JHEP08(2017)034 = 2 N erstring Springer July 16, 2017 July 24, 2017 May 26, 2017 August 9, 2017 : : : : n the presence = 2 supergravity N Revised Received Accepted Published lar field distances in field t of he scalar forces combined. emal black holes in linear combination of moduli riations, which is behaviour of forbidding towers of stable er field distance grows at best hat there must exist a particle s particle stronger than gravity. n distances. eases exponentially as a function rsit¨at, Published for SISSA by https://doi.org/10.1007/JHEP08(2017)034 . 3 1705.04328 The Authors. Black Holes in String Theory, Models of Quantum Gravity, Sup c

We propose a generalisation of the Weak Gravity Conjecture i , [email protected] E-mail: Institut Theoretische f¨ur Physik, Ruprecht-Karls-Unive Philosophenweg 19, Heidelberg, Germany predicted by the Refined Swampland Conjecture. In the contex gravitationally bound states. Iton amounts which to the the statement gauge t forceWe also acts propose more that the strongly scalarThis than force implies gravity itself that and should generically act t the onof mass thi of the this scalar particle decr field expectation value for super-Planckian va supergravity, but can be understood more generally in terms Open Access Article funded by SCOAP Abstract: Eran Palti The weak gravity conjecture and scalar fields Keywords: the Weak Gravity Conjecture boundspace. can be Guided tied by to this, bounds wein present on any a sca Calabi-Yau general compactification proof of thatlogarithmically string for any theory with the the prop moduli values for super-Planckia Vacua, Effective Field Theories ArXiv ePrint: of scalar fields. The proposal is guided by properties of extr JHEP08(2017)034 3 6 1 3 9 10 12 13 15 19 23 15 21 (1.1) quantum ]. It states 1 , there must exist a ] for the most recent g st force 24 – ng 2 tement that the particle ture (WGC) [ servation. It was argued tionally bound states. Such s (QFTs) can arise as effec- lly interesting and can have m gravity. Understanding the ly, see [ y than the U(1) force. The WGC, m . ≥ ] by the requirement that extremal black holes p 1 – 1 – gqM which satisfies the inequality m ) was motivated in [ 1.1 ]) that such a tower of stable states would be problematic for and mass 2 q = 2 extremal black holes and the weak gravity conjecture N 3.1.1 Moduli of Calabi-Yau orientifolds ] (see also [ One such quantitative constraint is the Weak Gravity Conjec 1 3.2 Masses of towers of states 4.1 A scalar4.2 weak gravity conjecture The refined swampland conjecture and gravity as the weake 2.1 2.2 Generalisation to non-supersymmetric theories 3.1 Logarithmic growth of field distances state of charge For this particular state gravity acts equallyand or various more modifications weakl of it, have been studied intensive results. The bound ( should be able towith decay. the largest It charge wasstates to also would mass motivated ratio be through should stablein not the against [ form sta decay gravita by charge and energy con that in a theory with a U(1) gauge symmetry, with gauge coupli tive theories of ultravioletprecise physics nature of which the includesimplications constraints for quantu on phenomenological QFTs model building. is both theoretica 1 Introduction It is expected that not all consistent Quantum Field Theorie 5 Summary A Derivation of field space distance formula 4 Gravity as the weakest force 3 Distances in moduli space Contents 1 Introduction 2 The weak gravity conjecture with scalar fields JHEP08(2017)034 . t (1.2) (1.3) ], states 26 , , malised field, 25 J,µν ] about infinite the variation in ) 27 F en field variations there is an infinite ⋆ ( p he presence of super- was presented based tionally bound states M I µν ture, and is supported e (RSC) [ ds are present. We will F nderstood in terms of a ≥ ) t φ s play a role. In this paper ge force should be stronger ( ined. ement in [ uantities are related to prop- ] a general argument for the tates will play an important IJ , so that the inequality can be d to axions, rather than gauge 26 t ce ∆ R , ralises and can be understood as ess bosonic fields s regarding super-Planckian variations = 2 supergravity context through p ound states. In short, it amounts φ + In [ ∆ M N α 1 is the action of the instanton coupling − J,µν ]. e S F ) 29 φ – I µν ( F 27 m ) , t ( ≤ ) is an application of them to simple Reissner- 25 – 2 – ) IJ φ 1.1 I , and propose the general formulation of the Weak I + A j + ∆ t . The logarithmic growth is further tied to the exponen- , whose mass decreases exponentially fast as a function φ t µ ( ∂ m is the metric on the single saxion field space. This implies i t m tt µ ) if the mass of the tower of states behaved as a power law in g ∂ ) t 1.3 ] for the most recent work. ( = 1 supersymmetry. The WGC as applied to axions states that ij 32 – g N . Here p 30 − and gauge fields , M is the axion decay constant and i 2 R 15 t f ≤ the proper field distance, as measured by the canonically nor = t p tt 1 2 g M − ) √ ] it was pointed out that there exists a simple relation betwe g ≥ is a constant which is determined by the choice of direction of , where t p − 25 ( α M L Another conjecture, termed the Refined Swampland Conjectur In [ While the decay of extremal black holes and no stable gravita ≤ There are a number of papers studying closely related question 1 of the scalar field variation tial behaviour of the RSC ( tower of states, with mass scale written as grows at best logarithmically with that for to the axion. In theerties presence of of supersymmetry the both axion of scalar these superpartner, q the saxion denoted field space. The conjecture is based on an earlier weaker stat with scalar fields Gravity Conjecture for this. We first formulate it in an where fields, combined with on black hole physics.general principle Nonetheless, in the the RSC same has sense as not the yet WGC. and been the u RSC on onesymmetry. hand and Its the simplest WGC formulation on utilises the the other WGC which applie arises in t distances in moduli spaceprimarily by which evidence we from termRSC string the applied theory Swampland [ to Conjec fields which appear in a gauge coupling of a U(1) that once a scalar field varies over a super-Planckian distan consider the most general (two-derivative) action of massl Nordstrom (RN) black holes wherewe only will gravity be and interested gauge in field how the WGC is modified when scalar fiel properties of extremal black holes, butthe propose general that statement it forbidding gene stableto gravitationally the b statement that therethan must gravity exist and a the state forces on mediated which the by gau the scalar fields comb in string theory, see [ fS role in our analysis. gravity. The absence of such stable gravitationally bound s are general principles, the formulation ( JHEP08(2017)034 ] 33 (2.1) (2.3) (2.2) states is well . These are i . is the number z V J,µν n ) F ⋆ ( t indeed the WCG and I µν , where rowth of proper distances ase when scalar fields are F V f scalar fields. We do this ace is determined through t build up the intuition we = 2 identity which leads to ependent of supersymmetry, hmically at super-Planckian e is part of an infinite tower. IJ mes exponentially light as a , d exponentially on the scalar hat any linear combination of are the electric field strengths J R uperpartners. We consider the N gravity. We consider the possi- fold compactifications of string I hat requiring the scalar force to the statement of the RSC. Ap- tes, but find that the arguments ce for the RSC. F + , . . . , n F ⋆ IJ = 1 J,µν i F . − I are defined as i I µν J I it F = 1. The G F , and + IJ p V i IJ I b M R – 3 – + are complex scalar fields, with field space metric = j = i i z z z , . . . , n µ I L ∂ F i δ δ z = 2 supergravity have been extensively studied. See [ = 0 µ − I ∂ N ij = g I G − 2 R = which are holomorphic functions of the scalar fields L = 2 supergravity where the framework of extremal black holes

