Lectures on the Swampland Program in String Compactifications
Marieke van Beest, José Calderón-Infante, Delaram Mirfendereski, Irene Valenzuela
LLectures on the Swampland Programin String Compactifications
QFT & Geometry Summer School
July 2020
Marieke van Beest , Jos´e Calder´on-Infante , Delaram Mirfendereski ,Irene Valenzuela Mathematical Institute, University of Oxford,Andrew-Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK Instituto de F´ısica Te´orica IFT-UAM/CSIC, C/ Nicol´as Cabrera 13-15, 28049 Madrid, Spain Physics Department, Bo˘gazi¸ci University, 34342 Bebek / Istanbul, TURKEY Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
The Swampland program aims to determine the constraints that an effective field theory mustsatisfy to be consistent with a UV embedding in a quantum gravity theory. Different proposalshave been formulated in the form of Swampland conjectures. In these lecture notes, we providea pedagogical introduction to the most important Swampland conjectures, their connectionsand their realization in string theory compactifications. The notes are based on the series oflectures given by Irene Valenzuela at the online
QFT and Geometry summer school in July2020. a r X i v : . [ h e p - t h ] F e b ontents N = 2 Type II Calabi-Yau Compactifications . . . . . . . . . . . . . 466.5.2 5d N = 1 M-theory and 6d F-theory Duals . . . . . . . . . . . . . . . . 506.5.3 4d N = 1 Effective Field Theories . . . . . . . . . . . . . . . . . . . . . 516.6 SDC with Potential and Implications for Inflation . . . . . . . . . . . . . . . . . 526.7 AdS Distance Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.8 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 AdS Instability Conjecture 59
10 Final Remarks 70
Not everything is possible in quantum gravity (QG), in the sense that not every model con-structed following the rules of quantum field theory will be consistent once gravity is consideredat the quantum level. This reflects the fact that obtaining a consistent quantum gravity theoryis not so simple, and can still hide many surprises for the physics at lower energies.The Swampland program aims to determine the constraints that an effective field the-ory must satisfy to be consistent with a UV embedding in a quantum gravity theory. Theyare dubbed swampland constraints , and the different proposals are formulated in the form ofSwampland conjectures. The goal is to identify these constraints, gather evidence to prove(or disprove) them in a quantum gravity framework, provide a rationale to explain them ina model-independent way and understand their phenomenological implications for low energyphysics. Although the notion of the swampland is in principle not restricted to string the-ory, the swampland conjectures are often motivated by or checked in string theory setups.Indeed, string theory provides a perfect framework to quantitatively and rigorously test theconjectures, improving our understanding of the possible string theory compactifications onthe way. Interestingly, we have recently revealed that many of these conjectures are actu-ally related, suggesting that they might simply be different faces of some more fundamentalquantum gravity principles yet to be uncovered.The swampland constraints may also have important implications for Particle Physics andCosmology. They can provide new guiding principles to constructing Beyond Standard Modeltheories and progress in High Energy Physics. They may also induce UV/IR mixing thatbreaks the expectation of scale separation and potentially provides new insights on the natu-ralness issues observed in our universe. Hence, the existence of a swampland is great news forPhenomenology!In these lectures, we provide a pedagogical introduction to the most important Swamplandconjectures, their connections and their realization in string theory compactifications. Thenotes are based on the series of lectures given by Dr. Irene Valenzuela at the online
QFT andGeometry summer school [1] in July 2020.These lecture notes are not intended to be a review, so many references belonging to thisresearch field will be missing. We will sometimes provide concrete references to particularpapers within each topic, but we will mainly be using the more pedagogical references, whichare most well-suited for students. For a more complete list of references of the Swampland3rogram, we refer the reader to [2], where the Swampland program is also reviewed andintroduced. Other key references for lecture notes about the Swampland program are [3, 4].
Effective field theories (EFTs) have proven to be very useful in the history of physics. Theyprovide a description of the physical phenomena which is valid up to a certain energy scaledenoted as the cut-off scale Λ. Above this scale, the EFT breaks down and needs to bemodified, but below Λ it should provide a valid description of the physics, both in agreementwith the full theory and with the experiments. Λ CC − Λ QCD M EW H M KK M s M p Λ (GeV)
Figure 1: Each EFT has an energy cut-off up to which the description is valid. Typicalreference scales in high energy physics are indicated here, including the cosmological constant,QCD, electroweak symmetry breaking, and the Planck scale M p . Also indicated is a possibleestimation of the Hubble scale H for inflation, the compactification scale M KK and the stringscale M s , at which stringy effects become important.It is well-known that, given some UV theory, it is always possible to integrate out theUV degrees of freedom, and thereby obtain the low energy EFT. In this way, the effectiveLagrangian of a d -dimensional theory can be divided into the renormalizable part and a towerof non-renormalizable operators (over the Planck mass M p ) L eff = L ren + ∞ (cid:88) n = d O n M n − dp . (2.1)The question we are trying to answer with the Swampland program is whether this processcan always be run in reverse; whether, starting from some quantum EFT weakly coupled toEinstein gravity, it can always be UV completed to a consistent theory of QG. The answer tothis question turns out to be: no, we cannot UV complete any EFT in a way that is consistentwith QG. This answer has its origin in the fact that not everything is possible in string theory.Even if the string landscape is huge, as we will discuss, it does not cover everything. The nextquestion is, thus: if this cannot always be done, what are the conditions that an EFT has tosatisfy in order to make this possible?In a sense, the fact that we can have a consistent EFT which becomes inconsistent whencoupled to a gauge field or gravity is not a new thing. For instance, we know that theremay appear new anomalies that need to be cancelled to guarantee consistency of the theory.To give an example, consider an EFT with an odd number of fermions in the fundamentalrepresentation of an SU (2) flavour global symmetry. This EFT is perfectly sensible on itsown, but when it is coupled to a dynamical gauge field that gauges the SU (2) symmetry, thetheory becomes inconsistent due to the presence of a Witten anomaly [5]. We therefore have4o require that an EFT coupled to a gauge field must be anomaly free. When the theoryis coupled to gravity new anomalies will in general appear, but, interestingly, the absence ofgravitational anomalies is not enough to guarantee quantum consistency of the theory. Thereare other constraints that one needs to satisfy, that simply seem to express the fact that ar-riving at a unitary theory of quantum gravity is not so simple. These additional constraintsare called quantum gravity or Swampland constraints . This brings us to our first definition ofthe lectures:
The Swampland
Those apparently consistent (anomaly free) quantum EFTs that cannot be embeddedin a UV consistent theory of quantum gravity [6].Hence, we say that an EFT belongs to the Swampland when it does not satisfy the Swamp-land constraints, meaning that the theory cannot be UV completed in QG. The name waschosen in contrast to the
Landscape , which include those EFTs that are consistent with quan-tum gravity considerations. The goal of the Swampland program is to determine where theboundary lies that separates the Swampland from the landscape. From the point of viewof string theory, the program attempts to classify the possible string geometries, i.e. stringcompactifications, to understand what can and cannot arise from string theory.An important point to note is that these constraints, in order to be Swampland constraints,should disappear as the Planck mass goes to infinity and we decouple gravity. This also impliesthat they are more constraining as the energy at which the EFT should be valid increases,picking out in the end a (possibly unique) theory of QG. This is illustrated in figure 2, wherethe blue cone represents the set of consistent theories, which we call the landscape, andthe Swampland surrounds the cone. The Swampland criteria become stronger as the energyincreases and we get closer to the QG scale, as illustrated by the cone shape, and terminatein a consistent theory of QG. It is tantalizing to conclude, following the previous reasoning,that there is a unique QG theory in the UV, meaning that all consistent QG theories shouldbe somehow connected to string theory. This is known as String Universality, which will bediscussed in more detail in section 4.3. As of today, it is an open question and one of themysteries that we address in the Swampland program.As explained above, for each EFT there exists some cut-off Λ
EFT at which the effectivedescription breaks down, typically because of the presence of new light degrees of freedom.Above this cut-off the EFT should be modified by “integrating in” these new degrees offreedom, resulting in the definition of a new EFT which is valid above that energy scale upto some new cut-off. However, this process cannot be repeated indefinitely: any EFT has acut-off (dubbed the QG cut-off Λ QG ) above which it cannot be amended to give a consistentQFT weakly coupled to Einstein gravity. In other words, it is not possible to “integratein” the new light degrees of freedom while preserving the QFT description. This occurs, forexample, if an infinite tower of new light states appears, they cannot simply be “integratedin”: quantum gravitational effects become important and the EFT completely breaks down.It is then necessary to drastically change what we identify as the “fundamental degrees offreedom” and go to a string theory description, grow an extra dimension, etc.. Therefore,an EFT belongs to the Swampland, not only if it has a feature which makes it impossible tobe UV completed in QG, but also if it is defined up to an energy scale which lies above its5 w a m p l a n d C o n s t r a i n t s Swampland Landscape
Not consistent withQuantum Gravity Consistent withQuantum Gravity
Figure 2: The Swampland and Landscape of EFTs. The space of consistent EFTs forms acone because Swampland constraints become stronger at high energies.QG cut-off. This is represented in figure 3, where the same EFT used to describe a physicalprocess at an energy E belongs to the landscape while it belongs to the Swampland if thecharacteristic energy of the process is E .To sum up, whether an EFT is in the Swampland depends on the energy scale at whichwe claim that it is valid. The higher this energy, the more difficult it is to be consistent withstring theory (the landscape is smaller). The Swampland constraints will sometimes indicatefeatures that the EFT must have, while others will define the QG cut-off at which the EFTshould drastically break down.An interesting ramification of the Swampland program is that it has brought about ashift in attitude in the string phenomenology community. Instead of trying to find out, from aphenomenological point of view, which vacuum we are living in – a daunting task given that thelandscape is huge – we can try to predict features of this vacuum by understanding what is notpossible, i.e. where we do not live. Thus, Swampland constraints can have phenomenologicalimplications and give rise to new guiding principles for constructing Beyond Standard Modeltheories as well as cosmological models. Furthermore, the Swampland breaks with the logic ofnaturalness, which is based on scale separation, because if QG imposes non-trivial constraintsin the IR, this constitutes UV/IR mixing which could explain the hierarchy problems thatwe observe. We will give some examples of this in the lectures, although the focus will be tounderstand the string theory and geometric realizations of the Swampland constraints. Of course, the Swampland constraints, that distinguish between what is in the landscape andwhat is in the Swampland, are not as well-understood as the gauge or gravitational anomalies.6igure 3: The Swampland and Landscape of EFTs. Cross section of figure 2. The effectivefield theory (EFT ) is in the landscape/swampland if it is used to describe a process ofcharacteristic energy E /E .At the moment, we only have proposals. These proposals go under the name of Swamplandconjectures because none of them are completely proven. However, in the last few years asignificant amount of evidence has been gathered for several of them, which are currentlywidely accepted in the string theory community. The differences in the status of the evidencefor each conjecture will become clear as this is discussed throughout the lectures. Figure 4shows the Swampland conjectures as well as some connections between them. Transplanckian CensorshipNo Global Symmetries Weak Gravity ConjectureDistance ConjectureNo Free Parameters Completeness HypothesisTrivial Cobordism Scalar WGC AdS Instability ConjectureAdS Distance Conjecture dS Conjecture
Figure 4: Map of the Swampland conjectures. The conjectures in black are at the core of theSwampland program, and we will discuss them in detail in the following. The conjectures inpurple will also be discussed throughout the lectures, but sometimes in less detail.In these lecture notes, we will go through the map and describe some of the most importantconjectures. One of the most interesting things about this map are all the connections betweenthe conjectures which are currently being discovered. This development is exciting becauseit might be indicating that some of the conjectures are actually different faces of the sameunderlying QG principles that are starting to appear. The hope is that they will all be unifiedin the future, improving our understanding of QG.Formulating the conjectures is only the first step in the Swampland program. Most of7he work actually comes afterwards, as several tasks need to be accomplished before claimingsuccess. This is illustrated in the scheme of figure 5. First, one tries to identify universalpatterns from the data we have from string theory or from black hole (BH) physics andformulate some criterion. Then a lot of work must go into testing the conjectures and, ifneeded, refine and sharpen them. This usually implies that the conjecture is tested in stringtheory (or in AdS/CFT), since it is a consistent quantum gravity framework and provides aperfect playing ground to quantitatively check the conjecture. However, this is not sufficientto claim that a conjecture applies in general for QG. Although the string theory tests can bevery rigorous, the proofs are typically restricted to specific classes of string compactificationsso they are not completely universal. Hence, one should also be able to provide some physicalrationale for the conjecture and explain what could go wrong otherwise. This is typicallydone using BH physics, consistency of the S-matrix, positivity constraints or some othermodel-independent approach. These arguments are, in general, only able to support mildversions of the conjectures, leaving several ambiguities. However, combined with the stringtheory tests, they can constitute strong evidence for the conjecture. Finally, it is interestingto study the phenomenological implications of the conjectures, both for Particle Physics andCosmology. In these lecture notes, we will focus on explaining the conjectures themselves,their connections and how they are realized in string theory compactifications.
Identify universal pa � ernsand formulate criteriaTest the conjecture Explain underlyingQG principle Phenomenologicalimplications Figure 5: Road map of the Swampland program.
The first conjecture, which is the oldest and the most widely accepted, is the statement thatthere are no global symmetries in QG. In order to discuss this conjecture, it is importantto first define what we mean by a global symmetry in this context. Because, of course, thisconjecture is not meant to imply that there are no symmetries in QG, just that they have tobe associated to gauge degrees of freedom. A useful reference on this topic is [7].8 .1 No Global Symmetries Conjecture
Conjecture 1: No Global Symmetries
There are no global symmetries in quantum gravity (i.e. any symmetry is either brokenor gauged).A global symmetry can be defined as a transformation described by a unitary local operatorthat commutes with the Hamiltonian and acts non-trivially on the Hilbert space of physicalstates. The latter requires the existence of at least one charged local operator, so we saythat the symmetry acts faithfully. The condition of commuting with the Hamiltonian isguaranteed by a local energy conservation condition in QFT implying that the stress-energytensor is neutral. The local operators are required to satisfy a group law, although it is a veryinteresting question to understand whether this can be relaxed and whether other topologicaloperators are likewise absent in QG. Analogously, we will also require that a global symmetryshould map local operators into local operators, although it seems that this requirement mightsometimes also be relaxed, as we will discuss in section 3.5.
Definition 1 (Global Symmetry) ∃ unitary local operator U ( g ) , g ∈ G that: • Satisfies a group law: U ( g ) U ( g (cid:48) ) = U ( gg (cid:48) ) , • Acts non-trivially on the Hilbert space: ∃ a charged local operator O ( x ) s.t. U † ( g ) O ( x ) U ( g ) (cid:54) = O ( x ) , • Commutes with the Hamiltonian: U † T µν U = T µν , • Maps local operators into local operators.
While most of the evidence for this conjecture takes all these conditions into account, partof proving and understanding why global symmetries are not allowed in QG implies tryingto properly define what the disallowed global symmetry is, and whether one can relax any ofthe above conditions. The approach is to use evidence from string theory and AdS/CFT toproperly and rigorously state what does and does not occur.Note that the global part of a gauge group is not included in the above definition of a globalsymmetry, i.e. this is perfectly allowed in QG. The reason is that the global part of a gaugesymmetry (and also its large gauge transformations) violate the second condition above, i.e.there are no gauge invariant charged local operators O ( x ). We illustrate this in the followingexample. Example 1.
Consider, as a typical example of a (zero-form) global symmetry, the shiftsymmetry of an axion φ → φ + c . (3.1)The current for this action is j = d φ , (3.2)and, if the theory has a Lagrangian description, we can write the corresponding local operatordescribing this symmetry as U = exp (cid:18) i α (cid:90) ∗ j (cid:19) . (3.3)9here are local operators charged under the symmetry given by O ( x ) = exp (i qφ ) . (3.4)This constitutes a global symmetry which is not allowed in QG. To adhere to the conjecturewe must either break the symmetry or gauge it. Gauging the symmetry amounts to couplingthe current to some gauge field (cid:90) A ∧ ∗ j → L ⊃ (d φ − A ) . (3.5)Then the theory is invariant under the combined transformation of the axion and the gaugefield φ → φ + c ( x ) , A → A + d c ( x ) . (3.6)Thus, once the symmetry is gauged the operator O ( x ) is not gauge invariant, which violatesthe second condition of definition 1, so this setup is perfectly allowed in QG. Example 2.
The conjecture of No Global Symmetries also applies to p -form global sym-metries. These are generalizations of global symmetries whose charged operators are supportedon p -dimensional manifolds [8]. We can take the example of a gauge theory invariant underthe transformation A p → A p + λ p . (3.7)with λ p a closed p -form. This provides a ( p + 1)-form current J = F p +1 which is conserved,d ∗ J = 0, where F p +1 = d A p is the gauge field strength. The symmetry operators read U g ( M d − p − ) = exp (cid:18) i α (cid:73) M d − p − ∗ J p +1 (cid:19) , (3.8)and its topological nature follows from the conservation of the current. The charged operatorsare Wilson lines operators supported on p -dimensional manifolds γ p as follows, O ( γ p ) = exp (cid:32) i n (cid:73) γ p A p (cid:33) . (3.9)We can either break this symmetry (e.g. by adding electrically charged states so that d ∗ J (cid:54) = 0)or gauge it by coupling the current to a ( p + 1)-form field, (cid:90) B p +1 ∧ ∗ J → L ⊃ (d A p − B p +1 ) , (3.10)in analogy to (3.5). In this latter case, there are no gauge invariant charged operators anymoreso the gauged symmetry is consistent with QG. In string theory, the breaking and gauging ofthese symmetries is typically guaranteed by the presence of Chern-Simon terms. Example 3.
