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+iφ/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: φi 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)− φi 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+iφ/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 • 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 • 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 | ⌘ ~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 • 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 | ⌘ ~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 • 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” ~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 • 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 ◆ 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 • 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 ◆ pN N-flation “1” WGC ~z = “1” pN | | zk pNδk 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.