I = 2 supergravity framework we also show that there must exist 1 2 N − = 2 supergravity and show that the same ,F ) N I g N X − (  = 2 extremal black holes and the weak gravity conjecture denotes the Hodge star. The denotes the Ricci scalar and we set ⋆ N R In this paper we generalise this simple argument and show tha Within the , which have components ij of U(1) fields and the magnetic field strengths Here for a review. The action takes the form the RSC are, at leastframework for certain of scalar fields, in some sense s Extremal black hole solutions of in the context of understood. We then propose ainspired generalisation of by the the WGC, black ind hole physics. 2.1 2 The weak gravityIn conjecture this with section scalar we fields present. propose a Since generalisation the of WGCneed the is to WGC tied to to consider the extremal a c black large holes, class to of firs black holes in the presence o for this require further workact to stronger establish than clearly. gravity We suggests show thatfields t the for WGC super-Planckian states depen scalarplying field it variations, to axionic which fields is leads to evidence that the WGC stat on which the scalar field forcesbility themselves that act this stronger is than a general statement relating to bound sta theory has a propervalues. field distance We which also grows furtherresult at identify of best a this logarit tower logarithmic of behaviour, states providing which new beco eviden the general expression of thein WGC field also leads space. to As ageometric an bound moduli application, on this fields the in will g Calabi-Yau allow or us Calabi-Yau to orienti prove t g of vector multiplets.the The periods geometric structure on the field sp The indices are ranged such that where JHEP08(2017)034 . 2 | Z | (2.6) (2.7) (2.5) (2.8) (2.4) (2.9) (2.10) (2.11) (2.12) , and the F I X ∂ . In those cases =

i I F 2 , z ] 1 .  34 , . ! . I ). The black hole electric = I I ′ , , ! q p Q I j I L ysis [ 2.1 through

ψ = X q X . ) MQ F N I · K T K  1 . X i I Q X ⋆F − Q ≡ z Z. from reduction on three-cycles. The right
M ay to understand this structure within the 2 1 ∂ j F alabi-Yau manifolds. For example, in type I I ∞ ( JL , I Kahler moduli and dilaton do not appear in X S F D through F 2 p I , which by supersymmetry determines Z X − ! Z on such that their F-terms are equal to 4
p J ≡ − I i I I 1 π Im + MN − p ′ 1
X 4 − D − F k 1 I , p p I IK ij I − IJ F ψ = 1 formula for the flux induced scalar potential in g q QQ F I Im X j ik N I N + X – 4 – , is the central charge q ·R· Im has weight 1). 2  = i , 1 | + Γ i Z R −RI Z I 2 − I K Z j 1 ( | I F ln e RI + 2 MQ P − ψ 1 p i T = z − = − = IJ ]. The Kahler potential for the scalar field-space metric

∂ Q 2 RI I = F Z 1 2 IJ Q = = −I + F N = , as defined through the integration over a sphere at infinity, j K I !
∞ ψ I S ≡ − I I i IJ

