Swampland Conjectures and Their Phenomenological Applications

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Swampland Conjectures and their Phenomenological Applications

Gary Shiu University of Wisconsin-Madison String Field Theory & String Phenomenology

String 4D Low Energy Field Effective Field Theories Theory String Pheno String Theory Landscape String Theory Landscape

Anything goes? An even vaster Swampland? An even vaster Swampland?

END OF LANDSCAPE SWAMPLAND 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] Based on work with:

J. Brown W. Cottrell P. Soler M. Montero

Y. Hamada S. Andriolo D.Junghans T. Noumi J. Brown, W. Cottrell, GS, P. Soler, JHEP 1510, 023 (2015), JHEP 1604, 017 (2016), JHEP 1610 025 (2016). M. Montero, GS and P. Soler, JHEP 1610 159 (2016). W. Cottrell, GS and P. Soler, arXiv:1611.06270 [hep-th]. Y. Hamada and GS, JHEP 1711, 043 (2017). S. Andriolo, D. Junghans, T. Noumi and GS, arXiv: 1802.04287 [hep-th]. Outline

• What is the Weak Gravity Conjecture?

• Phenomenological applications of the WGC

Axions, large ﬁeld inﬂation, and CMB B-mode

QCD axion

Relating Neutrino masses and type with the CC.

• Evidences for the WGC

• Conclusions Quantum Gravity and Global Symmetries QG 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

• Many phenomenological ramiﬁcations, e.g., mini-charged DM comes with a new massless gauge boson [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 extremal 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

“...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 extremal 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:

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 where m is the quantum ﬁeld theory of some matter, the bare quantities are GN , the bare NewtonsL constant, ⇤,thebarecosmologicalconstant,anda, b, c, the higher dimension irrelevant operator bare couplings. These terms are set to cancel the one loop divergences in the theory due to matter couplings [49]. For simplicity we will take the matter sector to consist of N minimally coupled scalar ﬁelds, with only quadratic Lagrangians,

N 1 = @ @µ + m22 . (2) Lm 2 µ j j j j j=1 Why is thisX a⇥ conjecture?⇤ This is sucient for our purposes. Generalizations to other matter sectors are straightfor- ward, and as we will discuss below, do not change the essential conclusions. q QExt • The one-loop contributions to the e↵ective action from the matter sector“1” will generically Heuristicexhibit argument quartic, quadratic suggests and logarithmic ∃ UVa state divergences w/ [49]. The quartic UV divergence m ⌘ MExt is the usual divergent contribution to the cosmological constant. If we truncate the matter theory to the quadratic Lagrangian (2) it may or may not appear depending on the regulator. • One oftenThe quadraticinvokes divergences the areremnants the wave-function argument renormalizations [Susskind] of the kinetic terms for in the (1), WGC 2 but the andsituations include renormalizations are different of the additional (ﬁnite “R ”termsintheaction. vs inﬁnite mass range).

Ταβ (φ) Ταβ (φ)

γμν • Perfectly OK for some extremal BHs to be stable [e.g., Strominger, Figure 1: One loop graviton vacuum polarization diagram. Vafa] as q ∈ central charge of SUSY algebra. • No q>mTo compute states the one possible loop integrals, one(∵ firstBPS needs bound). to regulate the divergent terms. We do so by introducing a system of Pauli-Villars regulators for every matter field in (2). The • scheme is conceptually the same as in flat space, where one introduces a regulator for every Moredivergent subtle loop. If thefor cuto theories↵ is above the inversewith curvaturesome scale q we∈ cancentral start with acharge locally flat region of space, introduce the regulators with minimal coupling to gravity. Because there are • The WGCfive distinctis a typesconjecture of required counterterms, on the reflecting finiteness five di↵erent divergences,of the # one of needs stable five regulators for each matter scalar [52], denoted by i (i =1,...,5) and coming with di↵erent states thatstatistics, arei (where not protectedi = 1forcommutingandanticommutingfieldsrespectively).The by a symmetry principle. ± regulator masses mi are much larger than the matter ones in order to formally cancel the UV divergences, and define the UV cuto↵, µ. The choice of the regulators and their statistics are determined by the requirements

5 5 2 i =0 and imi =0 (3) i=0 i=0 X X 4 WGC for Axions Axions and ALPs

The QCD axion [Wilczek, ’78]; [Weinberg, ’78] was introduced in the context of the Pecci-Quinn mechanism and the strong CP problem.

An axion enjoys a perturbative shift symmetry.

String theory has many higher-dimensional form-ﬁelds:

e.g.

