Quantum Gravity and the Swampland
Gary Shiu University of Wisconsin-Madison Conjecturing is part of the scientific process in fundamental physics, e.g.
Drawing lessons from the measured spectral lines of atomic transitions, and the proposal that energy is quantized (Planck), Bohr conjectured: L = n~,nZ 2 This led to the Bohr atomic model. This conjecture thus gives rise to other predictions which have been subsequently tested against experiments. His conjecture was eventually understood as a consequence of quantum mechanics, but nonetheless was a stepping stone for its advent. The Swampland Conjectures were made in a similar spirit.
While we don’t currently have strong experimental hints on what lies beyond the Standard Model(s), we have lots of theoretical data.
These theoretical data suggest some universal features of UV complete theories. They are the basis of these conjectures.
These conjectures led to interesting predictions in cosmology (e.g., inflationary B-mode, dark energy) and particle physics (e.g., Higgs physics, neutrino masses and type, axions, and other BSM physics).
Testing these conjectures and seeking the fundamental reason behind them is an active research area. The analogue of quantum mechanics is still out there, but that makes this research program particularly exciting. Landscape vs Swampland
Landscape Landscape vs Swampland
Swampland Landscape Landscape vs Swampland
Swampland Landscape
We refer to the space of quantum field theories which are incompatible with quantum gravity as the swampland. [Vafa, ’05]; [Ooguri, Vafa, ’06] Landscape vs Swampland
The Swampland program is about extracting possible universal predictions from string theory, and the rules governing them. The Landscape might be huge, but it is small compared to the Swampland
Energy scale String Theory (Quantum Gravity) Set of consistent low- energy effective Quantum Field Theories
Theory space
Theory space String Theory and the Swampland
The primary challenges include: • No completeWithin string definition theory, this apparentof String freedom, Theory manifests — how as the can Landscape we be sure? String Theory (Quantum Gravity) Energy scale
• Extra dimensions geometry • Branes, fluxes
• …
Theory space
Theory space
• String theory constructions live on a spectrum, where to draw the line?
“String Constructed” “String Inspired” + quantum corrections under control String Theory and the Swampland
To make progress, we combine inputs from different fronts: Practically the question is approached by combining three inputs
String Theory Constructions
A Swampland Conjecture
Quantum Gravity (Black Holes) New Microscopic Physics Web of Conjectures
No global symmetry Distance Conjecture [Banks, Dixon,’88];[Harlow, Ooguri,’18] [Ooguri, Vafa,’06] Lecture 1
g ≠ 0 Lecture 4 g → 0
Weak Gravity Conjecture (WGC) Tower WGC [Arkani-Hamed, Motl, Nicolis, Vafa. ’06] [Andriolo, Junghans, Noumi, GS, ’18] Lectures 1 and 2 sLWGC [Montero, GS, Soler ’16];[Heidenreich et al, ’16]
≧ → ＞ [Ooguri, Palti, GS, Vafa,’18] Lecture 4 non-SUSY Non-SUSY AdS dS Conjecture [Ooguri, Vafa,’16] [Obied, Ooguri, Spodyneiko, Vafa,’18] Lecture 3 Quantum Gravity and Global Symmetries Quantum Gravity and Global Symmetries
• Global symmetries are expected to be violated by gravity:
Q, M
Q, Mp • No hair theorem: Hawking radiation is insensitive to Q.
➡ Infinite number of states (remnants) with m . Mp ➡ Violation of entropy bounds. At finite temperature (e.g. in Rindler space), the density of states blows up. Susskind ‘95
• Swampland conjecture: theories with exact global symmetries are not UV- completable.
• In (perturbative) string theory, all symmetries are gauged [Banks, Dixon ’88].
• There is a recent proof using holography [Harlow, Ooguri, ’18].
• Many phenomenological ramifications, e.g., mini-charged DM comes with a new massless gauge boson [GS, Soler, Ye, ’13]. Quantum Gravity and Global Symmetries
• Pheno application: mini-charged DM
What if DM was not electrically neutral?
1 = F F µ⌫ + A (J µ + ✏ J µ ) L 4e µ⌫ µ SM DM
6 • Experimental constraints: ✏ . 10 • The SM charges are quantized (in units of e ). DM charges are proportional to ✏ e
• Imagine ✏ is irrational: charges are not quantized.
SM 0 1 2 3 4 5 6 7 8 9 Quantum Gravity and Global Symmetries
• Pheno application: mini-charged DM
What if DM was not electrically neutral?
1 = F F µ⌫ + A (J µ + ✏ J µ ) L 4e µ⌫ µ SM DM
6 • Experimental constraints: ✏ . 10 • The SM charges are quantized (in units of e ). DM charges are proportional to ✏ e
• Imagine ✏ is irrational: charges are not quantized.
SM 0 1 2 3 4 5 6 7 8 9 DM Quantum Gravity and Global Symmetries
• Pheno application: mini-charged DM
What if DM was not electrically neutral?
1 = F F µ⌫ + A (J µ + ✏ J µ ) L 4e µ⌫ µ SM DM
• Problem: the theory contains an exact global symm.
• Gauge symmetry: • Global symmetry:
Aµ Aµ + e @µ↵ Aµ Aµ ! i↵ ! SM e SM SM SM ! i✏↵ ! i e DM e DM DM ! DM ! Every gauge invariant operator, is invariant under global -transformations Quantum Gravity and Global Symmetries
• Pheno application: mini-charged DM
What if DM was not electrically neutral?
1 = F F µ⌫ + A (J µ + ✏ J µ ) L 4e µ⌫ µ SM DM
• Conclusion: simplest minicharged DM models Swampland 2 • Possible ways out:
- Small rational charges (“FCHAMP”): ✏ Q Naturalness??? 2 - Gauge extra global symmetry = New massless gauge boson! ) 1 1 = F F µ⌫ F˜ F˜µ⌫ + A (J µ + ✏ J µ )+A˜ J µ L 4e µ⌫ 4˜e µ⌫ µ SM DM µ DM
Holdom ’86; GS, Soler, Ye ‘13 The Weak Gravity Conjecture The Weak Gravity Conjecture
• We have argued that global symmetries are in conflict with Quantum Gravity
• Global symmetry = gauge symmetry at g=0
• It is not unreasonable to expect problems for gauge theories in the weak coupling limit: g → 0
• When do things go wrong? How? … The Weak Gravity Conjecture
Arkani-Hamed, Motl, Nicolis, Vafa ‘06 • The conjecture:
“Gravity is the Weakest Force”
• For every long range gauge field there exists a particle of charge q and mass m, s.t.
q M “1” m P
• Seems to hold for all known string theory models. The Weak Gravity Conjecture
Arkani-Hamed, Motl, Nicolis, Vafa ‘06 • The conjecture:
“Gravity is the Weakest Force”
• For every long range gauge field there exists a particle of charge q and mass m, s.t.
q QExt MP “1” MP m ⌘ MExt
• Seems to hold for all known string theory models. The Weak Gravity Conjecture
• Take U(1) gauge theory and a scalar with m>qMp F F F F e + g g + e
•1Stable Introduction bound states: the original argument EBH .... we did some really awesome things in the... past and here is another... awesome paper... 2m>M2 > 2q 3m>M3 > 3q Nm > MN >Nq M Q 1 ! 1 2 Remnants Revisited • All these BH states are exactly stable. In particular, large bound states The(charged Weak Gravity black Conjecture holes) do was not originally Hawking motivated radiate by once a desire they to reach avoid ‘remnants’.the In theextremal words of [1]: limit M=Q, equiv. T=0.
“...there should not exist a large number of exactly stable objects (extremal black holes) whose stability is not protected by any symmetries.”