As a final example we can consider the case of discrete symmetries. Theseshould also be either broken or gauged, the latter implying that the discrete symmetry can beembedded in a larger gauge group or be part of the diffeomorphisms of the geometry. Hence,a discrete gauge symmetry is allowed in QG as it is actually a redundancy of the theory.10onsider for example the E × E heterotic string theory. The EFT has a Z symmetryinterchanging the two E gauge sectors, but this is a discrete gauge symmetry as it has aclear geometric interpretation: the E × E heterotic arises from compactifying M-theoryon an interval with the E gauge sectors living at the endpoints, and the Z symmetry isequivalent to the geometric action of flipping the interval, so it is part of the diffeomorphismtransformations. The same occurs for SL (2 , Z ) of Type IIB. It is gauged, as all duality groupsin string theory. The particular case of SL (2 , Z ) can be understood by going to F-theory,where it becomes part of the diffeomorphisms of the torus. The Swampland conjectures usually come with both evidence from string theory and somepiece of motivation from a more bottom-up perspective, like black hole physics. In these notes,we will mainly emphasize the former, precisely because the arguments based on BH physicsshould be taken as (heuristic) motivation and not as proofs. As such, they are useful forgaining intuition for the consequences of a given conjecture, in the present case, to illustratewhat would go wrong if one allowed for global symmetries.If a BH is charged under some global symmetry, it will evaporate loosing mass but not theglobal charge, since Hawking radiation is blind to the global charge; ending up as a remnant ofPlanckian size. Since this can occur for BHs in any representation of the symmetry group, theresult is an infinite number of stable remnants. The remnants are arbitrarily long lived, sinceany combination of particles carrying such large representations of the global symmetry willbe heavier than the remnant. Gravitational bremsstrahlung cannot carry away global chargeeither, as gravitons are neutral under the global symmetry. Hence, in a theory with a globalsymmetry, starting with a charged particle we can actually end up with an infinite number ofstates by constructing black holes by throwing particles together and letting them evaporateuntil they reach Planckian size. Having an infinite number of states below a finite energyscale sounds at the very least problematic [9]. However, it is difficult to rigorously prove afundamental inconsistency as we are dealing with Planckian size objects and the semiclassicaldescription of gravity breaks down. It has also been argued [7] that the existence of theseremnants would violate the covariant entropy bound. According to this bound, the entropyassociated to the remnant should be finite, in contradiction to the fact that there are infinitelymany different states that can give rise to the same black hole geometry, suggesting insteadinfinite entropy.Another argument against global symmetries makes use of the No-hair theorem for blackholes [10], which states that a stable black hole can be completely characterised only in termsof its mass, gauge charge and angular momentum. This implies that, given a black hole of acertain mass, there is no way to determine its global charge. This yields an infinite uncertaintyto an observer outside of the black hole which can again be associated to an infinite entropy.However, this would violate the expectation that the black hole entropy should be finite andgiven by the Bekenstein-Hawking formula, i.e. proportional to the square of its mass.
String perturbation theory: 11uch of the evidence for this conjecture comes from string theory. In particular, it isproven in Polchinski’s book [11] that there are no continuous global symmetries in the targetspace of perturbative string theory. In other words, any global symmetry of the worldsheetcorresponds to a gauge symmetry in spacetime, and viceversa. In the following, we brieflyrevisit the proof for the case of bosonic string theory. Associated to any global symmetry inthe worldsheet, there is a worldsheet charge given by Q = 12 πi (cid:73) ( dzj z − d ¯ zj ¯ z ) (3.11)by Noether’s theorem, where j z , j ¯ z are the symmetry currents. This charge must be conformallyinvariant, which implies that j z transforms as a (1,0) tensor and j ¯ z as (0,1) tensor. We canthen form two vertex operators of the form j z ¯ ∂X µ e ikX , ∂X µ j ¯ z e ikX . (3.12)These vertex operators precisely create massless gauge vectors in the target space coupling tothe left and right moving parts of the charge Q . Hence, any global symmetry gives rise to agauge symmetry in spacetime, and not to a global symmetry. The same type of proof appliesto superstring theory.As of today, there is not a single counterexample of the conjecture, even beyond pertur-bative string theory.AdS/CFT:The conjecture has been proven in AdS/CFT under certain assumptions. It was shownin [12, 13] that a global symmetry in the bulk would lead to a contradiction in the CFT,in the case where the symmetry is splittable on the boundary. This result is derived usingentanglement wedge reconstruction, where the boundary is divided into several subregionswhich have access to some subregion of the bulk obtained using the Ryu-Takayanagi formula(see e.g. [14]). That the symmetry is splittable implies that it splits into a product of operators,where each operator is localized in one of the different subregions of the boundary. If thesubregions are small enough, they are not sensitive to a charged operator in the bulk, so noneof the operators localized in these regions can be charged under the global symmetry. Butwithout charged operators, then there is no global symmetry in the boundary, which leads toa contradiction. On the other hand, if the symmetry is gauged in the bulk, this implies thatwe can insert a Wilson line which can carry information of the bulk operator to the boundary.This setup is illustrated in figure 6. Hence, global symmetries in the boundary correspond togauge symmetries in the bulk, and the conjecture is satisfied. Approximate global symmetries in the IR:One could ask how this conjecture is useful from a bottom-up perspective, i.e. what areits phenomenological implications. In this regard, however, the conjecture that there are noglobal symmetries in QG does not give the strongest constraints. This is because, to satisfythe conjecture, one can either gauge the symmetry, which implies that there are degrees of12igure 6: Bulk and boundary region of AdS/CFT. In the center of the bulk sits a chargedsymmetry operator. The boundary is divided into subregions in which symmetry operatorslocalize. When the symmetry is gauged a Wilson line, shown in green, can carry informationof the symmetry to the boundary.freedom associated to the symmetry, or simply break it at very high energies. So globalsymmetries are not ruled out in the IR, so long as they are approximate global symmetries,which are broken at some high energy scale. Therefore, if this breaking is highly suppressed,the effects in the IR might be negligible and with no phenomenological consequences.In subsequent sections we shall discuss other related conjectures, which precisely try toquantify how approximate the global symmetries can be.New defects:There are certain occasions in which the absence of global symmetries can have implicationsfor a subsector of the theory. Sometimes this conjecture can be used to predict the existenceof new defects in order to make sure that some global symmetry is broken or gauged. Thesedefects can lead to new states or constrain some property of the theory. Examples of this willappear when discussing the triviality of cobordism classes in the next section.
There is a generalization of the notion that there are no global symmetries, where we includetopological global charges, which states that all cobordism classes must vanish. We will explainthe notion of a cobordism class here, and will discuss it further in the context of instability ofnon-supersymmetric vacua in section 8, where it also plays an important role.Cobordism is an equivalence relation on compact manifolds of the same dimensions:
Cobordism
Two given manifolds are in the same cobordism class and called “cobordant” if theirunion is the boundary of another compact manifold of one dimension higher.This has been illustrated in figure 7. It has an abelian group structure and a trivialelement, which is a manifold that is a boundary by itself. The following conjecture was thenproposed in [15]: 13 onjecture 2: Triviality of the Cobordism Classes
Consider some D -dimensional QG theory compactified on a d -dimensional internal man-ifold. All cobordism classes must vanish [15]Ω QG d = 0 . (3.13)Otherwise they give rise to a ( D − d − M ] ∈ Ω QG d .From the point of view of the lower ( D − d )-dimensional theory, the fact that two d -dimensional compactification manifolds are cobordant to each other, is seen as a ( D − d − d -dimensional manifolds that cannot beconnected with other ones, i.e. there are no domain walls interpolating between the associatedEFTs. This in turn implies that there is some global topological charge [ M ] ∈ Ω QG d , which wecannot get rid of. The global charges will give rise to some ( D − d − M EFT EFT Domain Wall M Figure 7: Cobordant manifolds M and M , and the EFTs arising from compactification on M and M . The higher dimensional manifold of which M , are the boundary is associatedto a domain wall interpolating between the two EFTs.The way to understand this conjecture is that a QG theory must contain the requireddefects to guarantee triviality of the bordism group. All these defects and any required addi-tional structure are encoded in the superscript “QG” in Ω QG . If at low energies the theoryhas a non-zero bordism group, it should be taken as an indication that the theory is missingsome ingredient that should be manifest at higher energies and make the cobordism groupvanish. This is why in [15], this conjecture was used to predict the existence of new non-supersymmetric defects in string theory. So there are consequences of this conjecture, butthey are not necessarily visible in the IR. For instance, if the manifold admits a spin structure, the relevant cobordism group is Ω spin . If there arefermions charged under some gauge field, one typically has to look at Ω spin c .
14o give an example, if the compactification includes internal fluxes, there can be non-trivialcobordism classes characterized by (cid:82) X d F d where d is the dimension of the compactificationmanifold. According to the conjecture, this theory by itself would be inconsistent with QG,as Ω d = Z . However, by adding branes acting as a source for the flux, one recovers a trivialcorbordim group since (cid:82) X d F d is not an invariant anymore. Hence, consistency with QG wouldtell us that, in the presence of internal fluxes, the existence of charged branes is required, asoccurs in string theory. In this case, triviality of the cobordism group could also be achievedby adding a Chern-Simons coupling that forces d F p (cid:54) = 0, so there is no conserved chargeanymore. This is also very common in string theory. The next conjecture we will discuss is the Completeness Hypothesis. Like the previous one,the Completeness Hypothesis is a very old and widely accepted Swampland conjecture. Some-times it can be difficult to ascertain who was the first to propose a conjecture, because in somecases the conjectures are like common lore in the community, but it can certainly be foundboth in [7] and in Polchinski’s book [11].
Conjecture 3: Completeness Hypothesis
A gauge field theory coupled to gravity must contain physical states with all possiblegauge charges consistent with Dirac quantization.A consequence of the Completeness Hypothesis and No Global Symmetries is that contin-uous gauge groups must be compact. Consider the example where the gauge group is R , whichallows for irrational electric charges. Then a particle with irrational charge cannot decay toparticles of rational charge (because of charge conservation), which implies that there is someglobal charge associated to this. This would be inconsistent with the statement that there areno global symmetries. This conjecture thus lets us understand why continuous gauge groups instring theory are always compact. Importantly, this does not apply to discrete gauge groups,which can be non-compact, as is the case with duality groups. Consequence
All continuous gauge groups must be compact.
BH physics:The conjecture states that in a theory with a gauge symmetry coupled to gravity, the fulllattice of allowed gauge charges must be populated by physical states. This is not necessary inQFT because a charged particle can always be decoupled from the theory by sending the massto infinity. This ensures that the particle in question is no longer part of the QFT. However,in QG one cannot decouple a particle in this way, because this process will simply result ina BH which is part of the spectrum in a complete theory of QG. Because BHs can have any15harge, we expect every possible gauge charge to be populated by some physical state (someof which could be BHs).Breaking global symmetries (connected gauge groups):There are some very interesting connections between completeness of the spectrum and theabsence of global symmetries in QG. Recall from section 3 that the latter includes the case ofhigher form global symmetries (i.e. p -form global symmetries) associated to higher dimensionalcharge defects, with operators that live on higher dimensional manifolds. A common way ofbreaking these higher form global symmetries is simply by having charged states. But only ifthe spectrum of charged states is complete, can we break the full group.Consider the Example 2 given in section 3. The current of the p -form global symmetry isgiven by the field strength of the gauge field F p +1 = d A p . Then a very straightforward way tobreak this global symmetry is precisely to introduce states that are charged under the gaugegroup, d ∗ J p +1 = d ∗ F p +1 = ∗ j electric . (4.1)Since the conservation equation for the current is tantamount to the equation of motion forthe gauge field, this will be non-zero if there are charged states, d ∗ J p +1 (cid:54) = 0.If the spectrum is not complete, there will be some discrete symmetry that remains intact.Consider e.g. only the existence of states with even charge; this leaves a Z discrete symmetry.Therefore, a complete spectrum is required to break any discrete remnant of a higher formglobal symmetry, as argued in [8]. Ongoing research is trying to understand to what extendthe two conjectures, i.e. No Global Symmetries and Completeness, are different or followfrom each other. The answer seems to depend on whether we are talking about connected ordisconnected gauge groups. In the latter case, it seems that the Completeness Hypothesis isrelated to the absence of more general topological operators. The evidence for the Completeness Hypothesis is very similar to that of the No Global Sym-metries Conjecture. It comes from string theory and AdS/CFT. In particular, in [16], it wasshown that the Completeness Hypothesis follows from solving the Factorization Problem inAdS/CFT, which pertains to a rephrasing of the Completeness Hypothesis where we requirethat the theory must contain enough charged states such that all Wilson lines can break.This is exactly what is happening in the gauge theory example above: there are some Wilsonline charged operators defined in (3.9) which can break because of the presence of electricallycharged states, which implies that the p -form global symmetry is broken. There is a generalization of the Completeness Hypothesis, where we also require completenessin terms of BPS states. In particular, the BPS Completeness condition states that if a chargecan support a BPS state, then there should be a BPS state populating that charge. Of courseit is generally difficult to understand which BPS states can exist in a theory, and which chargescan support BPS states. However, the idea is that if we were able to determine the lattice of16PS states at the same level of detail and generality that we understand it for gauge groups,then all these states should be populated.
BPS Completeness
If a charge can be populated by BPS states, then a BPS state with this charge shouldbe part of the physical spectrum.Interestingly, in the past two years there has been a lot of progress in obtaining constraintson the possible gauge groups in QG, using this simple generalization. Certain string theorycompactifications admit BPS strings, and anomaly constraints in the world-volume of thesestrings can be utilized to obtain constraints on the gauge groups allowed by QG. For instance,in [17, 18] this was used to rule out the following gauge groups: •
16 supercharges: E × U (1) , U (1) in 10d, rank r G > − d for d < • r G > k + k (cid:48) − k, k (cid:48) are Chern-Simons levels.To get some of the constraints on the rank of the gauge group in lower dimensions it isnecessary to combine the BPS Completeness with additional conjectures like the DistanceConjecture, which will be discussed in section 6. These bounds apply e.g. to the N = 4 SuperYang-Mills theory. There also exist some bounds on the allowed number of Abelian gaugegroup factors, i.e. the rank of the Mordell-Weil group, in 6d N = (1 ,
0) theories [19].One of the goals of this program of constraining the gauge groups is to find out whether wecan get everything from string theory that could be consistent with QG. In principle, e.g. ina 10D theory with 16 supercharges, in addition to the E × E and SO (32) gauge groups thatwe observe in string theory, one could have had other anomaly-free gauge groups. However,any other possible gauge groups have turned out to be inconsistent with BPS Completeness,so this might be the explanation for why we do not observe them in string theory, i.e. thatthere is no known string theory construction of these gauge groups.This bring us to a very deep and important question: how universal is string theory? Thisquestion is coupled to the Swampland program, since much of the evidence that we have forthe conjectures comes from string theory. Therefore, at some point we must ask whether ourresults are just an artifact of the lamppost we are looking under, or whether the conjecturesare more general and are actually reflecting inconsistencies in QG. If we can show that theanomaly free theories which do not appear in string theory, do not appear for some underlyingQG reason, e.g. they are inconsistent with a Swampland conjecture which is expected to bemore general, this could imply that we really get everything we can get in string theory. Atpresent, this seems to be the case for highly supersymmetric setups. The idea that everythingthat can possibly happen (i.e. that is not inconsistent with QG) does happen in string theoryis called String Universality or the
String Lamppost Principle . Certainly, it would be veryinteresting indeed if this is the case and any consistent QG theory is somehow connected tostring theory! Therefore, a significant amount of work in the Swampland program goes intostudying those anomaly-free theories that we do not know how to realize in string theory, andwhether there is always an additional inconsistency which prohibits them.17
The Weak Gravity Conjecture
We have seen that the absence of global symmetries and completeness of the charge spectraare at the core of the Swampland program. However, they lack phenomenological impactunless we can constrain how approximate a global symmetry can be, and whether there isany upper bound on the mass of some of the charged states. Otherwise, they only constrainthe full theory but not the low energy EFT. In particular, it is phenomenologically relevantwhether all charged particles can be super heavy and even correspond to BH’s, or there is somenotion of completeness of the spectrum that survives at low energies. Most of the Swamplandconjectures discussed in the rest of the lectures precisely deal with these questions; they aimto sharpen these statements and quantify how close we can get to the situation of recoveringsome global symmetries. For example, we can a priori continuously restore a U (1) globalsymmetry by sending the gauge coupling to zero, which should not be allowed in QG. Tryingto understand how string theory forbids this, and what goes wrong if one tries to do this,can provide information about the constraints that an EFT has to satisfy to be consistentwith QG. We will see that the Weak Gravity Conjecture forbids this process by signaling thepresence of new light charged states invalidating the EFT description. By doing so, it alsoprovides an upper bound on the mass of these charged states in terms of their charge. The Weak Gravity Conjecture (WGC) [20] contains two parts: the electric and magnetic ver-sion.