3-form ﬂux 2-form gauge potential:

gauge symmetry:

Integrating the 2-form over a 2-cycle gives an axion-like particle (ALP):

The gauge symmetry becomes a shift symmetry, that is broken by non-perturbative (instanton) eﬀects. WGC and Axions WGC and Axions Brown, Cottrell, GS, Soler Brown, Cottrell, GS, Soler • Formulate• T-duality the provides WGC in a asubtle duality connection frame wherebetween the axions and instantonsinstantons andturn particles into gauge ﬁelds and particles, e.g. Type IIA Type IIB

D(p+1)-Particle Dp-Instanton (Gauge bosons) (Axions)

d Rd R S˜1 S1

T-dual

d 1 1 d 1 1 R S R S˜ ⇥ ⇥ model-dependent, calculable

• The WGC takes the form f S (1) M · instanton  O P Primordial Gravitational Waves

Planck Collaboration: Constraints on inﬂation 55

PLANCK 2015 7 1 BK+P 1 B+P 6 ]

K+P 2

0.8 0.8 K µ 5 Joint 0.6 BICEP-Planck 0.6 4 peak peak

L/L L/L 3 0.4 0.4 2 @ l=80 & 353 GHz [ d

0.2 0.2 A 1

0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 A @ l=80 & 353 GHz [ K2] r d µ r

Fig. 54. Marginalized joint 68 % and 95 % CL regions for ns and r0.002 from Planck alone and in combination with its cross- correlationMany with BICEP2/Keck experiments Array and/or BAO data compared including with the theoretical predictions BICEP/KECK, of selected inflationary models. PLANCK, ACT, further improving on the upper limits obtained from the different Table 17. Results of inflationary model comparison using the data combinationsPolarBeaR, presented in Sect. 5. SPT, SPIDER,cross-correlation QUEIT, between BICEP2/Keck Clover, Array and Planck EBEX,. This QUaD, … By directly constraining the tensor mode, the BKP likeli- table is the analogue to Table 6, which did not use the BKP like- hood removes degeneracies between the tensor-to-scalar ratio lihood. -2 and othercan parameters. potentially Adding tensors and running, detect we obtain primordial B-mode at the sensitivity r~10 . Inflationary Model ln B0X r0.002 < 0.10 (95 % CL, Planck TT+lowP+BKP) , (168) wint = 0 wint , 0 whichFurther constitutes almost experiments, a 50 % improvement over the Plancksuch as CMB-S4,R + R2/6M2 ...PIXIE,+0.3 LiteBIRD, DECIGO, TT+lowP constraint quoted in Eq. (28). These limits on tensor n = 2 6.0 5.6 Natural 5.6 5.0 modes are more robust than the limits using the shape of the -3 CAli,TT spectrum .. alone may owing to improve the fact that scalar perturbations further theHilltop sensitivity (p = 2) 0.7 to0. 4 eventually reach r ~ 10 . ` Hilltop (p = 4) 0.6 0.9 cannot generate B modes irrespective of the shape of the scalar Double well 4.3 4.2 spectrum. Brane inflation (p = 2) +0.20 .0 Brane inflation (p = 4) +0.1 0.1 . . 13.1. Implications of BKP on selected inflationary models Exponential inflation 0 100 SB SUSY 1.8 1.5 Using the BKP likelihood further strengthens the constraints Supersymmetric ↵-model 1.1 +0.1 Superconformal (m = 1) 1.9 1.4 on the inflationary parameters and models discussed in Sect. 6, as seen in Fig. 54. If we set ✏3 = 0, the first slow-roll parameter is constrained to ✏1 < 0.0055 at 95 % CL by Planck TT+lowP+BKP. With the same data combination, concave po- BKP reduces the Bayes factor of the hilltop models compared 2 tentials are preferred over convex potentials with log B = 3.8, to R , because these models can predict a value of the tensor-to- which improves on log B = 2 obtained from the Planck data scalar ratio that better fits the statistically insignificant peak at r 0.05. See Table 17 for the Bayes factors of other inflationary alone. ⇡ Combining with the BKP likelihood strengthens the con- models with the same two cases of post-inflationary evolution straints on the selected inflationary models studied in Sect. 6. studied in Sect. 6. Using the same methodology as in Sect. 6 and adding the BKP 2 likelihood gives a Bayes factor preferring R over chaotic in- 13.2. Implications of BKP on scalar power spectrum flation with monomial quadratic potential and natural inflation by odds of 403:1 and 270:1, respectively, under the assumption The presence of tensors would, at least to some degree, require of a dust equation of state during the entropy generation stage. an enhanced suppression of the scalar power spectrum on large TT The combination with the BKP likelihood further penalizes the scales to account for the low-` deficit in the C` spectrum. We double-well model compared to R2 inflation. However, adding therefore repeat the analysis of an exponential cut-off studied B-mode and Inflation

If primordial B-mode is detected, natural interpretations:

✦ Inﬂation took place at an energy scale around the GUT scale

1/4 r 2 E 0.75 10 M inf ' ⇥ 0.1 ⇥ Pl ⇣ ⌘ ✦ The inflaton field excursion was super-Planckian r 1/2 M Lyth ’96 & 0.01 Pl ⇣ ⌘ ✦ Great news for string theory due to strong UV sensitivity! Large field inflation and UV Sensitivity