Arkani-Hamed et al. ‘06 The proposed conjecture in [1] was thus designed to eliminate the infinite tower of ex- tremal black holes from the spectrum of stable objects. However, string theory itself contains alarge,infact,infinite, number of exactly stable black holes and so it is not immediately clear why these should be acceptable while, presumably, some other class of remnants is not. To understand this issue better, let us review the arguments against remnants presented in [2]. Consider the spontaneous production of states in Rindler space:
ds2 = ⇢2dt2 + d⇢2 + dx2 (2.1) ? 1 An object of mass M centered at ⇢ feels an e↵ective temperature of T =(2⇡⇢) . If there are eS ‘species’ of this object then standard thermodynamics tells us that the equilibrium density is: 2⇡⇢M+S N e (2.2) ⇠ Under the ‘remnant hypothesis’, black holes of arbitrary initial mass decay to Planck scale remnants where the entropy of the initial configuration is stored. In this case, the number of ‘species’ eS must clearly be infinite and the remnant density diverges at any ⇢.On the other hand, Rindler space is simply a coordinate transform of Minkowski space, which has zero vacuum energy density. Thus, such theory does not transform sensibly under di↵eomorphisms and is not a good candidate for describing quantum gravity. The salient feature of this argument is that it eliminates remnants, even while accom- modating an infinite tower of black holes. To see how, note that the number of ‘species’
SBH associated with a 4d Schwarzschild black is given by e , where SBH is the Beckenstein- Hawking entropy, A/4G. Thus, the density is:
2⇡⇢M+4⇡GM 2 N = e (2.3)
1 The Weak Gravity Conjecture
• Take U(1) gauge theory and a scalar with m>qMp F F F F e + g g + e
• Stable bound states: the original argument ...... EBH 2m>M2 > 2q 3m>M3 > 3q Nm > MN >Nq M Q 1 ! 1
• All these BH states are exactly stable. In particular, large bound states (charged black holes) do not Hawking radiate once they reach the extremal limit M=Q, equiv. T=0.
• In order to avoid a large number of exactly stable states one must demand the existence of some particle with q Q 1 ext = m Mext Mp Evidences for the WGC Evidences for the Weak Gravity Conjecture
Several lines of argument have been taken (so far):
• Holography [Nakayama, Nomura, ’15];[Harlow, ‘15];[Benjamin, Dyer, Fitzpatrick, Kachru, ‘16];[Montero, GS, Soler, ’16];[Bae, Lee, Song, ’18];[Montero, ’18] • Cosmic Censorship [Horowitz, Santos, Way, ’16];[Cottrell, GS, Soler, ’16];[Crisford, Horowitz, Santos, ’17]
• Entropy considerations [Cottrell, GS, Soler, ’16] [Fisher, Mogni, ’17]; [Cheung, Liu, Remmen, ’18];[Hamada, Noumi, GS, ‘18] • IR Consistencies (unitarity & causality) [Cheung, Remmen, ’14] [Andriolo, Junghans, Noumi, GS,’18];[Hamada, Noumi, GS, ’18] Evidences for stronger versions of the WGC:
• Consistencies with T-duality [Brown, Cottrell, GS, Soler, ‘15] and dimensional reduction [Heidenreich, Reece, Rudelius ’15]. • Modular invariance + charge quantization suggest a sub-lattice WGC [Montero, GS, Soler, ‘16] (see also [Heidenreich, Reece, Rudelius ’16]) WGC and Holography WGC and Holography
• The WGC in AdSd+1 gives a bound on the conformal dimension of operators in the dual CFTd.
• In particular, the enhanced symmetry algebra for d=2 CFT imposes strong constraints on the spectrum of operators.
• Modular invariance and charge quantization imply invariance under spectral flow: k L → L + Q + , Q → Q + k, L˜ → L˜ 0 0 2 0 0 • Acting the spectral flow on the vacuum L 0 = L ˜ 0 = Q = 0 imply such a bound for a sub-lattice of charges (sublattice WGC). c Q2 Δ ≤ + [Montero, GS, Soler, ’16] 12 2k which has been verified using modular bootstrap [Bae, Lee, Song, ’18] Completeness Conjecture Completeness Conjecture
• All charged states allowed by Dirac quantization are populated in theories that couple consistently with gravity [Polchinski, ’03].
• A heuristic argument can be made with charged black holes: A S = 4G • The BH entropy has a statistical interpretation of BH microstates. As the BH has arbitrary charge Q, all allowed charged states should exist in the spectrum [Banks, Seiberg, ’10]; [Brennan, Carta, Vafa, ’17].
• A less heuristic argument can be made by requiring factorization of correlators in AdS/CFT [Harlow, ’15].
• Make no reference to the masses of these states: may not be accessible in the low energy effective field theory. Some WGC Generalizations WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘
“1” WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘ ~z “1” | BH | “1” WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘ ~z “1” | BH | “1” ~z (“1” 0) p1 ~z (0 “1”) p2 WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘ ~z “1” | BH | “1” ~z (“1” 0) p1 ~z (0 “1”) p2 WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘
~z BH “1” ~z ( “1”p20) | | p1 “1” ~z p2 ( 0 “1”p2) ~z p1 (“1” 0) ~z (0 “1”) p2 WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘
~z BH “1” ~z ( “1”p20) | | p1 “1” ~z p2 ( 0 “1”p2) ~z p1 (“1” 0) ~z (0 “1”) p2 WGC for Multiple U(1)’s
[Cheung, Remmen ’14];[Rudelius ’15];[Brown, Cottrell, GS, Soler ’15]
• Consider two U(1) bosons: there must be at least 2 particles i=1,2, so that BH’s can decay (we will see how this generalizes to axions in Lecture 3).
