# The Weak Gravity Conjecture the Weak Gravity Conjecture

The Weak Gravity Conjecture The Weak Gravity Conjecture

Arkani-Hamed et al. ‘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 M 1 P ⌘ The Weak Gravity Conjecture

• Take a U(1) and a single family with q < m ( WGC )

F F F F e + g g + e M 1 P ⌘ The Weak Gravity Conjecture

• Take a U(1) and a single family with q < m ( WGC )

F F F F e + g g + e

BH • Form bound states Extremal ...... BH

2m>M > 2q 3m>M3 > 3q Nm > MN >Nq M Q 2 1 ! 1

• All these (BH) states are stable. Trouble w/ remnants Susskind ‘95 M 1 P ⌘ The Weak Gravity Conjecture

• Take a U(1) and a single family with q < m ( WGC )

F F F F e + g g + e

BH • Form bound states Extremal ...... BH

2m>M > 2q 3m>M3 > 3q Nm > MN >Nq M Q 2 1 ! 1

• All these (BH) states are stable. Trouble w/ remnants Susskind ‘95

• Need a light state into which they can decay q Q “1” Ext m ⌘ MExt The Weak Gravity Conjecture

Arkani-Hamed et al. ‘06 • For bound states to decay, there must ∃ a particle w/

q QExt where QL is 22-dimensional vector and QR is 6-dimensional“1” vector. The charges are quan- tized, lying on the 28-dimensional even self-dualm lattice with ⌘ MExt

Q2 Q2 2 (21) Strong-WGC:L −satisﬁedR ∈ by lightest charged particle Moving around in moduli space corresponds to making SO(22, 6) Lorentz transformations on the charges. Weak-WGC: satisﬁed by any charged particle

M Q = M E.g. Heterotic spectrum:

M 2 Q2 +2N 2 / L L Q

Figure 4. The charge M of the heterotic string states of charge Q approaches the M = Q line from below. The yellow area denotes the allowed region.

The extremal black hole solutions in this theory were constructed by Sen [8]. For Q2 R − 2 QL > 0, there are BPS black hole solutions with mass 1 M 2 = Q2 (22) 2 R where we work in units with M =1.ForQ2 Q2 > 0, the black holes are not BPS; still, Pl L − R the extremal black holes have mass 1 M 2 = Q2 . (23) 2 L We can compare this with the spectrum of perturbative heterotic string states, given by 1 1 M 2 = Q2 + N = Q2 + N 1(24) 2 R R 2 L L − where NR,L are the string oscillator contributions and where we chose units with α′ =4. The 1, coming from the tachyon in the left-moving bosonic string,iscrucial.Notethat − this spectrum nicely explains the BH spectrum of the theory, as the highly excited strings 2 2 are progenitors of extremal black holes. Consider large QL,QR ,withQR >QL.Then,

13 The Weak Gravity Conjecture

• Suggested generalization to p-dimensional objects charged under (p+1)-forms: Q “1” Tp

• p=-1 applies to instantons coupled to axions:

S m+i/f e inst = e = fm “1” )

• Seems to explain difﬁculties in ﬁnding f>MP

• Is there evidence for the p=-1 version of the WGC?

Brown, Cottrell, GS, Soler WGC and Axions Brown, Cottrell, GS, Soler • T-duality provides a subtle connection between instantons and particles 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˜ ⇥ ⇥ WGC and Axions Brown, Cottrell, GS, Soler

Type IIA Type IIB

Ai C3 Axions: i C2 Gauge ﬁelds: (i) ⇠ ⌃(i) ⇠ ⌃ Z 2 Z 2 (i) Instantons: D1 on ⌃(i) Particles: D2 on ⌃2 2

pg33 m˜ k = mk 2⇡ ls i 1 WGC Sinstk mk + i(fk) i i i 1 p2 ⇠ q˜k =(fk) 4⇡ ls “Couplings”: “Couplings”: gs g g˜s = s pg33 MP M˜ P = MP pg33 WGC and Axions Brown, Cottrell, GS, Soler

4d Type IIB 4d Type IIA 5d M-theory D1-instantons D2-particles M2-particles

(5d) m m˜ i mi M mi i ⇠ i ⇠ 1 (5d) 1 fi q˜i f Q f ⇠ i i ⇠ i gs 1 g˜s 1 R ⌧ M !1

• Apply the WGC to 5d particles:

(5(5dd)) ((IIBIIB)) QQ (5(5dd)) MMPP Q 2 MM == “1”“1” MP = (5(5dd)) PP pp22ff mm ⌘ M 3 MMii ii ii ✓ ◆Ext5d r WGC and Axions Brown, Cottrell, GS, Soler

4d Type IIB 4d Type IIA 5d M-theory D1-instantons D2-particles M2-particles

(5d) m m˜ i mi M mi i ⇠ i ⇠ 1 (5d) 1 fi q˜i f Q f ⇠ i i ⇠ i gs 1 g˜s 1 R ⌧ M !1

• Apply the WGC to 5d particles:

(5d) (IIB) Q (5d) MP Q 2 M = “1” MP = (5d) P p2f m ⌘ M 3 Mi i i ✓ ◆Ext5d r WGC and Axions

• For each axion (gauge U(1)) there must be an instanton (particle) with

S m+i/f e inst = e

p3 f m M · 2 P

Brown, Cottrell, GS, Soler

For a RR 2-form in IIB string theory. Similar bounds for axions from other p-forms in other string theories have also been obtained. Multiple Axions/ Multiple U(1)’s WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

• Consider two U(1) bosons (axions): there must be 2 particles (instantons) i=1,2, so that BH’s can decay.

