JHEP08(2015)032 factor) Springer N July 7, 2015 √ May 25, 2015 : August 7, 2015 : : Accepted Received Published 10.1007/JHEP08(2015)032 a,b doi: , Published for SISSA by and Irene Valenzuela b [email protected] , . 3 Angel M. Uranga a,b 1503.03886 The Authors. c Black Holes in String Theory, D-branes, Models of Quantum Gravity, Global

We discuss quantum gravitational effects in Einstein theory coupled to periodic , [email protected] axions can still achieve a moderately superplanckian periodicity (by a N gauge symmetry. Finally we discuss the realization of these instantons as euclidean E-mail: [email protected] Departamento Facultad de Te´orica, de F´ısica Ciencias, Universidad28049 Aut´onomade Madrid, Madrid, Spain Instituto IFT-UAM/CSIC, de Te´orica Universidad F´ısica Aut´onomade Madrid, 28049 Madrid, Spain N b a Open Access Article funded by SCOAP ArXiv ePrint: Z D-branes in string compactifications. Keywords: Symmetries potential, which spoil superplanckian inflatonaxion field models range. with A lattice similarwith alignment result (like holds the for multi- Kim-Nilles-Pelosowith model). no Finally, higher theories harmonicshold in in the this case axion due potential. to the The absence Weak of Gravity some Conjecture instantons, fails which are to forbidden by a discrete periods (aka decay constants) andThe their possible effects application we to study largecharged field under correspond inflation the to models. axion, and theof) our nucleation results the are of Weak essentially euclidean Gravityperiods compatible Conjecture, gravitational with contain (but instantons as gravitational independent follows: instantons inducing single sizable axion higher theories harmonics with superplanckian in the axion Abstract: axion scalars to analyze the viability of several proposals to achieve superplanckian axion Miguel Montero, Transplanckian axions!? JHEP08(2015)032 ] for a recent string- 1 27 28 11 4 7 4 11 25 12 6 ], there has been an intense activity in building 30 4 – 1 – ], and still allowed by the joint Planck/BICEP2 upper 23 2 15 20 8 , raised by the initial BICEP2 detection claim of B-mode polarization r 25 21 1 ]). Due to the correlation, in single field inflation models, of this ratio and 31 3 12 [ . 0 r < Such large-field inflation models are sensitive to an infinite number of corrections to 6.2 D-brane instantons6.3 and gravitational instantons Charge dependence6.4 in flat multi-axion D-brane setups instantons and alignment 5.1 The string 5.2 Putting everything together 6.1 Generalities 4.1 Axion-driven multiple4.2 field inflation Gravitational effects 2.1 Dependence2.2 on the spectrum Weak of gravity the conjecture theory and axions 3.1 The instanton 3.2 Consequences for a transplanckian axion the inflaton field range by theinflation Lyth models bound in [ which the inflaton travels transplanckian distancesthe in inflaton field potential space. evenstruction if requires they accounting are for suppressed corrections by due the Planck even mass to scale, quantum so gravity effects. their con- Given The present precision era inactivity the in measurement model of building CMB ofmotivated observables early is review). universe triggering inflationary a The models vigorous tensor (see most to [ scalar recent ratio examplefrom is primordial gravitational provided waves [ by thebound possibility of a sizable 1 Introduction 7 Conclusions 6 Aspects of string theory realizations 4 Gravitational instantons for multiple axions 5 Two is more than one 3 Gravitational instantons for a single axion Contents 1 Introduction 2 Remarks on the Weak Gravity Conjecture JHEP08(2015)032 ] for (1.2) (1.1) 19 – , in any 10 inφ e ] (see [ 9 , 8 ]) for the existence of addi- 20 P f ] for similar work). Because the presence nM πf. 22 ∼ , + 2 gravitational instanton action is suppresed by E 21 φ – 2 – S n ∼ φ ] for recently suggested exceptions). The ‘empirical’ 6 with an approximate continuous shift symmetry, broken by to a discrete periodicity (see [ φ f iφ/f as e f ] (which provide a template for a quantum gravity framework), moti- 7 ], assuming a cosine gauge instanton potential, and requiring axions with trans- 5 Hence, axions with parametrically transplanckian periodharmonic contributions receive to unsuppressed their higher potential fromically gravitational prevent instantons. achieving These transplanckian gener- inflationary field ranges. For multiple axion models, effectiveachieved transplanckian for axion certain periods axion can linear in combinations. principle be However for models based on lattice For single axion models, thethe charge- axion period In this paper we undertake a direct approach to this problem, by considering euclidean These models are formulated in purely phenomenological field theory terms, and there- • • Einstein gravity and explicitlyvant constructing axions, gravitational and instantons evaluating coupling theirare to contributions wormhole to the the solutions, rele- axionshould and potential. provide for The good configurations sufficiently effectivequantum large descriptions theory axion containing of Einstein charge the gravity. have corresponding Our low results configuration can curvature in be and any summarized UV as follows: of gravity is essential inby the gauge argument, instantons these in contributionstions the may could phenomenological be spoil model. beyond the transplanckian those If fieldaxion captured range sufficiently period. by strong, introducing Unfortunately, additional such the maxima contribu- WGCof along arguments these the are effects. insufficient to quantify the strength fore there remains thetheories question including of quantum whether gravitational their corrections.(under features In the survive fact, name in there of are actualtional indirect the embeddings instanton arguments Weak in effects Gravity leading Conjecture toconsistent (WGC) higher theory [ harmonics of in quantum the gravity axion (see potential, [ i.e. vated the construction of models within multiple axions which and some potentials axion from gauge linearbasic instantons, combination period, effectively even hosts if a the transplanckianrecent original field works). periodicities range are in taken its sub-planckian [ This idea was originallyflation proposed [ at theplanckian phenomenological decay level constant incase-by-case so-called realization natural that in- thiscompactifications [ requirement is not obviously realized in string theory tecting the model against suchbased corrections; on therefore, axions, many i.e. large-field scalars non-perturbative inflation effects models are the difficulties in addressing the latter, a sensible approach is to invoke symmetries pro- JHEP08(2015)032 ] ]. 40 54 – 37 ] for a discrete , related 27 N , 5.1 Z 26 ] for a 4d phe- 53 , and explore the – 51 ], see [ 4.1 ), and its implications 25 – 3.1 23 reconciles the appearance ]), see [ 5 50 – ] for a possible exception). 46 36 discrete gauge symmetry, reflected we extend the analysis to multi- . Section N 4 Z we describe the quantum gravitational 4.2 we review the Weak Gravity Conjecture, factor) corresponding to travelling along 3 2 ], see also [ N ]. However, in non-supersymmetric situations 45 √ – 34 – 3 – 41 ). In section strings. We show that this result can be used to N axions, it is still possible to achieve a superplanckian 3.2 Z N ] for recent attempts to embed such models in string theory), is ] also investigates the consequences of the Weak Gravity Conjecture 33 – 22 28 ]. These considerations should be implemented to analyze actual string ], parametrically large field ranges also receive unsuppressed higher har- 8 35 absence of the missing instantonsin is the due existence to a generalize of the certain Weak Gravitygauge Conjecture symmetry. by including the effects of a monic contributions and the resultsinstantons are jeopardize similar to inflationary the transplanckian single field axion ranges. case:Finally, gravitational for theories with periodicity (by an enhancementa of diagonal a in the field space, with no higher harmonics in the axion potential. The alignment [ The recent work [ The paper is organized as follows. In section Before concluding, let us mention an alternative to natural or aligned inflation, the so- The gravitational instantons are effective descriptions of configurations in concrete • for transplanckian axionsaxion (section models. We discussimplications different of alignment gravitational corrections mechanisms in inof section section moderately superplanckian axionGravity Conjecture. periods The in systems certain contain alignment charged strings, models discussed with in the section Weak potential which could lead to detectable signatures in thefor primordial axion power inflationary spectrum models [ in string theory along linesand similar its to application our to own. corrections axion to the scalars. scalar potential In in section single axion models (section with a multivalued potentialor (and via monodromy potentials from arising flux eithernomenological backgrounds approach. via [ For brane the purposes couplingsthe of [ axion the period present paper, suffices the toIn subplanckian protect nature any these of models event, against gravitational the instantons effects would described still in this give paper. rise to periodic modulations of the saturated (by insertions of susy breakingscalar operators), potential and [ their contributions descendtheory to realizations the of natural and aligned inflation (see [ called axion monodromy inflation, which uses a single axion with subplanckian period, but that the corresponding instantonsNon-BPS may instantons not are correspond not tosetups often BPS they discussed instantons do in in not the the lead(higher vacuum. literature, derivative to or because corrections multi-fermion terms) in to [ the supersymmetric like inflation, superpotential, their but extra rather fermion to zero higher modes F-terms are lifted or equivalently the external legs are models of quantum gravityaligned like inflation, string similar effects theory. cancorresponding For arise to instance, from euclidean in gauge D-brane realizationsreview). instantons instantons of An (dubbed beyond natural important those exotic aspect, or the or which literature we stringy (see emphasize [ and [ has seemingly been overlooked in JHEP08(2015)032 ), 2.1 (2.2) (2.1) -form p ], its rationale is at the -form field. These black p 20 ). We will see an example 2.1 ), the relation between D-brane 6.1 times ( ) and applications to aligned axion j ). For instance, black branes could be F N . ] which effectively multiply the coupling 6.2 2.1 ∧ ∗ N 56 i g G F √ ij – 4 – G . ]. There are two posssible loopholes to the above 1 2 T 55 contains our conclusions. = 7 L -form field to its sources, and it has dimensions of p -dimensional object with tension p discusses some aspects of single and multiple axion models in string 6 discrete gauge symmetries [ ). Finally, section , in the context of instantons and 0-forms, but let us advance the main N 5 ) comes from the tension of an extremal black brane of Planckian size. To 6.4 Z encodes the possibility of kinetic mixing between the two U(1)’s. In this . The rationale for the conjecture is that one expects to be able to build 2 2.1 G D/ . Section 1 membranes electrically charged under the abelian − is the coupling of the − , so that the right WGC bound would be +1 5.2 g p p N Consider a theory with two U(1)’s in 4 dimensions, with lagrangian The essence of the WGC can be reduced to the statement that, for any abelian by The metric theory, we will be able to build Reissner-Nordstrom black holes charged under both U(1)’s. black hole is largercharged than under the right handg side of ( of this in section idea in the usual WGC setup of U(1) gauge theory in 4d. 2.1 Dependence onThe the bound spectrum ( of thedetermine theory this tension isunder a consideration, very and nontrivial it question is possible which to depends find on theories the in which specific the theory tension of the extremal ever, this is onlysmaller charge. possible The if extremal thereand black hence is hole the has an constraint. tension object Otherwiseremnants, given we with by lighter would the the than end associated right up the trouble handconjecture, with [ side black a which of large we brane ( number now and of discuss Planck-sized with in some a detail. Here, [mass] black branes will generically evaporate via Hawking radiation, radiating their charge away. How- heart of our approach. field, there must be a charged 2 Remarks on theWe Weak will Gravity study Conjecture the possibilityfrom of a transplanckian field semiclassical ranges perspective, inusing a focusing the quantum on theory Einstein-Hilbert those ofindependent action gravity effects of for which the the can validity be gravitational of safely field. the described Weak Although Gravity our Conjecture approach [ is section theory, including some generalinstantons considerations (section and gravitationalmodels instantons (section (section to discrete gauge symmetries which imply a generalization of the WGC, as discussed in JHEP08(2015)032 is + (2.3) (2.4) Q in the j ∧ ∗ ). A 2 ,Q 1 . For the smallest ) is only nontrivial Q 2 are admissible. Q 2.3 A = 1 Q . j ). Indeed, ( = 0 amounts to saying that Q . That this is the right bound can . Notice that the generalized Weak i g 2.4 g − ) (2.5) Q . constitute a symmetry of the theory, 2 2 2 1 Q is nondynamical does not mean it has √ √ ! π − ij 2. The WGC would seem to give us a A P has rank one. For instance, take P − G √ + + A M 1 1 1 1 M G q θ 1 g/ P (which we take as integers in order to comply

A ∼ ]) would have resulted in the more stringent given by ( ( → 2 2 M 1 2 – 5 – g 1 θ G Q 2 22 = ), the charges under the diagonal and antidiagonal , , = 2 1 = 21 + Q . For all intends and purposes, it seems we have a extremal ,Q G + A 1 M ) with F Q ) gives again 2 ). Requiring that extremal 2 2.3 ∧ ∗ . M 2.3 Q + ± M F 2, that is, a particle with charge 1 and lighter than this should 1 2 ] (see also [ 1 g √ = 1, ( Q 57 2 g/ = ( Q P 2, which as discussed above does not hold. ± = M √ Q 1 ∼ g/ Q P M discrete gauge symmetry. Since the allowed charged matter is of even charge, = 0. In other words, only states with no charge under 2 . j extremal Z M M Another way to rephrase the above is that the nondynamical antidiagonal U(1) imple- This has dramatic consequences for the bound of the WGC. There is no physical reason For a black hole with charges ( The fact that the antidiagonal gauge field Now, consider a degenerate limit in which That is, only one particular linear combination of the two U(1)’s, namely bound ments a global gauge transformations ofeven the of form its nonperturbative sector. black hole of charge 2,a the bound first twice one as knownalso for large, be sure i.e. to seen be also in byfor the looking black theory. at holes As ( charged ablack along result, hole the we with diagonal get U(1),Gravity that Conjecture is, of with [ even. In other words,which the removes effect all of the oddly the charged antidiagonal black U(1) holes is from toto the add impose spectrum. an that extra the selectionin black rule the hole spectrum of to charge 1 begin must with. be The able best to decay we if can said do black is hole to is repeat not the argument with the our case would correspond toaction. the black If holes), the there is gaugecondition a field coupling is of nondynamical, the form it acts as acombinations Lagrange are multiplier enforcing the theory with just abound single U(1) withexist. coupling This is however not correct, as we willno now effects explain. on the physics. On the contrary, in any theory with charged matter (which in is dynamical. The normalizationThe has been kinetic chosen term so is that black just holes have integer charges. with Dirac quantization) has mass The WGC would demandsmaller the than existence that of of the charged above particles extremal with black a holes, for charge-to any mass values ratio of ( The extremal black hole with charges JHEP08(2015)032 = 0 p ]. Therefore, for which the 60 α – 58 ] that in a quantum ]. Although a single 56 61 ] that the axion cannot have a 20 and the charge quantum (which can be an is nothing but the inverse of the axion decay . There is a unique choice of α g g/α – 6 – ]. ) exists, there is no similar argument for the axion, , we have many such instantons with small action, → 20 P α 2.1 M  f . This is usually taken to imply [ P 2 the charge can take only even values, i.e. the charge quantum is 2. f M √ . g/ S , and so the above conjecture formally implies the existence of an instanton f 0; whereas in the latter we do have stable black hole (in general, black brane) . This changes the coupling = 0, that is, an axion, the coupling αA p > A similar argument works also for Euclidean theories. As discussed above, one can As an example, consider Maxwell-Einstein theory with no charges. One can produce, Although this seems to suggest that the WGC for axion fields is not on such a firm Of course, all of the above discussion can be rephrased as a redefinition of the gauge p 2. Applying the usual WGC bound then yields the right result. Our main point is that → √ expected to addinstanton up does all not fulfill geometriesat this with condition, infinity, the since an its rightinstanton axionic instanton- asymptotics amplitude electric anti in charge [ instanton the may path beconfiguration pair integral, measured mediating does. it a is transition difficult Once between to two exclude we different the allow states; single instanton one the as could instanton- a always anti cut a the full theory. build a solution tois the an euclidean instanton. equations The ofwhich these question motion instantons now with are is nontrivial excludedcannot wether axion of it the be charge. is path avoided. This integral, possible or to on This build the a seems contrary their consistent unlikely, presence theory since in in a quantum theory of gravity we are theory of gravity one expects to have all possiblein charged a objects. very strong electricthe fact field, that pairs the of theory Reissner-Nordstromwith has the black nonsingular quantum holes configurations pair-production [ with process, nonvanishing charge, should together ensure the existence of these objects in ground, it does highlight the factto that have in a gravitational consistent instantons quantum coupled theorythe of to Euclidean gravity the one equations expects axion. of motion Theseasymptotic for charge. can the be This axion-gravity is found system, in just while accordance by with specifying solving the the considerations right in [ remnants unless an object satisfying ( since there is nogravitational would instanton. be When stable blackand hole computing solution: their the effectinconsistency electrically will of charged the be kind object a described is challenge; in a [ however, there does not seem to be any For constant with action parametrically flat potential.and However, there is a very strong difference between This is equivalent to ag theory in which thein charge order quantum to is apply 1the these and theory the bounds is gauge correctly, knowledge coupling essential. of is the spectrum2.2 of allowed objects in Weak gravity conjecture and axions coupling/charge normalization. One may rescaleA a U(1) gauge fieldarbitrary by real an arbitrary number) factor, getscharge multiplied quantum by is 1,with i.e. the coupling charge takes integer values. What we have just seen is that JHEP08(2015)032 = 2 ∗ (3.1) (3.2) ) is 3.1 , dualizing to a s ...... + +   dφ dφ ∧ ∗ ∧ ∗ dφ dφ 2 2 2 2 f f – 7 – + − RdV RdV 1 1 πG πG − − 16 16 -dimensional space with metric signature   n Z Z . The Minkowskian action for the theory is π ]. 62 -forms in + 2 k φ ∼ for φ s ) k − n ( k Throughout the paper we assume that the axion which is to act as the inflaton somehow We first study the consequences of gravitational instantons for a single axion with Probably, the smallest such instanton will involve Planckian geometries, and so it 1) − ( three-form and Euclidean rotation dois not obtained commute. by first The dualizing bestdualizing to semiclassical back a to approximation three-form, a then scalar; performing asof euclidean a this result rotation, the point, and extra then see minus [ sign arises. For a detailed explanation Notice that naive Wickaxion. rotation This would extra have minus signto produced for it the an being opposite axion dual as kinetic to compared term a to a three-form; for standard duality the scalar involves field the is Hodge related star operator and since periodicity where the dots stand for any extra light fields. The euclidean counterpart of ( has an adequate potential tofrom provide a successful inflation. gauge In sectorpotential typical is models coupled obtained: this to we comes will thecan e.g. take spoil axion. it it. as given We and will then ask not ourselves address if gravitational in effects this paper how this do not interfere with inflation.setup to Hence, address gravity questions coupled pertainingof to the the axions inflationary possible provides potential. gravitational the corrections Second,general appropriate we to as would the possible, like flatness to toWeak keep Gravity our explore Conjecture framework the in as general interplay minimal gravitational and of theories, gravitational in instantons, a axions, model-independent way. and the partners of the axionsetc. to turn The them motivation into forinflationary complex this dynamics requires fields, is the fermions twofold: axion or toin first, be fact, other for the the superpartners, applications only main (non-gravitational) challenge to dynamicallies in single-field field; realizing in inflation, inflation giving in the hierarchically UV complete large theories, masses like to string theory, all fields except for the inflaton, so that they studied even in the absence of an UV completion. 3 Gravitational instantons forIn a this single section axion we starteffects, constructing in gravitational theories instantons of gravity coupling coupled to to axions, axions. and These their do not include other fields like scalar defines a state within theconnecting theory, it so to it the makes vacuum sense (that to is, consider a finite singleshould action instanton). not configurations be takenthe very reach seriously. of However, semiclassical some Einstein instantons gravity. may Their turn out effects to on be axion dynamics within can be safely time slice between the instanton and the anti-instanton. This constant time configuration JHEP08(2015)032 (3.6) (3.7) (3.8) (3.9) (3.3) (3.4) (3.5) . 4 − P ). Similarly, r M ( 2 f .  