MP 1 2 ~z i Qi Qi ⌘ Mi WGC ~z “1” “1” | EBH |⌘ ~z “1” ~z = “1” | BH | | | “1” \ ~z (“1” 0) Convex Hull ~z ,~z p1 { p1 p2} ~z (0 “1”) p2 From this result we can compute the magnetic charge and the ADM tension:1
ep;dVp+1 p/2 Vp+1 p/2 Q˜ = p(r+r ) p = p(r+r ) p , (19) 2⇡d gp;dd
Vp+1 p p p T = 2 [(p +1)(r+ r )+2p r ] . (20) 2d
The extremality bound r+ r corresponds to the inequality 2 ˜2 2 2 gp;dQ < T . (21)
We can apply electromagnetic duality to obtain the analogous electrically charged black holes. 2(d p 2) ↵2 2 The metric takes the same form as above, with = , = , and d⌦d p 1 in k d 2 ? d p 2 2 2 places of d⌦p+1exist.. Furthermore, These are expected for electrically to have smaller charged instanton black holes, action and hence to play a dominant role in determining the axion potential. Similarly, if the Lattice WGC is true, then the extremal d p 2 gravitational instantons play the role ofr extremal black holes, Ælling out the large-charge regions f (r) 1 ± . (22) of the charge lattice. But± the⌘ small-charge r regions must still be Ælled by instantons that are not well approximated by gravitational⇣ instantons,⌘ just as the small-charge points in the charge The Øux and dilatonlattice of pro aÆUles(1) aregauge now theory are occupied by low-lying particle or string states. Without further ado, let us turn to the detailed version of the arguments and calculations we d p 2 have just summarized. We will o er a few concluding remarks and 2 thoughts on the next steps ↵ (d p 2) ep;d p (r+r ) ? e = fto( pursuer) in §6. ,Fp+1 = (d p 2) d p 1 ⌦p dr , (23) d r ^
1 p 1 where ⌦ = dt dy ... dy denotes the volume form along the brane. The extremality p 2^ Weak^ ^ Gravity for p-branes and circle compactiÆcations bound becomes WGC2 2 for2 2 p-branes 2.1 Conventions ep;dQ < T . (24) For instance, for d =4, p =1, and ↵ =0,weÆnd that =2, recovering the familiar result for • Before we address physics, let us Æx our conventions. We will work in a mostly-plus signature The WGC is about the2 2 existence2 2 of super-extremal objects, and should have Reissner-Nordströmfor the black metric. holes: We take2e theQ M actionPl 2.3 The general Weak1 Gravityd Conjecture1 2 1 d ↵ 2 S = d xp g ( ) d xp ge p;d F . (5) 22 Rd 2 r 2e2 p+1 We have taken our working statementd Z of the✓ Weak Gravity◆ Conjecturep;d Z to be that we demand the existencewhich ofNote a superextremal includes that the dilaton the particle dilation in this or expression as brane it appears that is not allows canonically in anystring extremal normalized; theory charged & weKK will reduction. black refer hole to it as or black brane“conventionally to decay. That normalized.” is, a charged Here objectFp+1 of= tensiondAp is theTp andÆeld quantized strength for charge a p-formq should gauge Æeld A , with exist that• violatesConsideringµ1 (24)....µp It ischarged useful to black state this p-brane with the solution explicit value leads of to : the WGC: 2 1 µ1...µq Fq Fµ1...µq F . (6) 2 ⌘ q! ↵ p(d p 2) 2 2 2 d 2 WithWeak this de GravityÆnition, :?F 2 = F+ ?F . The gaugeTpÆeld eAp;dq Mhasd . dimension p, so that(25) the q 2 ^ d 2 µ1...µp integral A over the worldvolume of a charged (p 1)-dimensional object is dimensionless. ⌃p p 2 Strictly speaking,• WhileThe our this coupling derivation extremality constant of the ep bound;d extremalityhas dimension is only bound 2( validp +1) is valid ford 1 onlywhereas ≤ p for ≤↵1 p d;d is -p dimensionless. 3, d it is3 , interesting but it The to R Ricci scalar has dimension 2, so 2 has dimension 2 d. We will also use the reduced is interestingexamine to examine the howbehaviorRd the formula for all behaves p if wed fornaively all p ifextrapolate we naïvely extrapolate it. it. Notice d-dimensional Planck mass Md and Newton constant Gd deÆned by 1 See for• instanceFor �=0, (2.8) inthe [29] extremality for the ADM tension bound of abecomes black brane. degenerate for p=0 (instantons) and 2 1 1 As noted above, p;d(↵)= d p 2;d( ↵), hence there is no need tod distinguish2 between electromagnetic p=d-2 (D7-branes in 10d ). 2 = = Md . (7) duals in computing . d 8⇡Gd • We Thewill subscriptsdiscuss pinand Lectured are useful 3 the when interesting matching case theories of in the di erentWGC dimensions, for 0-forms but will(axions) whosesometimes charged be dropped objects for convenience.are instantons10 In the (where case of four �≠ dimensions0 naturally). we will sometimes write MPl rather than M4. d We denote the volume of a unit d-dimensional sphere S by Vd and the corresponding volume form by !d, i.e. d+1 2⇡ 2 Vd = !d = d+1 . (8) d ZS 2 7 In the next two subsections we will show that the general bound is well-behaved under compactiÆcation on a circle, either preserving or decreasing the rank of our gauge Æeld. 2.4 Dimensional reductionWGC and on a circle, KK Reduction preserving p Suppose that we begin with the action (5) in D dimensions and compactify down to d = D 1 • dimensionsIs the on a WGC circle robust of radius againstR. We dimensional parametrize the reduction?D dimensional-metric g in terms of the d-dimensional• Compactify Einstein to frame d= D-1 metric dimensionsgˆ and radion on modea circle by: of radius R: (x) 2 d 2 2 (x) 2 ds = e dsˆ (x)+e dy . (27) We consider• Then the thep-form dimensionally in d dimensions reduced descending action: from the p-form in D dimensions, with the dimensionally reduced action 1 1 d − 1 1 S = ddx −ĝ R̂ − (∇ϕ)2 − (∇λ)2 − e−aϕ−bλF2 2 d 2 p+1 ∫ [ 2κd ( 2 4(d − 2) ) 2ep,d ] d 1 1 2 d 1 2 1 ↵ p 2 S = d x gˆ ˆ ( ) ( ) e p;D d 2 F . (28) 22 Rd 2 r 4(d 2) r 2e2 p+1 Z where " d ✓ p ◆ p;d # p (a, b) = (αp,D, ) for p = P reduction ⊥ brane d − 2 All indices in this expression are raised with gˆ. One factor of e d 2 arising from raising an p + 2 − d for reduction brane index on F with gµ⌫ is compensated(a, b) = (αP,D by, the factor) in pp =gP,− so1p factors remain∥ from the other d − 2 2 indices,• givingA linear rise combination to the dependence of � and in the λ exponent is decoupled multiplying from F thep+1 . p-form gauge At this point we recycle an idea from §2.2 and [28]: we rewrite and in terms of two field. The orthogonal combination, when canonically normalized (call it �) conventionally normalized Æelds, one of which (call it ) is decoupled from F 2 and can be has an effective coupling −αp;d ρ 2 p+1 e Fp+1 2 ↵p;d ⇢ set to zero in the solution while the other (call it ⇢) couples to Fp+1 via e . If we deÆne a ˆ d 1 conventionally normalized radion via 2(d 2) , its coupling to the gauge Æeld is given by ⌘ ˆ p;d 2 e Fp+1 where q 2p2 . (29) p;d ⌘ (d 1)(d 2) s A computation completely analogous to the one that led to equation (16) tells us that 2p2 ↵2 = ↵2 + 2 = ↵2 + (30) p;d p;D p;d p;D (d 1)(d 2) is the coupling of the e ective dilaton ⇢. This can be rewritten as: ↵2 p(d p 2) ↵2 p(D p 2) p;d + = p;D + , (31) 2 d 2 2 D 2 or p;d(↵p;d)= p;D(↵p;D). Using (cf. (26)): 1 2⇡R d 2 D 2 2 = 2 ,Md =(2⇡R)MD , (32) ep;d ep;D we conclude that the extremality bound (25) is unchanged after compactiÆcation on a circle. 12 From this result we can compute the magnetic charge and the ADM tension:1 ep;dVp+1 p/2 Vp+1 p/2 Q˜ = p(r+r ) p = p(r+r ) p , (19) 2⇡d gp;dd Vp+1 p p p T = 2 [(p +1)(r+ r )+2p r ] . (20) 2d The extremality bound r+ r corresponds to the inequality 2 ˜2 2 2 gp;dQ < T . (21) We can apply electromagnetic duality to obtain the analogous electrically charged black holes. 