Conjecture 4: Weak Gravity Conjecture – Electric
Given a gauge theory, weakly coupled to Einstein gravity, there exists an electricallycharged state with Qm ≥ Q M (cid:12)(cid:12)(cid:12)(cid:12) extremal = O (1) (5.1)in Planck units. Here Q and M are the charge and mass of an extremal black hole, and Q = qg , (5.2)where q is the quantized charge of the state and g is the gauge coupling.The electric version of the conjecture requires the existence of an electrically charged statewith a charge to mass ratio greater than the one of an extremal BH in that theory, which istypically an order one factor. The order one factor depends on the theory under consideration.The simplest case corresponds to a Maxwell theory coupled to Einstein gravity in which thereare no massless scalar fields. In that case, we can construct Reissner-Nordstr¨om BH solutions,so given a p -form gauge field in d dimensions, the WGC implies the existence of a ( p − p ( d − p − d − T ≤ Q M d − p . (5.3)Here, T is the tension of the brane, and instead of an unspecified O (1) number, we have giventhe precise bound for the case of a Reissner-Nordstr¨om BH. Hence, for the case of particles in18our dimensions, the order one factor in (5.1) is 1 / √ F grav ≤ F EM , (5.4)will imply that the charge is greater than the mass, so we arrive as the same condition asabove. This is no longer true in the presence of massless scalar fields, as discussed in section5.7.We now turn to discuss the magnetic version of the conjecture, which states that the EFTcut-off Λ is bounded from above by the gauge coupling, so that, given some EFT coupled toEinstein gravity, its cut-off is smaller than the Planck mass if the gauge coupling is small. Conjecture 5: Weak Gravity Conjecture – Magnetic
The EFT cut-off Λ is bounded from above by the gauge coupling:Λ ≤ gM ( d − / p . (5.5)For a p -form gauge field: Λ ≤ (cid:16) g M d − p (cid:17) p +1) . (5.6)The magnetic version of the conjecture can be derived in two ways. We can apply (5.4)to the magnetic field (which is why it is called the magnetic version), which implies that thetheory should have a monopole whose mass is smaller than its charge. Since the magneticcharge is proportional to the magnetic gauge coupling, which is the inverse of the electricgauge coupling, this means that the mass of the monopole must be smaller than the Planckmass over the electric gauge coupling: m ≤ M p /g . (5.7)The mass of the monopole will typically be at least at the order of the cut-off of the theoryover the gauge coupling squared (which is just the electrostatic energy integrated up to thecut-off) m ∼ Λ /g , (5.8)which implies that the cut-off of the theory is bounded byΛ ≤ gM p . (5.9)We arrive at the same conclusion if we require that the theory has some monopole that is nota BH, so that the mass of the unit charge monopole must be smaller than the Schwarzschildradius m ≤ M p R . (5.10)19ince this radius provides a cut-off for the EFT R ∼ Λ − , (5.11)this again implies that Λ is smaller than gM p . Note that the gauge coupling is always evaluatedat the given energy scale g = g (Λ).These are intuitive arguments for why this cut-off should be imposed. The EFT is breakingdown, because at this energy scale it becomes sensitive to the monopole degrees of freedom,so these degrees of freedom can no longer be treated as solitonic objects. However, this isnot necessarily a drastic breakdown of the EFT due to some QG effect. Below, we shall seea genuine QG cut-off appear in some stronger versions of the WGC, which really hints at adrastic breakdown of the EFT. The motivation for the WGC is twofold. First, it provides a QG obstruction to restoring aU(1) global symmetry by sending g →
0. If a gauge coupling goes to zero, according to theWGC, this results in new light particles; and the cut-off of the theory goes to zero accordingto (5.5), invalidating the EFT.Given some EFT, how small the gauge coupling can be depends on how high the energyof the process you want to describe with that EFT is. The smaller the energy of the process,the smaller the gauge coupling can be, as dictated by (5.5). Conversely, if you want to keepthe EFT valid up to a very high cut-off, then the gauge coupling cannot be too small. This isan example of a Swampland constraint becoming stronger for higher energies (as illustratedby the cone in figure 2). Needless to say, a theory with vanishing gauge coupling (i.e. a globalsymmetry) is inconsistent as the EFT cut-off is then also zero.The other main motivation for the WGC is that it is a kinematic requirement that allowsextremal BHs to decay. Charged BHs must satisfy an extremality bound in order to avoid thepresence of naked singularities, as implied by Weak Cosmic Censhorship. For a given charge Q , this extremality bound implies that the mass of the BH M must be greater than the charge M ≥ Q , (5.12)for the BH to have a regular horizon. Here, we are setting the extremality O (1) factor in(5.1) to one for simplicity. For an extremal BH the charge is equal to the mass, so the onlyway it can decay is if there exists a particle whose charge to mass ratio is at least 1. To seethis, consider the decay of an extremal BH, where one of the decay products has a chargesmaller than its mass or saturates the extremality bound, so Q ≤ M . Then the other decayproduct cannot have a charge smaller than the mass, i.e. Q ≥ m . This is simply a kinematicrequirement. Since the second decay product violates Weak Cosmic Censorship it cannot bea BH, so it must be a particle. The system is illustrated in figure 8.The above kinematic requirement can be derived by simply imposing mass/energy conser-vation and charge conservation as follows. The mass of the initial black hole state must begreater than the sum of the decay product masses M i , and the charge of the initial black hole Q must equal that of the decay products Q i , M ≥ (cid:88) i M i , Q = (cid:88) i Q i . (5.13)20 < M Q > M Particle Q > m Q = M Figure 8: Decay of an extremal BH. One decay product can have a charge to mass ratio Q /M < Q /m >
1. Therefore, the latter cannot be a BH but must be a particle.Then M Q ≥ Q (cid:88) i M i = 1 Q (cid:88) i M i Q i Q i ≥ Q (cid:18) M Q (cid:19) min (cid:88) i Q i = (cid:18) M Q (cid:19) min . (5.14)Hence, we have shown that the mass to charge ratio of the original BH must be grater thanthe ratio of the decay product with minimal mass to charge ratio, which is why we need someparticle satisfying the WGC.To sum up, if a theory does not satisfy the WGC, then extremal black holes cannot decay.The question one could now ask is: what is the problem with having stable extremal BHs? TheWGC is a kinematic requirement that allows extremal BHs to decay, but why is this actuallynecessary? There is a heuristic argument, which is very similar to the argument for the absenceof global symmetries, which suggests that we run into trouble with BH remnants. Starting withlarge BHs of all possible charges, they will evaporate until reaching the extremality bound. Ifthe gauge coupling is very small, they will all reach a very similar mass, but still having verydifferent charges. This will result in a large number of remnants (how large depends on howsmall the gauge coupling is) N ∼ /g , (5.15)that are almost degenerate in mass at weak coupling. The difference in mass is set by thegauge coupling ∆ M ∼ gM p . (5.16)So N → ∞ and ∆ M → .3 Evidence The motivations from BH physics are interesting because they constitute more general ar-guments that allow us to at least understand some difference in the physics of an EFT thatsatisfies the WGC and one that does not. If we only base the conjecture on evidence fromexamples in string theory, we risk being fooled by the lamppost we look under. However,these motivations are very far from proving anything. The reason why the WGC is one of themost important Swampland conjectures nowadays is not because of these BH arguments, butbecause of all the evidence that we have gathered for it in the past years. To go through allthe evidence in detail is beyond the capacity of these notes, instead we will give a summaryof the different works in this direction, and some useful references discussing these arguments.One of the nice things about the Swampland program in general, and in particular with thisconjecture, is that there are many people working from a wide range of areas of research tryingto prove the conjecture. Hence, the assumptions that go into a particular piece of evidencecoming from one corner of research is different from what another person assumes in a differentcorner. This suggests that there may indeed be something behind it all, and increases our con-fidence that some statement along the lines of the WGC is indeed a universal criterium of QG.Higher derivative corrections to BHs:One piece of evidence comes from studying higher derivative corrections to the BH chargeand mass. It turns out that these higher derivative corrections could be enough to show thatthe charge to mass ratio of small black holes is greater than the classical extremality bound.The latter is defined as the charge to mass ratio of large extremal black holes of large charge,for which quantum or higher derivative corrections become negligible, Q M ≥ Q M (cid:12)(cid:12)(cid:12)(cid:12) ext . (5.17)If that is the case, then the WGC could already be satisfied by small BHs. The pattern foundin certain setups is the following. If we plot the mass to charge ratio, as in figure 9, withthe straight line indicating the extremality bound for a large BH of large charge, then smallBHs happen to lie below this curve upon taking into account higher derivative corrections. Ifthis holds in general, it would be a way to prove a very mild version of the WGC, where theWGC-satisfying states are BHs instead of light particles. One could expect, though, that if weextrapolate the pattern to lower masses, small black holes will eventually turn into particles,so there will also be some particle with a charge to mass ratio bigger than one. In fact, itcan be interpreted as motivation for having an infinite tower of particles satisfying the WGC,where the heavier states are actually black holes. However, this extrapolation to lower energiesgoes beyond the regime of validity of the computations.For this story to work out, there is an assumption about the possible higher derivativeterms that can occur built into this argument. Interestingly, this pattern of higher derivativecorrections have been found to hold in certain string theory setups (see e.g. [21]), and there isa significant amount of research dedicated to investigate whether such assumptions on higherderivative terms can be related to other IR consistency criteria, as explained next.Positivity constraints from unitarity and causality:22 M Extremality boundBHs with corrections
Figure 9: Graph of charge Q versus mass M , where the line indicates the extremality boundfor a large BH, and the marks represent smaller BHs.The above evidence relies to some extent on specifics of the theory in which the BHs live.But there are several works trying to derive model independent results, based on the conceptsof unitarity and causality, to check if positivity constraints can provide any definite answeras to the direction of the corrections to the BHs, see e.g. [22, 23]. Interestingly, it seems thatpositivity constraints indeed imply (5.17) under mild assumptions of the UV theory which arerealized in string theory. Along a different but related avenue, interesting relations betweenthe WGC and analyticity and causality appear when studying the difference in the low en-ergy theory between integrating out charged states that satisfy or do not satisfy the WGC [24].Relation to Weak Cosmic Censorship:The WGC has interesting relations to Weak Cosmic Censorship, which forbids the pres-ence of naked singularities not hidden by a horizon. The latter has been extensively tested inthe context of general relativity in four dimensions, although there are some known possiblecounterexamples involving an electric field that grows in time without bound. A few yearsago, it was shown in [25] that if that theory also includes states that are charged under thegauge field and satisfy the WGC, then the theory does not violate Weak Cosmic Censorshipany more. Some work is currently going into trying to understand the connections betweenthe WGC and Weak Cosmic Censorship in more detail by generalizing the above setup. Itwould certainly be interesting if it turns out that we always need to satisfy the WGC or someother Swampland conjecture to preserve Weak Cosmic Censorship.AdS/CFT:Evidence for the WGC has also been obtained in the framework of AdS/CFT. Specifically,the WGC has been derived from imposing modular invariance of a 2d CFT [26,27], so it appliesin the context of AdS /CFT but also for any gauge field with a worldsheet description instring theory. It would be very interesting to find similar arguments for higher dimensionalCFTs. Further evidence has been found in AdS/CFT by studying thermalization properties23f the CFT [28, 29], as well as from quantum information theorems related to entanglemententropy [30]. Finally, and very importantly, there is a plethora of examples and works in the context ofstring theory compactifications, which is the main focus of these lectures. Some of these ex-amples will be discussed in more detail in section 6.5, since they provide evidence both for theWGC and the Distance Conjecture. However, we will already outline the basic features herein an attempt to give some intuition for how the WGC is usually realized in string theory. Asummary of the string theory evidence can also be found in [2]. SO (32) heterotic string on T :The most typical example, that already appeared in the original paper [20], is a toroidalcompactification of heterotic string theory. Compactifying the SO (32) heterotic string on T yields a theory with 16 supercharges, i.e. N = 4 supergravity in 4d. Since we have aperturbative description of this theory, the full perturbative spectrum is readily available, andit is known to form a self-dual lattice, charged under the U (1) gauge group. The extremalBH solutions of this theory are also known. The non-BPS extremal BHs satisfy M = 2 α (cid:48) | q | . (5.18)If one studies the non-BPS states in that theory, they do not saturate this extremality bound,rather they have m = 2 α (cid:48) (cid:0) | q | − (cid:1) . (5.19)So if we plot the mass versus the charge, as in figure 10, with the straight line indicating theextremality bound, i.e. where the extremal BHs live, the states (i.e. the particles) indeed liebelow this curve. Hence, the theory satisfies the WGC, and the states approach the extremalvalue for very large charges. This piece of evidence is particularly nice, because it concernsnon-BPS states and not just supersymmetric particles.Closed string gauge fields in Type II:In Type II compactifications, there are abelian gauge fields arising from dimensionally re-ducing the RR and NS 10D gauge potentials. They always come together with charged statescoming from wrapping D-branes and NS5-branes. In supersymmetric compactifications, some-times it is somewhat automatic that the WGC is satisfied because the BPS states themselvessaturate the WGC. In this way, the WGC can be thought of as an anti-BPS bound, althoughthis will be explained in more detail in section 5.8. For example, the supersymmetric mass M D p of a brane wrapping the cycle γ p is given, according to the DBI action, by the stringscale over the string coupling times the volume that it is wrapping V γ M D p ( γ p ) = 2 π M s g s V γ . (5.20)24 q √ α M Non-BPS extremal BHNon-BPS states
Figure 10: Graph of charge q versus mass M , where the line indicates the extremality boundfor non-BPS BHs, and the marks represent non-BPS states of the theory.In 4d Planck units this becomes M D p ( γ p ) = √ πM p V γ V , (5.21)where V is the overall volume of the compactification space. Notice that any time we arechecking a Swampland conjecture, it is very important to go to the Einstein frame, and writeevery mass in Planck units. Hence, when talking about light states in the Swampland program,we always refer to light with respect to the Planck mass. One might think that, since themass is proportional to the volume of the cycle, maybe we can violate the WGC by makingthis cycle very large, so the state becomes very massive. However, this will also change thegauge coupling as the gauge field comes from dimensionally reducing C p +1 on the same cycle γ p . In fact, if the state is BPS, in the asymptotic weak coupling limit the gauge coupling hasthe same dependence than the mass in Planck units, M D p ( γ p ) M p ∼ g (5.22)saturating the WGC. A proper analysis requires the inclusion of scalars in the extremalitybound, which is postponed to section 5.7. It will also be discussed in detail in section 6.5.1.Open string gauge fields:If we instead focus on open string gauge fields, one could try to play the same game. So,given some branes with a gauge field living on them, we can have strings suspended betweenthe branes, giving rise to charged states, see figure 11. One could try to violate the WGC inthis setup by increasing the mass of these charged states. This can be done by increasing thedistance between the branes as the mass of the charged strings is proportional to the distance.Unlike in the previous setup with closed string gauge fields, here the gauge coupling remainsconstant, as it is not sensitive to the distance between branes. However, in a compact space,it is not possible to increase the distance between the branes as much as we want withoutalso increasing the overall volume of the compactification space. Increasing the overall volume25ecreases the gravitational strength, since it results in a larger value for the Planck mass.For this reason, there is a maximum value of the mass measured in Planck units that canbe attained by increasing the distance between branes, implying that beyond that point, thecharge to mass ratio in Planck units remains constant QM p m ∼ const. . (5.23)Figure 11: Open string suspended between a set of D-branes.Upon compactification to lower dimensions, there is a richer set of possible limits thatone can take varying both the mass and the charge. We could think of violating the WGCby sending a gauge coupling to zero and parametrically decreasing the charge of any state.Consider for example the open string gauge fields from a stack of D7-branes wrapping somedivisor (co-dimension 1 subvariety) C A in an F-theory compactification. We can send thegauge coupling to zero by sending the volume of this divisor to infinity,1 g D ∼ vol( C A ) → ∞ . (5.24)However, we are going to run into the same above problems of increasing the overall volumeand decreasing M p , unless we are able to engineer a so-called equi-dimensional limit, in whichthe overall volume does not change. This is possible by simultaneously decreasing the volumeof another cycle C B while increasing the volume of C A , so that the overall volume V is keptfixed, i.e. vol( C A ) → ∞ , vol( C B ) → , s.t. V ∼ const. (5.25)However, as shown in [31], the curve C B will have a non-zero intersection with C A , and aD3-brane wrapping C B will give rise to a string charged under the D7-brane gauge sectorwhose tension is proportional to the volume of C B . The mass of the string excitation modeswill then go to zero, M D ( C B ) ∼ vol( C B ) → g D →
0, in a way consistent with the WGC. See figure 12 for an illustration. Hence, onceagain, the WGC is satisfied; in this case, thanks to the presence of the string modes arisingfrom a wrapping D3-brane. This example will be discussed in more detail in section 6.5.2.These examples are intended to give some intuition for how string theory typically defeatsour attempts to violate the WGC. We will get deeper into these examples and the geometricstructures underlying the relations between the charges and masses in section 6.5.26 A C B D7-branes D3-branes
Figure 12: Equi-dimensional limit. In order for the overall volume to remain finite while thevolume of the divisor C A goes to infinity, there must exist another curve C B which intersects C A and whose volume goes to zero. The gauge coupling of the D7-brane gauge sector wrapping C A will go to zero while the tension of a D3-brane wrapping C B will also go to zero. So far we have discussed the WGC in the presence of a single gauge field, but how does oneproperly define the conjecture in the presence of several gauge fields? One could have guessedthat it is enough to require the existence of a particle for each gauge field with a charge tomass ratio bigger than the extremality bound, but this turns out not to be sufficient. Instead,the correct generalization is a requirement known as the Convex Hull Condition.
Convex Hull WGC
For multiple U (1) gauge fields, the WGC is satisfied if the convex hull of the charge tomass ratio (cid:126)z = (cid:126)Q/m of all the states contains the extremal region [32].Consider the example of two U (1)s, shown in the plot in figure 13. If we determine theextremal region, i.e. the region in this plane that allows for BHs to exist, and plot the chargeto mass ratio (cid:126)z of all the states, then the requirement is that the convex hull of the charge tomass ratio of the states in the theory includes the extremal region. This ensures that there isalways some particle with a charge to mass ratio bigger than the extremality factor along any rational direction of the charge space.Here we see that having a set of states saturating the WGC separately for each gauge fieldwould cut into the extremal region, see figure 14, so it is not enough to satisfy the ConvexHull Condition. Therefore, the Convex Hull Condition is clearly a stronger statement thanrequiring the WGC to be satisfied for each U (1). Essentially this condition allows BHs with any charge under both gauge fields to decay. This puts non-trivial constraints on the EFTs.The extremal region is sometimes called a unit ball, since for simplicity we tend to setthe charge to mass ratio of an extremal black hole to 1. This terminology can be somewhatconfusing because, first, the extremal region is not necessarily of radius one (it is some O (1)constant), and, second, when scalar fields are present it is not necessarily even a ball. Theextremal region can then be an ellipse or have straight lines, as discussed in section 5.7.27 xtremal region z z Figure 13: Plot of charge to mass ratios (cid:126)z = ( z , z ) of a set of states for two U (1) gauge fields.The extremal region is indicated in gray. It is contained in the convex hull of the charge tomass ratios of the states, indicated by the black lines. The WGC is satisfied. z z Figure 14: Plot of charge to mass ratios (cid:126)z = ( z , z ) of two states saturating the WGC for the U (1) (1) and U (1) (2) gauge fields, separately. The extremal region is shown in gray, and theconvex hull of the charge to mass ratios of the states is indicated by the black line. The WGCconvex hull condition is not satisfied. One of the main open questions in the WGC is: how do we identify which states satisfythe WGC? This question is very important in order to be able to derive phenomenologicalimplications of the conjecture. The formulation of the conjecture given so far contains nostatement, or even indication, of whether there should be one or several states satisfying theWGC, if they need to be very light, have a very small charge, or the biggest charge to massratio, etc.. In this sense, there is a mild version and many strong versions of the WGC.The mild version simply says that there must be some state (any state) that satisfies theWGC. It can even be above the EFT cut-off, so in this form the conjecture does not haveany implications at low energies. Contrary to this, the strong versions specify which state(s)should satisfy the WGC. Such refinements typically imply particles below the cut-off, leadingto potential phenomenological constraints.For this purpose, a lot of the recent research has precisely been dedicated to understand-28ng this question, by investigating the structure of the states satisfying the WGC in stringtheory setups. In fact, there was an attempt in the original paper to get a strong version ofthe conjecture, by identifying the WGC-particle with the lightest one, the one with smallestcharge, or the one with the biggest charge to mass ratio. However, these stronger versions arenot well-defined in the presence of several gauge fields. At the moment the available strongversions that are consistent with everything we know and any string theory example are theSublattice WGC [26, 27, 33] or the Tower WGC [24], which require the states satisfying theWGC to form a (full dimensional) sublattice of the charge lattice or a tower, respectively.The two proposals are very similar, however it is slightly stronger to require a sublattice ofsuperextremal states than requiring a tower. It is also expected that the conjectures might besatisfied by unstable resonances but not by multiparticle states.