3 . P 6 Ω a f r d M 2 3 2 f  2 n = 3 π 2 2 π 6 surfaces behave like proper n ψ, θ, φ. 1 3 r . r = = 2 = µν  . , g dφ 2 2 2 2 ) ∗ ) G ) f , i π 0 n φ 0 , 4 ii 3 µ 3 2 g  π fφ 2 S fφ 2 n ( 3 ( ) φ∂ 2 2 0 . We thus get a system of two coupled , which corresponds to the maximum ds . Thus the single parameter controlling µ G . f r 4 2 2 ≡ ∂ 2 / πG r / π 1 fφ ( 1 r 1 4 ( 2 1 a πGr 2 + − f − 1 a f = 8 2 2 − = , a 2 2 3 n = 4 φ 3 π − dr φ r µ ) r dr 4 0 ν – 8 – ) = = r ) f = f f ( r φ∂ R ∼ φ∂ r ; it has to be larger than the Planck scale for the ( f ii µ − µ a f φ ∂ ∂ − = 2 (1 p ) which is asymptotically flat and with an axion profile 2 = , since asymptotically ,T 2 2 3 2 n + 1 r f ds ) 3.2 µν 2 0 πGf n f T φ π 2 ( − 2 2 − = 8 f f 1 2 ) = r = ( πGT rr gives the typical size of the instanton; at this radius the metric seems T = 8 4 / , dφ R 1 4 a a r 1 − 1 ) = r ( Einstein’s equations yield The stress-energy tensor of an axion field is With everything discussed above, we only need to find the right gravitational instanton. f is perfectly regular there.curvature at the Notice throat of that the for wormhole, wether the solution cansolution be to trusted be is reliable within the context of effective field theory. Notice also that the The parameter to become singular. This is a coordinate singularity, as the curvature scalar The solution is The only nonvanishing components of these are where we have redefined the radialspheres. coordinate so that The constant- onlythe dependence axion of profile the will be metricsecond-order taken is ODE’s as through which a function is the only easy radial of to factor analyze in detail. This means it has electric charge This will be a sphericallytake a symmetric spherically solution symmetric with ansatz the for right axionic the charge. metric, To do this we We are looking for awhich solution at of ( large distances from its core has the asymptotic form 3.1 The instanton JHEP08(2015)032 i θ | 1 (3.11) (3.12) (3.13) (3.10) , the metric . , so that 2 t 2 n ) P + f ∂φ a M ( 2 π = f 3 16 4 − √ r = = T dr 2 ; in other words, our instanton )) R , and r . , ( 0 × .  A φ i 3 3 ( 2 S ∧ πGT 3 S CS = 1; we need wormholes with higher r ds n 2 A | / n = 8 θ 1 2 in an earlier version of this manuscript. + ∧ / − 2 CS R ) A r in dt 3 ( e 1 4 R f – 9 –  Z 4 CS from infinity these areas reach a minimum, then / 2 n X are degenerate in energy, the true vacuum is a 1 3 ∞ 1 r a ]; in this case, the half-wormholes would generate a a = Z = = CS 66 i 2 = n – 2 θ f | CS r 2 64 ds n π = φ µ is of order Planck for φ∂ 4 µ / 1 a dV ∂ ∼ Z ] for pointing out a missing factor of 1 2 67 2 f -states” of Coleman [ α ], these half-wormholes are regarded as describing the nucleation of baby uni- = 63 ]. As discussed there, it is up to us to choose which geometries is the wormhole E S 63 The only other relevant quantity of the solution is its action. This can be computed Since two states with different The analog of the Chern-Simons number in the gravitational case would be the num- In [ Since we have defined our radial coordinate in terms of the areas of 3-spheres, what We thank [ 1 where in the firstinstanton equality is we only have half divided a by wormhole. two to take into account the fact that our potential for the axion indiscuss exactly the the effect same of way the as instanton the in instantons the in axion gaugeby potential theory. noting further We that will below. the equations of motion imply ber of connected componentsundergoes of a space. Big A Crunch;two half-wormhole after connected nucleates the a components. wormhole baby has Theanalogy universe closed, instanton which with we thus the end increases gauge upso-called this case, with “ number a one by Universe could one. with think Again that by the actual ground state is one of the vacuum of the form verses, which then evolve onsimilar their to own a towards afields, gauge in Big instanton, particular Crunch. which the The Chern-Simons changes interpretation number the is of asymptotic the very vaccum configuration of the gauge euclidean wormhole connecting two pointstween of them; spacetime, thus, with we might ana better arbitrarily wormhole long think (instanton), tube of and half be-(anti-instanton). a our solution solution describing with the negative “entrance” axionic of charge describing the exit which is conformally equivalent tois a actually flat a metric wormhole.hole on [ In fact,connecting: we have two just asymptotically rederived flat the regions Strominger-Giddings is worm- clearly unphysical. We might think of an axionic charge. is happening is thatstart as growing we again. go Inbecomes to fact, by making the change of coordinates wormhole radius JHEP08(2015)032 ]. 22 (3.14) (3.15) (3.16) (3.17) ) i n φ . h i d n ∗ φ h d d ( ∧ ∧∗ ϕ 2 dϕ + − ) i dϕ n φ , h has an imaginary part. This ) i∧∗ d n P nφ ) φ h φ , since it is at this scale when the ∧ ∗ ). In the dilute-gas approximation, d , 4 i P 0 ϕ cos( n , to + ( V ) M φ E a Giddings-Strominger wormhole; we d i h ϕ S dϕ n ∼ d ). For convenience from now on we will − φ ∗ ) e h = 2 ∧∗ d d half P i nφ |P| ∼ 3.17 g n ∗ dϕ φ d − h + ( cos( d i √ – 10 – ∧ n x E ) does not have an interpretation as a potential. 4 φ S ] and of Coleman and DeLuccia’s bounce solutions. ∧ ∗ h d − nϕ d e 70 2 3.16 dϕ Z is the fluctuation. The axion kinetic term thus yields ’s satisfying |P| i∧∗ = ϕ φ n = 2 ∼ − . φ i ]. We will take h n inst d φ S 69 h , d , where ) = is purely imaginary. However, as discussed above, the instanton 68 ϕ ϕ ∧ ∗ P + + i i dϕ n n φ φ + h ( d dϕ ≈ h ], it is argued that the Giddings-Strominger wormhole has exactly one negative ∧∗ φ ) 72 is a prefactor whose computation whose precise computation has only been , ϕ i ∧ ∗ is the required height of the inflaton potential in our model. Similarly, the bounce as n 0 + to be real and discuss the physics in terms of axion potentials. The reader should 71 n φ P i V h n P d Although the physical effects of an instanton and a bounce are very different, the dis- In [ However, it is also essential to know whether or not We now have to argue that this gravitational instanton can indeed generate the right φ h ( d where will initiate a decay towe the restrict true ourselves vacuum, to spoiling thetake inflation same and the field Universe range withkeep ( it, in unless mind the possibility that we are actually talking about bounces. tinction is not very important ifaxions we are as concerned candidates only with for the inflation.down possibility of the The transplanckian effective instanton field would range generate to a huge potential, cutting under consideration would bewould rotate something back like to anthe Euclidean negative mode solution survives after this cutting; theprefactor, however, wormhole since we throat. we do have It not not computed is know the not wether instanton clear this that solution is truly an instanton or a bounce. negative eigenmode. InThe this instanton case, is ( notthat mediating Minkowski space transitions is between an different unstablespirit configuration vacua, which of rather can Witten’s decay it bubble to is of something else, nothing telling in [ us the eigenmode, and thus where achieved in a few caseseffective description [ breaks down. This is in accordance withwill previous be literature, the see [ case if the spectrum of fluctuations around the instanton background has a summing over all thesethe instantons path and integral of fluctuations the around form them (switching gives back from a contribution to where we dropped the total derivative term.term, with We the thus instanton get supported the on usual the axion-instanton form coupling ( potential for the axion. Letcharge us expand the axion field in the background of an instanton of The last two terms may be rearranged as JHEP08(2015)032 , P M (3.18) . This ∼ , where . Thus, P 1 f f f M − 0 5 ) to be very V  ≈ n 3.16 P P f M . πM 1 16 3 − 0 √ ], and so they are in a ∼ < V 5. n 73 f / P P M M ), this means n √ ∼ 1 4 √ ) 3 3.13 π (3 < 1 − Λ since the wormhole throat is of Planckian size, 1, or in other words – 11 – ) which means that in terms of the canonically n ∼ will not be suppressed. Although strictly speaking nφ ⇒ to be available for inflation, we need ( E n S f 1. By looking at ( 1 is ruled out. In any case, this kind of effects do rule out − . 0  P 6 E M S < V 4 10 / 1 , the effective field range is just ∼ f fφ < a 1 = − ζ Λ ] was shown for several examples that in order to have a transplanckian decay 7 The above calculations assumed the validity of Einsteinian gravity up to the Planck Axions arising from string theory compactifications typically couple to extended instan- This bound is quite similar to those obtained from the WGC, where it was argued that The effective field range for the action is determined by the first instanton contribution is the scale of the inflationary potential. Thus, we have the bounds 0 4 Gravitational instantons for multipleThese bounds axions on transplanckian decay constants comingarguments from purely agree quantum gravitational with theaxion. difficulties In found [ inconstant string for theory an to axion, provide one a is transplanckian always forced to go beyond the safe and controlled regime V instanton solution the wormhole throatmoduli should become be light larger or thanabove we some computations start scale neglect to Λ the see at contributioninstantons stringy which of extra to physics the in axion the potential some stress-energy comingcomes way; of from at from the the the the solution. gauge same vicinity As time of most the the of throat, the we contribution must to demand the that action it is larger than further on this point in section scale. This isbreaks not down generically at true the inthe compactification string instanton scale. compactifications, calculation for To is which be bounded this more by description specific, both the sides: range of in validity order of to trust the gravitational allow the “decay” of the former. However, under some circumstances thetonic two objects may coincide. in theregards theory, for D-branes instance euclidean as D-branes.sense extended, The the extremal supergravity stringy perspective black versionD-branes hole of are the solutions often instantons [ claimed discussed to above. be On the the instantons required other by hand the euclidean WGC. We will ellaborate normalized axion there should be instantonswhich in makes the the theory field which rangeinstantons creates discussed subplanckian, a here for are potential all not with intends those higher demanded and by harmonics purposes. the WGC; The the gravitational latter are postulated to it is reasonable to say parametrically flat axion potentials, within the regime of lowto energy become effective relevant. field theory. Thisgenerates is a around potential of the form cos( If we want the full range ofsuppressed. the axion In other words, effects of gravitational instantons constrainthe the gravitational effective instantons axion with decay low constant.we cannot If trust our computation for low 3.2 Consequences for a transplanckian axion JHEP08(2015)032 – 20 (4.2) (4.3) and it turns 2 i /f 4 in which the kinetic = Λ . 2 i )) (4.1) m i  i . The shift symmetries will φ i /f φ i ˆ ζ cos( ~ φ ji R − kinetic basis R ] for recent attempts). While the  = (1 33 or 4 i ] the same question was addresed in ~ ζ – cos Λ 74 28 . The effective lagrangian in the kinetic − , i N =1 i 1 ζ X i R  f
