factor is h0 = with large axion decay constant in string theory, these 4 2 2 2 L /(r0u) , gθθ = u and the wrapped volume is V2ls = axions cannot be used for large field inflation because 2 4πu , taking u ∼ rUV , the axion decay constant becomes: the corresponding string scales are too small. Large field 10 inflation for observable tensor modes implies that infla- tion occurs at least at GUT scale M ∼ 10−2M (see f 2 6561 nk g l2u inf P l = s s . (52) eq. (1)), so the string scale should be at least around M 2 32π2(171 − 10 log 10) L2r −1 P l 0 Ms ∼ 10 MP l, for a supergravity analysis to be valid during inflation11. In fact, as is well known, dimensional The possibility now emerges that small r < u could 0 reduction gives the following relation between the string correspond to super-Planckian values [40]. scale and Planck scale in any string model: The condition that the backreaction of the probe D5- 2 brane be much smaller than that of the N D3-branes (26) 2 2 gs can now be written as: Ms = MP l w . (58) 4πV6 2 1 n L r > −1 Thus a string scale Ms ∼ 10 MP l is difficult to achieve 2 sin θ1  1 . (53) 2 N ls u within the limits of supergravity. This renders question- able all models of large field inflation in perturbative As the probe brane is close to the tip and the source string theory, even those with sub-Planckian decay con- N D3-branes, this should be a good indication of the stants like axion monodromy12.

4 11 10 This result differs from that in [40] since we use h = L valid Strictly speaking, a hierarchy of scales Minf < Mc < Ms . MP l, (ru)2 4 where Mc is the compactification scale, is needed in order to con- for θ 6= 0, in contrast to h = L , used in [40]. This latter warp 2 r4 sistently work in the four dimensional, low supergravity factor is also used to compute the warped volume of the throat limit of string theory [36, 57], and moduli must also be stabilised. up to the UV cutoff in [40]. 12 Factors like (2π) might help in general to achieve higher string 8

V. DISCUSSION ity Conjecture13. Furthermore, the Generalized Weak Gravity Conjecture, in its mild form, states that among We have shown that single open string axions coming any instantons, i, coupling to an axion with coupling fi, i from D-branes in warped geometries can enjoy super- there must exist at least one with action Scl . MP l/fi. Planckian axion decay constants consistently with the The strong form states, additionally, that this instanton Weak Gravity Conjecture and within the limits of the must be the one with smallest action. supergravity approximation. The most explicit realisa- However, we have seen that for open strings in warped tion is the model proposed by Kenton and Thomas [40] geometries a large axion decay constant is generated for Natural Inflation, using the position modulus of a within the limits of supergravity, where instanton effects wrapped probe D-brane moving in a compact angular must be exponentially suppressed. Indeed, the known in- direction near the tip of the warped resolved conifold. stantons that couple to the Dp-brane Wilson line and po- Here, initial estimates of the backreaction of the brane, sition open string moduli are Euclidean Eq-branes wrap- which is close to the source branes at the tip, also indicate ping (q + 1)-cycles on the internal space. These instan- that the probe approximation is reliable, though it would tons usually couple to the D-brane moduli with axion −1 be important to verify this with explicit computation of decay constant f, and their actions go as gs and the the backreaction. volume of the wrapped (q + 1)-cycle14. Thus they have However, none of the constructions considered here can Scl > MP l/f, and contributions are exponentially sup- be used for large field inflation, as their string scales are pressed within the supergravity approximation. There too low compared to the inflationary scale (1). Indeed, then seems to be three likely possibilities: (i) In the pres- we stress that one of the biggest challenges in realising ence of warping, the Eq-instantons couple to the open large field inflationary models and observable gravita- string axions with a suppressed effective axion coupling tional waves within string theory is to identify appropri- f 0 < f, such that they satisfy the Generalized Weak ate compactifications with sufficiently high string scale, Gravity Conjecture with large action. (ii) There are within the limits of four dimensional supergravity (or, new stringy instanton effects that couple to the D-brane alternatively, to evade the Lyth bound). Within the moduli with a suppressed effective axion coupling f 0 < f, perturbative supergravity limit of string theory, dimen- which satisfy the Generalized Weak Gravity Conjecture15 0 0 0 sional reduction of the 10D Einstein-Hilbert term relates with large action, Scl . MP l/f , either mildly (Scl > Scl) 0 the string scale and Planck scale as in (58). Assuming or strongly (Scl < Scl). (iii) The Generalized Weak w 0 gs . 0.3 and V6 & 1 for the weak coupling and α ex- Gravity Conjecture is incorrect. We hope this letter will pansions, implies Ms . 0.08MP l. Moreover, assuming a help to shed light on these issues. 2 0 curvature scale l such that (ls/l) . 0.3 for a reliable α expansion, implies Mkk . 0.04MP l. Although numeri- cal coefficients like factors of 2π could help, the required ACKNOWLEDGMENTS hierarchy Minf  Mkk  Ms, seems very difficult to achieve for high scale inflation Minf ∼ 0.01MP l. We could try to increase the string coupling and/or reduce We would like to thank Lilia Anguelova, Nima the internal volumes to drive the string scale up, but then Arkani-Hamed, Nana G. Cabo B., Zac Kenton, Carlos the relation (58) would not apply. In this case though, N´u˜nez,Fernando Quevedo, Gianmassimo Tasinato, Steve one could always dualize to an equivalent weak coupling, Thomas and Irene Valenzuela for useful discussions. IZ weak curvature description where (58) holds again, and thanks the Gordon Research Conference on String The- return to the bounds Ms . 0.08MP l, Mkk . 0.04MP l. ory and Cosmology for partial support, hospitality and Although the open string axions with large decay con- the stimulating atmosphere that inspired this project. stant studied here cannot be used for large field inflation, She also thanks the Physics Department of the Univer- they may give some insight into the Generalized Weak sity of Guanajuato, Mexico for hospitality while part of Gravity Conjecture. The D-brane axions have interpre- this work was done. The research of SLP is supported tation as (or are related to) gauge fields in higher dimen- by a Intra European Fellowship within the sions, and so would be directly subject to the Weak Grav- 7th European Community Framework Programme.

scales within the perturbative limits, but explicit constructions the Weak Gravity Conjecture and dimensional reduction and the enhance this problem. For example, in the original axion mon- Generalized Weak Gravity Conjecture for axions. odromy proposal [26], increasing Vw by using throats within 6 14 See [62] for an explicit computation of the instanton generated throats to prevent brane anti-brane annihilation and suppress scalar potential for open string moduli in toroidal orbifold com- brane backreaction [58, 59] will drive the string scale down. The pactifications with magnetized D-branes. Large Volume Scenario used in the D7-brane chaotic inflation 15 This is similar – but slightly different – to the loophole discussed model of [25] would also make a high string scale difficult to in [14] to achieve Natural Inflation consistently with the strong achieve. See also [60]. form of the Generalized Weak Gravity Conjecture. 13 See the contemporaneous paper [61] for a further discussion on 9

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