Z 2 P Q D N ,P π Q I 1 = Re [

4 Q IJ M ≡ R . J with Kahler weight are arbitrary constants, F j I ) can then be understood as the ] and I ψ ∂ p IJ 2.8 = N and IJ F I q We are interested in black hole solutions to the action ( It is useful to introduce = Im [ From the perspective of string theory there is an intuitive w 2 IJ and the notation I terms of the superpotential.the The superpotential important and point therefore is obey that a the no-scale type relati There is an identity which will play a central role in our anal on an object takes the form general expression for the symplectic matrix takes the form where and magnetic charges Here context of flux compactificationsIIB of both type the II U(1) stringhand field theories side strengths on and of C three-form ( NS fluxes arise In certain cases it is possible to go to special coordinates the periods are also determined in terms of a prepotential related through a symplectic matrix are related to quantised symplectic charges The covariant derivative acts as JHEP08(2017)034 oles = 2 (2.14) (2.15) (2.16) (2.13) N have an such that
I m , p l infinity. We I q = 0. There are Z es i D that they are at best ], are where the values rametrisation invariant ist for the black hole to 35 le with mass s as re is a scalar field spatial ds have a constant spatial . erties are highly restricted. fields on the horizon of an r BPS states. re the scalar field values at ore it satisfies eld space termed curves of les [ e the ADM mass of the black ting is sufficiently clear. since the particle must itself be ibly non-supersymmetric, cases. ADM Z. M j j m . ∂ = 2 supergravity, but the presence of j D . ∂ Z i N m ∞ ADM i , we should think of it as the mass as a | D ∂ ij Z M | ij i ) is saturated. However, in the next section g g ∂ ADM = + ij ) is non-vanishing we find a stronger constraint M g – 5 – 2.16 + 4 2 2 2.15 ADM + 4 ADM m M M ) as ≥ 2 = 2 ADM 2 2.14 Q ) arises from RN black holes which, in the electric case, M ) leads to the natural generalisation Q = 1.1 2.8 2 ) is positive definite. Therefore, for these extremal black h Q , rather than just in the vacuum. i z 2.15 . We would like to write a generalisation of this for general ) and ( denotes the evaluation of the fields at their values at spatia 2 2.13 ADM M ∞ = 2 q . However, if the last term in ( 2 = 2 supersymmetric extremal black hole solutions with charg 2 g m N We would now like to follow the logic that particles should ex It is informative to rewrite ( The last term of ( The WGC as stated in ( ≥ 2 be able to decay. This can be applied in the case of marginal stability. When they do decay, it can only be to othe extended supersymmetry means that theThe black supersymmetric hole decay black prop marginally holes stable, are and BPS sometimes states only and over this special means loci in fi infinity differ from thosegradient. on The the second type, black denoted hole double horizon extremal so black ho that the therefore two types of extremal black holes. The first are whe a BPS state, its mass is given by the central charge and theref we will utilise it as an inequality for more general, and poss to be able to decayQ we can impose thatthat there the must particle exist must a be partic strictly super-extremal. Indeed, at infinity are equalprofile. to those Such double on extremal thehole black horizon relative holes and to therefore its the maximis charge scalar at fiel infinity. The attractor mechanism statesextremal that black the hole values are of fixed the in scalar terms of the black hole charge satisfy function of the moduli The Here, since we have derivatives acting on ADM mass given by the central charge In fact, since it is BPS, the inequality in ( The subscript will often drop this index leaving this implicit when the set extremal black holes.quantity. To Using do ( this we must construct a field repa JHEP08(2017)034 l ) ) y ). cle, 2.18 2.15 2.18 (2.17) (2.18) ), but 1.1 in ( m in ( . gq  with . The gauge force i ]. This is suggestive z 2 38 he ADM mass is given m ADM 2 Particle , ) is the force due to the the hysics captured by ( ational and scalar forces M Q 37 ated as the existence of a  te a relation between ( s and in the background of le to write for any extremal netic terms as a function of 2.16 e. This can be written as e particle should indeed only , lation between the functional ≤ = 0 [ W ) holds for black holes which are j o the classical force coupling there is a ). If this scalar potential can be ), at least not utilising only charge W ange from relativistic to non-relativistic ∂ i ADM ∂ 2.15 W 2.14 i M ), using black hole decay. 2.16 ∂ ln ij j g 2.18 ¯ ∂ + 4 2 ADM – 6 – = 2 supersymmetry, it is unclear how to make a W M ). Further, as we change the values of the scalar N = appearing in ( ln i 2 . While the particle is required to be super extremal, 2 . For BPS states the inequality becomes an equality ∂ 2.16 Q m Q ij IJ ) can be phrased as the statement that the U(1) force g = ), rather than ( R induced by the cubic coupling of these fields to two WGC and ) would require a replacement of i BH 2.16 z 2.16 The scalar contribution can be simply calculated in the usua V , and that it should form a field reparametrisation invariant 3 2.16 = 1 + 4 ADM also changes, and in order for the particle to maintain a deca are the relevant expressions for the black hole and the parti  M ADM ) is tied to extremality rather than supersymmetry. M 2 BH ADM and on the horizon the fields solve 2 ADM -angle matrix 2 Particle Q M is substantially more complicated than the simple case 2.15 θ ) holds generally, it does not imply ( M ∞ Q 2  Q W| 2.15 and and not = is a real function, termed the ‘fake superpotential’, then t ). The inequality ( ). However, in the absence of m 2 BH W Q 2.16 ADM 2.16 Even if ( To further motivate a general statement we can consider the p Note that in going from the cubic coupling in the Lagrangian t M 3 contribution channel its mass needs tofield also dependence change of appropriately. the black This hole re and the particle can motiva the particular expression ( by quantity, which motivates a form ( which is the expression of the no-force condition. states from the mass term. where a non-vanishing exchange of the scalar fields way as the potential induced through thethis exchange is diagram due of to it being a general expression for dyonic object and ( and ( between two WGC statescombined. acts This at can least be as seen strongly by as noting the that the gravit last term in ( involve fields at infinity It is natural to expect that the charge to mass relation for th and energy conservation.particle The with requirement a for larger decay charge can to be mass ratio st than the black hol that the bound ( only half-BPS or evenblack not hole supersymmetric a at black all.the hole It scalar scalar is fields. potential possib which Thiswritten is as is the precisely gauge ki factor of two timeswavefunction the normalisation. mass of the external states due to the ch with respective masses Here It is interesting to consider if an analogous bound to ( 2.2 Generalisation to non-supersymmetric theories sharp general argument for ( JHEP08(2017)034 ) ay it is 2.19 (2.20) (2.19) ], and the 1 he particle. . If we write the i t ) to hold over all of e cture more generally. ation of the statement rticle may satisfy other tates was argued to be 2.19 the gauge field repulsion = 2 setting to gain some e those with the smallest must be positive definite. al stability. If the charges ally bound state is stable. s consequences. The scalar ij e metric. The conjecture is N er of particles which is the es not necessarily mean that stable gravitationally bound g ustifying this requirement as ys to ensures that they satisfy tion. ravitationally bound states of f minimal charge [ b-locus of the full field space. s in field space. A more refined , and massless real scalar fields ity of the sum over all the scalar . This is because the scalar force IJ R . j ) only over sub-loci of the field space. µ i , satisfying a bound and µ m ij IJ 2.19 m . g I i t ∂ + 2 ) is motivated by considering the particle with – 7 – = m i µ 2.19 ≥ ). There need not be any relation, including the index 2 Q 1.2 then i t ) should indeed hold for these stronger constraints on the as in ( 2.19 ) over all field space. Therefore, it natural to expect that ( ij g ]. The requirement of the absence of such bound states will pl 2 2.19 ]. In string theory it appears that both of these are true, and ) ensures that there are no gravitationally bound states of t 17 , ] and [ 2.19 1 11 are the (non-relativisitic) couplings of the WGC state to on i µ A natural question which arises is whether we should take ( In the context of the WGC this can be understood as a generalis Given this general understanding we can formulate the conje with field space metric i The inequality ( mass as a function of the fields Here However, the BPS nature of the statesan that equality the version black of hole deca ( should be taken to holdthe over extremal all black of hole field can space, decayconclusion but to can that the be particle this reached at do all by point thinking about the existence of properties of the particles. the scalar field space or only certain loci? We can utilise the There are stronger versionscriteria. of For the example, WGC thatgeneralisation which it state of should that this hold theLattice that for pa WGC it the [ particle should o certainly hold conceivable for that an ( infinite tow The existence of athe particle smallest satisfying ( mass to charge ratio in which case a gravitation ranges, between the gaugethen kinetic that matrices there and must the exist a field particle, spac with mass a property of quantum gravity,field but contribution will is assume positive definite itbetween on and equal the study charged side it particles of is gravity forces attractive. is The then positiv ensured by the fact that the field space metric that gravity is the weakestthe force. WGC It states. amounts Such tomass states forbidding will to g be charge stable ratio.problematic if the in WGC [ The states existence ar a of central role such in a our tower analysis, of however, we stable will s not focus on j intuition. A BPS blackof hole the can black only hole decayThis are over suggests relatively loci prime that of then perhaps margin this we is should a impose strict ( su t Consider a theory with gauge