The Weak Gravity Conjecture the Weak Gravity Conjecture

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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