UV sensitivity of large field inflation: Axions & Large Field Inflation

Natural Inflation [Freese, Frieman, Olinto] Pseudo-Nambu-Goldstone bosons are natural inflaton candidates. Axions & Large Field Inflation

Natural Inflation [Freese, Frieman, Olinto] Pseudo-Nambu-Goldstone bosons are natural inflaton candidates. They satisfy a shift symmetry that is only broken by non-perturbative effects:

decay constant Axions & Large Field Inﬂation

Natural Inflation [Freese, Frieman, Olinto] Pseudo-Nambu-Goldstone bosons are natural inflaton candidates. They satisfy a shift symmetry that is only broken by non-perturbative effects:

Slow roll: f>MP decay constant

(n+1) (1) (k) k ⇤ S V ()=1 ⇤ cos + ⇤ 1 cos if e inst << 1 f f ⇤(n) ⇠ ✓ ◆ k>X1  ✓ ◆ Axions & Large Field Inﬂation

Natural Inflation [Freese, Frieman, Olinto] Pseudo-Nambu-Goldstone bosons are natural inflaton candidates. They satisfy a shift symmetry that is only broken by non-perturbative effects:

Slow roll: f>MP decay constant

(n+1) (1) (k) k ⇤ S V ()=1 ⇤ cos + ⇤ 1 cos if e inst << 1 f f ⇤(n) ⇠ ✓ ◆ k>X1  ✓ ◆ The WGC implies that these conditions cannot be simultaneously satisﬁed. WGC and Multi-Axion Inﬂation

• Thorough searches for transplanckian axions in the string landscape have not been successful. Banks et al. ’03 …

• Models with multiple axions (e.g., N-ﬂation, KNP-alignment) have been proposed but they do not satisfy the convex hull condition [Brown, Cottrell, GS, Soler];[Cheung, Remmen]

p “1” N “1”

pN “1” “1” “1”