− 4 i 2 showing a loophole in the conjecture, similar i physical ). In [  Λ = f 5 ~ P φ – 12 – i ˆ ν ζ N =1 M i X ∂ ( .  diag − O G T 2 2.1  . The periodicities of the axions define a lattice in the  R i ~ φ π ˆ ζ µ = µ ∂ ∂  G + 2  i 2 1 being the number of axions, which survives to the gravitational . The effective lagrangian of the system takes the form 2 1 φ π = N = → L . This can lead to some difficulties when trying to embed the model i L i and the rotation angles. The maximum physical displacement in field φ with i f . We can go now to the , this theory gives rise to N fields with mases N i √ lattice basis ]. We will clarify this issue in section = Λ for all i 57 , space (in the absenceIf of Λ monodromy) is givenout by that half the of lightest the fieldin is diagonal order also of to the this obtain onetune new succesful the with lattice. inflation parameters the Λ along largest the available field largest range. direction In in general, field space we need to basis reads If we diagonalize now thekinetic potential eigenvalues we get a new lattice which depends non-trivially on the as the terms are canonically normalized by making a rotation to diagonalize the kinetic term followed by a field redefinition given by field space of side length 2 where in general the kinetic term is not neccessarily canonical. We will refer to this basis 4.1 Axion-driven multipleLet field us inflation consider an effectivebe field broken theory by containing non-perturbative Nunder axions effects discrete inducing shifts a scalar potential which is still invariant the presence of gravitational instantons,proportional we can to still have a certaineffects. numerical This enhancement is in apparent22 contradiction to the generalizedto Weak the Gravity Conjecture example [ discussed in section of any of thesefailure models to yet get in asame string single question theory transplankian when (see axion there [ the in is computation string more theory than of one isgravitational the constrainsts axion quite that previous is our widely instanton section not accepted, imposesfor clear to over the inflation. the at the effective all. field We case range will available Here of we see multiple generalize that axions while and parametrically flat study directions the are inconsistent with potential, become relevant atthe scales context of F-theory,decay finding similar constants. harmonics that Peoplemultiple would have axions forbid in tried parametrically which to large youan evade might enhanced hope these effective to field difficulties engineer range. by a However direction considering there in models is the not of moduli a space completely with succesful embedding of the effective theory. New contributions, usually in the form of higher harmonics to the JHEP08(2015)032 we (4.5) (4.6) (4.7) (4.4) N f . For later 1 ˆ ζ ∼ · · · ∼ 1 f (right), in which the ! ) ) 1 α α sin sin 2 1 . This relaxes the requirements φ φ i . + − 2 N ! Nf α α f α √ α . + π cos cos 1 sin cos f 1 2 ··· 2 φ φ α ( ( √ α + 1 2 pointing along a diagonal of the lattice, such f f π 2 1 the kinetic metric is not assumed to be diag- sin 2 i f cos

f – 13 – − q = π

~ φ ∆Φ = = R (left)). The maximum displacement is then given by a R ! 1 ) points in the direction of the diagonal of the new lattice, ∆Φ = 2 can always be tuned such that inflation would occur along 0 2 f i , so the original lattice is mapped to a new rectangular lattice 1 0 i f > f . ] φ i 1 ]

f N f kinetic alignment 14 8 = √ = (see figure ˆ ~ ζ i i ˆ ζ πf ] 9 (assuming with the largest eigenvalue ) as models the kinetic metric is assumed to be diagonal already in the lattice i 1 are the eigenvalues of the kinetic matrix and we have parametrized the rotation ζ ζ 4.2 2 4, obtaining , f π/ 1 f N-flation [ Kinetic alignment [ Lattice alignment [ = In models based on The different ways to achieve large field inflation with multiple axions can be divided • • • α N-flation eigenvector ie. The situation ofinflationary perfect trajectory kinetic (red arrow) alignment coincides is with the depicted direction determined in by figure matrix of ( It can be easily checked that the maximum physical displacement can be achieved if the two axions. The kinetic basis is given by where onal when written ineigenvector the lattice basis.that In the fact, maximum displacement oneto is can get then used large given field this inflation by kinetic freedom with matrix to N large, axions, choose since while the it the would others be could enough remain to small. have one For entry concretenes of let the us consider In the favorable case inget which an all enhancement the of decay constant are of similar order In basis. Then boththe lattice field and redefinition kinetic baseswith are side proportional lengths to 2 eachcollective mode other travelling and along related the by diagonal, assume that the parameters Λ the direction of interest.shortened Our upon focus including will the be effect in of proving if gravitational instantons. that flatin direction three survives main or proposals: it is in string theory. However, since this issue is very model dependent, we will ignore it and JHEP08(2015)032 ,  2 ∝ in g ' ξ α (4.9) (4.8) 2 = (4.10) (4.11) g 1 g − 1  g 2 2 θ g !
1.0 + 2 2 1 g 2 1 θ f
g 2 2  g + ! + 2 0.5 cos α α f 2 2 f − 2 sin 1 g 2 and the rotation angle 2 cos 1 α  g 2 g 2 2 4 2 f ! 0.0 − Λ , f cos 1 2 1 +
claims to achieve a parametrically
− f 2 ). A slight misaligment 2 2 /g /g α g 2 f 1 1 1 2
4.9 g 2 2  − g = 0 for the original models of N-flation. 0.5 1 2 sin /f /f 2 + 1 θ g 0.5 0.0 0.5 1.0 1 1 2 1 f + α 2 . Interestingly it turns out that one of the f 2 +

2 f g α f α 1 1 2 = θ f ] 2 g 2 1 8 1 g sin  – 14 – g M and sin α − 2 2 1 lattice alignment cos f

is related to the lattice basis as follows cos ~ θ , 1.0 − + i f, g 2
) θ . Perfect lattice alignment occurs if in addition 1 2 2 α M 2 2  f g f 2 4 1 = 1 − − = Λ ~ cos φ 2 1 1 1 0.5 2 g 1 f − f ( f 2 ) =

= 2 θ . This effective decay constant can be done a priori parametrically f 1 = µ 0. The problem is that by doing this we are also making the kinetic − / ∂ we can get the kinetic eigenvalues 2 0.0 ( G 1 M g → 2 1 G 1  + − +
2 2 T f ) 4 for perfect kinetic alignment and 1 0.5 θ p M . Different choices of basis for N-flation (left) and kinetic alignment (right) models. π/ 0.5 1.0 0.5 0.0 2 µ g ∂ = ( = which is orthogonal to the one appearing in ( 1 2 = α 2 G f θ = eff The Kim-Nilles-Peloso proposal of f − Figure 1 L 1 1 θ g By diagonalizing terms of the initial decay constants introduced above. The KNP basis In the lattice basis the kinetic term is not canonical and is given by gives rise to aby nearly flat directionlarge in by field sending spacemetric with diverge. effective field Let range parametrized us reformulate the model in terms of the lattice and kinetic basis lagrangian proposed in the original paper [ Consider for simplicity leading to a flat direction in the potential corresponding to the linear combination where flat direction in thelarge potential direction, starting although from hiddenbeginning subplanckian in and decay some constants. can choice be of However basis made this for manifest the by lattice, going is to there the from kinetic the basis. Let us consider the use the metric in the lattice basis can be written in general as JHEP08(2015)032 4 and (4.12) π/ → 2.0 α 1.5 is transplanckian. 1 f 1.0 0.5 µν g for the case of N axions with  0.0 3 ~ φ µ . Hence an effective transplanckian 1 so the rotation angle ∂  0.5 G ~ ∼ φ → µ ) 2 . Therefore the maximal displacement is 0.5 0.0 0.5 2.0 1.5 1.0 ∂ f α ξ  1 2 ≡ 1 − ) (to go to the kinetic basis) it can be checked ˆ ζ . Hence we neglect it from now on as was done – 15 – ~ φ ν 4.2 . To recap, a priori we could get a parametrically ∂ 3.2 6 } G 2 ~ φ µ , φ ∂ 1 4 0 which is equivalent to increase parametrically the value φ = { → while the other as in the limit of lattice alignment is pictorially translated into µν  2 , so the large scale is hidden in the model from the beginning T / 1 M 1 f 0 we also have that tan( ∼ 0 → 1 f  is consistent in a quantum theory of gravity. 2 1) is possible only if one of the kinetic eigenvalues is transplanckian. ) with the crucial difference that now the eigenvalue 1 f ), and the transformation required to go to the KNP basis. In the right 4.7 } 4  < 2 6 4 2 0 2 4 we illustrate the relation between the different choices of basis which play a , φ 1 . Different choices of basis in KNP models. The red line always correspond to the 2 φ { In the following we turn to study the effects of the gravitational instantons over these fields, but has no relevantas effect was in discussed the at resolutionin the of the end the single of Einstein’s field section gravitational case. equations The stress-energy tensor of the axion system is now multiple field inflationary models. 4.2 Gravitational effects Let us generalize thegeneric non-canonical computation kinetic terms. performed The in gauge potential section sets the periodicities of the axionic large flat direction by doing of the kinetic eigenvalue and is not atransplanckian consequence of a clever change of basis. The remaining question is then if a In figure role in the discussion.(basis In thefigure left we figure take the we KNPdegeneracy show basis of the as the lattice canonical matrix and generatedthe show by almost its the relation degenerate with vectors potential the other basis. The the system approaches perfect kineticdoing alignment. the field In redefinion fact,that by explained diagonalizing in the the ( kinetic metricdirection and eigenvector that with KNP the usesstill for largest given inflation, by eigenvalue ie. ( corresponds indeed to the flat inflationary trajectory. eigenvalues scale as field range ( Besides in the limit Figure 2 JHEP08(2015)032 . 4 ~n 2 f π/ = (4.17) (4.13) (4.15) (4.14) (4.16) = . Dirac G π α G correspond to the ~n 1 2 3 − 16) in order to compare π , ~n G upon setting 3 1 π/ T 3 ψ, θ, φ. 3 ~n ~n = 0 (left) and √ = ≡ α , i ii , a for ) ~n ~ φg r 1 6 r ) (4.18) 3 ( i r ∂ − f φ G G i ~n ~ ~n φ T n 1 2 1 r 1 ~n 4 r − ∂ − π ) G p ~n G cos( 2 4 r 1 T P ( (the action ommiting π 1 E = − ~n f 4 S M E G 2 ~ φ − ˜ S – 16 – π d = e − 0 3 ∼ 3 and it is rotated with respect to the fundamental 16 ~ φ 1 and the gravitational instantons induce a non- S P √ ~ r φ = Z G ∂ ≤ = ii = G ~ . The axis lengths of the ellipse are determined by twice φ V E E 2 r S S π ~ φ, T 3 16 r . The red line represents the would be inflationary trajectory √ , ∂ ∂ α 1, the corresponding gravitational induced potential will spoil 4 G a r = ~ φ are adimensional and define a lattice of periodicities 2 ∼ 1 r − i ∂ E E φ ˜ 1 S S = is characteristic of the gravitational instanton under consideration, so it is the same rr ) = π T 3 r 16 ( √ f , the effect of gravitational instantons on the realization of large field inflation to be small. G E some vector of integer entries. This implies that at large distances the axions must S ~n The situation of two axions is depicted in figure The prefactor 2 the eigenvalues of thelattice kinetic by matrix an angle given by both for one ordirectly the multiple constraints axions. for multiple Hence axions we with those plot for a single axion. (right), the latter corresponding toThe the perfect figure kinetic illustrates alignment the situationinstantons dual described above. lattice, with so integer for chargesthe instance (1,0) the region and vectors for (0,1) which negligible respectively. potential for The the blue axions ellipse represents for which the action inflation. Since we have encodedlattice all in the information aboutreduces the to periodicties see of if the theimply physical ‘tricks’ used to get large displacements in multiple field inflation also Notice that the fundamentalinstantons periodicities are of the the same potential as generated those by the of gravitational the original lattice. Thus if there is some vector which reduces to theThe solution action of of a the single gravitational field instanton obtained becomes in section and induces a potential for the system of axions given by By solving the Einstein’s equations we get with have have an asymptotic profile of the form Recall that the fields quantization then implies with nonvanishing components JHEP08(2015)032 ~n = 1 ˆ ζ (4.20) (4.19) 1.5 is not the E S 1.0 0.5 which enhances the 2 larger. This result