Sublattice/Tower WGC
For every site (cid:126)q of the charge lattice, there is a positive integer n such that there is asuperextremal state with charge n(cid:126)q satisfying the WGC. This integer can depend on (cid:126)q for the Tower version, while it is universal for the Sublattice.The main motivation for these proposals comes from demanding that the WGC is consis-tent under dimensional reduction [33]. Starting from a theory with a single particle satisfyingthe WGC, when this theory is dimensionally reduced it can result in an EFT that does notsatisfy the WGC anymore. On the other hand, starting with a sublattice (or an infinite tower)of states satisfying the WGC ensures that the dimensionally reduced theory will also satisfythe WGC. So it is exactly this consistency under dimensional reduction that requires the ex-istence of infinitely many states satisfying the WGC in the original theory. Arguments fromunitarity and causality [24] motivate the weaker version in which an infinite tower (instead ofa sublattice) suffices, which also relates to the Distance Conjecture as discussed in section 6.4.However, up to now, there is no known string theory counterexample to the Sublattice WGC(see though [34] for a potential candidate). In the following, we will see two key examples thatwere important in motivating these stronger versions, based on consistency under dimensionalreduction.Kaluza-Klein circle compactification:Consider a theory with a U (1) gauge field and some particle that satisfies the WGC, sothat it has charge to mass ratio z = q m ≥ . (5.27)Dimensionally reducing the theory by compactifying on a circle S R of radius R , will give riseto a lower dimensional U (1) gauge field, as well as an extra Kaluza-Klein (KK) U (1) gaugefield. The zero mode of the original particle with charge to mass ratio z will still satisfy theWGC for the U(1) gauge field in lower dimensions. There will also be states charged underthe extra U (1), namely the KK modes with m KK = m + q KK R , g KK = 1 R (5.28)29nd q KK = n with n being an integer. So here we see that, if there are massless states in theoriginal theory (i.e. m = 0), the KK modes of these massless states are going to saturatethe WGC for the KK gauge field, since their mass will be equal to their charge. Thus, it isautomatic in a KK compactification, that the KK modes of e.g. the graviton saturate theWGC.However, we have learned that in the presence of multiple U (1) gauge fields, we mustimpose a stronger condition than satisfying the WGC for each U (1). We must require thatthe Convex Hull Condition is satisfied. In the lower dimensional theory, the charge to massratio vector of the KK modes of the original WGC-satisfying particle is given by (cid:126)z = ( z U (1) , z KK ) = ( q , q KK /R ) (cid:113) m + q KK /R , (5.29)as they are charged both under the original U (1) gauge field and the KK photon. If we plotthis charge to mass ratio in the two directions spanned by the KK photon and the originalgauge field, we get figure 15. We can see that the states lie on an ellipsoid satisfying z U (1) /z + z KK = 1 , (5.30)outside the extremal region. However, since they only populate a finite region of the ellipsoid,they are not always enough to satisfy the WGC as their convex hull can cut the extremalregion as represented in figure 15. It can be checked that the convex hull WGC will only besatisfied if the following relation holds, (cid:0) m R (cid:1) ≥ z (cid:0) z − (cid:1) . (5.31)This implies that the Convex Hull Condition is violated unless the radius is very large or westarted with a very superextremal particle. Otherwise, the dimensionally reduced theory willviolate the WGC, even if it was satisfied in higher dimensions. · · · z U (1) z KK Figure 15: Plot of charge to mass ratios (cid:126)z = ( z U (1) , z KK ) of KK modes of a particle chargedunder a higher dimensional U (1) gauge field and originally satifying the WGC z > q KK = n ∈ Z .The WGC is not satisfied in the lower dimensional theory.The solution that was proposed in [33] was to start with a tower of states satisfying theWGC already the original theory in higher dimensions instead of a single particle. This tower30ill then give rise to lower dimensional states that will exactly lie on the extremal region,filling the entire ball and satisfying the Convex Hull Condition in lower dimensions.It is an interesting proposal, because this is in fact the way in which the WGC is satisfiedin typical string theory compactifications. Take e.g. the case of Type IIA string theory ona circle, which has the C gauge field as well as a KK photon. To satisfy the Convex HullWGC we need to have not just one particle but a charged tower under C in the original10d theory. This tower of particles is precisely the tower of D0-branes, which are the KKmodes from M-theory. So precisely because Type IIA already has this tower of charged states(which signals the extra M-theory direction when taking the strong coupling limit), the WGCcontinues to be satisfied under dimensional reduction to 9d. There are many more examplesof this phenomenon in string theory, where not only one state but towers of them satisfy theWGC. Furthermore, if one focuses on toroidal compactifications, one finds that there is a su-perextremal state for each point of the entire charge lattice. This provides a stronger versionof the Completeness Hypothesis. However, in more general situations, only a sublattice ispopulated by superextremal states, as explained next.Toroidal Orbifold Compactifications:Consistency under dimensional reduction was first used to formulate a Lattice WGC (sincetoroidal compactifications always give rise to a lattice), but in [27] it was shown, by studyingtoroidal orbifold compactifications, that only a sublattice of states satisfies the WGC. Considera toroidal orbifold e.g. T / Z × Z (cid:48) , (5.32)where each Z acts freely, so we mod out by a “rototranslation” symmetry. Compactificationon the three-torus gives rise to a set of KK U (1) gauge fieldsKK photons of T : W µ , Y µ , Z µ . (5.33)However, due to the orbifold, one of the KK photons, say W µ , is projected out. The corre-sponding KK modes stay, which has non-trivial implications. In fact, the KK modes associatedto odd charge n Y under Y µ receive a contribution from the KK modes of W µ , so that these oddcharge states are actually subextremal. Hence, it is only a sublattice of states (with even n Y charge) that satisfy the WGC. This was the first example to suggest the idea of the sublattice,confirmed by the results using modular invariance of the CFT [26]. When there is not just one but infinitely many weakly coupled states, e.g. a sublattice ortower of states, becoming light, this implies a drastic breakdown of the EFT by quantumgravitational effects. Therefore, there is a natural QG cut-off associated to a tower of statesknown as the Species Bound [35–38].The basic idea is that gravity becomes strongly coupled if there are too many light fields.Assuming that perturbative techniques work, one can check that the graviton propagator getsrenormalized due to the presence of light degrees of freedom, so the QG cut-off is actuallygiven by Λ QG = M p N /d − , (5.34)31here d is the spacetime dimension and N is the number of light species. Hence, the QGcut-off is smaller than the Planck scale, so quantum gravitational effects become importantearlier than one would naively expect. The same species bound has been motivated usingnon-perturbative arguments based on black hole physics [36, 37].For a tower of states, the number of light species, i.e. how many states there are belowΛ QG , is related to the mass gap of the tower as follows, N = Λ QG ∆ m . (5.35)Plugging this into (5.34) one finds that the species scale behaves asΛ QG = M d − d − p (∆ m ) d − . (5.36)When the full tower is getting light, they are also getting more degenerate in mass, so the QGcut-off goes to zero. If the states in the tower saturate the WGC, the mass gap is proportionalto the gauge coupling ∆ m ∼ g , (5.37)so the QG cut-off scales with a power of the gauge coupling itself,Λ QG ∼ g d − M d − d − p . (5.38)Therefore, we see that this QG cut-off goes to zero as the gauge coupling goes to zero. Thissituation is similar to what we encountered in the magnetic WGC, but here the cut-off is notmotivated by the presence of monopoles, but by the infinite tower of states that are becominglight. Unlike for the magnetic cut-off, this truly signals a drastic break down of the effectivefield theory by quantum gravitational effects. Λ QG M ∆ m Figure 16: Tower of states separated by a mass gap ∆ m in a theory with cut-off Λ QG . Among all the possible phenomenological implications of the Weak Gravity Conjecture andits refined versions, let us discuss one that triggered the renewed great deal of activity onthe Swampland program in the past five years. We are talking about the implications for32arge field inflation, more specifically in the context of natural inflation. This was triggeredby the following works [39–41] in response to the increased amount of research on large fieldinflationary models originated from the BICEP2 announcement about primordial gravitationalwaves.Recall that the WGC should apply not only to particles, but to any kind of state that ischarged under a p -form gauge field. According to this, for any p -form gauge field there shouldexist a ( p − p −
1) black brane. One can apply this for axions, that are 0-form gaugefields. The WGC for axions implies that there must exist an instanton with quantized charge q whose action satisfies S (cid:46) q M p f . (5.39)Here, we have used that the equivalent of the mass for an instanton is its action, and that thegauge coupling of an axion is the inverse of its decay constant, f . The twiddle in the inequalityis due to the lack of extremal solutions for instantons. Since there is no known way to define anextremality bound in this case, the previous bound actually contains an undetermined orderone factor.In natural inflation scenarios, an axion slowly rolls down a non-perturbative potentialgenerated by instantons. It takes the form V ( φ ) ∼ e − S cos (cid:18) πφf (cid:19) , (5.40)where we have assumed that the leading contribution comes from a charge one instanton andwe have neglected the higher harmonics contributions, which are suppressed with respect tothe fundamental one. Looking at this expression, we see that the field range available forinflation is of the order of the decay constant. Notice that, if the leading contribution comesfrom an instanton with quantized charge q greater than one, this field range is smaller, so ingeneral ∆ φ (cid:46) f .It is important to take into account that (5.40) applies in the diluted instanton gas ap-proximation, which requires that S (cid:38) S (cid:38) S (cid:38) φ (cid:46) f (cid:46) M p , (5.41)which means that transplanckian axionic excursions, and consequently, natural (large field)inflation, is disfavoured by the WGC. Of course what we are considering here is only thesimplest case which includes a single axion. One can use the convex hull WGC to placebounds on models with several axions, including N-flation and alignment models.A loophole in this argumentation is to be noted. It could be the case that the instantonsatisfying the WGC is not the one contributing the dominant piece of the potential for inflation.This could happen if both the action S and the quantized charge q of the WGC-satisfying33igure 17: Non-perturbative potential for the axion. The plain line is the fundamental har-monic contribution while the dashed one includes the first higher harmonic correction. Left:The action is larger than one so the correction does not modify the field range available forinflation. Right: For small action the corrections spoil the monotonicity of the potential,shortening the field range for inflation.instanton are larger than the ones corresponding to the instanton generating the inflationarypotential. In this way, the exponential suppression in (5.40) is greater for the WGC instanton,but the field range is instead bounded by∆ φ (cid:46) f (cid:46) q M p , (5.42)where we have considered the best case scenario in which the instanton generating the potentialhas unit charge. So a transplanckian field range is allowed if the charge of the WGC-satisfyinginstanton is q >
1. However, this requires a somewhat unnatural scenario in which the chargeto mass ratio of the WGC instanton is bigger than for the light instanton generating thepotential, which goes against the monotonicity found in string theory examples and which isexplained in figure 10. Nonetheless, we cannot discard this scenario.Thus we have seen that the mild version of the WGC, in which it is not specified whichinstanton should satisfy the WGC, is not enough to rule out large field inflation. In order todetermine meaningful constraints, some strong version of the WGC is needed. For instance, ifthe full lattice of WGC states is present, the unit charge one will imply an important constraintfor large field inflation. But if only a sublattice is required, as the index of the lattice grows, thequantized charge of the first state satisfying the WGC grows and the constraint gets weaker.If this index is very large, the implications stop being relevant for constraining large fieldinflation. This is why constraining the index of the sublattice is one of the most importantopen questions of research in the WGC. All known string theory examples have an index oforder one, but we are lacking a fundamental universal bound. This issue is highly linked tothe question of the maximum rank for discrete symmetries; a large sublattice index hints ata large discrete symmetry. However, this rank is expected to be bounded in order to avoid aglobal symmetry. 34 .7 The Weak Gravity Conjecture with Scalar Fields
The presence of massless scalar fields strongly affects the WGC bounds. As discussed in sec-tion 5.1, without scalar fields, there are two different but equivalent interpretations of theWGC. We could either think of it as a super-extremality condition or a repulsive force con-dition in the sense that the gauge repulsion is stronger than the gravity attraction over twoequal states. However, these two conditions will not be equivalent any more in the presenceof massless scalar fields. Clearly, the second condition gets modified due to the presence ofadditional scalar forces, and the extremality bound is also sensitive to the contribution frommassless scalars. The current convention is to still identify the WGC with a superextremalitycondition, while the the second interpretation receives the name of the Repulsive Force Con-dition. • WGC (superextremality condition):
Given some p-form gauge field, ∃ a superextremal state satisfying Q ≥ γ ( φ i ) M , (5.43)where the massless scalars { φ i } can also contribute to the extremality factor γ ( φ i ). Recallthat this factor is the charge to mass ratio of an extremal black hole in that theory. Forinstance, a famous example is a dilatonic black hole, which is an extremal black holesolution in an Einstein-Maxwell-dilaton theory in which the gauge kinetic function ofthe gauge fields is parametrized by (canonically normalised) massless scalars as follows, L ⊃ g e α i φ i F p +1 . (5.44)The scalars induce an additional contribution to the charge to mass ratio of the extremalblack holes, so the WGC becomes Q M d − p ≥ (cid:16) | (cid:126)α | p ( d − p − d − (cid:17) T (5.45)with Q = qg . By comparing to (5.3) we can see that there is an extra term proportionalto the scalar couplings (cid:126)α := { α i } in (5.44).The generalization to multi-fields will again involve a convex hull condition as in section5.4. However, notice that the extremal region will no longer be a unit ball but will de-pend on the scalar contribution. For states that are mutually BPS, the extremal regionbecomes an ellipse, while for states that are non-mutually BPS it develops straight lines.The take-home message is that, in order to check the WGC, it is essential to first knowthe charge to mass ratio of extremal black holes in that theory, which can vary extremelyfrom one theory to another if massless scalar fields are present. • Repulsive Force Condition (RFC): ∃ a state which is self-repulsive, namely the gauge re-pulsion between two copies of the state acts stronger than the sum of the gravitationaland scalar interactions [42, 43], i.e. F gauge ≥ F gravity + F scalar , (5.46)In a d-dimensional theory, each of these forces for a particle of mass m and gauge charge Q take the following form F gauge = Q r d − , F gravity = m M d − p r d − d − d − , F scalar = µ M d − p r d − , (5.47)where µ is the scalar Yukawa charge. The scalar Yukawa force emerges whenever themass m of the state is parametrized by a massless scalar φ , as we can always expand themass term as follows, L ⊃ m ( φ ) χ = (cid:16) m + 2 m ( ∂ φ m ) φ (cid:17) χ + . . . , (5.48)where the scalar Yukawa charge is then given by µ = ∂ φ m . For the case of a particle infour dimensions, the RFC in (5.46) reads Q M p ≥ m + g ij ( ∂ φ i m )( ∂ φ j m ) M p , (5.49)where we have allowed for the presence of multiple scalars with inverse field metric g ij .This condition was first proposed in [42] as the proper interpretation of the WGC in thepresence of scalar fields, and later named as the RFC in [43].For a p-form gauge field, we can generalize (5.47) and derive the existence of a ( p − T satisfying f ab q a q b ≥ g ij ( ∂ φ i T )( ∂ φ j T ) + p ( d − p − d − T , (5.50)where we have written explicitly the gauge charge in terms of the inverse gauge kineticfunction f ab and gauge quantized charges q a .Note that (5.45) and (5.50) are identical apart from the contribution from the masslessscalars. In the WGC, the scalars contribute through the dependence in the gauge kinetic func-tion; while in the RFC, they enter through the behaviour of the mass/tension. Consequently,it seems we now have two different bounds, so which one is realised in quantum gravity? Thisis an open question, although there is no known counterexample for any of the two conditionsin string theory, so it might be that both are realised. In fact, although seemingly different,they coincide in many cases. For instance, string theory evidence shows that they are fulfilledby the same states at the weak coupling limits g → .8 Sharpening the WGC and BPS states A sharpening of the WGC was proposed in [46]:
Sharpening of the WGC
The WGC can only be saturated by BPS states in a supersymmetric theory.This has important implications for the stability of non-susy vacua, as discussed in section8.1. The idea is that only supersymmetry can guarantee an exact equality relating the massand charge of a state. Otherwise, one should expect quantum corrections preventing thesephysical quantities from saturating the WGC bound. In the following, we will summarize someexamples from string theory compactifications that support this sharpening. Unfortunately,the evidence is scarce. This is due to the difficulties in computing the mass of non-BPS states. • The first typical example is the toroidal compactification of heterotic string theory, dis-cussed in section 5.3.1. In this setup, we can compute the whole spectrum perturbativelyincluding non-BPS states. These states do not saturate the WGC as their mass is strictlysmaller than the charge, m = 2 α (cid:48) (cid:0) | q | − (cid:1) < α (cid:48) | q | = M , (5.51)as given in (5.19). Indeed, by comparing to the charge to mass ratio of the extremalnon-BPS black hole solutions of the theory, these states are superextremal. This is whythey lie below the extremal curve in figure 10. • Another example are the exictation modes of strings becoming tensionless at the weakcoupling limit of gauge field theories in F-theory CY compactifications. The string arisesfrom wrapping a D3-brane on a shrinking curve. This example was briefly discussed insection 5.3.1 and we will have a more detailed explanation in section 6.5.2. The stringspectrum is given by non-BPS states satisfying g q = m + 4 jg > m , (5.52)where m and q are the mass and quantized charge of the state, g the gauge couplingand j some integer depending on some intersection number defined below (6.34). Again,these states are superextremal with respect to the black hole extremality bound, whichin the weak coupling limit coincides with a no-force condition. • Higher derivative corrections to non-BPS black holes in heterotic string theory alsomotivate the sharpening of the WGC. The corrections go in the direction of increasingthe charge to mass ratio of small black holes, so they satisfy the strict inequality of theWGC. • Consider some supersymmetric theory with massless scalar fields, so they contributeto the extremality bound as given in (5.45). Upon compactification, if supersymmetryis preserved, the scalar contribution | α | changes in such a way as to compensate thevariation of the second term in (5.45) due to the change in the space-time dimension.This way, a particle saturating the WGC in higher dimensions will also saturate it37n lower dimensions. However, if supersymmetry is broken while compactifying, thescalars will typically get a mass, so they will not contribute to the extremality boundanymore, since α = 0. Hence, the charge to mass ratio of extremal black holes decreases,and a previously extremal particle will become superextremal with respect to the newextremality bound in the lower dimensional non-supersymmetric theory. More generally,since the scalar contribution to the extremality bound is positive, this helps to make thestates satify the strict inequality when supersymmetry gets broken and the states areno longer BPS.All the previous items are examples of non-BPS states satisfying the strict inequality of theWGC, and not saturating it, in agreement with the sharpening of the WGC above, accordingto which, only BPS states could exactly saturate the WGC. If that happens, then the BPSstate is saturating both the WGC and the RFC (as it feels no force), so the scalar contributionsin (5.45) and (5.50) coincide. One should keep in mind, though, that it is not necessary fora BPS state to saturate the WGC; there are examples of non-extremal BPS states near finitedistance singularities of the moduli space, e.g. conifold singularities. Contrary, BPS statesbecoming light at the infinite field distance singularities are typically extremal [45].If the sharpening of the WGC is true, it predicts interesting conditions on the geometry ofstring theory compactifications. For instance, take M-theory compactified on K G N = 1, the particles are non-BPS. According tothe sharpened WGC, these states should satisfy only the strict inequality of the WGC. Here, to conclude the WGC, we mention some of the key open questions as well as somegeneralizations and other current avenues of research that we did not cover in these lectures: • WGC version for discrete gauge symmetries. Some proposals have appeared, see e.g. [47]and Z K Weak coupling conjecture in [48]. • Bottom-up motivation of the WGC for axions and low codimension objects. There areno extremal black hole solutions of such dimensionalities, so the motivation for the WGCbased on black hole physics is missing in these cases. • Very little evidence in non-susy setups. • Better understanding of the interplay between the extremality bound, the BPS boundand the repulsive force condition. • What is the exact strong version of the WGC realized in string theory? If it is thesublattice WGC, is there any upper bound on the sublattice index?38
Possible loophole: WGC under Higgsing, see e.g. [49]. The WGC might not be satisfiedin the IR upon Higgsing even if satisfied in the UV, unless the amount of Higgsing isalso constrained by quantum gravity. • Can we rigorously prove any fundamental inconsistency with having stable non-BPSblack holes? • WGC in AdS d +1 /CFT d for d >
2. The evidence for the WGC using modular invarianceonly applies for d = 2. • WGC in de Sitter space, see e.g. [50].
We now turn to discuss the Distance Conjecture, which plays a central role in the Swamplandprogram as it comes with many interesting relations to the other Swampland conjectures.