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘

“1” WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

• Consider two U(1) bosons (axions): there must be 2 particles (instantons) i=1,2, so that BH’s can decay.

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘ ~z “1” | BH | “1” WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

• Consider two U(1) bosons (axions): there must be 2 particles (instantons) i=1,2, so that BH’s can decay.

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘ ~z “1” | BH | “1” ~z (“1” 0) p1 ~z (0 “1”) p2 WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘ ~z “1” | BH | “1” ~z (“1” 0) p1 ~z (0 “1”) p2 WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘

~zBH “1” ~z ( “1”p20) | | p1 “1” ~zp2 ( 0 “1”p2) ~zp1 (“1” 0) ~z (0 “1”) p2 WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ WGC ~z “1” “1” | EBH | ⌘

~zBH “1” ~z ( “1”p20) | | p1 “1” ~zp2 ( 0 “1”p2) ~zp1 (“1” 0) ~z (0 “1”) p2 WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ 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 WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆

KNP “1” WGC ~z = “1” | | “1” \ Convex Hull ~z , ~z { p1 p2} WGC and Axions

Cheung, Remmen ’14 Brown, Cottrell, GS, Soler ’15 Multiple axions/U(1)s Rudelius ‘15

MP 1 2 MP 1 2 ~zi Q Q = 1/fi 1/fi i i p2 m ⌘ Mi ✓ i ◆ pN

N-ﬂation “1” WGC ~z = “1” pN | | zk pNk i i “1” “1” \ Convex Hull ~z , ~z { p1 p2} pN WGC and Multi-axion Inﬂation

• Generally, given Whena set P=N,of instantons define: that gives a super-Planckian ``diameter” in axion ﬁeld space When P=N, define:

Eigenvector , largest eigenvalue

Eigenvector , Bachlechner et al ‘15 welargest showed eigenvalue that the convex hull generated by these instantons does not contain the extremal ball, to have parametric control. • Our conclusions agree with the gravitational instanton diagnostics of [Montero, Uranga, Valenzuela ’15] in some instances but go beyond theirs in other cases. Is there a way around this?

Loophole suggested in Brown, Cottrell, GS, Soler Loophole suggested in Brown, Cottrell, GS, Soler, “Fencing in the Swampland”, arXiv:1503.04783 [hep-th] A possible loophole

• The WGC requires f∙m<1 for ONE instanton, but not ALL

m M V = e 1 cos + e 1 cos F f ✓ ◆ ✓ ◆

With 1

• The second instanton fulﬁlls the WGC, but is negligible, an “spectator”. Inﬂation is governed by the ﬁrst term. A possible loophole

• In the presence of “spectator” (negligible) instantons that fulﬁll the WGC, dominant instantons can generate an inﬂationary potential

• These scenarios generically violate the Strong-WGC: “The LIGHTEST charged states satisfy Q/M > 1 “

• Oscillations in power spectrum [Choi, Kim];[Kappl, Nilles,Winkler] Further Evidences for the WGC

• ∞ manly stable remnants in a ﬁnite mass range lead to pathologies [Susskind]; WGC is a stronger requirement but • It is suggestive based on analyticity, unitarity of scattering amplitudes [Cheung, Remmen]

• It seems compatible with holography [Nakayama, Nomura] • For WGC to be compatible w/ dim. reduction, an even stronger form (lattice WGC) [Heidenreich, Reece, Rudelius], than our original strong form, was proposed. • Attempts to evade the WGC (including our loophole) amount to hiding our ignorance about the UV. • Realizing large ﬁeld inﬂation in string theory to tame UV sensitivity; burden of proof on claim of counterexamples. Discrete symmetries and domain walls ✤ The integer k in the Lagrangian Axionµ2 Monodromy d4x F 2 + db kC 2 | 4| k2 | 2 3| Z • correspondsAxion to a isdiscrete mapped symmetry to a massiveof the theory gauge broken ﬁeld. spontaneously• Possible once tunnelinga choice of four-formto different ﬂux isbranches made. of the potential: This amounts to choose a branch of the scalar potential e.g. D7 on M6 k=4 S2

S3 BiCuLe taFen BLJH 3aGJKeL & 4aQLence T14 2⇡f k D5 on 3-cycle • Suppressing this tunneling can lead to a bound on ﬁeld range (hence r) Brown, Garcia-Etxebarria, Marchesano, GS, in progress Conclusions Conclusions

• Inﬂation is sensitive to UV physics. Large ﬁeld inﬂation requires even more input from quantum gravity.

• F-term axion monodromy inﬂation

• We have made the WGC precise for (a large class of) axions which can be dualized to U(1) gauge ﬁelds.

• Constraints on multiple axions in terms of convex hull (bound on the “diameter” of axion space):

• KNP, N-ﬂation, kinetic mixing,… Conclusions

• Exciting interface between BH in quantum gravity and inﬂation.

• Profound connections to quantum gravity promise to illuminate black holes, inﬂation, and string theory. Conclusions

• Exciting interface between BH in quantum gravity and inﬂation.

• Profound connections to quantum gravity promise to illuminate black holes, inﬂation, and string theory.

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