kinetic matrices Therefore, we see thatmust the overcome absence both of the a gravitational bound and state scalar means field attrac that JHEP08(2017)034 ]. hat 17 , How- 11 ) should 2.19 to be light, so send ith the picture of the ssless scalar fields. In lds. We then deduce a ht states. This imposes s this expectation. The its interactions with any ace it could happen that WGC state it should not e that the weak coupling theory consistent with the ossibility that the repulsive uch that their convex hull y that it would modify the or scalar fields is below the of marginal stability. In our o mass ratio. Then now we be very light. th noting that in any single ay there is a possibility that impose that at any point in s it is natural to expect that h is the Lattice WGC [ ity will always beat a massive article since they could decay s of the force mediators much . Certainly one possibility is to ould not form a gravitationally s do not play a role. een the gauge and scalar forces just the whole field space, since t. It is unclear how to formulate ar below the Planck scale. It is -dimensional space. In the absence ) should hold. N ), it is interesting to consider how it U(1)s, is that the relation ( 2.19 N ) but that this may not be the same state 2.19 2.19 – 8 – ) should hold for at least one state at any point in 2.19 ) implies that as the strength of the gauge coupling goes to 2.19 ]. The unit circle is the configuration space of RN black hole. 3 ). Now we give the gauge or scalar fields a small mass. It seems t forming a cutoff mass scale of the theory. g 2.19 particles with charge vectors spanning an 0 is leading to quite drastic behaviour. This is consistent w N → 0, we recover a constraint on the charges and couplings of lig g The minimal requirement, in the presence of The analysis so far has applied to massless gauge fields and ma If we consider the state with the largest charge to mass ratio However, it is unclear if an analysis of bound states support Note that the bound ( → non-trivial constraints on fieldalso theories interesting at to energy note scalesto that a f if large there degree, is then a it cancellation forces betw the mass scale of the state to m magnetic WGC of mass scale of the WGC state we may expectclassical that long ( range forcesmaller analysis than that relies of the on WGCforce states. taking carrier We the can has consider mas a first the mass. p At sufficiently large distances grav would be modified for massiveWGC force should mediators. behave. Consider how Weconstraint a start as with in ( massless gaugeas and long as scalar this fie modify mass the if mass sufficiently of smallercoupling the than to WGC the state. scalar mass fields. Similarly, of it the Therefore, seems if unlikel the mass of the gauge scalar field. Thislimit appears to be a strong statement in the sens ever, for more general black holesgenerally this a space natural stronger is statement very than differen require the that minimal a one WGC state exists for every charge choice, whic zero not only does the mass of the WGC state vanish but also all the charges are not relatively prime, and so theseapply subtletie to over all of field space.the transition In between the WGC presence statesanalysis of should we multiple will correspond gauge assume to that field loci field indeed ( space, but noteit that in should terms only of holddirection requiring over in black sub-loci charge hole of space dec field the space. loci of It marginal is stability also are wor bound state since it woulda be different stable. particle As becomescan we the move allow one around for with field gravitationally the sp to bound largest the states charge new of t particle.field the space original Therefore, there p the is one natural state conclusion which is satisfies ( to states. The particle with the largest charge to mass ratio sh of scalar fields, theincludes the charge unit (over circle [ mass) vectors needed to be s light of the possible infrared implications of ( JHEP08(2017)034 ] 25 ] the ) and 25 s which ) should 2.19 2.19 f these states ) captures the 2.19 be very interesting to ssible to form a black nstraint on the struc- ine violating ( y acts weaker than the alar fields, we now turn lsive force can act. It is , and how, ( lications for the infrared is will apply equally then. oduli fields of Calabi-Yau mond and Neveu-Schwarz ctifications, it is indeed the ic that ( expression involving a fake inverse mass. However, we d mass conservation, so it is ss to the carrier then as long s grow logarithmically in the Having stated this, by taking ngth scale will increase as we s. Such states would only be it is logarithmic in. In [ This is the topic of the Refined . However, there are no general is case the gauge field mass, is ) is violated in which case their e behaved logarithmically in the . 4 ian distances. Of course, logarith- 2.19 It is then natural to wonder what the 4 and holomorphic three-form Ω on cycles, or in – 9 – J ). There is a closely related conjecture made in [ 1.3 ), should be general for extremal black holes. 2.17 ). ) we do have bound states but they are over very large distance 2.19 2.19 we will also consider repulsive scalar forces, and the analys 4 ) would be. In this case it would ensure that the length scale o 2.19 It is well known that in simple setups, such as torodial compa Of course, a natural question which arises is whether gravit It is therefore difficult to reach a conclusive statement on if It is interesting to invert the reasoning and utilise the log If the attractive scalar force gains a mass then we could imag In section 4 case that field distances grow logarithmically in the moduli fields obtained by reducingother the words, Kahler in the volume of the cycles. mically is a meaningless statementfields without were specifying closed string what axions, andfield the obtained proper field by distanc reducingfields the on higher a dimensional cycle.compactifications, Ramond-Ra and aim In to this show section that proper we field will distance primarily consider m 3 Distances in moduliHaving space studied how theto WGC the is seemingly formulated unrelated in topicSwampland of the Conjecture distances presence as in stated of field in sc space. ( that field distances grow logarithmically for super-Planck fact that thereture should of be extremal no blacksuperpotential, holes. stable as This bound in appears ( states to to suggest that impose an a co be modified when thetry gauge and or understand scalar fields thisconsequences have of better a ( as mass. it It could would have important imp still not forming boundcould states form at bound scales states muchclassically larger at bound than arbitrarily through the a smallnot barrier distance clear rather scale if than they by are charge problematic. an as we satisfy ( role of ( will behave like thebuild inverse up mass charge. of This thelength is carrier. scale in can Also contrast be to this arbitrarily the le small. case when So ( if we give a small ma non-vanishing we will always have bound states. force carrier. Therefore, if the repulsive force mass, in th scalar field forces themselves. We explore this in section seems less problematic from aa quantum sufficiently gravity large perspective. numberhole of with the a WGC radiusunclear states much to it larger us appears than what to the the be implications scale of po at this which are. the repu JHEP08(2017)034 to i = 2 ρ (3.1) (3.2) (3.4) (3.3) N the field ) for a possible the purpose of 2.8 nces and the WGC the WGC to . 3.2 result for moduli fields in such fields it is clear that . tates whose mass depends, ntractable. If we consider eral result exists then it is A pecial geometry. This mean an look to ( both the Kahler and complex Yau compactifications. Given ction to be of the form s discussed in the introduction, form a guiding principle for the ces is not sufficient to prove the varies from its initial values . dρ , ρ k 1 2 . X − 0 j
. j i X X t h i = 1 i . We would like to analyse a proper field ) in appendix i ij h X t |} g i i 1, the prepotential takes the form 3.3 i h h ijk X {| – 10 – ≫ K f = i ρ 6 1 i t ρ ρ = 2 framework with a cubic prepotential. In the ). With the relation between the WGC and field − Z Max ) in such cases is not feasible. = = 2.8 N 3.3 φ F ∆ ) is a difficult task. Even for a small number of fields inverting 3.3 . We present a derivation of ( f are arbitrary constants but we can, with full generality for ρ i is the metric on the field space and the field h ij g ] it was pointed out that a simple relation between field dista In the previous section we showed that the generalisation of Evaluating ( 25 analysing distances in field space, normalise them such that Here the The proper distance along this direction can be written as distance along a direction in field space. Consider this dire where structure moduli span athat moduli they space can supporting beso-called so-called large described s volume in regime, which an is systems is baseddistances on discussed the at identity the start ( result of on this field section distances. inCalabi-Yau compactifications mind, We we of would c string primarily theory. like In to this obtain case a RSC. One still needs toas show a that power there law, is on an the infinite moduli. tower of This s will be3.1 the topic of Logarithmic section growth ofConsider field a distances field space spanned by real fields arises in the presence ofanalysis supersymmetry. in This this relation section. will Logarithmic growth of field distan its final one the complexity of alikely typical that Calabi-Yau there moduli is an space,in underlying if [ general a reason for gen this. A results for more complicated constructions such as Calabi- space metric is athat non-linear a problem typical Calabi-Yau whichforcing moduli quickly an space becomes explicit can calculation i of contain ( hundreds JHEP08(2017)034 . f 6= to for ρ ij (3.7) (3.6) (3.9) (3.5) (3.8) term ρ I ≤ (3.10) 2 K M . Let us F − ρ e 2 M 1 2 ρ ≤ . − i , ρ ! 1. 0 regime. Since j , by ) implies that as  b f ≥ i ), and setting the j ρ b → Z. t . 3.9 φ i j κ i ly vanish at singular t ρ ly drop the 2.8 + ≤ b  3 D is holds for any linear . ρ ij f Z M i − g k correction to the metric, ρ ρ t 4 1 ogarithmically with ≤ ′ D . ij refore ( though it appears unlikely  i i es further evidence for the α i ij κ ijk b 1 b ρ g = 2 system.  = 0 in ( K κ