√ 2 N ) 0.0 2 √ n in the lattice basis. Along n 1 2 so this eigenvector is 2 f + √ 2 0.5 n 1 P n = ( > f M 1 2 2 1.0 π n ) is a factor f 1 3 f 16 ≡ + √ 4.7 2 n ) 1.5 = 2 n 0.5 0.0 0.5 1.0 1.5 1.5 1.0 ~n 1 − − ), leading to 1 G n T ( – 17 – 4.8 2 ~n 1 1.5 f all the eigenvalues remain subplanckian and the p P 2 2 2 bigger comparing to the one obtained in the single f 1.0 = 0 and the decay constant entering in M √ 2 1 1 π f α 3 2 for the cases under consideration. But let us take a closer 16 √ 0.5 3 = = ~n 1 E − 0.0 S is close enough to the inflationary trajectory. This is not the case G T ~n kinetic alignment ~n 0.5 or 1.0 ), while the maximum displacement ( 2, corresponding to a direction with √ / 3.13 ) 1.5 2 . Instanton action for a model of N-flation with two axions (left) and a model of kinetic φ 0.5 1.0 1.5 1.0 0.5 0.0 1.5 N-flation 4. The kinetic metric is given by ( + π/ 1 The same results apply for N-flation, where the only difference is that both kinetic and In φ ( = 1 effective field range, also appearsinduced in gravitational the potential. instanton action suppressing the amplitude oflattice the basis are proportionallargest so one but the Pythagorean sum of all of them. Notice that this actionfield is case a ( factor can be generalize to N axions obtaining that the same factor The maximum displacement takeslargest place eigenvalue. along For thef concreteness, direction let of us the consider the eigenvector inflationary with trajectory, the the action for the gravitational instanton becomes enhancement is generated by travellinglet along us a consider diagonal two of axionsα the in field the space. most favorable For situation simplicity of perfect kinetic alignment, ie. inflation if the vector if the metric eigenvalues remain subplanckianas and can therefore be no deduced point fromlook lies figure within to the understand ellipse, how this works in general. Figure 3 alignment (right). The red line corresponds to the direction ofin the each inflationary trajectory. case.lies If inside the the physical ellipse, lattice then is the such gravitational instantons that will some induce point a corresponding potential which to spoil a vector JHEP08(2015)032 1. It can . 1 (in order to E S  < 6 . In the limit of lattice alignment 4 4 2 – 18 – will have a nearly zero action and will induce a 0 the misalignment parameter, so if 2  n , no matter how big its components, will have nearly − 2 ~n is however different. We have seen that a large effective = being 1 2 n  (all of them if there is no misalignment). This implies that 6 4 2 0 2 4 ∼ 2 n E − S = 1 . Hence the inverse of the metric has a nearly zero eigenvalue and the n lattice alignment G . Gravitational effects for a model of lattice alignment. The blue ellipse represents the Our results indicate that having multiple axions might help to relax the gravitational The case of is partially lost. Oneeffects, should which really is keep clearly trace notdoes of an not all easy provide these task. any non-negligible improvement In with non-perturbative this respect sense to a a model modelconstraints based of found in a lattice for single alignment axion. the effective field range of a single axion, as long as there are not indeed generate a potentialthe along points the lying inflationary within trajectoryreduces the and dramatically ellipse spoil the can inflation.which field be all Each range. seen of these as contributions Onethe a are advantage could higher of suppressed hope harmonic invoking enough an contribution to axion to which engineer in get order a succesful to concrete inflation, have controll model but over in then higher order corrections the ellipse is verythe thin points in one with axisthe and instantons very with long charges inhuge the potential other for one, thethe so direction axions. it of captures the In eigenvector most with addition, of the the largest eigenvalue, inflationary implying trajectory that these is instantons identified with have a transplanckian effective fieldbe range) then checked the that action the isso kernel automatically there direction is is no indeedvanishing way action the as out. nearly long flat as AThe it direction vector result is used becomes close for enough even to more inflation, one the clear kernel by of direction looking the and at will figure kinetic spoil eigenvalues inflation. tends to zero while the other diverges. This implies that decay constant in thekinetic proposal matrix of KNPeffects implies of a the gravitational parametricallythe instantons large action are will eigenvalue huge. scale of as In the more detail, along the kernel direction Figure 4 region for which theThe instanton red action line is corresponds to small the and direction its of effect the over inflationary the trajectory. axion can not be neglected. JHEP08(2015)032 (4.21) we are still , since one 3 mf , but before that, = 5 0 1 reads f axions running in the . Since the dual lattice L 1. The answer is that all N m > ⊂ m 0 P S f m > ), the potential is also periodic = 1 actually would correspond we are missing some instantons n nM L φ/f ∼ 1. Thus a natural question would be E , but we are missing the information cos( m ∝ ) that by redefining ,S m > ≥ V 0  n 4.21 and 0 – 19 – f , cancelling the Pythagorean enhancement from , by considering to be a constant. However, the Planck mass will nφ m ∗ mf N  P S = √ , M ]. It has been argued that suitable cancellations can 0 S cos . Hence although they might modulate the inflationary f can be any positive integer. This is indeed what happens E 28 S πf , m 2 9 − . However these latter instantons induce a potential periodic e , which do not respect the periodicity of the potential but ∗ ) with ∼ 0 L φ > . It is clear from ( < m where V 0 will grow as m n mφ/f P = πmf 0 M n cos( = 2 but not in ∝ φ ∗ V S , but they are also compatible with any sublattice L is contained in the dual of ∗ S Finally, here we have assumed First, we have assumed that the fundamental periodicities of the axions are the ones , so for N fixed one can never get a parametrically large field range. Even so, one ⊂ N ∗ also receive quantum correctionsinternal from loops, diagrams so involvingtravelling the along the diagonaloccur [ such that evenget if a a sufficient parametrically large large field field range range to is provide not inflation. possible, Studies one might regarding still the diameters of easily generalized to morea than lattice one axion. TheL periodicities set by thepresent potential in generate instead have a bigger period. capturing the effects ofabout all instantons with instantons with only under bigger shifts ∆ potential in arelevant non for negligible our way discussion if since they the do instanton not action reduce is the effective small field enough, range. they This are can not be The instanton to which we wereto assigning that the of unit charge charge potential is what changes in the studyof of our results the still gravitational apply instantonsinstantons in if the which same could way, but modify weThe the are generalization profile missing of in of the addition the some potential potential gravitational induced but by not the reduce instantons the if field range. let us comment about two assumptions we have madedetermined in by our the scalar computation. potential.under However, shifts if ∆ in most of the attempts to embed alignment inflation in string theory, where the induced could think thatcould this always result integrate is outto in all the contradiction massive inflationary degrees with oftransplanckian trajectory, the field freedom, ending results range except up but of theTherefore this again section one we time with must corresponding without be a a missingdegrees gravitational single some of instanton inflationary information freedom. to when model constrain The integrating solution it. with out to naively this a the ‘paradox’ massive is clarified in section always go toencoded the in lattice the kinetic basis matrix,is such and the check that that case all the onea eigenvalues the remain can diagonal√ subplanckian. information still of If about have the this an field the enhancement space. length of scales However the this is decay enhancement constant is by given travelling at along most by a factor of intrinsic transplanckian scales hidden by any change of basis. To check that, one can JHEP08(2015)032 3 1). , and (5.2) (5.3) (5.1) (1 P 2 1 discrete that in √ = Λ and 4 i N ] discussed Z 14 f < M . As discussed ], showing the . P 21 M  2 2 ζ  √ i  . )) i cos 2, which is where inflation φ  0) to the diagonal √ , 2 1 ζ cos( π/ √ any integer. As discussed there,  − ∼ , n 1 (1 cos . In the basis in which the lattice is R ζ 4 i ~ φ Λ − ! . The lagrangian will be 1 2 0 N =1  g G , with i X 4  2 0 f − f 1 2 ˆ ζ  √