Alignment [Kim, Nilles, Peloso, ’04] N-ﬂation [Dimopoulos et al, ’05] WGC for the QCD Axion QCD instanton action is QCD instanton action is SQCD =4ln• TheM QCD/⇤QCD instanton160 ,action (3) QCD instanton action is ⇤ ⇡ SQCD =4lnM /⇤QCD 160 , (3) ⇤ ⇡ M M where is an ultraviolet scale such as GUT.S Using=4ln the generalM /⇤ bound160 (2), wewhere predictM = UV scale, e.g.,MGUT(3) ⇤ where M isQCD an ultraviolet scaleQCD such as MGUT. Using⇤ the general bound (2) we predict ⇤ ⇤ ⇡ 16 16 where M is an ultravioletfQCD• . 10 scaleGeV such, as MGUT. Using thef general10 boundGeV , (2) we(4) predict (4) ⇤ The WGC implies a bound: QCD . with mild logarithmic uncertainties. Experimentalf observation1016 GeV of a, QCD axion with decay (4) • withWhile mild weaker logarithmic thanQCD the. uncertainties. commonly quoted Experimental cosmological observation bound of a: QCD axion with decay constant well above the GUT scale wouldconstant contradict well above the WGC. the GUT scale would contradict the WGC. with mild logarithmic uncertainties. Experimental observation12 of a QCD axion with decay12 A common claim is that early-universeA cosmology common claim requires is thatfQCD early-universe< 10 GeV cosmologyto avoid requires dark fQCD < 10 GeV to avoid dark matter overproduction,constant but well alternative above thematter cosmologies GUT overproduction, scale [36–38]would contradict butor initial alternative the conditions WGC. cosmologies [39, 40] [36–38] allow or initial conditions [39, 40] allow A common claim is that early-universe cosmologyf requires f < 1012 GeV to avoid dark larger f . The absence of QCD axionslargerscenariosf withQCD that. decay The allow absence constants larger of QCD QCD above have axions the been GUTQCD with proposed, decay scale constants is e.g. a [Wilczek, above the’04] GUT scale is a QCD matter overproduction,nontrivial, but alternative falsiÆable cosmologies prediction of [36–38] the WGC. or initial It can conditions be tested in [39, terrestrial 40] allow searches for axion nontrivial, falsiÆable prediction of the WGC. It can be tested in terrestrial searches for axion larger fQCD. The• absencedarkQCD matter axion of QCD [41–43]. with axions decay Axions withconstants with decay decay above constants constants the GUT above above scale the the GUT GUTcan scalebe scale tested: also is cause a superradiant dark matter [41–43].nontrivial, Axions with falsi decayÆableinstabilities prediction constants above inof black the WGC.the holes, GUT It detectable can scale be also tested through cause in astrophysical terrestrial superradiant searches observations for axion including gravitational instabilities in blackdark holes, matter detectable [41–43]. through Axionswave• laboratory signals astrophysical with decay [44–47]. searches constants The observations new e.g., above window ABRACADABRA the including on GUT gravitational scale gravitational also physics cause that superradiant LIGO has opened can also wave signals [44–47].instabilities The new inwindow black holes, ontest• gravitationalgravitational the detectable WGC! through physicswave observatory astrophysical that LIGO has observationse.g., opened LIGO (via can including black also hole gravitational super- test the WGC! wave signals [44–47]. Theradiance new window, [Arvanitaki, on gravitational Dimopoulos, physics Dubovsky, that Kaloper, LIGO hasMarch-Russell, opened can ‘09] also) test the WGC! 2.3 AdS/CFT 2.3 AdS/CFT The AdS/CFT correspondence encompasses many quantum gravity theories. The simplest 4d 2.3 AdS/CFT CFT translation of the WGC in AdS5 is the existence of a charge Q, dimension operator with The AdS/CFT correspondence encompasses many quantum gravity theories. The simplest 4d The AdS/CFT correspondence encompasses many quantum gravity theories.Q The simplest 4d CFT translation of the WGC in AdS5 is the existence of a charge Q, dimension operator, with (5) CFT translation of the WGC in AdS5 is the existence of a chargep12cQ, dimensionpb operator with Q with c the central, charge and bQthe coecient of the current(5) two-point function. However, the Øat-spacep12c  WGCp may be modiÆed due, to the curvature of AdS [48] (cf. [49](5) on AdS3/CFT2). We b p12c  p have checked that the naive analogb of the LWGC based on (5) holds for =4super-Yang-Mills with c the central charge and b the coecient of the current two-point function. However, the N with c the central chargetheory. and Furtherb the coe explorationcient of of the the current LWGC in two-point conformal function.Æeld theory However, should the be rewarding. Øat-space WGC may be modiÆed due to the curvature of AdS [48] (cf. [49] on AdS /CFT ). We Øat-space WGC may be modiÆed due to the curvature of AdS [48] (cf.3 [49]2 on AdS3/CFT2). We have checked that thehave naive checked analog that of the the naive LWGC analog based of the on (5)LWGC holds based for on=4 (5) holdssuper-Yang-Mills for =4super-Yang-Mills 2.4 Geometry N N theory. Further explorationtheory. Further of the explorationLWGC in conformal of the LWGCÆeld in theory conformal shouldÆeld be theory rewarding. should be rewarding. In some string theory compactiÆcations, the masses and charges of particles or branes are related to geometrical properties of the compactiÆcation manifold. This means that the LWGC, 2.4 Geometry2.4 Geometry if true, has corollaries in pure mathematics. (A related discussion of the “geometric weak gravity In some string theoryIn some compacti stringÆcations, theoryconjecture” compacti the masses appearsÆcations, and in the charges [34], masses though of and particles they charges have or not of branes considered particles are or the branes implications are of the stronger related to geometricalLattice properties WGC.) of the Compactifying compactiÆcation M-theory manifold. on a Calabi-Yau This means three-fold that the LWGC, gives a 5d supergravity related to geometrical properties of thetheory compacti withÆcation BPS particle manifold. states This corresponding means that to theM2-branes LWGC, wrapping holomorphic curves in if true, has corollariesif true, in pure has mathematics. corollaries in pure (A related mathematics. discussion (A related of the discussion “geometric of theweak “geometric gravity weak gravity conjecture” appears inthe [34], three-fold. though These they have BPS states not considered generate the the “e implicationsective” cone in of the the charge stronger lattice, inside which conjecture” appears in [34], though theythe have BPS not bound considered requires Q~ the implicationsM in Planck of units. the stronger Simultaneously satisfying this bound as Lattice WGC.) Compactifying M-theory on| a| Calabi-Yau three-fold gives a 5d supergravity Lattice WGC.) Compactifying M-theorywell on as a the Calabi-Yau LWGC requires three-fold a BPS gives state of a 5dQ~ supergravity= M at every lattice point within the cone. theory with BPS particle states corresponding to M2-branes wrapping| | holomorphic curves in theory with BPS particlethe three-fold. states corresponding TheseGeometrically, BPS states to generate M2-branes this means the wrapping “e everyective” eective cone holomorphic 2-cycle in the charge class curves of lattice, a compact in inside Calabi-Yau which should contain a holomorphic curve representative. This conjecture is similar in spirit to Vafa’s inference of the three-fold. Thesethe BPS BPS states bound generate requires theQ~ “eective”M in cone Planck in units. the charge Simultaneously lattice, inside satisfying which this bound as ~ | |  the BPS bound requireswell asQ the LWGCM in requires Planck a units. BPS state Simultaneously of Q~ = M satisfyingat every lattice this bound point within as the cone. | |  | | 5 well as the LWGCGeometrically, requires a BPS this state means of everyQ~ = eMectiveat 2-cycle every lattice class of point a compact within Calabi-Yau the cone. should contain | | Geometrically, this meansa holomorphic every e curveective representative. 2-cycle class of This a compact conjecture Calabi-Yau is similar in should spirit contain to Vafa’s inference of a holomorphic curve representative. This conjecture is similar in spirit to Vafa’s inference of 5 5 WGC and Particle Physics The Standard Model in the Deep IR

• The deep IR of the SM, below the electron mass scale, is simple:

• Bosonic dof: photon (2) and graviton (2)

• Fermionic dof: �’s (6 or 12 for Majorana/Dirac �’s)

• The mass scale of neutrinos: 1 2 m 10 10 eV ⌫ ' • The only other known IR scale is the cosmological constant:

11 4 2 4 ⇤ 3.25 10 eV =(0.24 10 eV ) ' ⇥ ⇥ • This coincidence (?) has been a source of inspiration/speculations:

⇤ m4 ' ⌫ WGC for Branes

• We have seen the applications of the WGC to particles (and instantons). Analogously, the WGC for branes is: “T Q ” p  p • A stronger form [Ooguri,Vafa, ’16]: this bound is saturated only for a BPS state in a SUSY theory.

• A corollary of this strong form: non-SUSY AdS vacua supported by ﬂuxes are unstable.

• In AdS space, a brane with T < Q leads to an instability (AdS fragmentation) [Maldacena, Michelson, Strominger, ’99].

• This brane gets nucleated and expands. It reaches the boundary of AdS within a ﬁnite time and dilute the ﬂux. AdS Instability

• Instability if there exists a T

AdS vacuum

AdS with less ﬂux

• A stronger form of the Ooguri-Vafa conjecture: “all non-SUSY AdS (in theories whose low energy description is Einstein gravity coupled to a ﬁnite # of ﬁelds) are unstable”

• How do we test this conjecture? AdS Instability

• Instability if there exists a T

AdS vacuum

AdS with less ﬂux

• A stronger form of the Ooguri-Vafa conjecture: “all non-SUSY AdS (in theories whose low energy description is Einstein gravity coupled to a ﬁnite # of ﬁelds) are unstable”

• How do we test this conjecture? The Higgs Potential

• After the Higgs discovery, we know that there is an additional Higgs vacuum at high scale, other than the EW vacuum:

1 ×1068 c6=0 Mt=171.3GeV 67 c6=150 8 ×10 Mt=171.00390 GeV 6 ×1055 67 c =160 6 ×10 M =171.00410 GeV 6 ]

] t 4 4 67 4 ×1055 4 ×10 Mt=171.00430 GeV GeV GeV 67 [ [ 2 ×10 55 V

V 2 ×10 0 -2 ×1067 0 -4 ×1067 0 1 ×1018 2 ×1018 3 ×1018 4 ×1018 0 5.0 ×1014 1.0 ×1015 1.5 ×1015 2.0 ×1015 h [GeV] h [GeV]

• This high scale vacuum can be AdS4, M4, or dS4 depending on theFigure top 1: quark The 4-dimensional mass and Higgs the potential higher-dimensional as a function of the physical operators Higgs field. h, Eq. (1). In the left panel, we put c6 = 0. The potential has AdS, flat or dS vacua depending on the value of the top mass. In the right panel, the c term is included while M is fixed. • 6 t ApplyingWe again have this AdS, conjecture flat or dS minima to the corresponding SM landscape, to the value we of c 6can. constrain the top mass, Higgs potential, and BSM physics. [Hamada, GS].

the mostly positive metric convention. In our universe, we have ⇤4 3.25 4 ' ⇥ 10 meV . If we consider the high scale vacuum in four dimensions, ⇤4 can all take other values. We also add VS1 , the one-loop Casimir energy, for later convenience. The remaining terms include the Higgs boson, fermions and the SU(3) SU(2) gauge fields. Since⇥ the radius of S1 is denoted by L, the volume of the compactified space is 2⇡L, and so the momentum is quantized as 2⇡n/L. The metric of this S1 compactification is

2 2 2 2 2 ds = gij + L AiAj dxidxj +2L Aidxidx3 + L (dx3) , (5) where x3 is the compactiﬁed dimension, 0 x3 2⇡, Ai is the graviphoton, and i, j =0, 1, 2. Then, we have the following decomposition: 1 1 det ( g )=L2 det ( g ) ,R= R(3) 2 2L L2F F µ⌫, (6) µ⌫ ij Lr 4 µ⌫

(3) where µ, ⌫ =0, 1, 2, 3, R is the Ricci scalar constructed from gij. The dimensional reduction yields

3 (3) 1 2 (3) 1 2 1 2 µ⌫ all S = d x g (2⇡L) M R 2 L L F F ⇤ V 1 2 P Lr 4 µ⌫ 4 S Z p  ⇢ 3 (3) 1 2 (3) 1 2 µ⌫ all = d x g (2⇡L) M R L F F ⇤ V 1 , (7) 2 P 4 µ⌫ 4 S Z p  ⇢ where the total derivative is omitted in the last equality. Performing the redef-