In string theory every coupling, mass, etc., is controlled by the vacuum expectation value(vev) of some scalar fields. They are called moduli as they are massless before adding fluxesor other ingredients to the compactification manifold. From the 10d perspective, they controlthe size and shape of the extra dimensions. This motivates the common lore that there areno free parameters in string theory, but all of them are dynamical .Different effective field theories can be explored by moving in this moduli space. Forexample, one may try to move towards a point in which a global symmetry is restored, bye.g. sending some gauge coupling to zero. This case is particularly interesting, since globalsymmetries are forbidden in Quantum Gravity, so something dramatic is expected to happenwhen approaching such a limit. Leaving its definition for later, let us recall that the modulispace is naturally equipped with a Riemannian metric. The obvious way in which the theoryis protected against restoring a global symmetry in this way is that, according to this metric,weak coupling limits are at infinite field distance in moduli space. But this is not the end of thestory. One would expect the effective field theory description to continuously break down asthe approximate global symmetry looks more and more exact as we move towards the infinitedistance loci. This is precisely the behaviour predicted e.g. by the magnetic version of theWGC or the species bound for the WGC tower, since the EFT cut-off will fall down to zeroas the gauge coupling goes to zero.It is worth noting that the cut-off decreasing is purely a quantum gravity mechanism.From the Quantum Field Theory perspective, even if the point with vanishing gauge couplingis unreachable since it is at infinite distance, there seems to be nothing wrong with being asclose as possible. In fact, it is precisely in these weak coupling limits where we have goodcontrol of the EFT, so a tiny gauge coupling is very appealing from a bottom-up perspective.However, restoring a global symmetry is strictly forbidden in quantum gravity and, as aconsequence, the effective field theory description should break down in a continuous way byquantum gravitational effects when taking these infinite distance limits. This is actually related to the absence of ( −
39e could try to run a similar argument when taking other limits restoring other typesof global symmetries. Or more generally, we could wonder what happens when approachingany infinite distance boundary in moduli space. Does the EFT always break down? And ifso, what is the mechanism introduced by QG so that it happens? This is quantified by theSwampland Distance Conjecture (SDC), to which this section is devoted.
Consider a D -dimensional effective field theory coupled to Einstein gravity and with somemoduli space M parametrized by massless scalar fields. The action includes the term S ⊃ M D − p (cid:90) d D x √− h (cid:18) R − g ij ∂ µ φ i ∂ µ φ j (cid:19) , (6.1)where g ij is identified as the metric in moduli space.The first statement of this conjecture is that the moduli space is non-compact. Startingfrom a point P ∈ M , there always exist another point Q ∈ M at infinite geodesic distance d ( P, Q ). Note that a point is at infinite distance if every trajectory approaching the point hasinfinite length. One typical example in string theory compactifications is the decompactifica-tion limit. The second and more important statement describes what happens if we try toapproach some point at infinite field distance:
Conjecture 6: Swampland Distance Conjecture
There is an infinite tower of states that becomes exponentially light at any infinite fielddistance limit as M ( Q ) ∼ M ( P ) e − λ ∆ φ when ∆ φ → ∞ , (6.2)in terms of the geodesic field distance ∆ φ ≡ d ( P, Q ) [51].Figure 18: Left: Pictorial representation of quantum gravity moduli space with several asymp-totic limits at infinite distance. Right: Quantum gravity cut-off falling exponentially with thegeodesic distance as we approach an infinite distance point, due to the tower of states predictedby the SDC. 40he exponential rate λ , apart from being positive, is not specified by the conjecture. Itis expected to be an O (1) constant, as otherwise it could spoil the exponential behaviour,but its origin is not known. In fact, determining how small this parameter can be is themain open question about the SDC, and we will see that significant progress on this has beenachieved in certain string theory setups. Concrete lower bounds have also been proposed inthe literature [45, 52–54] and will be discussed later. Clearly, having an unspecified factor inthe conjecture is not satisfactory; we would like to be able to compute it from first principlesor effective field theory data. This was the case for the WGC, in which the order one factoris specified in terms of the extremality bound for black holes.The infinite tower of states is weakly coupled and signals the breakdown of the effectivefield theory, as it is impossible to have an effective field theory description weakly coupledto Einstein gravity with infinitely many light degrees of freedom. Hence, there is a quantumgravity cut-off associated to the infinite tower of states, which decreases exponentially in termsof the proper field distance, Λ QG = Λ e − λ ∆ φ , (6.3)as represented in figure 18. For simplicity, here we are taking this cut-off to coincide with thefirst state of the tower. But a more accurate way of defining it is via the species bound cut-offgiven in (5.36), whose exponential rate will differ from λ by an order one factor depending onthe space-time dimension.An immediate consequence is that effective field theories are only valid for finite scalarfield variations. From (6.3) and taking into account that Λ ≤ M p one gets∆ φ ≤ λ log M p Λ , (6.4)which is telling us that the maximum field variation actually depends on the cut-off of theeffective field theory. This means that the higher the cut-off or the process changing the vev ofthe scalar, the smaller is the maximum field distance that can be described within the effectivefield theory. This statement has direct implications for inflation that will be discussed later.Notice that this correlation between the cut-off and the maximum field range is intrinsicallyquantum gravitational, as these two quantities are a priori unrelated from a QFT perspective. The prototypical example of how the SDC is realized is a KK circle compactification of stringtheory to d space-time dimensions. Taking r to be the modulus controlling the radius of thecircle one finds: S ⊃ M d − p (cid:90) d d x √− h (cid:18) R − d − d − ∂r ) r (cid:19) (6.5)where we show only the Einstein-Hilbert term and the kinetic term for the modulus r . Thereare thus two limits at infinite distance, small radius r → r → ∞ . Theproper field distance is given by the canonically normalized field∆ R = (cid:114) d − d − r . (6.6)41aking the decompactification limit r → ∞ we know that the KK tower becomes light. Themass of these modes is given by m KK = qr = q e − (cid:113) d − d − ∆ R , (6.7)so we indeed find an infinite tower of states becoming exponentially light with the distance,as required by the SDC. Moreover, in this case we have been able to compute the exponentialrate in terms of the space-time dimension of the effective field theory as λ = (cid:114) d − d − . (6.8)As expected, it turns out to be an order one constant.In the opposite limit at infinite distance r → Duality conjecture , in the sensethat it is predicting the existence of a duality at every infinite distance limit such that the42nfinite tower provides the new fundamental (weakly coupled) degrees of freedom of the dualdescription. In this fashion, trying to understand why the SDC is true from first principlesin Quantum Gravity is closely related to giving a reason why dualities exist in the first place.A bottom-up explanation of this feature or of the SDC is missing. Currently, the SDC justpoints out a universal feature observed in string theory compactifications, but there is noclear intuition why it should hold in general in quantum gravity. It is fair to say, though,that there is so much evidence in support of the conjecture in string theory, that the SDC iswidely accepted in the community. It has also been noted that a possible explanation arises byconnecting the Distance conjecture with the WGC and the absence of global symmetries. Thisconnection is natural when some global symmetry is restored in the limit at infinite distance,as was discussed in section 6.1. In the next section we will elaborate more in this direction.Before moving on, it is worth mentioning a refinement of the SDC regarding the natureof the tower of states, which also connects with known dualities. This is the Emergent StringConjecture [55]:
Emergent String Conjecture
Any infinite distance limit is either a decompactification limit or a limit in which thereis a weakly coupled string becoming tensionless.This means that the leading tower becoming light is either a KK tower (in some appropiatedual frame) or some string excitation modes, such that the resolution in Quantum Gravityof the EFT breaking down is given by growing an extra dimension or by considering a stringperturbation theory. The first case is associated to a T-duality while the second one fits withan S-duality.Let us show the typical example in which these two possibilities are realised in the samemoduli space. This is the case of type IIA in ten dimensions, whose action in the Einsteinframe contains S ⊃ M p (cid:90) d x √− h (cid:18) R − ( ∂s ) s (cid:19) , (6.9)where s is related to the string coupling g s and the 10d dilaton Φ via s = 1 g s = e − Φ . (6.10)We again find two infinite distance limits that correspond to strong ( g s → ∞ ) or weak ( g s → φ = √ log s .In the strong coupling limit, a tower of D0-branes becomes exponentially light with thedistance. Indeed, a D0-brane becomes massless polynomially with the string coupling as m D ∼ M s g s ∼ M p g / s ∼ M p e − √ ∆ φ , (6.11)which means that they do so exponentially with the proper field distance. Importantly, theseD0-branes are the KK modes of M-theory on S , and thus we see that this corresponds to adecompactification limit. 43ontrary, in the weak coupling limit, the string excitations are the modes becoming expo-nentially light. The mass of these states falls polynomially to 0 as g s → M s ∼ M p g / s ∼ M p e − √ ∆ φ (6.12)which again decreases exponentially with the proper field distance. We see that in this casethere is no extra dimension growing up, rather this is the perturbative string limit of Type IIA.In both cases, as dictated by the SDC, the ten dimensional quantum field theory descriptionbreaks down, and we need to either grow extra dimensions or consider a string theoreticaldescription of the theory.The motivation for this refinement of the SDC is example based. So, as with the SDCitself, there is no known quantum gravity reason why the leading towers should always be KKmodes or a weakly coupled string. For instance, one could a priori expect a limit in which thetower corresponds to a tensionless membrane. The absence of this type of limits is, though, inagreement with the lack of a perturbative description for a membrane theory, which has beenextensively searched for in the string theory community. As already mentioned in section 6.1, all limits in which a global symmetry is restored becausesome gauge coupling vanishes are at infinite field distance in moduli space. The oppositedirection has also been proposed to be true [45] although not proven in general. Interestingly,in all string theory examples studied so far, there seems to be a global symmetry restored atinfinite distance coming from a p -form gauge coupling going to zero (sometimes also an infiniteorder discrete global symmetry appears). The SDC, like the WGC, can then be understoodas a quantum gravity obstruction to restore a global symmetry.If there is a vanishing p -form gauge coupling in a certain infinite distance limit, chargedstates satisfying some tower or sublattice version of the WGC would be the natural candidatesfor the SDC tower of states becoming light. And in that case, the exponential rate λ will befixed by the black hole extremality bound [44, 45]. Note that even if this charged toweris not the leading one becoming light, it can be used to put a lower bound on the SDCexponential rate, as any other fast-decaying tower will have a bigger λ . Thus, this is a veryappealing situation in which the WGC and the SDC merges in one single statement in whichno undetermined order one factors are left.To give an example, this situation is realized in the previously discussed KK compactifica-tion on a circle of section 6.3. In this case we have a KK photon whose kinetic gauge functioncan be read from S ⊃ − M d − p (cid:90) d d x √− h e − (cid:113) d − d − φ F KK , (6.13)which supplements the Einstein-Hilbert term and the scalar kinetic term given in (6.5). Here, φ = log r with r the radius of the circle. This implies that the scalar contribution to the blackhole extremality bound in (5.45) is given by α = 2 (cid:114) d − d − . (6.14)44y plugging this into (5.45) one finds that the WGC requires the existence of some chargestates satisfying m = q r , (6.15)where q is the quantized charge and we have used that the gauge coupling for the KK photonis given by g = r M d − p . KK modes are charged under the KK photon and precisely satisfy(6.15). They also become exponentially light with the proper field distance, as shown in (6.7).Therefore, a KK tower satisfies simultaneously the WGC and the SDC with an exponentialrate given by λ = α/
2, obtained from comparing (6.8) and (6.14).This matching is actually quite general and will happen whenever the extremality condition(5.45) and the RFC (5.50) coincide, in the sense that extremal states feel no force. In thatcase, the second term on the left hand side of (5.45) is equal to the second term in (5.50),which is related to the exponential decay rate of the tower. Then, by equating both boundsfor the case of particles, one gets λ = α , (6.16)thus fixing the exponential decay rate in terms of the extremality bound. This merging of theconjectures has been seen to occur at weak coupling limits of several string theory setups [44,45]and it is expected to hold whenever there is a gauge coupling vanishing at infinite distance.If the tower of states of the SDC comes from the excitations of a weakly coupled string, theexponential rate can be fixed in a similar way by using the WGC for the two-form gauge fieldto which it couples.Hence, the generality of the lower bound for the SDC exponential factor in terms of theextremality factor in the WGC depends on whether there is always a gauge coupling vanishingat any infinite distance limit. This has been proposed to be the case in [45] and it is supportedby the string theory evidence and the Emergent String Conjecture. The latter suggests thatwe can always use either the extremality bound associated to a KK photon or to a two-formgauge field for the case of a weakly coupled string, since in both cases their gauge coupling isgoing to zero in the asymptotic limit.There is an alternative connection between the WGC and the SDC which does not requirea gauge coupling vanishing in each infinite distance limit. More specifically, this requires aversion of the WGC dubbed the Scalar WGC that does not even need the presence of a gaugetheory at all to be formulated. It was proposed in [42] and poses that gravity should be weakerthan any scalar force, even if both of them are attractive, so they are not actually competingwith each other. For a particle with mass m , controlled by the vev of some moduli, thistranslates into the following condition: Scalar WGC
Given some EFT weakly coupled to Einstein gravity with some massless scalar fields φ i , there must exist a state with mass m satisfying: g ij ∂ i m ∂ j m > d − d − m , (6.17)where g ij is the field metric and d the space-time dimension.45he above condition can be obtained from simply setting the gauge charge to zero in(5.49). Unlike the usual WGC, it is not related to any black hole extremality bound, as thereare no extremal black holes with only scalar charge by the no-hair theorem. Therefore, themotivation in terms of black hole decay is missing, although one could run a similar argumentin terms of gravitational bound states instead of black hole remnants.Notice that, in order to satisfy this condition at large field distance, the mass of the statehas to decrease exponentially as required for the SDC. Hence, using the scalar WGC one canalso argue for an exponential rate of order one. We would like to remark, though, that theevidence behind this scalar WGC is very scarce, so it is not on the same level of rigor as theusual WGC or the SDC. In fact, refinements involving second derivatives of the mass [56–60]have appeared, which suggests that the simplest form in (6.17) might not be the end of thestory, and a proper formulation of the conjecture (if any) is still a subject for further research. The evidence for the SDC comprises a plethora of examples in string theory. Those discussedin section 6.3 are so simple that the conjecture might look trivial from a string theoreticalperspective. However, in general there are many different infinite distance limits that one canengineer in more involved compactifications, and how string theory manages to always havesuch light towers of states is non-trivial. In fact, testing the SDC against different classes ofmodels has triggered research in several corners of string theory compactifications, uncoveringvery interesting connections to cutting-edge mathematics, such as algebraic geometry andmodular forms.In the following we will be discussing the setups in which most of the progress testing theSDC has been made, providing an introduction to the underlying mathematical structuresuncovered on the way. These are supersymmetric compactifications of the following types: • N = 2 from Type II. • N = 1 from M-theory and F-theory duals. • N = 1 from Type II.In all these examples the towers have been shown to satisfy both the WGC and the SDC. N = 2 Type II Calabi-Yau Compactifications
The moduli space in this class of compactifications is known to be the product of the K¨ahlerand the complex structure moduli spaces. As a consequence we can study them separately toleading order. Let us first focus on the vector multiplet moduli space, which is the complexstructure in Type IIB or K¨ahler in Type IIA.Infinite (geodesic) distance can only occur when approaching some singularity. Hence,testing the SDC reduces to studying the physics near singularities in the moduli space. In fact,it is known that there are always some
BPS states becoming massless at the singularities ofthe vector multiplet moduli space, so they are perfect candidates for the SDC tower. Note thatone can have different kinds of singularities in moduli space, and not all of them are necessarilyat infinite distance. For instance, the well-known conifold points are indeed singular but atfinite distance. Interestingly, in this case one does not find an infinite number of BPS states46ecoming light, but a finite amount of them. Then, for the singularities at infinite distancethe SDC is predicting that the number of BPS states becoming massless has to actually beinfinite, and this is not trivial at all.Following [61, 62], let us work with the complex structure moduli space of Type IIB CY compactifications. These works constitute a non-trivial test of the SDC, as they show theexistence of an infinite tower of BPS states becoming exponentially light with the distanceat any infinite distance singularity of any Calabi-Yau manifold. Notice that, upon mirrorsymmetry, the analysis translates to the K¨ahler moduli space in Type IIA [63]. First let usintroduce some background about the metric and the set of BPS states which will be relevantfor realizing the SDC.The vector multiplet moduli space of a Calabi-Yau three-fold is known to be a specialK¨ahler manifold. This means that the metric is determined by the K¨ahler potential, whichcan be written in terms of a holomorphic ( D, g I ¯ J = ∂ z I ∂ ¯ z J K , K = − log (cid:18) i D (cid:90) CY D Ω ∧ ¯Ω (cid:19) , (6.18)with D = 3 for a CY . Here, we have chosen a set of complex coordinates z I , known as thecomplex structure deformation moduli, that locally parametrise the moduli space. It is alwayspossible to find an appropiate symplectic basis of three-cycles, such that the K¨ahler potentialcan be written as K = − log (cid:0) − i D Π T η ¯Π (cid:1) , (6.19)where η is the symplectic intersection product and Π is a vector containing the periods of Ωover the basis of three-cycles, Π I = (cid:90) Γ I Ω . (6.20)BPS particles can be obtained by wrapping D3-branes on special Lagrangian three-cycles Γ I . Their mass is given by its central charge, which is again determined by the periods as M = | Z | = e K/ | q T η Π | . (6.21)Here, q is the vector of quantized charges of the BPS particle with respect to the gauge fieldscoming from the dimensional reduction of the Type IIB RR 4-form field, A I = (cid:90) Γ I C . (6.22)One could now start testing the SDC case by case, computing the periods Π in differentCalabi-Yau manifolds, but given the incredible amount of them this does not seem like thebest way of gathering evidence for this conjecture. Fortunately, we can do better and give asystematic approach which is valid for any Calabi-Yau, and may be even more general thanthat. In order to do so, we will be using some powerful tools of algebraic geometry and BPScounting. Namely, we will use theorems of limiting Hodge theory to describe the behaviourof the periods near any infinite distance loci, and the study of walls of marginal stability tocount the number of BPS states becoming light. In the IIA mirror, they correspond to bound states of D0 and D2-branes wrapping two-cycles. z I , for which thesingularity is located at z i → i ∞ , the periods transform asΠ( ..., z i + 1 , ... ) = T i Π( ..., z i + 1 , ... ) , (6.23)while the K¨ahler potential remains invariant. From the point of view of the effective fieldtheory, z i → z i + 1 is a discrete shift of an axion, so the monodromies correspond to axionicshift symmetries in the EFT.The monodromy around a singularity can be of finite or infinite order. That is, after goingaround the singularity several times we either get back to the same point, so T n = I for some n >