2(d p 2) ↵2 2 The metric takes the same form as above, with = , = , and d⌦d p 1 in k d 2 ? d p 2 2 2 places of d⌦p+1. Furthermore, for electrically charged black holes, d p 2 r f (r) 1 ± . (22) ± ⌘ r ⇣ ⌘ The Øux and dilaton proÆles are now d p 2 2 ↵ (d p 2) ep;d p (r+r ) e = f (r) ? ,F= (d p 2) ⌦ dr , (23) WGCp+1 and KK Reductiond p 1 p d r ^ 1 p 1 where ⌦ = dt dy ... dy denotes the volume form along the brane. The extremality p• In both^ cases:^ ^ bound becomes 2 2 2 2 2 2 αp;d pe(pd;d−Q p<−2T) . αp;D P(D − P − 2) (24) + = + For instance, for d =4, p =1, and2 ↵ =0,wed −Ænd2 that =22 , recoveringD − the2 familiar result for Reissner-Nordström black holes: 2e2Q2M 2 1 2πR 2.3 The general Weak Gravity Conjecture = , T = T for p = P e2 e2 p P Md−2 = (2πR)MD−2 and p;d p;D We have taken ourd working statementD of the Weak Gravity2 Conjecture2 to be that we demand the existence of a superextremal particle or brane{ thate allowsp;d = (2π anyR)eP extremal,D, Tp = charged(2πR)TP blackfor p = holeP − 1 or black brane to decay. That is, a charged object of tension Tp and quantized charge q should exist that• violatesWe see (24). that It is the useful WGC to state is thisstable with theunder explicit dimensional value of : reduction: 2 ↵ p(d p 2) 2 2 2 d 2 Weak Gravity : + T e q M . (25) 2 d 2 p p;d d Strictly speaking, our derivation of the extremality bound is valid only for 1 p d 3, but it • Moreover, if the dilation or the radion acquires a mass (setting �=0), the is interesting to examine how the formula behaves for all p if we naïvely extrapolate it. Notice WGC bound becomes monotonically weaker in the IR. 1See for instance (2.8) in [29] for the ADM tension of a black brane. 2 As noted above, p;d(↵)= d p 2;d( ↵), hence there is no need to distinguish between electromagnetic duals in computing . 10 3.1 The KK photon We begin by considering the case of a pure KK photon. The metric ansatz for reducing on a circle of radius R with a radion and a KK photon B1 is 3.1 The KK photon 2 (x) 2 (x) 2 ds = e d 2 dsˆ (x)+e (dy + RB ) , (37) We begin by considering1 the case of a pure KK photon. The metric ansatz for reducing on a circle of radius R with a radion and a KK photon B is 3.1 The KKwhere photonyWGC⇠= y +2⇡ Randand B 1KKis normalized Photons so that the KK modes carry integral charges. The 1 3.1 The KK photon (x) We begin by consideringdimensionally the case reduced of a pure action KK is photon. The metric ansatz for reducing on a 2 2 (x) 2 3.1 The KK photon ds = e d 2 dsˆ (x)+e (dy + RB ) , (37) 3.1 The KK• photon 1 circleKK reduction ofWe radius beginR by withalso considering a leads radion the to case anda ofKK a a KK purephoton. photon KK photon. ConsiderB1 is The metric reduction ansatz for on reducing a circle: on2 a We begin by considering the case of a pure KK photon.1 The metric ansatz for reducingd on a1 R d 1 We begin by consideringcircle the of case radius of aR purewith KK a radion photon. and The a metric KK photond ansatzB for1 is reducingˆ on a 2 d 2 2 S = d x gˆ d where( y =) y +2⇡eR and HB1 is, normalized so(38) that the KK modes carry integral charges. The circle of radius R with a radion and a KK2 photon (x)B1 22is (x) 2 ⇠ 2 circle of radius R with a radion and a KK photon B1 isd 2(2x) R 4(d 2) r 2 ds =2 e dsˆd(2x)+e (x) (dy + RB21) , (37) (x) ds = e d 2 dsˆ Z(x)+e (dy + RB ) , dimensionally reduced(37) action is 2 d 2 2 ( x) 2 1 2 (xds) =2 e dsˆ ( x(x)+) e (dy +2RB1) ,p (37) ds e d 2 ds x e dy RB , where y = y +2=where⇡ R ˆandH( 2)+B=1 dBis normalized1(. Thus,+ 1) the so that KK the photon KK modes gauge(37) carry coupling integral and charges. the radion–KK The photon coupling 2 where⇠ y = y +2⇡R and B1 is normalized so that the KK modes carry integral charges. The 1 d 1 R d 1 where• y = y +2⇡R and⇠ B1 is normalized so that the KK modes carry integral charges. The d ˆ 2 d 2 2 dimensionallyThe⇠ dimensionallyare reduced action reduced is action: S = d x gˆ d ( ) e H2 , (38) where y = ydimensionally+2⇡R anddimensionally reducedB1 is normalized action reduced is so action that the is KK modes carry integral charges. The 22 R 4(d 2) r 2 ⇠ d dimensionally reduced action is 1 2 1 2 d 2 2 2(d 1) Z d 1 d 1 d 1 p 1 d 1 1 d d 1 2 Rdd =11 R2 M2 R R, ↵ 2 = , (39) ˆ d ˆ 2 d 2 2d d 2 KK 2 S = 2 d Sx = gˆ d d x gˆ (ˆ d)2 e ( H 2 ) , wheree (38)dHwhere22 , H2 = dB(38)1. Thus, the KK photon gauge coupling and the radion–KK photon coupling S = 2 d x gˆ d e d 1 2( ) e H2 , d 2 (38) 1 2d 2 2dR 1 4(d 2) rRR 4(KK2d 2) r 2 d ˆ2 d 2 R 4(d d2 2)2 r 2 r S = d x Zgˆ d d Z ( ) e H , (38) 22 Rp Z4(d 2) rp 2 2 are where H2 =ddB1. Thus, the KK photon gaugep coupling and the radion–KK photon coupling d 1 whereZ H2 = dB 1. Thus, the KK photon gauge coupling and the radion–KK photon coupling ˆ 1 1 2(d 1) are• whereWe canH2 = readdBpwhere1 .off Thus, the the lattercouplings KK is photon deÆ nedof gaugethe by canonically the coupling coupling and normalized to the the radion–KK normalized fields: photon radion coupling = 2(d 2) . 2 d 2 where H = dB . Thus,are the KK photon gauge coupling and the radion–KK photon coupling = R M , ↵KK = , (39) 2 1 1 1 2(d 1) 2 d are 2 d 12 1 2(d 1) eKK 2 d 2 are From2 = (25),R Md we, Æ↵ndKK = the2 d WGC2 , bound for the KK(39) photon: q r e 2 = R M d , 2↵KK = , (39) 1 1 KK 1 2 1 2(2 d rdd1)2 2(d 1) 2 d 2 eKK 2 d 2 ˆ d 1 = R M , 2↵KK== R M d , , ↵KK2 =r d 1 (39), where the latter is de(39)Æned by the coupling to the normalized radion = . where the latter is2 deÆned by thed coupling to the normalized radion↵ ˆ = d .3 2(d 2) eKK 2 eKK 2 d 2 KK 2(d 2)d 2 2 d 21 2 d 2 where the latter is deÆned by ther coupling to the normalized+r radion mˆ = e q. M . (40) From (25), we Ænd the WGC bound for the KK photon: 2(dFromKK2) (25),d we Ænd the WGC bound for the KK photon: ˆ 2 d q1 d 2 d 1 q where the latterwhere is deÆned the byFrom latter the (25), coupling is de weÆÆnednd to the the by WGC normalized the coupling bound radion for to the the KK= normalized photon:2(d 2) . radion ˆq= . • The WGC for KK photons:2 2(d 2) ↵KK d 3 2 2 2 d 2 2 From (25), we Ænd the WGC bound for the+ KK photon:m2 eKKq Md . q (40) ↵KK d 3 2 2 2 d 2 From (25),Thus we Ænd KK the2 =1 WGCd /22 boundand↵KK ford the3 KK2 photon:2 2 d 2 q + m e q M . (40) + m eKKq Md . (40) 2 KK d 2 2 d 2 ↵KK d 3 2 22 2 2 dd2 2 2 q Thus KK =1/2 and + m e q M . (40) ↵KKKK 2 dd 3 becomes2 2 2 dm2 . (41) 2 d 2 2 q+ m e q M . 2 (40) Thus KK =1/2 and m . KK d R(41)Thus KK =1/2 and 2 R2 d 2 2 Thus =1/2 and 2 q 2 KK Conversely, the KK spectrumm of a. particle with mass m0 in the higher-dimensional(41) theory is2 q Conversely,• the KK spectrum of a particle2 with mass m0 in the higher-dimensional2 theory is Thusthus KKthe=1 KK/2 modesand 2of Bq saturate the WGCR bound (no surprise). m 2 . (41) m . 2 (41) R Conversely, the KK spectrum2 of a particleq with mass2 m in the higher-dimensional2 theory is mR2 = m2 + , 2 q 0 2 2 (42) q 0 R2 m . m = m +Conversely,, the KK spectrum(41) of a particle with(42) mass m0 in the higher-dimensional theory is Conversely, the KK spectrum of a particle with mass m0 in the higher-dimensional2 2 theory is 0 2 2 2 R q R where the KK charge q Z speciÆes the quantized momentumm = mq/R+ along, the compact circle. (42) Conversely, the KK2 spectrum ofq a2 particle with mass0 R2m in the higher-dimensional theory is q2 Thus, massless particles in them higher2 = m dimensional2 + , theoryq generate KK modes0 which saturate(42) the q/R 2 2 where the KK0 charge2 Z speciÆes the quantized momentum along the compactm circle.