7 StandardCompactiﬁcationCompactiﬁcation Model Landscape of the of theSM SMto 3d to 3d

Compactiﬁcation• Upon compactiﬁcation, of the SM to 3d the SM gives rise to a rich landscapemassive of particles: 1 massive particles: 3d vacua Standard[Arkani-Hamed,Standard Model Model + GravityDubovsky, + Gravity on S onNicolis, 1 : S : Villadoro]. neutrinos,…neutrinos,… massless particles: 1 Standard Model + Gravity on S : graviton, photon 32⇡r3⇤ 3 r3 2⇡r ⇤4 4 r si • A competition betweenV ( RΛ)4 and Casimir4 energies+ (depends(2⇡R)(si 1)onn i⇢i(R) 3 3 V (R) 2 2 4 6 +6 (2⇡R)( 1) ni⇢i(R) 2⇡r ⇤4 r ' 'R R 720⇡R720⇡R V (R) 4 ✓ ◆i i ' R2 mass, 720 spin,⇡R6 b.c., and #dof; negligible✓ unless◆ mL <

Dirac Dirac neutrinosneutrinos

MajoranaMajorana neutrinosneutrinos

• The more massive the neutrinos, the deeper the AdS minimum. The moreThe more massive massive the neutrinos, the neutrinos, the deeper the deeper the AdS the vacuum AdS vacuum Barring the potential instability of the AdS vacuum [Hamada, GS], the WGC puts a bound on the neutrino mass and type. Figure 22:Summary The tree level stabilization of Results of ⌧ moduli. [Hamada, GS]

• Compactify the SM on S1 and T2 , starting from both the electroweak vacuum and the high scale vacuum, with general b.c. and WL. model AdS ﬂat dS 2.8 4 2.8 4 2.8 4 2.6 4 U(1), neutral ⇤4 . 10 Me ⇤4 10 Me 10 Me . ⇤4 . 10 Me U(1), charged – ' – – SM, ⌫M always – – 1 S SM, ⌫D, NH 8.4meV. m⌫,lightest m⌫,lightest 8.4meV 7.3meV. m⌫,lightest . 8.4meV SM, ⌫ , IH 3.1meV m m ' 3.1meV 2.5meV m 3.1meV D . ⌫,lightest ⌫,lightest ' . ⌫,lightest . SM, ⌫M , high scale – – – SM, ⌫ , high scale ⇤ (neutrino mass)4 – – D 4 ⌧ axion ⇤4 < 0 – – 2.1 4 2.1 4 2.5 4 2.1 4 U(1), neutral ⇤4 . 10 Me ⇤4 10 Me 10 Me . ⇤4 . 10 Me U(1), charged – ' – – 2 T SM, ⌫M always – – SM, ⌫D, NH 4.5meV. m⌫,lightest m⌫,lightest 4.5meV 4.5meV. m⌫,lightest . 6.5meV SM, ⌫ , IH 1.1meV m m ' 1.1meV 1.1meV m 1.55 meV D . ⌫,lightest ⌫,lightest ' . ⌫,lightest . axion ⇤4 < 0 – –

• TableCan 9: Aavoid summary AdS of the vacua analysis if in neutrinos this paper. Hereare theDirac periodic w/ boundary the lightest condition neutrino is taken.mass We≲ alsoO (1-10) impose themeV. current (also upper [Ibanez, bound on Martin-Lozano, the neutrino mass, Valenzuela]m⌫,lightest . ). 0.1eV[50,51].

37 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]

• Cosmic Censorship [Horowitz, Santos, Way, ’16];[Cottrell, GS, Soler, ’16];[Crisford, Horowitz, Santos, ’17]

• Entropy considerations [Cottrell, GS, Soler, ’16] (note however unjustiﬁed claims in [Fisher, Mogni, ’17]; [Cheung, Liu, Remmen, ’18]).

• IR Consistencies (unitarity & causality) [Cheung, Remmen, ’14] [Andriolo, Junghans, 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 Blackhole Entropy Entropy Corrections

• We computed loop corrections to the entropy of extremal blackholes using Sen’s entropy functional formalism [Cottrell, GS, Soler, ‘16].

• While corrections from neutral particles have been well studied (loops of massless particles give log (A) corrections to BH entropy), we found new features when charged particles are integrated out.

• Fermion spectral density in AdS2 x S2 is divergent for: Energy = p2qMP Magnetic WGC!

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 ⇠ a

• The entropy corrections formulae used in [Fisher, Mogni, ’17] cannot be applied to macroscopic black holes, nor away from extremality, which is where conﬂicts with the WGC were argued to arise.

• [Cheung, Liu, Remmen, ‘18] made a connection between the WGC and the positivity of entropy corrections. It is not known, however, if the latter follows from some fundamental consistency conditions. WGC and Positivity Bounds Einstein-Maxwell + massive charged particles

integrate out matters

IR eﬀective theory of photon & graviton

Positivity of EFT coeﬃcients follow from unitary, causality, and analyticity of scattering amplitudes.

Q. What does the positivity of this EFT imply? # 1-loop eﬀective action for photon & graviton M 2 1 = Pl R F 2 Le↵ 2 4 µ⌫

µ⌫ 2 µ⌫ 2 µ⌫⇢

+ ↵1(Fµ⌫ F ) + ↵2(Fµ⌫ F˜ ) + ↵3Fµ⌫ F⇢W + ...