~ – 20 – φ = 2Λ n T ν .  , we see that this field theory describes 2 fields with ∂ f   R 2 2 )  i = G ζ gζ √ , we may take the latter as an inflaton. The potential 2 ji  G . f ] for a more optimistic view). We just want to remark ~ φ √ f , we know that such a theory has gravitational instantons µ 2 πR 17 ∂  = 3 √  , then the potential becomes 2 g ˆ ζ ~ φ 1 2 , cos(2 R 1 = . If 2 2 =1 = fζ j L X 2 ~ ζ /f − √ 4 2 =   1 ˆ 4 ζ and Λ Λ and higher, as long as we are away of 2 2 /g is the rotation matrix which takes the vector (1 4 /g 4 R . We will consider a system with two axions However, the gravitational instantons contributing to the action can also be studied From the results of section So far, we have only described again the kinetic alignment mechanism of [ 4 these gravitational instantons reducethereby the spoiling effective inflation. field range, constraining before integrating out thethis heavy case axion, inflation and with we a know transplanckian from field the range results is of indeed section possible. What is going on? in the previous section.mass However, Λ we mayends. now integrate We will out be allhave left degrees an with of enhanced a periodicity freedom theory with of a single transplanckiancontributing axion potentials which of would the seem to form cos By changing to masses Λ along its direction has a periodicity where Indeed, if one defines We will couple this systemabove, minimally kinetic to alignment gravity, provides and successful also inflation. assume In Λ other words, we take Λ and diagonal, we will have a nontrivial kinetic term present in the IR,gauge might symmetries actually in not the exist UV, due as to we the proceed selection to5 rules explain of in some the following. Two is moreWe than now one want to point out what seems to be a paradox between the results of sections difficulties arising to haveoverall an volume enhancement small of (see athat also moduli the [ space gravitational diameter effects whilegetting which keeping transplanckian are the field usually ranges arguedcase in to of string be multiple theory behind axions. might the actually difficulties Some be from of suppressed the for the gravitational instantons which are assumed to be axion moduli spaces in Calabi-Yau manifolds have been carried out in [ JHEP08(2015)032 (5.5) (5.6) (5.4) 0 limit there is → g , in accordance with . /g ) n (2 f 1): P 2 , . That is, gravitational instan- M √  (1 and hence will be tremendously f n π . 1 2 ˆ ζ 3 16 √ = 1) /g √ , ]. Asymptotically, the stress energy n ~n 1 2 , = 81  1 – P − ~ φ. 78 , d M f 2 √ n – 21 – string diag(1 ) tells us that any instanton not in the direction Z π 2 3 C r 16 √ 4.17 = ν µ as we circle around the string. Solutions to this system ) = T . Notice however that configurations with actions of order n g ( π ~m is subplanckian) but will be retained nevertheless; otherwise,  f f is the angle around the string. The instanton couples electrically instanton E θ S ]. We will now study the precise way in which the IR theory forbids 77 – ] that of the solutions of Minkowskian Einstein’s equations corresponding 1 is the regime when the instanton throat, etc. are larger than the Planck , where 75 , 80  θ ~m 56 /f = P ~ φ will be large (since M 1) will pick up a contribution proportional to 1 , So, we are led to study solutions of the coupled Einstein-axion system with cylindrical Clearly, the effective field theory of one axion is missing some constraint coming from Let us compute the action of an instanton with A look at the instanton action ( /f P It turns out [ to the above mondromyvery unphysical only configuration one in which hasis there not locally is static. flat an event asymptotics, Physically, horizon what and outside happens which it the is corresponds metric that to the a constant stress-energy implied by an symmetry and monodromy 2 have been considered beforetensor in in cylindrical the coordinates literature is [ of the form profile to the axion field, whereas the string has a magnetic coupling of the kind axionic strings. 5.1 The string Gravity not only enablesallows us to us build to an build instanton the for dual any object: axion we a may have, string but around it which also the axion has the asymptotic axion established by the gauge potential. the UV. A selection rulesymmetry forbidding [ odd-order harmonics is highlythe reminiscent presence of of a these discrete oddanother charge instantons: set of it instantons can of do low so action because which in contribute the the path integral, namely, charged The instanton generates a potential oftons the only form generate cos even harmonics in terms of the periodicity of the effective transplanckian M we would not beAlso, able to analyze thelength potential and generated so our by effective the field gravitational theory instantons. computations can be trusted. of (1 suppresed. When computing instantonrion corrections to is the to path pick integral,in all a this the consistent section crite- configurations we withour will initial action drop statement smaller any that than instantons some whose preestablished action value; goes as 1 JHEP08(2015)032 . ~m . G (5.9) (5.8) (5.7) 0 T r ~m . =  2 0 µ r R dθ 2 2  ). which encloses the µσr 5 . Also, 1 ), but as usual when + strings with opposite + log ~mθ S σ 2 1 σ − = dr  two 1 . ~ φ µ − exp( is any √ σr µ 0 σ h after which the approximation , or r 1  √ √ r √ ~m 2 0 2 S r R r = π 8 = =  h r ]. The period of the imaginary time ~ φ = θθ θ log 82 g µ ∂  1 σ √ h r R ones. Physically, these would describe the = 2 1 + , where the can be related to the total length of the Dirac    ~ φ 0 r µ – 22 – π ~m log + ∂ 2  µ = 2 G √ dz σ will be of order of the radius of the closed string. h  Euclidean ~ φ r + ~ φ d √ 2 R µ 2 1 π ∂ S 8 dt  R g =  √ times twice the horizon radius, as usual. 0 1 2 r r σr π µdr  √ √ h log ) has a coordinate singularity at R r ], whith “wrong” kinetic term sign) 2 1 , i.e. 2 σ Z 5.7 h h 80 , r πr 2 1 + 78 π  8 ]. This was also crucial to ensure the existence of our instantons above. = ≈ 62 2 ds πR We have The metric ( On the other hand, the metric we have built for the euclidean string is not asymp- This metric will describe the near-field limit of a closed string. The Dirac domain wall We thus propose to consider a metric tensor for the string of the form (analytic con- Although it is not possible to have an asymptotically flat Minkowskian metric describ- string ) is no longer good; typically, E 2 S 5.7 And thus the action per unit length is one expects that anabove metric asymptotically is flat a well solutionfor approximation exists realistic near the solutions for core one a of should( the closed impose string. an string, However, upper and this cuttof means that that the continuating to the Euclidean theas space periodic is in much order nicer.coordinate to We is avoid must 4 a take conical the singularity time [ coordinate totically flat. This is not a problem, as it describes the fields near the core of the string; string in the singular gaugecontribute description a of trivial phase. monopoles. The Instantonsstring parameter crossing in this some wall contrived should way; we will leave itwill and therefore study be the a results disc, as whose a boundary function is of the actual string (see figure The axion string muststring. satisfy This immediately implies by symmetryNotice that the branch cut for the axion; this creates a “Dirac domain wall”, similar to the Dirac ones is the swapclidean of [ the sign of the kinetictinuation term of of [ an axion when continued to the eu- monodromies, so that asymptoticallyis the indeed stress-energy the vanishes. case)there that This might although suggests there be (and are asymptotically this nucleation no flat of asymptotically a flat string- Minkowskiancontributing anti solutions, to string vacuum pair, to vacuum or amplitudes.of one The well-behaved could crucial asymptotically have element closed flat ensuring the strings Euclidean existence as solutions loop when diagrams there are no Minkowskian as we go away fromcompensate the and get string, a which flatthe in solution core turn one of curves must the the put string, space large and at negative these sources asymptotic cause of regions. theing stress-energy losss To one at of such staticity string, of the it solution. may be possible to have one describing asymptotic profile with the above monodomy acts as a fluid with larger and larger stresses JHEP08(2015)032 σ < (5.10) (5.11) (5.12) but not /g P M ∼ , this imposes .   R have an action which 2  ) 2 0 2 r n R m 2 +  − 1 1 n m ( 2 + log . g  ~m 2 0 p + ) with 2 l p r G will be of order 2 2 1  ) T n M 0 2 r ~m − , the action of the string can be made m log σ 1 ) their action goes as s   + n 3 1  π 5.11 0 p = ( l m r 16 – 23 – (  2 ~n 1 & f should be greater than the Planck length, otherwise log h r r P r string E µ M S 3 √ 2 π π 16 , which corresponds to vanishing small mass per unit length. 8 & > . f → ∞ πR string σ E string E 2 S S ), so we end up with a bound (restoring the Planck mass) p l and are therefore suppressed. However, their dual strings (say, the string & 0)) are not suppressed; from ( /g , R P M ) so we arrive at . Pictorial representation of the euclidean string (thick black line) and its associated Dirac = (1 p /l ~m 0 With this criterion, instantons with The point here is that, as a function of r scales as with 5.2 Putting everythingNow together that we have the actioncircumstances of the do Euclidean they strings, we contributeabove, have to signficantly that ask to we ourselves under will thethose which neglect path which contributions depend integral. to on the Recall, path as integral discussed of action Also, the radius oflength the (i.e, string must be several orders of magnitude larger than the Planck The good thing about the logarithms is that they contribute things of order 1, 2 at most. arbitrarily small for So in principle thehowever action is of imposed the by the above factthe instanton that can above be solution made shouldlog( arbitrarilly not small. be A trusted. cutoff Since Figure 5 domain wall (blue shaded circle).extra When phase as we it move an crosses instanton the around domain wall. the dashed This circle, phase it must be picks an trivial for the theory to be consistent. JHEP08(2015)032 in P live, could (5.13) ~ φ 0) that n discrete , f/M 2 2, and that Z 1). , = (1 √ (1 ~m . Actually, the 2 n π πf/ = ~n 1). This is not a vector , (1 1 2 . If we move the instanton so = ~n ~n ) degenerates to a single straight gauge symmetry would have been 0; otherwise we have gravitational g , it is not consistent to integrate out 2 − ~m. 1 → · ˆ Z ζ f, with linking number 1 with the string is ( g 2 1 1 π ~n √ S = 2 ~ φ – 24 – d ) and · 0 limit the ). As a result, an extra lattice point appears on the ~n f, g 1 6 → ( S and an instanton with 2 1 Z g √ ~m . Consistency of the path integral thus requires that only πn symmetry, even if they are very suppressed. 2 Z 0) (see figure 0, the two antidiagonal corners of the axion lattice are at zero f, ( → 2 coupled to gravity; it also contains euclidean strings which change the 1 √ g symmetry: two is more than one. f , in the degenerate limit the extra lattice point cuts that periodicity by 2 2 f 2 √ Z √ π If we had actually taken the In the limit From the UV point of view, what is happening is clear. The instanton discussed in These strings constrain the allowed gravitational instantons, forbidding those of odd and retain the gravitational instantons which generate the even-harmonics potential for . If we want the latter, we must allow for any other configuration which goes as 2 1 ˆ ˆ would be the endwe of can the have story. aallowing for transplanckian By tranplanckian axion allowing inflationary a with regimes.axion small, highly Speaking lattice explicit loosely, suppressed in one breaking gravitational a mayaxion of degenerate contributions, precise is the the subplanckian, way but symmetry, so the that gauge the effects gravitational know it effects is “think” not. that the effective effective theory of the remainingan axion, additional but rather requires the inclusion of something else, exact and even thetheory would gauge contain a potential single used effective axion for of inflation subplanckian periodicity would 2 have had to obey it. The which is spanned byline the generated vectors by diagonal of the unit cell.of So length although 2 it indeedhalf. seems As that the a axion result,a lattice only has theory even-order a harmonics with diagonal are two allowed axions in in the which potential. one In becomes other words, very massive cannot be reduced to the symmetry operating in thein IR is the harder potential to isinstantons see. only breaking Notice strictly the that true the in absence the of odd limit harmonics distance from each other. The lattice in which the canonically normalized fields the above phase becomes just even-order harmonics contribute. the previous paragraph wouldof lift the to instanton an lattice, instanton and with hence the instanton is not allowed. However, the To write this amplitude inbe terms any of integer IR and quantities, still we must respect substitute the periodicity of the lattice. However, for charge. As discussed above, theration strings with generate a a string Diracas with domain to vector wall. cross Consider the aphase Dirac configu- the domain instanton gets wall, upon the moving phase along an should be (a multiple of) 2 ζ ζ the path integral, such asof the periodicity strings. Thequantum effective theory. theory in the IR is not a single axion Thus, when we compute the IR effective theory for JHEP08(2015)032