= m0 + 2 , (42) WGC bound,where whereas the massive KK charge particlesq doZ notRspeci satisfyÆes the the2 bound. quantized Since momentum2 the higher-dimensionalq/R along the compact circle. R Thus, massless2 particles2 in the2 higherq dimensional theory generate KK modes which saturate the where the KKtheory charge necessarilyqThus,Z speci contains masslessÆes a the particles massless quantized graviton, in the momentum higher the WGC dimensionalm boundq/R= malong for+ theory the the KK, compact generate photon is KKcircle. saturated. modes which saturate the (42) As usual, stabilizing2 the radion leads to a weaker WGC bound,0 whichR is2 then satisÆed but notwhere the KK charge q Z speciÆes the quantized momentum q/R along the compact circle. Thus, massless particlesWGC in the bound, higherWGC whereas dimensional bound, massive whereas theory particles generate massive do KK not modes satisfy particles which the bound. saturate do not Since the satisfy the higher-dimensional the bound. Since the2 higher-dimensional saturated. Thus, massless particles in the higher dimensional theory generate KK modes which saturate the WGC bound, whereaswhere massive thetheory KK necessarily particlestheory charge doq necessarily contains notZ satisfyspeci a massless theÆ containses bound. the graviton, quantized Since a massless the the WGC higher-dimensional momentum bound graviton, forq/R the the KKalong WGC photon the boundis compact saturated. for circle. the KK photon is saturated. We brieØAsy consider usual, stabilizing magnetic2 charges. the radion The leads Kaluza-Klein to a weaker monopole WGC [32, bound, 33] is which a smooth isWGC then satis bound,Æed but whereas not massive particles do not satisfy the bound. Since the higher-dimensional theory necessarilygravityThus, contains solution massless in a masslessd +1As particlesdimensions usual, graviton, in stabilizing that the the appears higher WGC as bound dimensional the a magnetic radion for the monopole KK theory leads photon in to generate the is ad saturated.-dimensional weaker KK modes WGC which bound, saturate which the is then satisÆed but not saturated. As usual, stabilizingtheory.WGC For the bound,d =4 radion, the whereas leads monopole to a massive mass weaker is WGC particles bound, do which not satisfy is then the satis bound.Æed but not Sincetheory the higher-dimensional necessarily contains a massless graviton, the WGC bound for the KK photon is saturated. saturated. We briesaturated.Øy consider magnetic charges. The Kaluza-Klein monopole [32, 33] is a smooth 2 As usual, stabilizing the radion leads to a weaker WGC bound, which is then satisÆed but not We brieØy considertheorygravity necessarily magnetic solution charges.We contains in d brie+1 TheMØ adimensionsKK;magy massless Kaluza-Klein consider= ⇡M that graviton,PlR, magneticmonopole appears the as [32, WGC a charges. magnetic 33] bound is a monopole smooth The for the(43) Kaluza-Klein in KK the photond-dimensional is monopolesaturated. [32, 33] is a smooth As usual,theory. stabilizing For d =4˜ the, the radion monopole leads mass to2 is a weaker2 2 WGC bound, whichsaturated. is then satisÆed but not gravity solutionand in itsd magnetic+1dimensions chargegravity is Q that=1 appears solution. Using the as a fact in magneticd that+1gKKdimensions monopole=4⇡ /eKK in, we the see thatd-dimensional that appears this saturates as a magnetic monopole in the d-dimensional theory. For dthe=4saturated. magnetic, the monopole WGC for mass the KK is photon: 2 We brieØy consider magnetic charges. The Kaluza-Klein monopole [32, 33] is a smooth theory. For d =4, theM monopoleKK;mag = ⇡M massPlR, is gravity solution(43) in d +1dimensions that appears as a magnetic monopole in the d-dimensional We brieØy consider magneticg2 Q˜22= charges.2M 2 . The Kaluza-Klein monopole(44) [32, 33] is a smooth MKK;magKK=KK⇡˜MPlR,d KK;mag 2 2 2 (43) gravityand solution its magnetic in d +1 chargedimensions is Q =1. Using that appears the fact that as agKK magnetic=4⇡ /eKK monopole, wetheory. see that2 in the this Ford saturatesd-dimensional=4, the monopole mass is MKK;mag = ⇡MPlR, (43) and its magnetic chargethe is Q magnetic˜ =1. Using WGC the for fact the that KKg14 photon:2 =4⇡2/e2 , we see that this saturates theory. For d =4, the monopole massKK is KK 2 the magnetic WGC for the KK photon: 2 ˜˜2 2 2 2 2 2 MKK;mag = ⇡MPlR, (43) and its magnetic charge KKgKK isQQ ==1dM. UsingKK;mag2 . the fact that gKK =4⇡ (44)/eKK, we see that this saturates 2 2 2 2 MKK;mag = ⇡MPlR, (43) KKg Q˜ = M . (44) ˜ 2 2 2 the magneticKK WGCd KK;mag for the KK photon: and its magnetic charge is Q =1. Using the fact that gKK =4⇡ /eKK, we see that this saturates ˜ 14 2 2 2 and its magnetic charge is Q =1. Using the fact that gKK =4⇡ /eKK,the we magneticsee that this WGC saturates for the KK photon: 14 2 ˜2 2 2 the magnetic WGC for the KK photon: KKgKKQ = dMKK;mag. (44) 2 2 2 2 KKg Q˜ = M . (44) 2 ˜2 2 2 KK d KK;mag KKgKKQ = dMKK;mag. 14 (44) 14 14 (p, q) strings wound around the circle. In this case, the KK modes are BPS states, and the mass formula (84) isKK exact. Photon with Other U(1)’s It’s interesting to compare the above discussion to the case where the radion and dilaton are stabilized after compactiÆcation. As a result of stabilization, the black hole region lies • 2 2 2 entirelyIf the BH within carries the ellipse chargesZF /Z0 + underZKK =1 bothon which the the KK KK U(1) modes and appear. another Since theU(1). axion is > generically stabilized by non-perturbative e ects at ✓ =0, for Z0 p2 (depending somewhat on ⇠ ↵) the convex hull condition is satisÆed for any stabilized radius R. However, if we also impose the WGC at local maxima of the scalar potential—the XWGC [14]—then putting ✓ = ⇡/q will violate the convex hull condition if R is too small, regardless of Z0, in the absence of additional charged particles. 4 The Lattice Weak Gravity Conjecture For the FigureÆrst time 4: The in CHCthis paper, for a theory we have with a encountered KK U(1) plus an another exampleU(1). where It is possible the WGC for the in charge-to-massD dimen- • The KKvector modes of every KKonly mode populate to obey Z~ a finite1 without density satisfying of the CHC,points, as shown and at so left. for Instead, if R the is sions does not necessarily imply the| WGCn| in d dimensions. This is an appealing property of too small,charge-to-mass the convex vectors must hull be condition su ciently large is that violated. the line segments connecting them lie outside the our earlierunit disk,examples, as shown and at right. it is tempting to postulate a stronger form of the WGC that is al- ways preserved under dimensional reduction. Given the discussion of the convex hull condition • inThis §3.3, led an obvious[Heidenreich, candidate Reece, is the following: Rudelius ’15] to conjecture: for a particle with mass m and charges QF and QKK under the two photons, where ✓ is the Q~2 2 The Latticevev of Weak the axion. Gravity The Conjecture black hole region (LWGC): is thenFor the every unit point disk, ZF +onZ theKK < charge1. The lattice, WGC there isin aD particledimensions of charge impliesQ~ thatwith there charge-to-mass is a particle ratio of charge at leastQF as= largeq and as mass thatm of0 asuch large, that D 2 1 2 q ~ ~ semi-classical,Z0 e non-rotatingDM 2 | | extremal1. Dimensionally black hole with reducing, charge weQBH obtainQ a. tower of KK modes, with ⌘ D m0 / 2 2 1 q✓ 2 masses m = m0 + 2 n and charges QF = q and QKK = n, such that Note that this condition impliesR 2 the⇡ convex hull condition, and is preserved under toroidal compactiÆcation, at least in the examples(µZ 0,xn) discussed in this paper. Moreover,q✓ there are certainly Z~(n) = ,µm0R, xn n . (81) 2 2 examples where the LWGC is true,µ such+ xn as for the⌘ KK reduction⌘ on a2 two-torus⇡ and cases related to this by string~ dualities. Although2 the2 LWGC2 implies an inÆnite number of charged The vectors Z(n) lies on thep ellipsoid ZF /Z0 + ZKK =1, outside the unit disk, so that each KK particles,mode many has of su these cient particles charge to will discharge have super-Planckian a subextremal black masses, hole and with can a proportional be interpreted charge as extremalvector. black holes. For this to succeed, it is important that Planckian corrections to the black hole extremalityHowever, bound this is reduce not su the cient mass to ensure of extremal that the black convex holes. hull condition Fortunately, is satis thereÆed, is because some evidence for this hypothesis [23]. 