⌧ ⌧ 1/2 1/2 1/2 1/2 R R R R ...

- positivity implies ↵1 + ↵2 0

...

g g g g g g

F F F F - ↵ i depends on mass and charge of particles integrated out R1/2 R1/2 F F g g ↵ = + (g2)+ (g0) ... i g g ... O O 1/2

... R gravitational eﬀects R1/2 F ... F - Cheung-Remmen found positivity implies z4 z2 + 0 qg ※ z = , is a UV sensitive ( z 0 ) coeﬃcient m/MPl O Figure 1: Fermionic/scalar(free parameter in loops the EFT with framework)Figure 2: Fermionic/scalar loops with external photons. external gravitons.

2.2 Causality Constraints We now study the IR consistency of the e↵ective Lagrangian (2.6) with the vanishing Chern-Simons term, M 1 = 3 R F F + C (F F )(F F ) , (2.12) 1 2 4 ij i · j ijkl i · j k · l

where Cijkl depends on the charge and mass of matters integrated out. An explicit form may be obtained by replacing ci in (D.63)byci+˜ci, wherec ˜i’s are the unknown coecients mentioned in the previous subsection. In 3D, it is convenient to introduce a dual scalar theory as (see Appendix C)

M 1 = 3 R @ @ +4C (@ @ )(@ @ ) , (2.13) 1 2 2 ij i · j ijkl i · j k · l where we dropped higher order derivatives. The photons and scalars are related to each µ⌫⇢ other by Fiµ⌫ = i✏ @⇢i up to higher derivatives.[TN: Do we need i here?] We then introduce a positivity bound on Cijkl by requiring the subluminality of ﬂuctuations around 5 nontrivial backgrounds. For this purpose, we expand gµ⌫ and i around their background values, denoted with a bar:

gµ⌫ =¯gµ⌫ + hµ⌫ , i = ¯i + 'i . (2.14)

Since the graviton is non-dynamical in 3D, we set hµ⌫ =0.

For simplicity, let us assume a constant electromagnetic ﬁeld background @ai = wia. Here and in what follows we take the local Lorentz frame and use the Latin indices for local Lorentz indices. The metric is given by ⌘ =( ++)in particular. In this frame, ab 5 To be precise, we need to discuss the global causal structure. It however turns out that the subluminality argument we do here practically reproduces the same condition. See [40] for more details.

8 Positivity of photon-graviton EFT implies z4 z2 + 0 → at lest one of the following two should be satisﬁed 1) WGC type lower bound on charge-to-mass ratio

in particular when =0 , WGC z 2 1 is reproduced! 2) not so small value of UV sensitive parameter

AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUC9S9OKxgrGVNpTJdtMu3U3C7kYoob/CiwcVr/4db/4bt20OWn0w8Hhvhpl5YSq4Nq775ZSWlldW18rrlY3Nre2d6u7evU4yRZlPE5GodoiaCR4z33AjWDtVDGUoWCscXU/91iNTmifxnRmnLJA4iHnEKRorPXQHKCWSS7dXrbl1dwbyl3gFqUGBZq/62e0nNJMsNlSg1h3PTU2QozKcCjapdDPNUqQjHLCOpTFKpoN8dvCEHFmlT6JE2YoNmak/J3KUWo9laDslmqFe9Kbif14nM9F5kPM4zQyL6XxRlAliEjL9nvS5YtSIsSVIFbe3EjpEhdTYjCo2BG/x5b/EP6lf1N3b01rjqkijDAdwCMfgwRk04Aaa4AMFCU/wAq+Ocp6dN+d93lpyipl9+AXn4xtANI+U > 0

In [Andriolo, Junghans, Noumi, GS], we discussed - multiple U(1)’s - implications for KK reduction and found qualitatively new features. Multiple U(1)’s

# for example, let us consider U(1) U(1) 1 ⇥ 2 a new ingredient is positivity of + + 1 2 ! 1 2 Im 0 implies z2z2 z2 z2 0 1 2 1 2

- z i = q i /m is the charge-to-mass ratio for each U(1)

- we set ( z 0 )=0 for illustration (same asγ = 0 before) O

the punchline here: positivity bound cannot be satisﬁed unless z2z2 =0 1 2 6 → requires existence of a bifundamental particle! Implications for KK reduction

# S 1 compactify d+1 dim Einstein-Maxwell with single U(1) into d dim Einstein-Maxwell with U(1) U(1) ⇥ KK d+1 dim charged particle (q,m)