0 f, 2 -cycles in the in- √ p axions. In this case, N ] for such axions as inflaton 50 , 49 , -form fields over p  46 , ). 2  g , resulting in a one-dimensional lattice with √ 2 43 g √ − ,... , , OP 0 2 2 f f f, √ √ 0, the two elements of the lattice basis move along ( ] for such axions as inflaton candidates); they   – 25 – N 1 → 54 √ , P g -cycles. p 48 , 47 , 44 , 42 – 37 gauge symmetry, which in the UV will come from the degeneration of N Z O . Basis of the axion lattice with canonically normalized axions. The diameter of the axion -form potentials integrated over In string theory, axions are ubiquitous. For instance they can arise very generically The above discussion generalizes in a straightforward manner to p ternal space (see e.galso [ arise from D-braneture positions limits in of toroidal Calabi-Yau compactifications compactificationscandidates). (see or For [ in concreteness and large forRR complex their struc- genericity, we focus on axions coming from the inflationary scenario. This typically requiresin a order relatively to large breaking getspecial of rid attention supersymmetry, to of often-forgotten the features saxion arising in superpartners. the absence Therefore,from of in the supersymmetry. KK our compactification consideration of we higher-dimensional pay 6.1 Generalities A first observation is that, intheory order to compactifications produce viable must single-field dealaxion(s) inflation models, massive with at the string the a hierarchically challenge large scale, of so rendering that they all do fields not interfere except with the the axion In earlier sections we have consideredto gravitational axions, instantons in as theories part of of gravitysection, the coupled effects we present focus in a oncompactifications complete the realizing quantum gravitational the interplay theory. axion between In models our this described. results and expectations in string theory there will be a all the lattice basis vectors to the form 6 Aspects of string theory realizations lattice is given by thethe red blue dashed dashed line. linehalf to As the degenerate periodicity. to the segment Figure 6 JHEP08(2015)032 In 4 ]), for so that 44 , ]. How- 3 42 85 ], see also [ 41 ]. 26 – 23 component 2-cycles, each of which can be wrapped N ) gauge theory is manifest in the F-theory lift, in which – 26 – N ]. Instantons with additional fermion zero modes induce 84 ]; but these are precisely the axions which we have assumed have been 83 D7-branes pinches off into N measure) [ θ 2 ]; more microscopically, the susy breaking sources can lift the additional d D3-brane instantons wrapped on the D7-brane 4-cycle (and with the same ] and references therein). Therefore, a general lesson in non-supersymmetric 35 5 87 , 86 ] for discussion). For instance, superpotential terms (and therefore contributions to 34 In supersymmetric backgrounds, the kind of superspace interaction induced by eu- In addition to these field theory non-perturbative effects, there are non-perturbative The non-perturbative contributions to the potential for these axions (the gaugino con- In general, many of these axions can get masses from fluxes in general compactifica- Actually, for axions made massive by fluxes, the corresponding instantonsThe fractionalization are of absent instantons due in an to SU( Freed- Actually, is is possible to consider models in which some axion receives a hierarchically small contribu- 4 5 3 which our analysis does not apply. Witten consistency conditions [ integrated out in our effective theories. the elliptic fibration over by an instanton (described as an M5-brane on the 2-cycle of the elliptic fiber fibered over the base 4-cycle). fermion zero modes of(see the e.g. [ instanton, to allow them to contribute to thetion to scalar its potential potentialit from induces the axion fluxes. monodromy In (e.g. those the cases F-term however axion the monodromy structure inflation of models the [ flux potential is such that ten like D-terms), namelyever, higher-derivative in terms the presence or of multi-fermionto supersymmetry operators contributions breaking [ to effects, the theseby higher superpotential contracting F-terms (or can the in descend additionaling non-susy external operators language, legs [ to with the insertions scalar of potential), the supersymmetry break- clidean D-brane instanton effects(see depends [ on thethe structure scalar potential) of are exact generated(to only fermion by saturate zero instantons the with modes exactlynon-perturbative two fermion corrections zero to modes higher F-terms (which locally in moduli space can be writ- tually, the non-perturbative gaugino condensateof superpotential ‘fractional’ can beworldvolume described gauge in background). terms D3-branecounterpart instantons are which dubbed do exotic not or have stringy a in gauge the theory literature [ effects from euclidean D-brane instantons.are such In D-brane general, instanton for effects anythe which axion above induce in example, corrections string in theory ofion), the there the type 4d corresponding IIB effective instantons action. withmagnetized, are euclidean inducing axion D3-branes D(-1)-brane from wrapped instanton the charge, on coupling RR 4-cycles to (possibly 4-form the (and universal the axion). universal Ac- ax- wrapped D-branes in the model.with axions For coming instance, from many the modelsevant RR gaugino are 4-form based condensate on on non-perturbative 4-cycles, typeon effects and IIB 4-cycles arise the theory (and from 10d possibly type gaugecoupling with IIB sectors to worldvolume axion. on gauge the The D7-branes backgrounds 10d rel- inducing type D3-brane IIB charge axion). tions. In applications toone inflation, is one left assumes with that a this low happens energy at theory a with highdensate a in scale, reduced phenomenological set models like ofvariants) natural axions. inflation are or realized its diverse in aligned terms multi-axion of non-perturbative effects on gauge sectors arising from JHEP08(2015)032 ]. ], 88 ) is 90 (6.2) (6.1) is an , 6.1 , where 89 P s /M susy M periodicity, π !# i φ We will often omit the very i n 6 , and with a worldvolume gauge i , and define i X Σ i

,... . n s i cos , normalized to 2 4 i = 1 P C /M φ − i i , 1 Σ i ), suggests the gravitational instantons may " susy Z is the D-brane instanton action. M = inst – 27 – 4.18 − i inst D . φ − 0 S D n − e S ) and ( ∼ P 6.1 inst − D V ]. It is tantalizing to propose that our wormhole solutions correspond to a 92 denote the universal axion. In this setup, the instanton generating ( , are integer charges under the axions 0 91 φ i n is the susy breaking scale. Since our main interest lies in applicactions to large-field An interesting observation is that BPS euclidean D-brane instantons (in general space- The close resemblance of ( For instance, in the example of axions arising from the RR 4-form (and the universal This is similar to the implicit assumption we are taking for gravitational instantons, which in the pres- 6 susy made massive in thestring string and setup). Einstein frames Notice are that equivalent,gravity so in solution the the becomes wormhole absence more nature of of physical. the dilation-like D-brane scalars, instanton the ence of supersymmetry may have extra fermion zero modes and contribute to higher-derivative terms [ by a dilaton-like partner), havemoreover precisely in the the axionic string charge structure framesee of they also our are [ solutions, described and bylow-energy version wormhole of geometries, the see D-brane instanton e.g.partners wormhole [ solutions, are once the removed dilaton-like (by scalar a hypothetical integrating out mechanism once these fields are respondence between gravitational blackfollowing hole we solutions provide and somelevel systems hints (which of is supporting presumably D-branes. the this best In interpretation, one the at can hope the for,time semi-quantitative in these dimensions) non-supersymmetric can setups). bemotion. described These as solutions solution source to only the the euclidean gravitational supergravity field equations and the of axion(s) (complexified an euclidean D3-brane wrappedbackground with on second the Chern 4-cycle class correspond to an effective descriptiondescribed of non-perturbative in effects terms which are of microscopically D-brane instantons. This would be much in the spirit of the cor- axion), we can introduce a basis of 4-cycles Σ and let where instanton dependent prefactor, and model-dependent prefactors depending on 6.2 D-brane instantonsString and theory gravitational thus instantons providesaxions, a and set inducing of non-perturbative non-perturbative contributions to objects the coupling scalar to potential the of the low-energy form potential; the only price to payM is the appearance of suppressioninflation, factors in in which the(e.g. inflation to scale remove is thecontribute high saxion to and degrees the supersymmetry of is potential freedom), broken of our substantially setup the is corresponding that axions. all D-brane instantons situations is that fairly generically all D-brane instantons induce corrections to the scalar JHEP08(2015)032 p (6.3) (6.4) (6.5) (6.6) . To . On , with f 0 0 ~n p S and ∼ P times on the M n inst − D S of the compactification
2 C