2 2 2 the KK modes only populate a Ænite density of points along the ellipsoid ZF /Z0 + ZKK =1 Theaway LWGC from is a from strengthened the poles, (0 form, 1) of, and the WGC the lines which between has not, consecutive to our knowledge, points can intersect been pre- the ± viouslyunit discussed disk, as in illustrated the literature. in Figure Possible 4. In particular, strong forms the of orthogonal the WGC distance were discussed to the origin already of the in [2] andline recently between theirZ~(n) possibleand Z~(n+1) importanceis both for understanding the WGC itself and for ap- plications to inØation was stressed by [9]. The latter paper suggested a strong form1/ of2 the WGC 2 that requires that the lightest1 statexn (possiblyxn+1 consisting2 of multiple2 2 particles)2 in any direction 1+ 2 1+ 2 2 1 xn+1 + µ xn + µ . (82) in charge space be" superextremal,Z0 whichZ0 µ would follow from the LWGC. We require!# a superex- ✓ ◆ ✓q ◆ tremal state for all points in the charge lattice, rather than all directions,p because we have seen For a Æxed value of ✓, the convex hull condition demands that this distance is at least one for that the WGC can fail after compactiÆcation otherwise. Single-particle statesq✓ are required so all n. The strongest constraint comes from 1 x 0 (when x = mod 1 ), so that that after dimensional reduction we can sensibly talk aboutn their associatedn 2 KK⇡ modes. These particles need not be perfectly2 stable or1 even weaklyn0(1 coupled;n0) a blackq✓ hole, for our purposes, (m0R) 2 2 + 2 ,n0 mod 1 . (83) is a single-particle state. One appealing4Z0 (Z0 aspect1) of thisZ0 form of the⌘ weak2⇡ gravity conjecture is Thus, for any value of Z0, there is some minimum radius Rmin below which the convex hull condition is not satisÆed! Since we have assumed25 an unstabilized radion (e.g. due to unbroken 23 Z the gµw modes of the metric which acquire an extra sign under the action of 2′ .Themetric moduli ϕ and ϕ have σ = 1forthesamereason. wy wz − Now we ask: for which sites of the charge lattice do we find a superextremal mode? The charge lattice is defined by integer values of (ny,nz). Depending on whether the charges are even or odd, we have three cases: Both charges even. In this case, modes of the graviton with n =0willsaturatethe • w extremality bound. ✓ Odd n ,evenn . If n is odd and n =0,amodewillbeprojectedoutbyZ unless • z y z w 2′ we have σ(φ)= 1. Hence, for these sites in the charge lattice, graviton KK modes − are subextremal but KK modes of the (broken) gauge field Wµ saturate the extremality bound (as do those of the massive moduli ϕwy and ϕwz). ✓ Odd n . For all fields, the first Z ensures that unprojected modes with odd values • y 2 of ny must also have(Sub)Lattice an odd value of nw.Inthiscasethemodegetsanadditional WGC 2 contribution (nw/Rw) to its mass squared. No superextremal modes exist at these • This points stronger of the form charge of lattice. the WGC✗ implies the convex hull condition, and is Wepreserved see that theunder sites toroidal in the charge compactification lattice for which on superext a circle.remal particles exist are those • withThe nLatticey even and WGCnz arbitrary. turns out This to be is a false proper [Montero, sublattice GS, of the Soler, full charge ’16]; [Heidenreich, lattice, so the LatticeReece, Weak Rudelius Gravity ’16] Conjecture. Rather, fails,what but seem a related to survive statement scrutiny is true. is This a weaker leads naturally form 8 toknown a revised as Latticesub lattice Weak GravityWGC. Conjecture: The Sublattice Weak Gravity Conjecture (sLWGC): For a theory with charge lattice Γ,thereexistsasublatticeΓ Γ of finite coarseness such that for each ⃗q Γ ,thereis ext ⊆ ∈ ext a(possiblyunstable)superextremalparticleofcharge⃗q. 9 • WeHere have we already define the seen “coarseness” the argument of the sublattice for thisΓext sLWGCΓ to be using the smallest holography integer ⊆ N[Montero,such that GS,N⃗q Soler,Γext ’16]for where any ⃗q theΓ.Inparticular,thisimpliesthatthesublatticeof index of the sub lattice is level k. ∈ ∈ superextremal particles has the same dimension as the full charge lattice. It also implies • One can derive the sLWGC for perturbative string theory using again that Γext has a finite index with respect to Γ,hencethequotientΓ/Γext is a finite group. Notespectral that theflowindex on ofthe the worldsheet sublattice is CFT the order [Heidenreich, of Γ/Γext,whilethe Reece, Rudeliuscoarseness ’16]is the. least common multiple of the element orders within Γ/Γ (equal to the largest element order • We will see in Lecture 2 that a similar strongext form known as tower WGC because Γ/Γext is Abelian). The two are not equal unless Γ/Γext is cyclic, and in general follows from unitarity and causality [Andriolo, Junghans, Noumi, GS,’18]. coarseness index coarsenessk for a k-dimensional lattice. The finite group Γ/Γ might ≤ ≤ ext 8The case D = 4 is special because the gauge coupling runs logarithmically in the infrared in the presence of light charged particles. When there are massless charged particles, as at a conifold transition in moduli space of type II string theory on an empty Calabi-Yau threefold, thegaugecouplingflowstozerointheinfrared, yet there is still a (logarithmically renormalized) long range force. In these cases, it may be necessary to revisit what is meant by a “superextremal” charged particle.Wedeferfurtherconsiderationofthistofuture work [34]; for D>4thelogarithmicrunningisabsentandthisissuedoesnotarise. 9Here “ext” is a shorthand for (super)extremal. –7– Swampland Distance Conjecture Maximum Field Range • In theories with gravity, there seems to be a maximum field range beyond which a given effective description breaks down, e.g. For large Δφ, new light states emerge, invalidating the EFT. Swampland Distance Conjecture(s) • Swampland Distance Conjecture [Ooguri, Vafa]: In a consistent quantum theory of gravity, for any point p0 in the field space,∃ point p1 arbitrarily far away. As the shortest geodesic d(p0 , p1) diverges, a tower of lights states with m: d(p ,p ) m 1 • Refined Swampland Conjecture [Klawer, Palti]: (1) – 23 – 2.6 New Physics from the Boundaries of Moduli Space Fix a point p .Inthelimitofinfinitedistancefromp ,thatisasd(p ,p)=T , 0 2 M 0 0 !1 there will be a tower of states in the EFT whose mass decreases exponentially with T , ↵T m e . (2.5) Boundary of Moduli Space⇠ we take τ i keeping τ0 finite, the geodesicIn the previous length criterion betweenwe saw that forτ anyand choiceτ of0 a startingis approximately point p and any → ∞ 0 2 M real number T>0, we will always be able to find a (in general not unique) point p • The geodesic distance indeed diverges logarithmically as : 2 M given by such that the distance between p and p is larger� than →Ti∞. So in general will have some 0 M non-compact directions. We want to ask now what happens when we go extremely far away T inlog(Im moduli spaceτ along/Im oneτ0 of) those. directions, or equivalently we take T to be extremely large ∼[67]. Heuristically, we can understand this by considering the point compactification of moduli • − 1 T Light stringy states (D-branes)space, appearˆ4, so that at ˆtheis a boundary finite manifold where of moduli the infinities space. of correspond to singular Since light stringy excitations of mass e M appearM at τ,thisexemplifiestheconjecturesM ∼points of ¯ . Now going to infinity corresponds to going to a singularity where generically M 1and2.