3 to the inverse field space metric I κ + ln 4 N 3 = 6 p κ 2 i IJ ry + M ij − − , ρ is positive and that I ρ 2 2 ρ = = ρ ) must tend to a minimum finite value. ]) , κ − ρ ) and ij = ( k i IJ 39 F t I j  F , h t = 1 Z + j ijk dρ 2 , for the interval D 5 K = (0 1 ρ ρ j Z h i f I – 11 – , does not depend on the = = ρ is modified. It would be interesting to see what M q ij D i ρ ij j g ! 2 g i Z ij h ρ j h g b ij + , κ g ij ij ≥ ) to obtain an upper bound on the proper field distance + i k g g t , g h 2 dρ j j 4 4 | t 3.3 h i  − M Z 1 , modifies the prepotential in such a way that t ij | j 3 ρ ], it can be checked that the first . ′ j g  κ i b ijk κ i α Z i M j 40 ) is that the last term is positive definite, this is all we need h κ ) in ( 2 j ) as b κ ρ 3 b i K 4 ρ 3 ij D ij 3.8 Z g 3.9 3.8 g = Z = − 4 i ] that field distances grow logarithmically for ∆ ≤ κ j − ij = 0 since the metric D , h 25 φ i κ ij 1 + 4 b ij κ 3 ∆ g  g 4 i