| 5 2 2 | | F 4 | ω dφ | | for some harmonic 4-form, in 4d + 6 -cycle 2 4 p X . The parametric dependence on f R 0 Z -dependent prefactor. ~n + 4 φ ω s 2Φ 0 4 n 1 g − α ω = e ∼ 4 g R ' C G − n 2 ), as we describe in a particular setup in the − √ P f is hidden in the 4d quantities 2 √ P generalizes straightforwardly to the multi-axion ). M – 28 – , i.e. a large multiple of a basic vector x , f 4.17 s s 0 M g g 10 ∼ d 4.17 p ~n 2 s E x g 6 4 S Z 4 = d ) on V 0 4 0 α 1)-brane wrapping a 1 ~n Z α 3.13 ' − ∼ ∼ 2 p P d ). The agreement on the parametric dependence is in general d 4 M S 10 S 4.17 gravitaional instanton action has a parametric dependence n ) on the charges of the axion lattice can be shown to survive the embedding into 4.17 . Let us consider the single axion case for simplicity. For gravitational instantons, s Consider an euclidean ( In some particularly simple instances, the action of D-brane instantons agrees with This analysis for the dependence on The above expression has the precise parametric dependence expected for a charge- An interesting hint in this direction comes from the dependence of the instanton actions g the action of the D-brane instanton associated to euclidean D-brane instanton (namely, a D3-brane multiply wrapped 0 In some very specifiction ( models, the specificstring theory. dependence of the gravitational instanton ac- manifold. Even if the brane is not BPS, one may estimate the action of the instanton via the precise dependence of thenext charges section. in ( 6.3 Charge dependence in flat multi-axion setups not exact, butaxion holds charge for vector large ofgeneral charges the grounds, as form we follows. expectS the Consider D-brane instanton an action euclideanis to straightforwardly D-brane scale reproduced with as from ( n corresponding 4-cycle). Weindicated take relation this between gravitational agreement instantons as and a D-brane suggestive instantons. indicationcase, of using the the above action ( where in the second equation we have ignored the with Hence the charge- with Φ the 10d dilaton.we get Defining the axion by on the dependence of themake action it explicit, ( let usKK consider reduction the of case the of 10dself-duality an action issues) for axion has gravity from and the the the structure RR RR 4-form, 4-form, and which perform (modulo the the familiar JHEP08(2015)032 (6.9) (6.7) (6.8) (6.12) (6.10) (6.11) -fields B ) p -cycle will be a flat − p n ) form exists such that p cannot be guaranteed ). Grouping the axions ω p ω p Z − , ∧ − ∧ ∗ n n n -cycle. Using the condition p . Then p p ω T 1)-branes wrapped on cycles − } − ( i n , with n p − β ω ~ . are harmonic. This means that φ ω { ∗ -forms. Namely, one expands H p , j ( . i G p p β ∗ ) α V p T ), and these may be expanded as p ~ − φ p s ∧ ∗ Z n g n i , µ i ω T n i β α , i ), and thus we may write p Z β p = T n p i n ) dual to − ( ∧ ∗ − T p φ − n p = n Z p Z n ω , ω ) − H dV ω X ∗ n p n = X C ) has a dependence on the charges of the axion − T is the dimension of the ambient space (typically ω i ∧ ∗ = Z ( n ( ∧ ∗ – 29 – j = p p n p ∗ p s ω 6.7 p -dimensional torus, and the p − g − µ , β H p − n i n p n C n − ∧ ∗ β ω ω is the (unique) covariantly constant representative of ≈ n h ω p = -cycle. We will now see that, under some (admittedly n ( i ω ) are naturally isomorphic via Poincar´eduality. There p − ∗ T p = ∗ α n , the Z Z DBI , p n ω ij dV S ( n s G T ∗ ). = n T ( T λ p p V . Here, − C 4.17 p n Z p is the basis of is a constant and H } dV ≈ = i λ is proportional to the volume form of the ˜ α ∝ { p p DBI − , one gets − n S ) and n via Poincar´eduality. Furthermore, the n ω Z , their kinetic term will be of the form i ω ∗ , , where dV . This is a hard thing to do in general, since ~ ∗ φ α n p p of the brane. Let us assume that a globally defined ( , where T = − p i ( n being the harmonic representatives of a basis of p p ˜ α λdV ω i } dV i H dV n = β . { p P n As discussed above, an easy way to obtain axions in the effective theory coming from In order to do this, it will be convenient to have an alternative expression for the volume ∧ ∗ − T n p = 6, although we may also consider submanifolds of the compactification manifold if we = -dimensional hiperplane. ω C satisfies that the image of ˜ the action of the instantons is, in this case The instantons coupled to theseof axions will be euclideanis ( a single flat hyperplane in each class of this string compactification is via the zero modes of RR with into a vector suitable to exist, buttake it the is ambient a spacep to particularly be simple thing to do for tori. In what follows, we will like). That is, dV The usefulness of the above expression depends largely on the possibility of finding a lattice similar to that of ( element ∗ n turned on the brane worldvolume, this is just proportional to the volumevery of specific) the circumstances, the action ( the usual Dirac-Born-Infeld action. In the particular case there are no gauge or JHEP08(2015)032 , k ]), 4 (6.15) (6.13) (6.14) , as it did n necessarily contain D- ) scales with . 4 p − n 6.12 ω , j n ∧ ∗ i p ) is constant, and hence n p − P 1 n − − ij f n ω G ω p M n T = ∼ Z ∧ ∗ p s p − 1 p V g n − by virtue of the duality between the two bases. n ω i ∼ – 30 – j ω ( ) = ∧ ∗ p ∗ , β i − p inst ) on homologous 4-cycles (i.e. in the same class [Σ β n − − n h n ω D ω S n ∧ ∗ T p branes around it, the action ( Z − n n ω , so that the D-brane instanton action , and consider D-brane instantons corresponding to the charges ( f ∗ 4 1), which in fact is of the form considered above for D-brane instantons − , is the inverse matrix to (1 i -cycle and wrap j p p , α i ) = α 2 h It is illustrative to consider a concrete setup. Several attempts to embed these align- Given the close relation between the axion charges of D-brane instanton and gravi- One expects the gravitational instanton solution to be approximately valid when the Notice that to recover the precise matching of the charge dependence it is crucial that In this way, the dependence of the gravitational instanton action on its axionic charges , n 1 n becomes small for any point inside the ellipse. ment models in string(possibly theory a multiple are number based of times onand carrying considering worldvolume two gauge stacks backgrounds with of slightly D7-branes different wrapped instanton numbers, ( in multi-axion setups. Thisplanckian decay effectively constant corresponds to a single-axion model with a trans- gauge non-perturbative axion potentials.the In job other carried words, out the by the D-brane gravitational instantons instantons take in up tational the instanton, description the in earlier analysisthe sections. is form simple. shown in For figure a lattice alignment model, we recast it in 6.4 D-brane instantons andLet alignment us now applyalizations the of D-brane transplanckian alignmentrealization instanton of models. viewpoint the to lattice In thebrane alignment instantons particular, analysis which models we of spoil described now string the in show theory naive section that transplanckian re- inflaton string field range allowed by the the internal geometric objectsand are integration actually amount constant,instanton to so can that be picking regarded the up in integrands theinformation volume general are is case factors. constant truncated as an to average constant assuming In pieces. the this internal geometric sense, the gravitational charges are largeparticular and the curvature atin the the single-axion throat case, and of at the the wormhole end of is the low. previous section. If we fix a since is reproduced in thea torus. stringy setup. Exactgeneric computation Strictly Calabi-Yau of manifold speaking the is however action a this of challenging result matter euclidean into only non-BPS which holds D-brane we instantons will for not in delve. a However, we now have where the volume factor arises because JHEP08(2015)032 . 4 /f . In P f M ) 0 , namely k 2 n − − k ( p = 1 n anti-D3-brane on Σ p , and k the maximum axion periodicity, max f discrete gauge symmetry. , i.e. we use the potential to define the 2 N ~v 2 Z n , since the instanton action decreases with + – 31 – f 1 ~v . The lowest-energy state in this charge sector is a 1 0 k n ). Actually, we are interested in considering the lattice 0 n, k , each with gauge instanton number is the number of axions and = ( 4 ) which in terms of the underlying homology lattice corresponds to 2 2 N ). The relevant D-brane instantons are given by ~v ~v − 1)-brane instantons, whose action is essentially given by ), 1 4.2 ~v − ( p n, k , where ) D( 0 k = ( )). In other words, the D-brane instanton is, at the level of charges, a superposi- max 0 − 1 D3-branes on Σ k ~v k Nf ( p − p √ k The last result suggests a generalization of the Weak Gravity Conjecture, in theories Our conclusions are as follows. In single-axion models, gravitational instanton effects The bottomline is that in string theory D-brane instantons provide the microscopic ( , p . In terms of the homology class of the 4-cycle and the class of the point, the non- 0 with appropriate discrete gauge symmetries.but This generalization to applies not general onlyobjects to theories axions, with of gravitational differentwhich and dimensions charged gauge (by objects are invoking interactions extended generalized for as discrete charged well). gauge extended symmetries, for sense there is notthat advantage kinetic with alignment respect can be to usedform the to achieve single moderately field transplanckian field case.corresponding ranges, of Finally, to the we travelling have shown alongmodel a along this diagonal transplanckian inthe direction, the transplanckian the range lattice. gravitational are instantons absent In which due would the to spoil a effective single-axion spoil transplanckian axion decay constant multiple axion setups, we haveshown considered that different in scenarios models of ofgravitational axion lattice alignment. instanton alignment, effects We the have parametrically which transplanckian jeopardize axion its suffers transplanckian field range, so in this axion, which induce corrections togrees the of axion freedom potential. is The actuallyin effective the inflationary theory right with applications. framework, these asGravity de- In they Conjecture, most should although be cases, the thepaper existence our only is of results relevant in the ones are principle euclidean in independent instantons of agreement considered the with in validity the this of the Weak WGC. In this paper we haveplanckian analyzed certain field quantum ranges gravitational effects inmonodromy), on single and the their possible and implications trans- multiple forhave large axion focused field models inflation on model (excluding gravitational building. setups instantons Concretely of with we the axion effective theory of gravity coupled to the description of the effectsin which have the been effective theory interpreted in in the terms earlier of sections. gravitational instantons 7 Conclusions the end of section are of the form (0 tion of with worldvolume instanton number set of perturbative effects on thesevectors gauge sectors correspond togenerated D-brane by instantons these with charge vectors,periodicities of namely the axions (usingto the including more more refined instanton lattice defined effects, by but the these homology do amounts not reduce the field range, as explaine at k JHEP08(2015)032 ]. ] ]. JHEP ] SPIRE , IN Phys. SPIRE , ][ IN [ (2008) 003 Phys. 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Lyth, D. Baumann and L. McAllister, [7] [8] [9] [5] [6] [2] [3] [4] [1] [10] [11] any medium, provided the original author(s) and source areReferences credited. 2012-ADG-20120216-320421 and the grantSevero SEV-2012-0249 Ochoa” Programme. of M.M. thesupported is “Centro supported through de by the Excelencia a FPU “La grant Caixa” AP-2012-2690. Ph.DOpen scholarship. Access. I.V. is Attribution License ( We thank Luis Ibanez, FernandoHerrero-Valea Marchesano, for Pablo useful Soler, Gianluca discussions. Zoccaratocussions We during and thank the Mario early Ander stages Retolaza ofFPA2012-32828 this for from work. the very MINECO, This illuminating the work dis- is ERC partially Advanced Grant supported SPLE by the under grants contract ERC- with axions, a most relevant topicpolarization given in the the near-future CMB prospects [ ofof exploration of the B-mode validity oftriggers effective further research field in theory these in two fronts. quantum gravityAcknowledgments theories. 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