• More generally, going to infinitiesextra massless corresponds degrees of freedom to appear. going to a singularity where new massless dofs appear. The metric (3.1) has a constant negative curvature, consistently with the conjecture 3. The curvature integral as well as the volume integral is finite, thanks to the quotienting by the S-duality symmetryThe newSL dofs(2, Z ).can It isbe known that the geodesic flow on compact spaces with constant negative massless curvatures particles, is ergodic [7]. In particular, if we consider a region or tensionless strings/branes. τ2 > 1/ϵ in the fundamental domain, generic geodesics will pass through it within finite geodesic time. This moduli space is simply connected since SL(2, Z)isgeneratedbyS and ST transformations (S : τ 1/τ,T: τ τ +1),bothofwhichhavefixedpointsinthe →− → fundamental domain, at τ = i and τ =exp(2πi/3)Figure respectively. 6. A schematic picture of the moduli space. We can illustrate this criterion in a very easy example. Consider the compactification of a EFT on a circle 1 . This theory has a modulus, which is the radius R of the circle. The Example iii) Type IIB Strings Compactified on SSR1 In this case the moduli space of compactification is characterized by the radius r of S1,inadditiontotheτ space discussed above. This example– is 24 similar – to the examplei, and we have T = log(r/r ) | 0 | in string frame in 10 dimensions. We get light Kaluza-Klein modes for r and light →∞ winding modes for r 0. In the 9-dimensional Einstein frame, this translates to the → existence of light mass scales for large T given by 8 6 ϵ(T ) r−1/7 exp( T ) exp T or exp T ∼ − ∼ !−7 " !−7 " Thus, both conjectures 1 and 2 are satisfied. Example iv) Theories with 16 Supercharges The classical moduli space of K3is (classical) = SO(3, 19; Z) SO(3, 19)/(SO(2) SO(19)). MK3 \ × 7 Lagrangian for this scalar field, in one lower dimension, will be given by Lagrangian for this scalar field, in one lower dimension, will be given by dR 2 = 2 + ... (2.6) Leff dRR eff = Z ✓ ◆ + ... (2.6) Lagrangian for this scalarTower field, inL one of lower Light dimension,R States will be given by Now, let us see how the criterion 5 and 6Z apply✓ to◆ this case. In this example, the moduli 2 Now,space let is usjust see 1-dimensional, how the criterion and we 5 can and see 6 immediately applydR to this that case. there In are this infinite example, distances. the moduli For Lagrangian for this scalar field, in oneeff lower= dimension,+ ... will be given by (2.6) space• isConsider just 1-dimensional, compactification and we can Lon seea circle: immediatelyR that there are infinite distances. For example, fix a radius R0 and pick a T>0.Z The✓ distance◆ from R0 to some other point R˜ will 2 ˜ example,then beNow, fixgiven let a radius usby see howR0 and the criterion pick a T> 5 and0. 6 The applydR distance to this case. from InR this0 to example, some other the moduli point R will eff = + ... (2.6) R˜L R then bespace given is justby 1-dimensional, and wedR can seeZ immediately✓ ◆ that there are infinite distances. For ˜ = log(R˜) log(R ) , ˜ (2.7) example, fix a radius R0 and pickR adRT>0. The distance from0 R0 to some other point R will •Now,The let moduli us see howspace the criterionis 1D.R Using0 5R and the 6 apply moduli˜ to this space case. metric: In this example, the moduli then be given by Z = log(R) log(R0) , (2.7) space is just 1-dimensional, and we can˜R see immediately that there are infinite distances. For and we see that we can alwaysZ findR0 R adR suitable R˜ to make this distance as big as we want. example, fix a radius R0 and pick a T>0.= log( TheR˜ distance) log(R from) , R0 to some other point R˜(2.7)will R ˜ 0 andSo we thethen see criterion be that given we number by can always 5 is satisfied. findZR0 a suitable Let us considerR to make the limit this distance of very large as big radius, as we to want. see ˜˜ Socriterion the criterionand 6 we at see work. number that As we the can 5 is radius always satisfied.R findR growsdR a Let suitable to us infinity, considerR˜ to make we the will this limit have distance of Kaluza-Klein very as big large as we radius, modes, want. with to see • The moduli space is non-compact= log(.R ˜As) welog( goR ) to, infinite distances: (2.7) mass givenSo the by criterion number 5 is satisfied.˜ Let us consider the0 limit of very large radius, to see criterion 6 at work. As the radius RR0 growsR to infinity, we will have Kaluza-Klein modes, with criterion 6 at work. As the radiusZ R˜ grows to infinity,1 we will have Kaluza-Klein modes, with massand given we by see that we can always find a suitablem R˜ to. make this distance as big as we want. (2.8) mass given by ⇠ R1˜ So the criterion number 5 is satisfied. Letm us consider1 . the limit of very large radius, to see (2.8) ˜ m ˜ . (2.8) On the• A other tower end, ofT KKlog( modesR) and ˜ therefore becomes⇠ we light.R˜ see that In we terms have some of the fields canonically with mass criterion 6 at work.⇠ As the radius R grows to⇠ infinity,R we will have Kaluza-Klein modes, with massnormalized given by field T, ˜the KK modes are exponentially light: On theOn other the end, otherT end, Tlog(log(R)R˜ and) and therefore therefore we see seeT that that we we have have some some fields fields with mass with mass ⇠ ⇠ m e1 , (2.9) m ⇠ . (2.8) ˜ T m ⇠ eR T, (2.9) m ⇠ e , (2.9) whichOn get the exponentially other end, T lightlog(R˜ when) and therefore we go to⇠ we infinity see that in we have. some fields with mass which get exponentially⇠ light when we go to infinity in M. which get exponentially light when we go to infinityT inM . This conjecture also implies the remarkablem facte that, a consistentM theory of quantum(2.9) gravity This conjecture also implies the remarkable⇠ fact that a consistent theory of quantum gravity must have extended objects in its spectrum. This conjecturewhichmust get have exponentially also extended implies objects light the in when remarkableits spectrum. we go to infinity fact that in a. consistent theory of quantum gravity M must have extended objects in its spectrum. ThisThose conjectureThose extended extended also objects implies objects can the can be remarkable be for for example example fact that membranes, membranes, a consistent strings, strings, theory etc. etc. of Therefore, quantum Therefore, by gravity this by this criterionmustcriterion have one can extended one can argue argue objects that that in quantum its quantum spectrum. gravity gravity cannot cannot be be a atheory theory of just of just particles. particles. We will We will Those extended objects can be for example membranes, strings, etc. Therefore, by this now shownow show this in this the in the same same easy easy example example we we used. Pick Pick now now the the same same reference reference point pointR0 butR0 but criterion one can argue that quantum gravity cannot be a theory of just particles. We will insteadinsteadThose of going of extended going to larger to objects larger radius radius can go be go forto to smaller example smaller and and membranes, smaller smaller radius. strings, radius. We