= 0, we obtain the relation κ 3 κ 4 h 3 i ln 6 2 κ b = − − − = 2 appears at 2 ) = = = N ρ ( ij K IJ is a norm for a vector with a positive definite metric, it can on g I F ). However, this term has a crucial role as a regulator for the 1, with a prefactor which is smaller than, or equal to, one. Th j Therefore the bound 0, within the large volume geometric regime, h Utilising the results in [ We can now utilise ( ≥ 3.9 We are free to set This also shows that this equality does not hold for an arbitra 6 . Since we are interested in an upper bound, we are free to simp ij . Therefore, by choosing the charges . 5 6 g φ φ → i ij ij Then we can write g also assume, for simplicity and with generality, that Supersymmetry relates the inverse gauge kinetic metric g Here we define results could be obtainedthat away this from regime the can large support volume large regime, field variations. where ρ use and so let us write ( ∆ axions to zero The crucial property of ( which for Let us denote the minimum value of Given this prepotential we have (see for example [ Therefore we find that the proper field distance grows at best l in ( ∆ points in moduli space away from the large volume regime. The combination of moduli forconjecture any made Calabi-Yau. in [ The result provid h JHEP08(2017)034 ), = 1 3.3 oten- (3.13) (3.12) (3.14) (3.11) (3.15) through → N D ng in ( ppendices B = 2 n sub-manifold = 1 theories and N . N = 2 analysis carries = 2 hypermultiplets, in again a logarithmic onal dilaton N N = 2 special Kahler man- ) = 0 an be written in a more cations. Another often- α omplex-structure moduli, he . N r type IIB duals (as well as ) CF . y the α more complicated because it is α olds. These compactifications ). We consider a linear combi- = 1 chiral multiplets (which is ve action and τ ng the hypermultiplets into two . CF 2.8 + 2 N ) = 2 special Kahler periods such that α . ˜ ) = Im ( ρ ξ ( i α 2Re ( α N τ G − = α p CX on the original α + = 2 structure. The truncation is imposed = 1 2 2 I α range then these are dual to type IIB O3/O7 ρ N + ,T range they are dual to type IIB O5/O9 orien- k k k = , τ l – 12 – α il 2 k |  ) = Re ( q ) ]. The action on the Kahler moduli is very simple, k k + α = k 39 ξ , p CF ρ CX k 2 1 q  lies in the ( | = I k = 2 structure. = Re ( = Re N sum to that of the  N are projections from the k is the appropriately truncated complex-structure Kahler p k l α α τ K CX ) then gives, after some calculation (utilising primarily a  and and ). k 2.8 k Im l , where denotes the norm of the linear combination vector as appeari is an angle defining the calibration of the special Lagrangia 3.10 K 2 θ | 2 1 ) ]), = 1 theory inherits a truncated = 1 supersymmetry but they are not completely general + α denotes a positive definite contribution. We therefore obta D 39 2 , p N N − ) k ) and is the compensator field which is related to the four-dimensi ρ iθ q ( ( − | C 2.4 e We consider orientifold projections in type IIA since they c The G = through universal manner. However, theyheterotic capture duals and the possible physicsprojection F-theory of are uplifts). thei described The in detail effecti in [ We can therefore write down a projection of equation ( tifolds. The fields ifold. If the 0 index value of the orientifolds, while if the 0 index is in the growth bound ( Here tial ( in the quaternionic space. and C of [ and nation of the moduli fields it is a truncation of the index range of the fields. Therefore t still support an underlying The projection of ( The index range of a projection from awhich special includes quaternionic manifold a spanned specialonto b Kahler a different submanifold Kahler spanned sub-manifold by spanned the by c the through unchanged. The complex-structure moduli sector is Here in fact special Lagrangian).type The of projection chiral acts multiplets whose by splitti scalar components we denote The results onstudied moduli class so of farpreserve compactifications applied are to Calabi-Yau Calabi-Yau orientif compactifi 3.1.1 Moduli of Calabi-Yau orientifolds C JHEP08(2017)034 . 7 0. J ). ≥ i 1.3 (3.16) (3.18) (3.19) (3.17) h ometric rather than is the largest 2 B M t in the RSC ( M t to the case h . But more generally α i 1 t is the number of fields . With this restriction, wer of the size of these e and therefore to the i n ey are complex-structure ∼ h control the sizes of cycles. tring theory which measure i t i KK , So, say for IIA, we therefore stance in the moduli, we can them. The precise dependence ) ond factor is the inverse length is logarithmic for any values of i M M ( 8 are Kahler moduli they directly t ρ 1 in the RSC since the mass dependence i . M , t ≥ ′ ′ h on ite tower of states in a string theory setup φ φ α , ∆ φ ≥ ∆ is expected to take the rough form of α e α i − i 1 − n e t
i α t ≥ the RSC can be avoided, only that the form its final value, and n i ) i f f h ρ ≤ ρ ∼ M ( , ρ t – 13 – ) ) 1 n ) i f , f ( ( M ( come from the reduction of the NS form i KK as the one for which the term t M ≥ t ) M i ) M i KK, of an infinite tower of states matches the RSC. KK, i f M M ( . We have therefore established that at least one of the M ( ) does not imply that 3.1.1 t φ t M i c M nh ρ ρ = . ′ 3.10 ≤ i ] for an analysis). The first factor comes from the modifica- φ f 1 + 41 ρ dependence on the is the dimension of the cycle. − d φ i KK ) only allows for a strict proof for positive M = ∆ depends on the extra dimensional geometry as well as which ge ′ i moduli in section t φ 3.10 k N are measuring. For a torodial setup we have . Therefore we have i ρ t (see for example [ denotes the initial value of 1 d . Then we can write ). This dependence will also determine the exponent factor ) i f − M ( ρ
t 1.3 i , to prove the Refined Swampland Conjecture we need to restric i t is an additional ambiguity. Within a string theory compactification setting the fields While we established that the dependence of ∆ The KK masses h It is worth noting that the result ( Note that the 1 2 ρ 7 8 increases exponentially in ∆ − i where one in quantity the measure the volumes of cycles andmoduli so then the they are mass mirror of dualthe KK to size modes. Kahler of If cycles moduli and th in therefore typeof have IIB the KK s KK towers mass associated on to let us denote the index choice cycles. For example, in type IIA string theory if the There is an infinite tower of modes whose mass decreases as a po t we can parameterise it as scale of the cycle, where tion of the string scale relative to the Planck scale. The sec The exponential dependence on ∆ so that However, they are stillsince expected they to contribute control to the the mass ‘stringy’ of volume an of infin the cycle. appearing in κ Having established the logarithmicturn behaviour to of the the fieldRSC dependence ( di of towers of states on the field distanc 3.2 Masses of towers of states This does not mean that for some negative of the inequality ( on where we define ∆ the JHEP08(2017)034 9 e ]. 1 ). ), 36 ≥ 3.4 i 3.19 (3.20) (3.21) ρ ]. They 17 . , for magnetic κ 4 11 3 , depending on 4 1 ompactification. m i r with small charge ≥ 1 6 m | α = Z or 0 we have that | i e very much if the other ). The mass of a small m 0 ≥ not clear which version d branes on the cycles. m i κ ifications. xponentially or that the . est bound on the effective h ′ 2.13 attice WGC [ r results. Within a string gued to be present generally harge φ 2 B. For odd cycles in IIA the ale ubic prepotential as in ( ∆ hem they can be described as for all practical purposes, the al statement on a lower bound − e he cycle, but we will henceforth neric large volume limit, heavier , m i n ed by an exponentially decreasing r there is an appropriate representative is holds and that there is indeed such a 4 2 3 κ wrapped branes as different states or not. κ vance here, it is more natural to expect an 3 r ion in field space we are in. Looking at two 4 ≤ r 1 and that for M = f ≤ ρ may not influence m i 4 nh M we have 3 M h t – 14 – κ s 0, it is expected that only one state becomes massless [ ≤ → ) , m f i M ( t i 3 . Therefore, we can not prove that the states become t t ) for electric (two-cycle) moduli and 4 f κ M ( 3 t 4 1 2 s ) are particularly interesting because they are in some sens r ≥ ≤ e 0 = α 3.20 i e m , have exponentially decreasing bounds. κ ) is a bound on the mass rather than on the mass variation as in ( , m 3.21 κ 3 4 = 2 supergravity perspective without referring to a string c r must appear at least linearly in N appears in = M 0 e t for general Calabi-Yau and Calabi-Yau orientifold compact M t m There is another infinite tower of states which are the wrappe Note that ( The towers of states ( α Note that there is a subtlety in whether to count the multiply 9 Since vectors. Looking at unit charged vectors, and applying the c (four-cycle) moduli. However, it isfor difficult to prove a gener While in the decompactificationinfinite limit, tower which of is states. the one of rele compactification setting, however, they are, atthan least the in KK a modes ge andfield so theory we comes expect from that KK practically towers. the strong The reason for this is that a variation of more general than the KK states.from This an is because they can be ar are therefore important with respect to the generality of ou In the last inequality we used the fact that number of probe wrapped branes is still given bywe the have central electric c and magnetic mass scales This is a questioncycle of in the the Calabi-Yau homology geometrytower class of to of states. do each with This wrapping.extreme whethe statement limits: We is assume in likely that the to th case depend of on a the conifold, reg in the large volume regime. Note that also, either the mass sc consider the particles case. Whena there classical is charged a extremal large number black of hole t with a mass given by ( These are particles for evenrelevant states cycles are in strings IIA whose and tension odd is cycles controlled in by II t expect a rough bound of Indeed, they can be associated to the towers of states of the L exponentially lighter, but only thatbound. their mass This is restrict isshould a be slightly weaker imposed statementbound generally: on than the the that mass RSC. scale the It decreases exponentially. mass is Of scale course, decreases e how moduli were much larger than JHEP08(2017)034 , . 43 (4.1)   ted by

′ ′ ). This ], could be Z Z 1 2.8 Z Z re strongly

  ) reads (see [ Im s was (
2.8 I = 2 results should p ′ I q N itude of the scalar and he RSC. It is therefore consider two states with − , which are mutually local I ′ ′ now would like to consider scale was very low to start C on the breakdown of the similar methodology to the e of a stable gravitationally p m I n gravity. We can therefore plications for the field space. een the forces acting on the itationally bound states. We q strict to mutually local states s vanish. However, we will see ection we consider the relative and vanishing vector interactions the 2 1 on generalising ( ion is much more complicated than rom being a black hole, as in [ − m we established connections between 3  

′ ′ l, before the collapse to a black hole. As is the Z Z Z Z

. The last term is only non-vanishing if the ′   = 0. In this case we see that the scalar forces ′ – 15 – QQ Re ′ QQ QQ =
| Z | j = 2 context. We will then propose general relations based on ∂

′ N Z But it is worth pointing out this subtlety.