0; or this is not possible, so T n (cid:54) = I for any n >
0. We will see that infinite distancesingularities are necessarily related to infinite order monodromies.If the monodromy is of infinite order, one can define a non-trivial nilpotent operator bytaking N i = log T i . In this situation, the Nilpotent Orbit Theorem of Schmid [64] gives anasymptotic expansion of the periods near the singular locus such that they are well approxi-mated by the following nilpotent orbitΠ( z ) = e z i N i Π + O ( e πiz i ) , (6.24)up to exponentially suppressed corrections, where Π only depends on the complex coordinateswhich are not sent to infinity.For simplicity, let us now stick to the case of a single modulus z → ∞ . By plugging (6.24)into (6.19) and using that N T η = − ηN , we can see that K = − log( p d (Im z ) + O ( e πiz )) nearthe infinite distance locus, where p d (Im z ) is a polynomial of degree d . This further impliesthe following leading term for the metric, g z ¯ z = d z ) . (6.25)Here, d is the effective nilpotency order defined as N d Π (cid:54) = 0 , N d +1 Π = 0 , (6.26)which is bounded by the complex dimension of the Calabi-Yau, so in the case at hand it cantake values d = 0 , , ,
3. Clearly, only for d (cid:54) = 0 the singularity will be at infinite distance,so an infinite order monodromy is a necessary condition for infinite distance. This nilpotencyorder can be used to classify different asymptotic limits and, interestingly, it also fixes theexponential rate of the tower for the SDC. Indeed, one can show that there is always aninfinite tower of BPS states behaving as required by the SDC and whose exponential rate isbounded by 1 √ d < λ < √ d . (6.27)Since d ≤ λ ≥ √ . (6.28)48he exponential behaviour of the mass is a consequence of the asymptotic behaviour of themetric. By plugging (6.24) into (6.21) the BPS mass behaves as a polynomial in the coordinates z i which, by using (6.25) to canonically normalise the scalars, depends exponentially on theproper field distance ∆ φ = (cid:114) d z . (6.29)The role of the monodromy transformations does not end here. The existence of an infinitetower of states (and not only a finite number of them) can be intuitively explained by it.The monodromy transformation acts on the charge of the BPS states, generating an orbit ofcharges q n = T n q . Monodromies are redundancies of the theory, which implies that even if thecharge of a single state varies, the full tower must reorder itself and remain invariant under themonodromy so that the physics does not change. In fact, these monodromies become actualglobal symmetries at infinite distance. Hence, the existence of a single state implies that theentire orbit must be populated by physical BPS states, which yields an infinite tower if themonodromy is of infinite order. In this way, the fact that there is an infinite tower of BPSstates tracks back to the monodromy transformations around infinite distance singularitiesbeing of infinite order. This intuitive reasoning must be accompanied by the analysis of wallsof marginal stability and a careful check of the existence of the seed charges q , but we referthe student interested in the details to the original paper [61]. The generalization to multiplemoduli limits can be found in [62].In this scenario we can relate the SDC with the WGC as already discussed in section 6.4.We find that in these limits there is always a gauge coupling going to zero for some gauge field A I of (6.22) and that the bound in (6.27) can be related to the black hole extremality bound.Hence, the SDC tower also saturates the WGC. These conjectures can then be understood asquantum gravity obstructions to restore some global symmetries at infinite distance, comingboth from the vanishing gauge couplings and from the monodromies, which can be promotedto continuous global symmetries of the theory at infinite distance.After having studied the vector multiplet moduli space, we are left with the hypermultipletmoduli space. In what follows, let us give a very brief review of the results that have beenobtained, referring to [65, 66] for more details. In this case, one does not find a tower of BPSstates but KK towers, charged strings from wrapping branes and instantons becoming light.Importantly, the fact that some instantons become light (in the sense that their action goes tozero) means that there could be relevant quantum corrections to the field metric, which couldeven obstruct certain infinite distance limits from being taken. What typically happens thenis that the trajectories approaching such limits get deviated towards another one in whichthere is a weakly-coupled string becoming light at the same rate as a KK tower, T string M p ∼ M KK M p → . (6.30)Since the KK modes have a less dense spectrum than the string excitation modes, i.e. m KK ∼ k M KK vs m ∼ kT str for k ∈ N , the leading tower fulfilling the SDC is given by stringexcitation modes in these limits. 49 .5.2 5d N = 1 M-theory and 6d F-theory Duals
We consider the K¨ahler moduli space of M-theory compactified on a Calabi-Yau three-fold,and their 6d F-theory duals. In this setup, the SDC is realized by a tower of particles comingfrom wrapping M2-branes on two-cycles, which corresponds to strings becoming tensionlessfrom the F-theory perspective.There have been two approaches to analyse and classify the possible infinite distance lim-its in this setup. One approach consists in borrowing the classification of singular limits onCalabi-Yau three-folds used to study 4d N = 2 theories in section 6.5.1 and applying it tocompactifications of M-theory to 5d on these spaces. Since the geometry is the same, the clas-sification based on the properties of the monodromies still applies, and only the microscopicinterpretation changes. Now the monodromies will not be related to shift symmetries of theaxions but to large gauge transformations of the one-form gauge fields obtained from reducing C on the two-cycles. They will act on the M2-brane charges, populating an infinite tower ofparticles.The other approach involves an equivalent classification of these singularities in terms ofthe fibration structure that the Calabi-Yau three-fold develops in the limit. This was workedout in [31, 34, 55, 67], where it was shown that the possible fibrations are T , K T .The limits studied in this series of works involve shrinking some of these fibers. Interestingly,this classification translates into certain properties of the intersection numbers which can bemapped one-to-one to the properties of the monodromy transformations and the previousclassification of singularities. In the following, we will focus on this second approach andexplain the appearance of the infinite tower from the F-theory perspective. To illustrate themain features we will restrict to the case of a K3 fibration.Consider F-theory compactified on an elliptically fibered CY . The 6d Planck mass is givenby M p = 4 π Vol( B ) , (6.31)where Vol( B ) is the volume of the base of the fibration. The theory contains a gauge theorysector coming from 7-branes wrapping some divisor of the base, C . The associated gaugecoupling is given by 1 g Y M = 12 π Vol( C ) . (6.32)The limit Vol( C ) → ∞ is a weak coupling limit at infinite distance in moduli space. There arenow two possibilities. It could be that, when taking the volume of the divisor to be large, theoverall volume of the base also goes to infinity, leading to a decompactification limit and thecorresponding tower of KK modes. The second possibility is to engineer an equi-dimensionallimit, in which Vol( B ) remains finite even if Vol( C ) → ∞ . To make this geometrically feasible,it can be proven that there is always another curve C that intersects C and whose volume isgoing to zero. D3-branes wrapping this shrinking curve will therefore give rise to strings thatbecome tensionless in the weak coupling limit, T ∼ Vol( C ) → . (6.33)For the SDC to hold it is still necessary to show that the string spectrum contains infinitelymany excitation modes in this limit. In this present case it is easy to show this by using In the K¨ahler moduli space, different properties of the monodromies, like the effective nilpotency order,translates into different patterns of vanishing and non-vanihising intersection numbers [63]. m n = 8 πT ( n −
1) (6.34)with j = C · C and n the excitation level of the string. In fact, one obtains a sublattice ofcharges with index 2 j such that for each charge q k = 2 jk , k ∈ Z , (6.35)there exists a state at excitation level n ( k ) = jk . These states satisfy g Y M q k = m k + 4 jg Y M > m k , (6.36)in agreement with the WGC. Therefore, one gets a sublattice of non-BPS states satisfying theSDC and the strict inequality of the WGC at the same time. This provides evidence for bothconjectures in the case in which the states are not necessarily supersymmetric, and it is alsoconsistent with the sharpened WGC in section 5.8. The analysis can also be extended to 4d N = 1 theories obtained from F-theory on Calabi-Yau four-folds [34].These works are again an example in which the swampland conjectures have served touncover the underlying geometric structures of string compactifications and new relations tomathematics. In particular, this has triggered research on modular forms, yielding interestingnon-trivial connections between algebraic geometry of Calabi Yau manifolds and modularproperties of quasi-Jacobi forms that allow the WGC and the SDC to hold. N = 1 Effective Field Theories
In this case there are no BPS particles in the spectrum, and that makes it more difficult to keeptrack of the states that could satisfy the SDC at the boundaries of moduli space. However,there are BPS charged strings and membranes that can be shown to become tensionless inthese limits [69]. Even though they are not necessarily the leading towers, they can still giveus valuable information about the physics at these boundaries [54, 70].The strings are charged under two-form gauge fields that are dual to axions in 4d. Thegauge kinetic function for the two-forms is equal to the inverse of the axionic field metric,which in N = 1 can be given in terms of the K¨ahler potential, g ij = ∂ K∂s i ∂s j . (6.37)Here, s i are the saxions which, together with the axions φ i , make up the complex scalars ofthe chiral multiplets. Thus, information about the gauge kinetic terms of the two-forms underwhich the strings are charged gets translated into information about the metric in modulispace. In the string theory setups analysed, the leading tower either correspond to KK modes (in some appropiatedual frame) or excitation modes of these BPS strings, in agreement with the Emergent String Conjecture.
51n the same way, membranes are charged under three-forms and their gauge kinetic function T ab is related to the scalar potential as follows, V = 12 T ab ( s, φ ) f a f b = 12 Z ab ( s ) ρ a ( f, φ ) ρ b ( f, φ ) , (6.38)where f a are the discrete internal fluxes dual to the three-form gauge fields. Interestingly,the flux potential of string compactifications can always be factorized between a saxionic anaxionic part, where the ρ a functions are shift invariant axion polynomials including the fluxes,see e.g. [71, 72]. Therefore, studying the behaviour of the three-form gauge couplings andcharges of BPS membranes in the asymptotic limits provides information about the scalarpotential.In conclusion, the asymptotic behaviour of the field metric and the scalar potential canbe translated into properties of BPS strings and membranes charged under two-form andthree-form gauge fields, respectively, in 4d N = 1 EFTs. This yields interesting relationsbetween different Swampland conjectures [54]. For instance, the exponential behaviour of theSDC tower becomes a consequence of having a BPS string satisfying the WGC. This allowsus to provide a lower bound for the SDC exponential rate in 4d N = 1 EFTs in terms ofthe extremality factor for the strings. Similarly, a potential described by a WGC-saturatingmembrane satisfies the de Sitter conjecture that will be explained in section 9. Up to now, the SDC has been a statement about the moduli space of the theory. Thismeans that it regards a set of scalars with an exactly flat potential (typically protected byextended supersymmetry) that parametrizes different vacua of the theory. However, it isphenomenologically relevant, e.g. for inflation, to understand what happens when a potentialis added so that this moduli space is lifted. Based on physical grounds, one would expect thatthe SDC should also apply to the valleys of the potential, i.e. to directions along which thepotential may not be exactly flat but the relevant energies are smaller than a given cut-off.The refined SDC [73] proposes that:1. The exponential behaviour with λ ∼ O (1) should be manifest when ∆ φ (cid:38) M p .2. The conjecture should also hold for scalars with nearly flat potential.This can have important implications for inflationary models. Consider the bound for thefield distance in terms of the cut-off in (6.4). In order to accommodate inflation in the effectivefield theory we need the cut-off to be above the Hubble scale, Λ > H . These two conditionstogether yield ∆ φ ≤ λ log (cid:18) M p H (cid:19) . (6.39)Thus, the SDC gives an upper bound on the field range of inflation in terms of the Hubblescale. In particular, when H is close to the Planck scale one finds that the field range isbounded by an order one number in Planck units. The SDC also places important constraints on relaxation models of the EW scale, since they typicallyrequire transplanckian field ranges.
52n slow roll inflation, the field range provides an upper bound on the tensor-to-scalar ratio r via the Lyth bound, ∆ φ ≥ (cid:16) r . (cid:17) / . (6.40)The tensor-to-scalar ratio can be related to the Hubble scale as M p H = (cid:114) π A s r , (6.41)where A s is the amplitude of scalar perturbations whose value has been measured experimen-tally to log(10 A s ) = 3 . ± .
015 [74]. Using (6.41) we can write the SDC bound in (6.39)in terms of r as follows, ∆ φ ≤ − λ (cid:18) log (cid:18) π A s (cid:19) + log r (cid:19) , (6.42)so it provides a lower bound on the tensor-to-scalar ratio. When written in this way, we notethat the Lyth and the SDC bounds are complementary and can constrain different models ofinflation [75]. To give an example, chaotic inflation [76] would be ruled out. This is shown infigure 20.Figure 20: Left: Allowed (blue) region of scalar field excursion during inflation against tensor-to-scalar ratio obtained by combining Lyth bound and SDC constraint for λ = 1, togetherwith the experimental bound put by Planck 18 [77]. The red point corresponds to chaoticinflation [76]. Right: SDC bound for λ = 0 . , , λ .In conclusion, we see that the SDC can constrain some models of large field inflation(including axion monodromy) but it does not rule it out, as a moderate transplanckian fieldrange might suffice for some models. Notice that the SDC constraint on large field inflationhighly depends on the exact value of λ . For this reason, it is really important to learn aboutthis exponential decay rate and how it can be computed from first principles or effectivefield theory data. It should also be noted that replacing H as the cut-off in (6.39) gives avery conservative bound, in the sense that the EFT will likely break down (or at least getsensitive to the infinite tower) before the mass of the first state becomes of order Hubble, sothe constraints might be stronger than represented here.53he implications of the SDC do not end here. Remember that the SDC applies to geodesicsin moduli space but, with the inclusion of a potential, the trajectory followed during inflationis not necessarily a geodesic from the perspective of the UV moduli space. However, it stillmakes sense to apply the SDC if it corresponds to a geodesic from the perspective of the IRpseudo-moduli space generated by the valleys of the potential. In such a case, for the SDCto hold, it should be impossible to generate a potential such that the trajectory is sufficientlynon-geodesic so that the exponential behaviour of the tower is violated [78]. This is to say, byrequiring consistency of the SDC at any energy scale, we can put constraints on the type ofpotentials that one can engineer in quantum gravity!The main evidence that we have for this statement, and that motivated the Refined SDC,comes from Calabi-Yau flux compactifications in string theory. Since the theory is 4d N = 1,the scalars come in chiral multiplets including an axion and its partner, the saxion. Beforeadding the scalar potential, purely saxionic trajectories are geodesics of the moduli space.When adding the flux potential, both saxions and axions get stabilized. One could think thata way of avoiding any tower of states in inflation could be by moving only along an axionictrajectory, while keeping the saxion fixed. This way, one would engineer an axion monodromymodel with a highly turning trajectory and very large field range, see figure 21. However, thestructure of the flux potential coming from string theory is such that the minimization of thepotential in the saxionic directions implies ∂ s i V = 0 = ⇒ s i = ˜ λφ i + · · · (6.43)for large φ i . That is, when trying to move in an axionic direction, the potential inducesa linear backreaction on the saxion. So one cannot move along large values of the axionwithout displacing the saxion in the same way. This implies that if there was a tower decayingexponentially with the saxionic distance, it will also decay exponentially upon taking intoaccount the backreaction in the axion monodromy model.Figure 21: Green: purely saxionic (geodesic) trajectory. Blue: a highly turning trajectory, likein axion monodromy. The black dot represents a locus at infinite distance, while the radialand angular coordinates correspond to the saxion and axion scalars respectively.This backreaction issue was first introduced in [79] and further checked in [80–82]. In thecontext of F-theory compactifications on Calabi-Yau four-folds with fluxes, this backreactionwas proven to hold more generally [82] due to an asymptotic homogeneity of the potential V ( αs i , αφ i ) (cid:39) α d i V ( s i , φ i ) (6.44)54ealised near the infinite distance loci. Finally, we are going to very briefly mention a generalization of the Distance conjecture toother field space configurations beyond the moduli space. In particular, one can define a notionof distance between different metric configurations of AdS spacetime. As discussed in [83],the flat spacetime limit Λ → →
0, one gets the following conjecture:
Conjecture 7: AdS Distance Conjecture
Any AdS vacuum has an infinite tower of states that becomes light in the flat spacelimit Λ →
0, satisfying m ∼ | Λ | α . (6.45)A strong version of the conjecture implies α = if the vacuum is supersymmetric and α ≥ for non-susy AdS and α ≤ for dS space. A consequence of the strong versionof the conjecture is that a supersymmetric d -dimensional AdS vacuum cannot exhibit scaleseparation between the AdS scale and the cut-off of the d -dimensional theory (typically givenby the KK scale), as the associated scale of the tower is of order the AdS scale Λ / . Mostof the AdS vacua constructed in string theory satisfy this conjecture, as they do not havescale separation between the internal dimensions and the AdS length. However, there isone exception associated to Type IIA compactified on a particular limit of a flux orientifoldCY [84], which has captured a lot of recent attention since the vacuum is in principle atparametric control, although its validity is still under debate (see e.g. [85, 86]). In any case,this latter example would only be a counterexample to the strong version of the conjecturebut not to (6.45). There are a number of interesting open questions about the SDC: • All the evidence for the conjecture comes from string theory compactifications. Hence,a botom-up explanation for the conjecture is missing. • The existence of the towers is linked to the manifestation of dualities at the asymptoticlimits. Why are dualities ubiquitous in string theory? Can we map the geometricclassification of infinite distance limits to different types of dualities? • SDC in AdS/CFT, see e.g. [87] and [88]. • Can the SDC always be understood as a quatum gravity obstruction to restore a globalsymmetry at infinite distance? Taking a background metric and considering perturbations around it, we will have a metric configurationspace on which we can define distances. More concretely, is there always some gauge coupling going to zero at infinite distance inmoduli space? If so, there is a WGC tower satisfying the SDC, which would bound theexponential rate, see [45], and would serve as a quantum gravity obstruction to restorea global symmetry. • What is the specific value of the exponential rate? Can we compute it or bound it fromfirst principles or effective field theory data? • Better understand the constraints on scalar potentials coming from the Refined SDC. • What is the nature of the leading tower? Could it ever originate from p -branes with p ≥
2? This would disprove the Emergent String Conjecture. • If the scalars have a non-trivial profile in the non-compact space-time dimensions, doesthe EFT also break down at large distances and how? This is known as the Local SDC,see e.g. [73, 89]. • What are the cosmological signatures of the tower during inflation or late-time cosmol-ogy?