→ KK tower with the charged-to-mass ratios

q n (z,zKK)= , 2 2 2 2 2 m + n mKK (m/mKK) + n ! p p in the small radius limit m , KK !1 the lowest mode (n = 0): (z,zKK)=(q/m, 0) KK modes (n ≠ 0): (z,z ) (0, 1) KK ' ※ no bifundamentals → positivity bound generically U(1) ` d+1 dim charged particles labeled by ` =1, 2,... (q, m)=(` q , ` m ) ⇤ ⇤ q s.t. z = ⇤ = (1) ⇤ m O ⇤ U(1) ` d+1 dim charged particles labeled by ` =1, 2,... (q, m)=(` q , ` m ) ⇤ ⇤ q s.t. z = ⇤ = (1) ⇤ m O ⇤

n U(1)KK

d dim charged particles

` z n (z,zKK)= ⇤ , 2 2 2 2 2 2 ` (m /mKK) + n ` (m /mKK) + n ! ⇤ ⇤ p p U(1) ` d+1 dim charged particles m bifundamentals: ` KK n labeled by ` =1, 2,... ⇠ m ⇤ (q, m)=(` q , ` m ) ⇤ ⇤ q s.t. z = ⇤ = (1) ⇤ m O ⇤

n U(1)KK

d dim charged particles

` z n (z,zKK)= ⇤ , 2 2 2 2 2 2 ` (m /mKK) + n ` (m /mKK) + n ! ⇤ ⇤ p p m U(1) KK =3 m ` ⇤ d+1 dim charged particles m bifundamentals: ` KK n labeled by ` =1, 2,... ⇠ m ⇤ (q, m)=(` q , ` m ) m 1 ⇤ ⇤ KK = q m 3 s.t. z = ⇤ = (1) ⇤ ⇤ m O ⇤

n U(1)KK

d dim charged particles

` z n (z,zKK)= ⇤ , 2 2 2 2 2 2 ` (m /mKK) + n ` (m /mKK) + n ! ⇤ ⇤ p p Tower WGC

[Andriolo, Junghans, Noumi, GS]

Consistency with KK reduction seems to imply a tower of d+1 dim U(1) charged particles

→ Tower Weak Gravity Conjecture!

※ a similar “(sub)lattice WGC” was proposed based on modular invariance or holography

[Montero, GS, Soler, ’16];[Heidenreich, Reece, Rudelius, ’16] Conclusions Conclusions

QCD instanton action is • Swampland conjectures have a variety of interesting applications SQCD =4lnM /⇤QCD 160 , (3) in cosmology and particle physics. ⇤ ⇡

where M is an ultraviolet scale such as MGUT. Using the general bound (2) we predict • The WGC when applied⇤ to ALPs constrains inﬂationary B-modes; 16 when applied to the QCD axion implies f QCD . 10 GeV , which can (4) be falsiﬁed by laboratory axion searches or GW detectors. with mild logarithmic uncertainties. Experimental observation of a QCD axion with decay • The WGC constantoffers well interesting above the GUT perspectives scale would on contradict how theΛ WGC. and the 12 neutrino massesA common are linked. claim is that early-universe cosmology requires fQCD < 10 GeV to avoid dark matter overproduction, but alternative cosmologies [36–38] or initial conditions [39, 40] allow • Further evidenceslarger fQCD for. the The WGC absence based of QCD on entropy axions with considerations decay constants above the GUT scale is a and IR consistenciesnontrivial, falsi. Æable prediction of the WGC. It can be tested in terrestrial searches for axion dark matter [41–43]. Axions with decay constants above the GUT scale also cause superradiant instabilities in black holes, detectable through astrophysical observations including gravitational wave signals [44–47]. The new window on gravitational physics that LIGO has opened can also test the WGC!

2.3 AdS/CFT The AdS/CFT correspondence encompasses many quantum gravity theories. The simplest 4d CFT translation of the WGC in AdS5 is the existence of a charge Q, dimension operator with Q , (5) p12c  pb with c the central charge and b the coecient of the current two-point function. However, the Øat-space WGC may be modiÆed due to the curvature of AdS [48] (cf. [49] on AdS3/CFT2). We have checked that the naive analog of the LWGC based on (5) holds for =4super-Yang-Mills N theory. Further exploration of the LWGC in conformal Æeld theory should be rewarding.

2.4 Geometry In some string theory compactiÆcations, the masses and charges of particles or branes are related to geometrical properties of the compactiÆcation manifold. This means that the LWGC, if true, has corollaries in pure mathematics. (A related discussion of the “geometric weak gravity conjecture” appears in [34], though they have not considered the implications of the stronger Lattice WGC.) Compactifying M-theory on a Calabi-Yau three-fold gives a 5d supergravity theory with BPS particle states corresponding to M2-branes wrapping holomorphic curves in the three-fold. These BPS states generate the “eective” cone in the charge lattice, inside which the BPS bound requires Q~ M in Planck units. Simultaneously satisfying this bound as | |  well as the LWGC requires a BPS state of Q~ = M at every lattice point within the cone. | | Geometrically, this means every eective 2-cycle class of a compact Calabi-Yau should contain a holomorphic curve representative. This conjecture is similar in spirit to Vafa’s inference of

5 Conclusions

QCD instanton action is • Swampland conjectures have a variety of interesting applications SQCD =4lnM /⇤QCD 160 , (3) in cosmology and particle physics. ⇤ ⇡