) is the gravitational force and the second is the force media i 10 ∂ ij 4.1 g 4 we reached the conclusion that the gauge forces should act mo and it is less clear if it holds. In section 2 2 + Re = 2 supergravity setting the key equation for the gauge force

′ N Z

| In the case of gauge forces we were able to argue that the It is analogous to the statement that forbidding a monopole f Z | ] for useful texts) 10 in section capturing the relevant physics, however, this generalisat The first term in ( resolved by some othercase states, for not an necessarily SU(2) gravitationa monopole. scalar forces act repulsively. the scalars. The gauge force is given by cancels the gravitational force. Therefore for states with 4.1 A scalarIn weak the gravity conjecture was interpreted as the selfinteractions between interaction different of states. the The WGC relevant relati states. We the gauge forcenatural statement to and explore distances ifWe a in show statement that moduli on indeed scalar space ties forces to and also the t has RSC im can be established. than the gravitational and scalarmagnitude forces of combined. the In scalargauge this and s force gravitational analysis forces. whereWGC states we We within follow first the establish relations betw 4 Gravity as theIn weakest force section 44 off with it wouldeffective only field place theory. a stronger restriction than the RS difference is not of great importance in that if the tower mass interacting states are notand mutually so local. take it We to hencefortha vanish. vanishing re Now gauge the force important between point them is that we can bound state requires thatconjecture that the if scalar we consider forces two WGC act states stronger of mass tha hold generally due tocan the attempt interpretation to asgravitational apply forbidding forces the grav in same the casethat logic when there also the is for gauge an vector important the force difference relative in magn this case. The absenc JHEP08(2017)034 in be 1). e is , (4.2) stence of ) ensures 4.2 ), the scalar forces tate of two orthogo- 1.2 difference is in how a rom the single charged sive. Then ( . The first fundamental how a mass for this field 2 = 2 formalism only makes er, this is unsatisfactory for se mass of the scalar field. the moduli are continuous. e charge of another state in h an arbitrary gauge kinetic stronger than gravity. This c state and a purely electric en by a single gauge field to N on onal charge choice. The role of the field is to stop in the case of multiple U(1)s. ), the arguments for this gen- ear even if we can not describe ence of a scalar field, which is h can be taken as a conjecture l theory ( . he bound state is purely electric ′ 4.2 state is more complicated than that of poses, the case of a single electri- Applying the same logic therefore mm 11 ) as ( s because a gravitational orbit will induce a ≥
′ 4.1 . regarding the meaning of the WGC when was regarding the implications of the pres- m 2 j 2 t ∂ ) – 16 – m i t ∂ ( ij g 0), and a single magnetically charged state, denoted (0 − , ) that if the gauge force is non-vanishing, then its magnitud 4.1 ) and see if a tower of stable gravitationally bound state can 1). And then considering an orthogonal dyonic state to it. Or , 4.2 ). Note that we observe already a striking statement: the exi 2.19 The second fundamental difference is in considering a bound s Having stated the natural generalisation of ( Note that we consider mutually local states because the One way to create a tower is to consider forming a bound state f Note however that the structure of the gravitationally bound 11 It can also be seen from ( nal states compared totower of a states bound could arise. statecally charged Consider, of state, for the denoted illustration (1 same pur particle. The can affect the setup.that The if idea a is bound thatThis state the is scalar exists analogous field then tothe can the its gauge be discussion radius field mas in is is section set massive. by the inver eralisation are far lessdifference is clear that than while in theence analysis the of in gauge massless section scalar casea fields, in much here secti stronger we statement. are requiring This the also pres means that we must state implies that again thealso scalar implies force that must thestop act presence stable repulsively of gravitationally and a bound dyonic scalar states. field is required ev generalising ( two U(1) gauge fields requiresthe the gravitational existence bound of a state scalar made field. from the possible orthog must act repulsively and at least as strong as gravity equal to the scalar and gravitational forces combined, whic one, then they will again notthem exert simultaneously any as gauge force. local This states is in cl a field theory. sense in such cases. However, if we consider a purely magneti and have vanishing gauge interactions, then, for the genera The same analysis appliesThen to let two orthogonal us electric violate states ( matrix this can beAnother done reason since is the that charges the are next quantised state while in the tower has twice th other words, performing an electric-magneticand rotation then adding so a t magnetic statea in number this of new reasons. frame to One it. reason Howev is that it is not clear that wit states, so of charge (1 constructed. two electric states whichrelative velocity are of orthogonal the with particle and multiple therefore U(1) a U(1) force JHEP08(2017)034 ) ) ]. ld 0) 1) 5 , – 4.4 4.2 ower 3 (4.3) ( (4.4) m, ). ] 34 2.19 = 2 setting, then another ]. We see that N 11 ), and what are . In the same way 4.1 ce of charges to be ) must hold once all IJ F nd state are precisely of ( 4.3 ow be forbidden for the consider a tower of (1 → ) holds then even if ( r and gravitational forces require understanding the om bound states of ( ecreases as we move up the y be less problematic from IJ d in what problems such a 4.3 er the increase in the charge er than the decrease due to ) or ( ld induce a gauge interaction. son is that the original bound condition as studied in [ N his could be overcome by the ing than that of ( tionally bound states. s can not be precisely extremal ring them to decay, or that the 4.2 re should exist a super-extremal Then the bound states are such ) as possibilities. However, the = 2 equation here is [ Z. he Lattice WGC [ j , 4.3 N D B j Z 1). However, a gravitationally bound µ i ) are not expected to be stable. If the , B i D ) but with µ (0 , n ij ij ) and ( n g 2.7 g − 4.2 + → 2 1) bound state itself ) | 2 B , – 17 – Z m , n | 0) and (0 ≥ m, ) = 2 B F Q ( 2 1) states are precisely extremal, as in the , ) for the (1 Q 2.19 ) as a bound on the sum of the scalar and gravitational forces, ) would imply a tower of stable states. More generally we shou 0) and (0 2.8 4.2 , ) is violated and we have a stable tower. Heavy states in this t 4.3 subscript denotes that this is a bound state. If ( ) is defined in the same way as in ( ) is violated and we have a tower, and also stable black holes, B ). By themselves each of ( F ( 4.3 , n 2 Q In summary, it is not clear what is a possible generalisation Note that if we retain the Lattice WGC, and consider the latti We are also interested in the relative magnitude of the scala If ( In the absence of scalar fields, if the constituents of the bou Another way to consider forming a tower of stable states is to with (0 structure of the tower and of the bound state better. state can be stable againstto this mass decay. ratio This depends duehaving on to a wheth the higher gravitational state binding in energy the is tower. larg Understanding this would charge to mass ratio in theladder tower then of states we of have the a Lattice decay WGC d channel (0 populated by states, then we must consider forming a tower fr where the for the interaction of a state with itself. The relevant Here the resulting implications.uncertainty We in considered the ( stabilitytower causes, of means the that tower their of generality is bound on states, less firm an foot that their radius isa limited quantum by gravity the perspective. mass of the scalar. This ma solution is to propose that thisparticle must decay for and each therefore charge the lattice,the this scalar is the fields reasoning offer for anthe t alternative, scalar fields we are can accounted demand for, that including ( massive ones. are extremal black holes alongtower this of charge stable direction. states is Requi but absent, must means be that slightly the super-extremal. constituent This is the convex hull require a constraint as in ( is violated there would still not be a tower of stable gravita that we interpreted ( tower to be stable, and itwill is have unclear a to dipole us U(1) moment how. due Yet to another its rea finite size, which wou the theory and so can decay to two such states. This must someh extremal, then ( bound states. These statesattraction. will If feel a the gauge (1 repulsion, but t then a violation of ( JHEP08(2017)034 ). (4.5) (4.6) 2.19 ) and all genvalue. 4.5 of them have ) as the Scalar for future work. V ), and that if it n 4.5 4.5 than that of ( bination of the Scalar . Therefore we deduce where is negative definite. The ), and ( o scalar superpartners. In ravity truly is the weakest weakest force we will often wever, in the absence of a > m ty acts more weakly. We GC state for which gravity uld lead to the stability of which violated ( plicity just one canonically | ns, which are therefore or- WGC. The second is that it eraction of the WGC states IJ rsion of ( 4.3 e fields, we can term this the calars and gravity act attrac- wo natural possibilities. The I calar fields only, the evidence m eral if the states are precisely ), but this is not clear. We can best approximately conserved, t ∂ y bound states, however, in the | 4.3 , e of them has gravity acting strictly 2 . ′ magnetic WGC states. ) or ( > m V mm n 4.2 ) ). Now say that one state was such that ≥ m