The so-called Emergence proposal [2, 16, 62, 63, 90, 91] provides a general rationale for why theSwampland conjectures, especially the WGC and the SDC, should hold true in any theory ofquantum gravity. Recall that these two conjectures relate the kinetic terms (i.e., gauge cou-plings, field metrics, etc.) to the mass of some new states. But it is well known that relationsbetween the mass of heavy states that are integrated out and couplings of the low energyeffective field theory Lagrangian are already guaranteed by the renormalization group equa-tions. So could it be that the relations provided by the Weak Gravity and Distance conjecturesare simply due to the renormalization group equations upon integrating out the WGC- andSDC-satifying states? The answer would be affirmative according to the Emergence proposal.
Emergence Proposal
All the kinetic terms in an EFT emerge from integrating out the massive states up tosome quantum gravity cut-off [2, 61].This implies that all fields are non-dynamical at the quantum gravity scale where gravitybecomes strongly coupled, hinting at some sort of UV topological description. That is, thereis no kinetic term to start with. It is only upon going to the IR and integrating out thetower of states predicted by the Weak Gravity and/or Distance conjecture, that we get somefinite kinetic terms. We will see that when the tower becomes light, it generates small gaugecouplings and parametrically large field distances. In other words, in the UV there are no tinygauge couplings or infinite distances; these, as well as the approximate global symmetries thatcome with them, are just artifacts of the IR description.A weaker version of the proposal [90, 91] allows for a classical piece of the kinetic termsof the same form as the quantum corrected piece, so the IR value is not purely coming fromquantum corrections. In that case, at the quantum gravity scale the fields are still dynamical56ut a sort of unification occurs, in the sense that gravity and the other gauge and scalar forcesbecome strongly coupled at the same quantum gravity cut-off scale.We started the section by promising that the emergence proposal provides a bottom-upexplanation for the Swampland conjectures. The WGC and the SDC imply the presence of aninfinite tower of states getting light, whenever the EFT allows for very small gauge couplingsand/or very large field distance (how light depends on how small the gauge couplings or howlarge is the field distance). The emergence proposal goes the other way around by saying thathaving an infinite tower of states becoming light at some point in the field space generates theinfinite field distance and the small gauge coupling there, see figure 22.Figure 22: Comparison of the emergence proposal and the WGC/SDC.We can perform computations in a field theory toy model to exemplify how the infinitetower can indeed generate the small gauge coupling and the infinite field distance. Considerthe IR EFT resulting from integrating out a tower of states charged under some gauge field,for simplicity we stick to the 4d case. We should integrate out the tower up to a speciesbound Λ QG , as in (5.34), which acts as a QG cut-off above which quantum gravitationaleffects becomes important and QFT computations are not reliable any more. Using (5.34)and (5.35), the number of states N of the tower running in the loops can be written purely interms of the mass gap ∆ m as follows, N = (cid:18) ∆ mM p (cid:19) − . (7.1)Computing the one-loop correction from integrating out the tower of states, the IR quantum-corrected gauge coupling is given by1 g = N (cid:88) k =1 q k log (cid:16) Λ m k (cid:17) (cid:39) N (cid:88) k =1 k log N k (cid:39) N (cid:39) m ) , (7.2)where m k = k ∆ m and q k = k are the mass and charge of the k -th state of the tower, and Λ QG is the species scale in (5.34). This result implies that the mass of the states that we integratedout is proportional to the renormalized gauge coupling in the IR, i.e. m k (cid:39) k g IR , (7.3)and this is precisely the WGC! 57t should be emphasized that, in order to obtain this result, it is essential that we aresumming an infinite tower up to its species bound. The dependence of the cut-off and N on∆ m qualitatively changes the result, transforming the logarithmic behaviour on the mass intopolynomial in (7.2). In contrast, we could not reach this conclusion by integrating out only afinite number of states and having a constant cut-off. This motivates the stronger versions ofthe WGC in which there is not only one but an infinite number of states.A similar calculation can be done for the Distance conjecture. If the mass of the tower ofstates is parametrised by some scalar field, quantum corrections from integrating out the towerwill modify the field metric of the scalar. The one-loop contribution to the scalar propagatoris given by g φφ = N (cid:88) k =1 (cid:0) ∂ φ m k (cid:1) = (cid:0) ∂ φ ∆ m (cid:1) N (cid:88) k =1 k (cid:39) (cid:0) ∂ φ ∆ m (cid:1) N (cid:39) (cid:16) ∂ φ ∆ m k ∆ m k (cid:17) , (7.4)where m k is the mass of the particle of the tower running in the loop. As before, it is importantto sum up to the species bound (7.1) where quantum field theory computations are not reliableany more. The proper field distance measured by using this renormalized field metric reads∆ φ (cid:39) (cid:90) φ φ √ g φφ (cid:39) log (cid:18) ∆ m ( φ )∆ m ( φ ) (cid:19) . (7.5)This gives rise to the exponential behaviour of the tower predicted by the SDC in terms ofthe proper field distance. If the full tower becomes massless at some point φ of the modulispace, so ∆ m ( φ ) = 0, then the field distance becomes infinite. Again, this only occurs uponintegrating out an infinite tower up to its species scale, since a finite number of fields willalways generate a finite distance. So from this point of view it is the tower itself that isresponsible for the infinite distance.Although these are just toy model computations, the emergence of the WGC and SDCbehaviours arising from integrating out the towers of states serves as motivation for the Emer-gence proposal. Unfortunately, the estimations of the species bound are not precise enough totrust the resulting numerical factors. There are some string theory setups, though, in whichthese toy model computations actually provide a good approximation of the physics and theEmergence proposal can be checked. Consider the type IIB Calabi-Yau compactifications ofsection 6.5.1. It is known that the geometry near the conifold singularity in the complexstructure moduli space can be reproduced by computing the quantum corrections of the BPSstate becoming massless at the singular point [92]. Thanks to supersymmetry, it is enoughto compute the one-loop correction to the gauge coupling and the field metric to reproducethe geometric result. Hence, the complex structure moduli space is quantum in nature, in thesense that it incorporates the effects from integrating out the BPS states. The singularities arethen generated because one is ‘incorrectly’ integrating out some BPS states that are actuallymassless. Analogously, one can compute the quantum corrections from integrating out thetower of BPS states near the infinite distance singularities and check that the resulting gaugecoupling and field metric in (7.2) and (7.4) matches with the results obtained from the ge-ometry. This was done in [62], providing evidence for the Emergence proposal. The difficulty Quantum corrections from integrating out only a finite number n of fields would generate a finite fielddistance given by ∆ φ (cid:39) n (cid:82) ∂ φ m = n ( m ( φ ) − m ( φ )) < ∞ . The conjecture first appeared a few years ago in [46, 93]. While [46] focused on the non-trivialcase of AdS spacetime, it was emphasized in [93] that it should apply to any non-susy vacuum.
Conjecture 8: No Non-susy Stable Vacuum
Any non-supersymmetric vacuum is at best metastable and has to decay eventually.In other words, supersymmetry is the only mechanism to protect a vacuum from decayingin quantum gravity. Typically, it is quite common to generate some instabilities when breakingsupersymmetry. But it is highly non-trivial to propose that this should always be the case; inparticular, it is false for a QFT in the absence of gravity.The instability can be perturbative or non-perturbative. In the latter case, it is called ametastable vacuum, as the decay rate is exponentially suppressed, see figure 23. The conjecturedoes not give any bound on the decay rate, so the vacuum can be very long-lived. Hence,the implications of this conjecture for dS could be negligible (however see the next section forstronger conjectures about dS vacua). On the other hand, the implications for AdS are huge,as the existence of the boundary implies that a mestastable AdS vacuum cannot live longerthan a Hubble time. We will discuss the implications for AdS/CFT in section 8.3. RegardingMinkowski vacua, they could in principle also be very long-lived, although it is actually notclear whether non-susy Minkowski vacua even exist as they require an extreme fine-tunningto keep a vanishing cosmological constant.Figure 23: Potential with a metastable vacuum (the left one). It can decay non-perturbativelyto the lower energy minimum on the right, which presumably should be supersymmetric ifstable.
The original motivation for the conjecture comes from the WGC, as it provides a decaymode for non-susy vacua supported by fluxes. Consider an AdS vacuum supported by fluxes,59eaning that some of the scalars are stabilized by a scalar potential, which is generatedby these fluxes. Specifically, fluxes can originate from gauge fields with field strength F p propagating in the extra dimensions, so the fluxes f are given by the integrals (cid:90) Σ p F p = f (8.1)over some internal non-trivial p -cycle. These fluxes induce a scalar potential which can stabilizethe scalars in a particular minimum, say an AdS vacuum. The fluxes are Hodge dual to topform gauge field strengths F d = d C d − where d is the spacetime dimension. We can now applythe WGC to the gauge field C d − which implies the existence of some electrically charged( d − The WGC requiressuch a brane to satisfy T ≤ QM p , (8.2)where T denotes the brane tension. These branes are magnetically charged under the fluxesand interpolate between different vacua with different values for the fluxes. Suppose that thevacuum state on one side of the domain wall has flux f . Then the charge conservation lawdetermines the flux of the vacuum state on the other side of the wall to be equal to f + q ,where q is the quantized charge of the brane (see figure 24). The presence of these domainwalls is somewhat expected in a flux landscape, but the novelty of the WGC is the upperbound on the tension. domain wall with charge vacuum with flux vacuum with flux Figure 24: A charged brane interpolates between vacua with different fluxes.Now, let us assume that the sharpened version of the WGC discussed in section 5.8 isvalid. If the vacuum is not supersymmetric, then the sharpening of the WGC requires thestrict inequality
T < QM p . But a codimension one brane with a tension smaller than itscharge corresponds to an instability in AdS! It is not necessary that all scalars are stabilized by fluxes, but rather changing the flux changes the vacuumwe are in. Here we applied WGC to the non-dynamical gauge fields dual to the fluxes in the lower dimensional EFT.Equivalently, from the higher dimensional perspective, we can apply the WGC to the magnetic duals (cid:63)F p ofthe gauge fields in the extra dimensions. It implies that there should be a charged ( D − p − D is the total spacetime dimension. Then, dimensional reduction of these branes on the dual cycles to Σ p gives( d − AdS with less flux
Figure 25: Bubble nucleation of an AdS vacuum with less flux inside. It will expand mediatingvacuum decay.More concretely, one can construct the instanton solution that mediates the vacuum decay.For this purpose, consider the following parametrization of the AdS d metric, ds = L (cosh r dτ + dr + sinh r d Ω d − ) (8.3)with L the AdS radius. Following [94], the instanton solution corresponds to a spherical braneof action S = L d − Ω d − (cid:90) dτ T sinh d − r (cid:115) cosh r + (cid:18) drdτ (cid:19) − Q sinh d − r . (8.4)Upon solving the euclidean Einstein’s equations, one gets that the radius for the sphericalbrane solution is given by tanh R = TQ , (8.5)in Planck units. In Lorentzian signature, this will be associated with the nucleation of abubble of radius R that will continue expanding at the speed of light. The strict inequalityon the WGC condition (8.2) implies tanh R < , (8.6)which admits solutions for finite values of the radius. On the contrary, if the vacuum issupersymmetric, we could satisfy the WGC by a BPS brane saturating the inequality (8.2),implying tanh R = 1 , (8.7) This computation was performed in the probe/thin wall approximation, neglecting e.g. the backreactionon the scalar fields. Although this is a good approximation far away from the object, it would be desirable tohave a more complete analysis including the scalar contribution to the WGC bound.
We have seen that the sharpened WGC provides a decay mode for non-susy vacua withfluxes. But what if there are no fluxes and, therefore, we cannot use the WGC? In principle,conjecture 8 should apply to any non-susy vacuum. It would be very interesting then to lookfor an universal instability whenever supersymmetry is broken, regardless of the ingredientsof the theory.A potential candidate for a universal instability, whenever there are extra dimensions, is a bubble of nothing . This was first constructed by Witten in [95], and we have recently learnedthat these instabilities are far more common than thought [96].
Bubble of Nothing
A bubble of nothing is a non-perturbative instability mediating the decay from thevacuum to nothing, i.e. the vacuum annihilates itself. This occurs when a compactdimension collapses to zero size at the bubble wall.Hence, it is literally a bubble with nothing inside (not even a spacetime) that pops up inthe vacuum and starts expanding at the speed of light, leaving nothing behind. Expressedthis way, it may sounds very exotic. However, it is a perfectly well defined smooth solution tothe equations of motion. The trick is that some extra dimension collapses to zero size at thebubble wall, allowing for the full solution to be regular and geodesically complete from thehigher dimensional perspective.The existence of bubbles of nothing was first brought up by Witten [95] in the case of aKaluza-Klein circle compactification of five dimensional Einstein gravity to four dimensions.The circle shrinks to zero size when approaching the bubble wall, as represented in figure26. More generally, we can have bubbles of nothing for higher dimensional compact spaces aslong as the compact dimensions are allowed to collapse. Bubbles of nothing have e.g. beenconstructed for
AdS × S / Γ with freely acting Γ and
AdS × C P , where no other decaymode was previously known, providing evidence for conjecture 8.Consider some compactification M D × C d with C d the compact space and M D the space-time. Whether this compact space is topologically allowed to shrink to zero size gets translatedto the question of whether it can be written as the boundary of another manifold of onedimension higher, i.e. C d = ∂B d +1 . For Witten’s case with C = S , the circle is theboundary of a disk (figure 26.a). In general the compact manifold will be a boundary if itbelongs to the trivial class of the cobordism group Ω d (figure 26.b). Recall that the bordismgroup was defined in section 3.5, so we refer the reader there for details. In a nutshell, wesay that two manifolds are bordant to each other if together they form a boundary of anothermanifold, and a manifold which is a boundary by itself belongs to the trivial class. Manifoldsof this latter type admit bubbles of nothing at the topological level. This can be summarized62 ubble of nothing Figure 26: (a) The bubble of nothing is expanding at the speed of light. The internal compactmanifold shrinks to zero size on the bubble wall. Here S is an example of internal compactmanifold that belongs to the trivial class, i.e. S = ∂D . (b) A compact manifold C d = ∂B d +1 that belongs to the trivial bordism class Ω d , can become a bubble of nothing.as follows:Compact manifold C d can shrink to zero size ⇐⇒ C d = ∂B d +1 ⇐⇒ C d belongs to trivial class of Ω d ⇐⇒ A bubble of nothing is topologically allowed.The euclidean instanton solution mediating the decay is a warped product of the bordismmanifold over a ( D − M BON = B d +1 × W S D − . (8.8)The decay rate is exponentially suppressed by the euclidean action, i.e. Γ ∼ e − S BON , where S BON can be computed to be proportional to the volume of the compact space C d at spatialinfinity.For the case of a KK circle compactification with fermions, C d = S preserves a spinstructure. We then have Ω spin1 = Z , implying that only the circle with antiperiodic boundaryconditions for the fermions belongs to the trivial class on bordism and admits a bubble ofnothing. Contrary, the circle with periodic (SUSY-compatible) boundary conditions cannotdecay via a bubble of nothing, even if the vacuum breaks supersymmetry.However, this topological obstruction can be absent in higher dimensions since e.g. Ω spin d =0 for d = 3 , , ,
7. Therefore, any spin manifold of these dimensions admits a bubble ofnothing! This was shown in [96] and used to construct new types of bubbles of nothing for atoroidal T compactification with periodic boundary conditions. This is very interesting as itimplies that a bubble of nothing can exist even if the boundary conditions are periodic andsupersymmetry is restored at high energies (as long as the vacuum breaks susy spontaneously).If the compactification includes other ingredients, like fluxes or charged fermions, theseneed to be added to the bordism group, which we will label in general as Ω d . This is wherethe relation to other Swampland conjectures arises. Recall from section 3.5 that a Swampland63onjecture was proposed in [15] stating that the cobordism group of a consistent theory ofquantum gravity must be trivial, i.e Ω QG d = 0, in order to avoid the presence of global sym-metries. Hence, the theory should always include the necessary defects or ingredients in orderto guarantee the vanishing of all cobordism classes. As explained in [96], this conjecture hasan important consequence for the discussion at hand, as it implies that all compactificationmanifolds belong to the trivial class of the bordism group. Therefore, any internal compactmanifold of a consistent QG compactification can shrink to a point, implying that there is notopological obstruction to construct bubbles of nothing in quantum gravity .It is very important to remark, though, that the fact that a bubble of nothing is topologi-cally allowed, does not mean that it will expand after nucleation, describing an instability. Forthis to happen, the decay must be dynamically allowed, in the sense that it must be energeti-cally favourable to decay. Otherwise, it will either collapse after nucleation or it will describe aflat domain wall instead of a bubble instability, as happens for a vacuum that preserves super-symmetry. In the latter case the radius of the euclidean solution will be infinite, describing anend-of-the-world brane. This dynamical obstruction is tied to whether a certain local energycondition is satisfied [96]. For instance, in the case of a manifold with covariantly constantspinors, the bubble of nothing will be dynamically allowed and the vacuum will decay onlyif the Dominant Energy Condition is violated. Interestingly, this condition is genericallyviolated when the vacuum breaks supersymmetry. For a theory with charged fermions, theenergy condition is modified and resembles the WGC. By studying the relation between theseenergy conditions and supersymmetry in more detail, we could end up proving or disprovingthe conjecture about the instability of any non-susy vacuum.