etc. We also Therefore, also find that find by this that this is this is now show this in the same easy example we used. Pick now the same reference point R but anothercriterionanother infinite one infinite distancecan distance argue in that inmoduli moduli quantum space, space, gravity as as cannot be a theory of just particles. We will 0 insteadnow of show going this to in larger the same radius easy example go to smaller we used. and Pick smaller now the radius. same reference We also point findR thatbut this is R0 dR 0 anotherinstead infinite of going distance to larger in moduli radius go space, to smallerlim as R and0 dR smaller, radius. We also find that(2.10) this is limR˜ 0 R˜ R , (2.10) another infinite distance in moduli space,! asZ R˜ 0 R˜ R diverges. Therefore, due to criterion 6, we! alsoZ R0 expectdR to have in this case some states with R0 diverges.mass Therefore, getting exponentially due to criterion low. However, 6,lim we such also statesdR expect cannot, to have be particle in this states case because some all states the with(2.10) R˜lim0 R˜ R, (2.10) R˜!0 ZR˜ R mass gettingparticle exponentiallystates will be given low. by However, KK modes,! suchZ and states those KK cannot modes be will particle be instead states very because massive all the diverges.particlediverges.in states the Therefore, small Therefore, will radius be due given due limit. to to criterion by The criterion KK only modes, way6, 6, we we also and also can expect those get expect light KK to objects to have modes have in in this in will the this case small be somecase instead radius somestates limit, very withstates ismassive with to have some extended objects which can wrap around the circle which thus become lighter massin the gettingmass small getting exponentially radius exponentially limit. The low. low. only However, However, way we such such can states states get cannot light cannot objects be particle be particle in the states small states because radius because all the limit, all isthe as we go to the small radius limit. So if our theory does not have extended objects, we do not particleparticle states states will will be be given given by by KK KK modes, modes, and and those those KK KK modes modes will will be instead be instead very massive very massive to havehave some any extended light states objects at all in thiswhich limit can and wrap we therefore around violate the circle the criterion which numberthus become 7. This lighter in the small radius limit. The only way we can get light objects in the small radius limit, is inas the we small gois for to example radius the small limit. what radius happens The limit. only in M-theory So way if our we when theory can we get doescompactify light not objects have on the extended in circle: the smallThe objects, M2 radius branes we do limit, not is to have some extended objects which can wrap around the circle which thus become lighter tohave have any some light extended states at objects all in this which limit can and wrap we therefore around theviolate circle the which criterion thus number become 7. lighter This as we go to the small radius limit. So if our theory does not have extended objects, we do not is for example what happens in M-theory when we compactify on the circle: The M2 branes as wehave go to any the light small states radius at all limit. in this So limit if andour we theory therefore does violate not have the criterion extended number objects, 7. This we do not have any light states at all in this limit and we therefore violate the criterion number 7. This is for example what happens in M-theory when– 25 we – compactify on the circle: The M2 branes is for example what happens in M-theory when we compactify on the circle: The M2 branes – 25 – – 25 – – 25 – Lagrangian for this scalar field, in one lower dimension, will be given by dR 2 = + ... (2.6) Leff R Z ✓ ◆ Now, let us see how the criterion 5 and 6 apply to this case. In this example, the moduli space is just 1-dimensional, and we can see immediately that there are infinite distances. For example, fix a radius R0 and pick a T>0. The distance from R0 to some other point R˜ will then be given by ˜ R dR = log(R˜) log(R ) , (2.7) R 0 ZR0 and we see that we can always find a suitable R˜ to make this distance as big as we want. So the criterion number 5 is satisfied. Let us consider the limit of very large radius, to see criterion 6 at work. As the radius R˜ grows to infinity, we will have Kaluza-Klein modes, with mass given by 1 m . (2.8) ⇠ R˜ On the other end, T log(R˜) and therefore we see that we have some fields with mass ⇠ T m e , (2.9) ⇠ which get exponentially light when we go to infinity in . M This conjecture also implies the remarkable fact that a consistent theory of quantum gravity must have extended objectsExtended in its spectrum. Objects Those extended objects can be for example membranes, strings, etc. Therefore, by this • Thecriterion SDC one also can argue implies that quantumthat a consistent gravity cannot theory be a theory of quantum of just particles. gravity We willmust havenow show extended this in the objects same easy (strings, example we branes) used. Pick nowin the the spectrum. same reference point R0 but instead of going to larger radius go to smaller and smaller radius. We also find that this is • Insteadanother infinite of going distance to infinite in moduli radius, space, as consider R→0: R0 dR lim , (2.10) R˜ 0 R˜ R ! Z • Thediverges. geodesic Therefore, distance due to criterion diverges 6, we also as expect well, to but have all in the this case particle some states states with are KKmass states getting which exponentially become low. However, massive such as states R→ cannot0. be particle states because all the particle states will be given by KK modes, and those KK modes will be instead very massive • Thein the only small way radius to limit. get The light only states way we can as get R→ light0 is objects to have in the small extended radius limit, objects is windingto have some around extended the objects circle which which can become wrap around light the circlein this which limit. thus become lighter as we go to the small radius limit. So if our theory does not have extended objects, we do not • Otherhave any than light strings states at allwrapping in this limit around and we therefore a circle violate giving the criterion light particles, number 7. This there areis for other example light what states. happens inFor M-theory example, when we M2-branes compactify on wrapping the circle: The a R M2→0 branes circle become light string states in 10d. – 25 – Summary Summary of Lecture 1 • In this lecture, I have introduced several swampland conjectures: • No global symmetry • Completeness Conjecture • Weak Gravity Conjecture • Convex hull condition • Sub-lattice/Tower Weak Gravity Conjecture • Swampland Distance Conjecture and presented some evidences and applications of these conjectures. • As we will see, these conjectures are related in such intricate way. Summary of Lecture 1 • Plan for the next few lectures: • Lecture 2: Proof of the (mild form of the) WGC from unitarity and causality and the tower WGC. • Lecture 3: Applications to inflation and particle physics. • Lecture 4: de Sitter vacua in string theory and the Swampland. Summary of Lecture 1 • Plan for the next few lectures: • Lecture 2: Proof of the (mild form of the) WGC from unitarity and causality and the tower WGC. • Lecture 3: Applications to inflation and particle physics. • Lecture 4: de Sitter vacua in string theory and the Swampland.