j ′ t 2.6 = 2 framework, cannot be directly deduced by ∂ m t – 18 – N )( = 2 case the spectrum of states satisfying this ∂ in (

m | ), or more generally ( i t N electric plus M m ∂ t 4.2 ( V ∂ | n ) reads ij g 4.2 ). In the ) generally as ), away from the strictly positive eigenvalues and one strictly negative ei 1.2 4.4 V 4.5 n 12 has ) holds generally, but leave building more evidence for this real scalar fields, and V IJ ) becomes a strict inequality for all the other states. 4.5 ) n F 4.5 , then all the other orthogonal states must have < m | The generality of ( We can denote the statements ( We can formulate ( There are 2 m t 12 ∂ is the weakest force. scalar forces acting strictly stronger thanstronger. gravity, and on The odd one outother is words, due we can to say the that graviphoton for which has each n scalar field there is one W that at most one state can violate a non-strict inequality ve matrix Im ( thinking about the existence oftively. bound It states is possible since that both it the can s be deduced from ( This means that if we consider the WGC states, there is a basis charge to the scalarsolid fields. argument for For example, the stabilityfor a of the shift bound General symmetry. states Scalar Ho and coupled WGC Gauge to remains WGCs s being rather the weak. single statement Due that to gravity is the the com General Scalar WGC. This isforce, the so more general for statementcan each that again g scalar motivate force it inabsence there terms of is of an a forbidding associatedsuch gravitationall state gauge states. symmetry on It it which is is possible gravi unclear that what one co could associate some, at WGC. We repeat, thatWith the respect argument to the forfirst generality them is of is that the it much conjecture,associated should less there to hold strong gauge are as fields. t ashould We statement can hold about term completely the this generally the scalar even Gauge-Scalar int in the absence of gaug does then ( propose that ( | the others satisfied it.extermal. It appears Consider that onlynormalised this electric scalar structure field states is so for gen that now, ( and for sim for the general theory ( This must holdthogonal for with all respect state to with the vanishing matrix gauge interactio gives information on their relative magnitude. The matrix was such that there was one (electric and one magnetic) state JHEP08(2017)034 . ) t ≤ al his ) 4.2 (4.8) (4.7) (4.9) m (4.10) j t ∂ separately. ght of as )( that ( = 2 i orce t m i t We also find N ∂ ( 15 ij g of gauge fields but to 14 + ). 2 and moduli i 4.7 nm b in this section holds for ss, of whether we are at It would be interesting to upling, the mass and the wo WGC states should be ng identity in y normalised scalar field t is possible to phrase it as 13 Z. the field displacement is the same i e (if we also ask that the lattice D uation ( tial equations. But to illustrate Z . i i t D ij arges so that the WGC state couples to just g on for the axions is positive and so its mass decreases. , + 2 α αt | − ) which would suggest Z > m . ) gives | e > t . | 4.7 | V = p m 4.8 – 19 – n | t . Indeed, to satisfy the inequality for arbitrarily ), it is informative to consider its implications. In ∂ m t | = 2 4.5 | the behaviour of the mass of the WGC states must be Z , then ( | t p j t D i = D m ]. Indeed, we see that the behaviour asymptotes to exponenti is opposite for states with vanishing gauge interactions. T ij g 27 α is the number of fields coupling to the WGC state. , which has important phenomenological implications. Note n α ]. ), which states that gravity is the weakest force, can be thou appears because the WGC states only couple to one combination 1, so that we recover the Refined Swampland Conjecture. the only appears in the coupling to the moduli 26 V 4.5 n V , where n t > 2 1. Therefore for large m j > t | the mass must be an exponential ∇ α i t | t Before proceeding it is worth mentioning another interesti ∇ We can formulate a Scalar WGC based on ( The behaviour where the power in a power law must increase with The factor ij 14 15 13 g 1 2 Consider a power-law form Then, for all but one of the states we have, The factor of all the moduli.one This linear can combination be of seen axions, by restricting and to calculating electric the ch relati quickly for a lower bound on implies that the signmeans of that there is always one state for which exponential, but this is preciselyof the charges Swampland is Conjectur populated) [ the key point consider a simple theory of a single canonicall general, these are complicated coupled non-linear differen see if there is a general interpretation for the physics of eq as that found in [ three-point coupling of thea WGC bound states that to the thelarger scalar four-point than fields. the coupling coupling for I of two two scalar gravitons to fields two to WGC t states. We can interpret this as a relation between the four-point co 4.2 The refinedThe swampland conjecture bound and ( gravity as the weakest f either version of the scalar conjectures. refer to them both jointly as the WGC. The rest of the analysis with large This will be violated for large enough differential equation in thethis point mass able of to the strongly motivate WGC ( states. Regardle JHEP08(2017)034
e I ds, GC , p I fields q ), where t ( olved can be nf states have gravity ) = ). Now if we take ) is certainly com- V t ( n n q, p 4.10 re not just related by 1. We can also expect m gh an infinite tower of RSC can be understood ce that the scalar field > -field case, that the mass . The conjecture is that | ch the infinite tower of the e not constrained by this.  α increases monotonically (or ′ | atement that gravity is the ble with periodic axions. We an be viewed as evidence for behaviour is present requires | ves as e Scalar WGCs and the RSC. , q ′ ot be tied to the period length m ss than one. The period could haviour ( 1, and so there is no required ns (so exponentially inside the between many states and field t m ∂ ] may be understood in terms of the , p | ′ requires the full charge lattice ≤ i 17 i b i b ,  b eld space for which 2 the one state on which this is not true, get 11 -arrangement of states is such that generically m to be the axions. Then the exponential i = b ) with only two states interchanging their
) to a state of charges ( 4.8 4.8 , p, q – 20 – i b ) holds, but this does not have to be the same state m ) we require such an infinite tower. If we allow for an 17 t ( 4.8 m is an integer denoting the state in the tower. Then since n being some integer, then we can instead consider a different n This means that effectively, by choosing different states as w ). Note that this argument ties the existence of periodic fiel it is not possible for the states to replace each other as the W Note that in string theory this is the relevant case for axion 16 ) such that ′ n 4.10 18 1 and , p ′ q ≤ = 2 framework and take the ′ i b N ) is incompatible with the axion periodicity. How this is res 4.10 , with ′ i b + is independent of | n It is interesting to consider how this conclusion is compati Therefore, we can not say for certain, even for the simple one We showed how axions escape the exponential behaviour throu In other words, we find some evidence towards the idea that the n I thank Arthur Hebecker for pointing this out. There is a subtle point here because the periodicity of the It also suggests that the Lattice WGC for gauge fields [ n m = is an arbitrary function and 18 16 17 t m i ∂ | states. A thorough proofmore or work analysis but, of having stated when the the caveats, the exponential exponential be stays constant) then it must be an exponential with exponent periodic structure of expectation values of line operators. pelling in terms of evidence towards a connection between th must be exponential. We can make the statement that if periods). However, the periodbe of made the longer oscillations by must includingnon-oscillatory be more case. and le more It states,distances. until is we Note interesting rea also to that see thein field an a distance axion connection excursion monodromy need type n scenario, and so such scenarios ar states. For a monotonic function role as the WGC state. which only appear through world-sheet instanton correctio for all values of the field. axions, with the WGC statesa populating scalar a version full of lattice. the Lattice This WGC. c understood as follows. Consider applying ( as the statement thatmediates acts there more must strongly existsupersymmetry, than but a gravity. they state So are the onweakest together RSC force. which forming and the WGC the for a general st state with charges ( there must exist a state for which ( b which includes the graviphoton charge.the This means states that for the re whichmixed gravity up. acts Nonetheless, weaker there than is the always a scalars, basis and at each point in fi exponential behaviour if the mass of the tower of states beha behaviour ( as the weakest force. exponential behaviour ( oscillatory function it is possible to satisfy ( can consider the f move around the axion field space, we can consider JHEP08(2017)034 east ]. The 38 , 37 ]. r of the mass of 26 , ity due to the latter property of the WGC 25 = 2 supergravity. This ts to directly imposing here must exists a state ence of scalar fields. We njecture. The results are on value, as in the RSC. in [ tion of moduli in Calabi- N scalar field spaces. As an ity, is independent of their e unclear how their stability only a single state forming a ge fields. We considered two a tower of stable gravitation- es. There are arguments for ey exhibit certain structures alar WGC. force should act stronger than r field distance grows at best limit. njecture can be phrased as the ind the RSC. We were unable rbidding gravitationally bound ciple is that the WGC particles ult. This general proof presents usly. The conjecture would also eresting to develop the quantum nce for the Gauge-Scalar WGC, nces. We also identified infinite otential’ formalism [ ns non-trivial even for states with ). The two statements together were termed the – 21 – 4.5 ]. We did not develop these further but took the absence 2 ) and ( , 1 = 2 identity. We showed that this identity can be utilised 4.2 = 2 black holes to formulate the WGC generally for arbitrary N = 2 supersymmetric context. There are examples and evidence = 2 supergravity setting that scalar field forces also act at l N N N = 2 setting, requires further work. N The WGC bound is marginally satisfied by BPS states in We showed that the Scalar WGCs naturally lead to the behaviou We also formulated a General Scalar WGC which states that the We showed in the logarithmically with the modulitowers for of states super-Planckian whose dista mass decreasesnew exponentially evidence as for a res the Refined Swampland Conjecture developed can be shown utilisingto an extract information onapplication the we behaviour presented ofYau a or field proof Calabi-Yau distances that orientifold in for compactifications, any the prope linear combina statements generalising this, ( as strongly as gravity on the WGC states associated to the gau is ensured and therefore is less motivated than thethe Gauge-Sc WGC states toThis be introduces exponential a in candidateto general the show physical scalar that principle field the beh expectati RSC is implied by the Scalar WGCs with general states, of having scalarconnection fields to act the strongeron gauge on which fields. them gravity than acts Sothat grav ‘gravity weaker that is than the for weakest the every force’.states, scalar but scalar It in can field. field the be absence motivated t of by This the fo gauge amoun symmetry it is even mor Gauge-Scalar WGC. However, the evidence for thisally in bound terms states of is weakertower. than in Establishing the the case existence where ofaway we a from consider tower, the or other evide follow from requiring thewhich decay are of present extremal in an black holes if th conjecture exhibits the interesting propertya that very it light remai mass, and therefore has an interesting infrared of such states astherefore an reliant on assumption this and assumption and used itgravity reasoning would it be for to very the int absence formulate of the such co towers more rigoro towards this possibility within the so-called ‘fake superp We studied aspects ofutilised the the Weak structure Gravity of Conjecture in the pres 5 Summary should not form awhy tower this of should be stable the gravitationally case in bound [ stat scalar field-space metric and gaugestatement kinetic that there function. must exist The a co gravity particle and on the which scalar the forces gauge combined. The underlying prin JHEP08(2017)034 1. > | α | . It implies that the at all the parameters inflation and therefore r super-Planckian dis- in assumptions, that the the tensor-to-scalar ratio ld forces means that this akest force acting on any possibility is if the WGC al Physics. onal restrictions on the ef- flation because it implies ld. It therefore allows for e larger than one itational waves that could ergence of supersymmetric here are ways to avoid the emly helpful discussions. I is to axions we argued that of states which play the role eties discussed in this work. or such a strong statement, s that, in contrast to gauge he Hubble scale during infla- scales above the mass of the e the initial mass scale of the d does not yet yield numbers of high scale supersymmetry. c coupling of the scalar fields, ntial equations for the mass. he gauge interaction vanishes. lds. Otherwise, there would be hat the WGC, or similar general ation of the exponential behaviour – 22 – ] for a quantitative study of a bound due to such physics. 29 This in turn limits the energy scale of the effective theory of 19 In M-theory there are no constant coupling parameters, so th If true, the Gauge-Scalar WGC has some striking conclusions In the single scalar field case we were able to show, up to certa 1. This conservative bound can be easily sharpened by additi See also [ 19 I would like to thankam Arthur supported Hebecker by and the Pablo Heidelberg Soler Graduate for School extr of Fundament but we simply wantpossibility to is point at out least that not the ruled existence out. of scalar fie Acknowledgments in four-dimensions are functionsfields, of every scalar state in fields.the the possibility This theory of mean must astate. couple Super to WGC, At some which least is scalar thisscalars. that fie could gravity hold Of is at course, the sufficiently there we short is distance no sufficiently strong evidence f Further, the couplings of thethrough gauge the fields properties are of tied the tovector WGC states. the multiplets. cubi We This are seeing leadsreasoning the to em about the quantum striking gravity, could possibility imply t the existence fective theory. Of course, this assumesto the the possible inflaton, applic which is subject to the assumptionsexistence and of subtl gauge fieldsnothing requires to the stop existence forming of gravitationally scalar bound fie states when t be produced. The mostcomparable conservative with application current of experiments. this boun Fortower example, of if states we to tak starttion at should the be Planck scale, lower than andr impose the < that tower mass t scale, then we find for places a direct bound on the magnitude of the primordial grav Even in the simple caseexponential of behaviour. a One single way scalar isof if field the there we WGC are showed state a that at largein t different number points fact in there field must space.mass be Applying is an th oscillatory. infinite tower of such states. Another being formulated as complicated coupled non-linear differe This bound cana have lower important bound implications howtances. for fast large the tower field of in WGC states becomes light fo exponential behaviour is such that the exponent is bound to b JHEP08(2017)034 j ij σ M ij ) fix (A.6) (A.7) (A.5) (A.2) (A.3) (A.4) (A.1) M A.4 ommons , = ], for the i t as 25 ]. M , SPIRE for some matrix IN j ][ ∂σ . The equations ( ]. . λ ij σ redited. M = 0 = SPIRE . ..., jλ l i IN + ∂t M ][ ∂σ and the j . . 2 k l ) l j The string landscape, black holes and ρ 0 hep-th/0601001 ∂σ , = [ . j ∂t ∂ρ . We consider the kinetic term i i M 0 jl ( i j l k t jλ l i ∂t 0 ∂t M 0 M ij i h 0 ), which appeared already in [ ij M g ij M ik l M g . We parameterise the matrix g l does not appear. Now from the definition we ). = k defined by M kl λ = il 3.3 − 0 M 1 g i ρ ρ g – 23 – 0 = ij i 3.3 σ ∂σ M = ij (2007) 060 g = g kl M − arXiv:1402.2287 0 ∂ρ g [ i 06 = Kin Weak gravity conjecture and extremal black holes and = − L M jλ Naturalness and the weak gravity conjecture = ∂ρ Kin ), which permits any use, distribution and reproduction in M . With the appropriate normalisation we then find i ]. 0 L JHEP = i 0 Kin , 0 L , leading to ( M M i ∂σ ij h g SPIRE (2014) 051601 = IN [ i into 0 CC-BY 4.0 i 113 M This article is distributed under the terms of the Creative C ∂σ . Note that this is not equivalent to the coordinate change ij denote kinetic terms where M ... is not constant in general. The kinetic terms now read ij are chosen such that to be proportional to M j gravity as the weakest force arXiv:1611.06270 Phys. Rev. Lett. l i l [1] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, [2] W. Cottrell, G. Shiu and P. Soler, [3] C. Cheung and G.N. 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