Here, we are going to discuss two implications of the AdS instability conjecture.Non-susy AdS/CFT:An instability in AdS, even if highly suppressed, has dramatic implications for the bound-ary CFT. Note that an observer at the boundary has access to an infinite volume of the bulkand hence can detect any bulk instability instantaneously. This implies that the AdS vacuumhas no dual unitary CFT. From the gravity side, the instability will reach the boundary andcome back in Hubble time, so this is the maximum lifetime of the vacuum.This can have important implications for non-susy holography. Notice, though, that theprevious discussion about AdS instabilities was performed in the context of a weakly coupledEinstein gravity theory, in which a semiclassical description of bubble instabilities is possibleand the WGC applies. Consequently, non-susy holography can be consistent but only if thenon-susy unitary CFT is dual to a gravity theory which is not AdS weakly coupled to Einsteingravity. In other words, a non-susy unitary CFT cannot admit a large N expansion and/or alarge gap to single-trace higher spin operators (spin greater than two).Compactifications of Standard Model of particle physics: The Dominant Energy Condition requires the vector − T µν n ν > n µ and T µν being the stress-energy tensor. in terms of the Dirac mass of the lightest neutrino m ν [97], whichroughly goes as Λ (cid:38) m ν . (8.9)Interestingly, this could explain the numerical coincidence Λ ∼ m ν observed in our universe.Furthermore, we can write the neutrino mass in terms of the Higgs vev, so that the previousbound gets translated to an upper bound on the electroweak scale (cid:104) H (cid:105) (cid:46) Λ / Y ν , (8.10)where Y ν is the neutrino Yukawa coupling. Amazingly, this relates the two naturalness prob-lems associated to the cosmological constant and the EW scale, so that if we happened to livein a universe with a small Λ / , the EW scale would also have to be small (of order TeV atmost) by consistency of quantum gravity. Hence, there is no EW hierarchy problem to startwith, as greater values of (cid:104) H (cid:105) would not lead to theories consistent with quantum gravity.Let us emphasize, though, that this is not a solution to the EW hierarchy problem yet.Clearly, we first need to prove the validity of the Swampland conjecture. But also, we areassuming that there is no other way to destabilize the lower dimensional AdS vacua beyondthose provided by the Standard Model, which can be criticized. Yet, this result exemplifiesthe potential of the Swampland conjectures to change our logic of naturalness, which breaksdown when taking into account quantum gravity constraints. The naturalness issues observedin our universe could simply be artifacts of not yet knowing the space of parameters that isconsistent with quantum gravity. Remember, not everything goes! For the time being, it is still an open question whether string theory admits a de Sitter vacuumor not. At the moment, there is no full-fledged top-down de Sitter construction in a control-lable regime of string theory. Although there is no universal no-go theorem either. This isa challenging and very important objective in string theory, due to the obvious phenomeno-logical implications for the cosmological expansion of our universe. Based on the relationsto other Swampland conjectures, and the difficulties of constructing a dS vacuum in stringtheory, the following Swampland conjecture has been proposed [98,99] asserting that de Sitter Majorana neutrinos (only 2 degrees of freedom) would be ruled out according to the conjecture, as theyalways generate an AdS vacuum.
Conjecture 9: de Sitter Conjecture
A scalar potential of an EFT weakly coupled to Einstein gravity must satisfy M P |∇ V | V ≥ c , (9.1)with c some O (1) constant. This was further refined by stating that the previousbound only needs to be imposed if the following condition on the second derivative ofthe potential is violated, min (cid:0) ∇ i ∇ j V (cid:1) ≤ − c (cid:48) VM P , (9.2)with c (cid:48) another O (1) constant. This way, only dS minima (and not critical points ingeneral) are ruled out.This conjecture is very strong and not free of controversy. Currently, it has only beenchecked in the asymptotic regions of the moduli space (i.e. near infinite distance singularities)where it is more widely accepted to be true. We will discuss this asymptotic version of the dSconjecture in the next subsection.Let us finally mention that there is another conjecture also constraining the scalar po-tential in de Sitter space; this is the Transplanckian Censorship conjecture (TCC) [52]. Inthese lectures, we will not discuss it in much detail, but we will at least give its definition andcomment on the differences with respect to the original dS conjecture. Transplanckian Censorship Conjecture
The expansion of the universe must slow down before all Planckian modes are stretchedbeyond Hubble size. It has two implications: • No dS minima can exist at the asymptotic boundaries of the moduli space. Inthe asymptotic regimes, one recovers a similar bound to (9.1) constraining theasymptotic behaviour of the potential, but with a fixed constant c given by |∇ V | V ≥ (cid:112) ( d − d − . (9.3) • A dS minimum can exist deep inside the bulk, but it must be short-lived. Thelifetime τ for a metastable dS vacuum is bounded from above by τ ≤ H log M P H , (9.4)where H is the Hubble scale.In a certain sense, the TCC is weaker than the dS conjecture as it does not completelyforbid the existence of dS vacua, but only does so asymptotically. It also provides a lowerbound on the exponential rate of the SDC tower, by assuming that the potential scales as m d d the space-time dimension [52] or as m [53]. From the landscape of string theory compactifications, we can gather evidence for an asymp-totic version of the dS conjecture. Although it has no official name, we will refer to it as theAsymptotic dS conjecture, see figure 27.
Asymptotic dS Conjecture
A scalar potential of an EFT weakly coupled to Einstein gravity presents a runawaybehavior when approaching an infinite field distance point, |∇ V | ≥ cV ; c ∼ O (1) . (9.5)We can also motivate this claim by using the Distance conjecture, as we will do below. Theparameter c is undetermined although the relation to the SDC suggests that it is proportionalto the exponential rate of the infinite tower.Figure 27: The runaway behavior of the potential as we approach an infinite distance pointin the field space according to the asymptotic de Sitter conjecture.This can be thought of as a generalization of the “Dine-Seiberg problem” for any directionin field space. Dine and Seiberg [100] pointed out that the string coupling cannot be stabilizedat weak coupling by classical effects. Classical limits are examples of infinite distance limits,and only away from these limits, when quantum corrections are important, can a minimum begenerated. The asymptotic dS conjecture basically implies the same conclusion for any scalardirection when taking a large field limit. So there are no dS vacua at parametric control.This is a concrete claim that we can check in string theory, as these asymptotic regimesare precisely those regimes in which we have control of the corrections and we can trust theEFT. In fact, the majority of models in string phenomenology are constructed in these regimesas we are often dealing with large volume, weak couplings, etc. So most of the time we areat some asymptotic regime in the field space even if we do not declare it explicitly. To beclear, every limit represents some perturbative expansion of the EFT in 1 /φ where φ wouldbe the scalar taken to be large. Whenever this perturbative expansion makes sense and thecorrections are subleading with respect to the leading term, we say we are in the asymptoticregime. For the particular case of the string coupling, the corrections can be understood as67uantum effects over a classical leading piece. In section 9.2 we will discuss the string theoryevidence in support of this asymptotic version of the dS conjecture.According to the Distance conjecture of section 6, the moment we are in some asymptoticregime of the moduli space, there is an infinite tower of states that becomes exponentiallylight. It was proposed in [99] that the SDC tower is responsible for the runaway behaviourof the potential. The argument goes as follows. The entropy S tower associated to the infinitetower of states depends on the number of states N as well as the de Sitter horizon radius R .According to the SDC, the number of states will increases exponentially with the field distance N ∼ e bφ . (9.6)For a quasi-dS space in which we have a nearly flat potential, the contribution of the tower tothe entropy cannot exceed the Gibbons-Hawking entropy, which is fixed by the value of R , S tower ( N, R ) ≤ S GH = R = 1 H , (9.7)where H is the Hubble scale. In the weak coupling limit, one would expect that the lightdegrees of freedom dominate the Hilbert space, so the above bound is saturated.Assuming that S tower depends polynomially on N and R , the previous two equationsimply an exponential behaviour of R in terms of the field distance. For a nearly flat potential, H = R − scales the same way as the potential, implying V ( φ ) ∼ H ∼ e − cφ . (9.8)The constant c will be proportional to b in (9.6) which is related to the exponential rate ofthe tower. More concretely, by parametrizing the entropy of the tower as S tower (cid:39) N γ H − δ (9.9)the bound (9.7) implies H (cid:46) N γ − δ (9.10)in Planck units. The coefficient c in the de Sitter conjecture is then given by c = 2 γ − δ b . (9.11)As a particular example, we can take a KK tower of a circle compactification, so theexponential rate λ is given in (6.8). We can estimate the number of states as the number ofspecies in (5.35), which can be written in Planck units as N = ∆ m − dd − (9.12)using (5.36). Hence, it implies b = d − d − λ = (cid:114) d − d − . (9.13) Infinite distane limits are associated with weakly coupled descriptions of the theory. H (cid:46) Λ QG (cid:39) N d − (9.14)in Planck units, which is equivalent to (9.10) upon identifying γ − δ = d − . An interestingcoincidence occurs when using this species bound result and the growth of a KK tower. Byreplacing (9.6) and (9.13) into (9.14), we get that the potential decreases exponentially in thedistance with the following rate c = 2 d − λ = 2 (cid:112) ( d − d −
2) (9.15)which is precisely the value of the TCC factor in (9.3). Hence, upon using the species bound,the exponential rate of a KK tower would give rise to the asymptotic behaviour of the scalarpotential predicted by the TCC.
Even before the formulation of this Swampland conjecture, there were several no-go theoremsfor de Sitter in the literature (see e.g. [101–104]), valid in the classical regimes of certain stringtheory compactifications. This has recently been extended to other setups, see e.g. [105,106] forrecent reviews, as well as more general infinite distance limits beyond the classical regime (i.e.beyond small g s ) [82]. Different no-go theorems apply to different compactification manifoldswith fluxes and types of localized sources: D-branes, orientifolds, etc. Many of them are basedon the asymptotic scaling of the potential with the moduli. As an example, we are going todiscuss the original no-go theorem regarding type IIA on an orientifold CY [102], as all theothers can, in a sense, be considered as generalizations of this one.Consider type IIA on a CY with fluxes, D6-branes and O6-planes. In this example, allscalars from the closed string sector can be stabilized by an appropriate choice of RR and NSfluxes. For the no-go theorem it is enough to provide the dependence of the scalar potentialon the 4d dilaton τ and the overall volume ρ of the CY . At weak coupling and large volume,the potential reads V = A NS ( φ ) ρ τ + (cid:88) p =0 , , , A RR ,p ( φ ) ρ p − τ + A D6 ( φ ) τ − A O6 ( φ ) τ , (9.16)where each term corresponds to a different contribution from NS fluxes, RR p -form fluxes, D6-branes and O6-planes. The coefficients A i ( φ ) are positive and depend on the axion partnersof the dilaton and the volume, as well as other ingredients, such as fluxes or other scalars ofthe compactfication. One can check that the derivatives of the scalar potential satisfy − ρ∂ ρ V − τ ∂ τ V = 9 V + (cid:88) p =0 , , , pV p ≥ V , (9.17)where the last inequality holds because (cid:80) pV p is always positive. This implies that we cannotstabilize the moduli in dS, as this equation has no solution for ∂ i V = 0 and V >
0. In fact,no dS critical point is allowed. 69ther no-go theorems will involve a similar inequality to (9.17) but with different numericalcoefficients coming from the moduli scaling of the different contributions to the potential. Moregenerally, the no-go theorems based on the moduli scaling take the following form, aV ( φ ) + (cid:88) b i ∂ φ i V ( φ ) ≤ , a > , b i (cid:54) = 0 , (9.18)where the values of a, b i will depend on the specific moduli scaling of the potential and thefield metric. If (9.18) is satisfied, then the dS conjecture is satisfied with an order one factor c given by |∇ V | ≥ cV ; c = a (cid:80) b i . (9.19)When leaving the asymptotic regime, the coefficients A i ( φ ) receive corrections which caneventually change the sign of the coefficient, invalidating the no-go theorem.Although many of these no-go theorems are formulated in the weak coupling limit of typeII compactifications, it is also possible to generalize them beyond weak coupling by studyingF-theory compactifications dual to M-theory with G flux [82]. Employing the techniques ofmixed Hodge structures, used to test the Distance conjecture in Calabi-Yau manifolds (seesection 6.5.1), we can determine the asymptotic form of the flux potential near any type ofinfinite distance limit in the complex structure moduli space of the CY , which also includesthe type II dilaton. This way, one realizes that the coefficients a, b i , and consequently c , aredetermined in terms of the type of infinite distance singularity under consideration, which isin turn fixed by the monodromy properties, as happened for the SDC. This supports that thedS conjecture is not associated to weak coupling but valid in more general infinite distancelimits, and that indeed the SDC exponential rate and the dS conjecture factor c are related,as they have the same geometrical origin.There is still much more work to do and many questions to answer regarding this conjec-ture. For instance, we need to have a better understanding of the nature of the factor c . Thesituation is analogous to the exponential rate of the SDC, where we can gather evidence fromthe geometry of different string theory compactifications, but a general derivation based onEFT data is missing. Interestingly, from a bottom-up perspective, the factor c may also berelated to the extremality bound of the membranes that generate the scalar potential V ( φ ).Recall from (6.38) that the fluxes can be dualized to top-form gauge field strengths, so V ( φ )becomes equivalent to the physical charge of a membrane whose quantized charges are equalto the fluxes. By applying the WGC to these fluxes, we find that membranes saturating theWGC generate a scalar potential that satisfy the de Sitter conjecture, as shown in [54].
10 Final Remarks
To conclude these lectures, we can take a final look at the web of Swampland conjectures repre-sented in figure 4. Now, the meaning of the different connecting lines between the conjecturesis hopefully more clear for the reader. Sometimes, they imply that a conjecture follows fromthe other in a concrete setup, or that a certain conjecture can inform another. In other cases,they imply generalizations or extensions of a given conjecture. Hence, proving one conjectureis not enough to prove the ones related to it. However, the uncovered relations between a pri-ori disconnected conjectures suggest that we are on the right path, and that some constraintsalong these lines should be require to guarantee consistency in quantum gravity.70n these notes, we have discussed in detail those conjectures for which the amount ofevidence has significantly increased in the past few years, and with it, our confidence in theconjectures. These are the Absence of Global Symmetries, the Weak Gravity Conjectureand the Distance Conjecture. We have also briefly explained the rest of the conjecturesrepresented in figure 4, although they can often be understood as extensions of the previousthree. Let us remark that there are a few other swampland conjectures which have not beenincluded in the figure, due to their recent formulation and lack of sufficient research by differentgroups. However, they might play an important role in the future, in the same way that someof the present conjectures might get disproven and disappear. The swampland program isvery rapidly evolving, so we cannot discard the possibility that the diagram of the relevantconjectures might have a very different shape in the future.Our job is to continue gathering evidence to prove or disprove the conjectures, as well asunderstanding their implications. In this sense, string theory is a perfect framework to sharpenand define the swampland constraints in a precise way. Even if our understanding of stringtheory is still in continuous evolution, we already have plenty of data from corners which areunder computational control. Fortunately, it is precisely at these corners, typically boundariesof the field space, where weakly coupled Einstein gravity theories arise and, therefore, wherethe swampland conjectures should naturally hold and can be quantitatively tested. Anyambiguity or undetermined order one factor in the conjectures should then disappear uponcollecting enough information about the precise statements that hold at these string theorycorners. A more difficult obstacle will be to abandon supersymmetry, although some firststeps to testing the conjectures in non-supersymmetric setups have already been taken.As a final remark, let us emphasize that it is at the infinite distance boundaries of field spacewhere approximate global symmetries, weakly coupled gauge theories, large field ranges andan Einstein gravity description arise. And it is precisely at these boundaries where quantumgravitational effects preventing too well-approximated global symmetries become important,and where the swampland conjectures seem to have implications for the IR physics. Ouruniverse shares these features, as we observe Einstein gravity at low energies as well as aweakly coupled photon. So it might be precisely in universes like ours where we cannot ignorethe UV imprint of quantum gravity at low energies. In fact, nature might be telling us with thenaturalness issue of the cosmological constant and the EW hierarchy problem that it is time tobreak with the expectation of separation of scales originating from a Wilsonian quantum fieldtheory approach. The UV/IR mixing induced by quantum gravity might be the missing pieceto understanding these naturalness issues. In particular, our notion of naturalness shouldbe modified if not the entire space of parameters is consistent with quantum gravity, asthe Swampland program defends. Maybe, after all, Quantum Gravity really matters at lowenergies, even if the Planck scale happens to be so high in energies. And this is, certainly, avery exciting possibility!
Acknowledgements
We would like to thank the organizers of the QFT and Geometry summer school 2020 fororganizing such a wonderful school in such difficult times. Many thanks also to Miguel Monterofor very helpful discussions and comments on the manuscript. The work of M.v.B. is in partsupported by the ERC Consolidator Grant number 682608 “Higgs bundles: Supersymmetric71auge Theories and Geometry (HIGGSBNDL)”. The work by J.C. is supported by the FPUgrant no. FPU17/04181 from the Spanish Ministry of Education. D.M. is supported byTUBITAK grant 117F376. The research of I.V. was supported by a grant from the SimonsFoundation (602883, CV).
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