Recent Developments in Analytic Bootstrap Annual Theory Meeting, - PowerPoint PPT Presentation

Recent Developments in Analytic Bootstrap Annual Theory Meeting, Durham, 2017 Alexander Zhiboedov, Harvard U

How do we compute when the coupling is strong? No small coupling expansion No Lagrangian No extra symmetries/integrability Bootstrap is an old idea of solving theories based on consistency.

Radical Bootstrap “Nature is as it is because this is the only possible Nature consistent with itself” G. Chew (In other words, a consistent theory of quantum gravity compatible with all known experimental data is unique) This is too ambitious! But for Conformal Field Theories (CFTs) it is almost true. Conformal Bootstrap is a method to solve them based on consistency.

Why CFTs? UV RG flow fixed points IR critical points in condensed matter systems [Sachdev et al.] non-perturbative quantum gravity in Anti-de Sitter (AdS/CFT) [Maldacena]

Plan 1. Basics of Conformal Bootstrap 2. Analytic Bootstrap: Spin = Expansion Parameter 3. Applications

Basics of Conformal Bootstrap

Conformal Bootstrap Conformal Bootstrap is based on symmetries and consistency conditions : Conformal Symmetry Unitarity and the OPE Crossing Equations As such it is suitable for strongly coupled theories.

Basics of Conformal Symmetry Poincare symmetry: translations and rotations P µ M µ ν Scale or dilatation invariance δ x µ = λ x µ D δ x µ = 2( b.x ) x µ − b µ x 2 Special conformal transformation K µ ∞ ∞ P µ K µ = − IP µ I 0 0 [ D, P µ ] = P µ , [ K ν , P µ ] = 2 δ ν µ D + 2 M µ ν , [ D, K ν ] = − K ν .

Observables The basic observables are correlation functions of local operators h O 1 ( x 1 ) O 2 ( x 2 ) ... O n ( x n ) i Each operator is characterized by Scaling dimension ∆ Representation under rotations (spin J ) Primary operators [ K ν , O ∆ , J (0)] = 0 1 � � ∂ x 0 d � � λ = � � ∂ x could not be written O ( x 0 ) = λ � ∆ O ( x ) � � as a derivative of smth Descendants P µ k ...P µ 1 O = ∂ µ k ... ∂ µ 1 O

Operators Simple examples present in every CFT are Unit operator ∆ = 0 J = 0 (lightest operator = dominates the OPE) ∂ µ T µ ν = 0 Stress energy tensor T µ µ = 0 (gravity dual) ∆ = d J = 2 Unitarity bounds ∆ ≥ d − 2 + J

Two- and Three-point Functions Correlation functions are invariant under symmetries. Conformal symmetry fixes 1-, 2-, and 3-point functions. critical exponents h O i ( x ) i = 0 (measured in experiments) h O i ( x 1 ) O j ( x 2 ) i = δ ij x 2 ∆ ij λ ijk h O ∆ i ( x 1 ) O ∆ j ( x 2 ) O ∆ k ( x 3 ) i = ∆ i + ∆ j − ∆ k ∆ i + ∆ k − ∆ j ∆ j + ∆ k − ∆ i ( x 2 ( x 2 ( x 2 12 ) 13 ) 23 ) 2 2 2 [Polyakov 70’] CFT data: ( ∆ , J ) λ ijk Goal: Find it!

Operator Product Expansion Operators form an algebra (OPE) λ ijk | x | ∆ i + ∆ j − ∆ k ( O k (0) + x µ ∂ µ O k (0) + ... ) X O i ( x ) O j (0) = k expansion in powers fixed of distance Consider now the four-point function of identical operators: G ( u, v ) h O ( x 1 ) O ( x 2 ) O ( x 3 ) O ( x 4 ) i = ( x 2 12 x 2 34 ) ∆ u = x 2 12 x 2 v = x 2 14 x 2 34 23 , x 2 13 x 2 x 2 13 x 2 24 24

Crossing Equations We can apply the OPE inside the correlation function O ( x 2 ) O ( x 3 ) O ( x 2 ) O ( x 3 ) O i X X O i = i i O ( x 1 ) O ( x 4 ) O ( x 1 ) O ( x 4 ) Nonperturbative!

Conformal Bootstrap Original Idea [Ferrara, Gatto, Grillo 73’] [Polyakov 74’] Realization in 2d [Belavin, Polyakov, Zamolodchikov 83’] Realization in 4d (based on results of ) [Dolan, Osborn 00’] [Rattazzi, Rychkov, Tonni, Vichi 08’] Numerical Solution of the Critical 3d Ising Model [El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin 12’-14’] Analytic Bootstrap [Fitzpatrick, Kaplan, Poland, Simmons-Duffin 12’] [Komargodski, AZ 12’]

O ( x 2 ) O ( x 3 ) Crossing Equations O ( x 2 ) O ( x 3 ) O i X X O i = i i O ( x 1 ) O ( x 4 ) O ( x 1 ) O ( x 4 ) v ∆ X ∆ , J g ∆ , J ( u, v ) = u ∆ X λ 2 λ 2 ∆ , J g ∆ , J ( v, u ) conformal block ∆ , J ∆ , J (known functions) Conformal block = contribution of the primary and its descendants ( O , ∂ O , ∂ 2 O , ... ) Conformal bootstrap = solve these equations Functional constraints on CFT data. Must be satisfied for all values of the cross ratios.

Conformal Blocks (Technical Details) Let us list few basic properties of conformal blocks: ✦ Eigenfunctions of the Casimir operator ˆ C g ∆ ,J ( u, v ) = ( ∆ + J )( ∆ + J − 1) g ∆ ,J ( u, v ) ✦ Small u<<1 limit twist τ 2 f τ ,J ( v ) , g ∆ ,J ( u, v ) ∼ u τ = ∆ − J expansion in powers ✦ Small v<<1 limit of cross ratios g ∆ ,J ( u, v ) ∼ log v

Conformal Map of the World twist Black Holes Regge Limit ∆ − J Thermal Physics Chaos DIS in AdS S-matrix Bootstrap EFT/cond-mat Numerical Analytic Bootstrap Bootstrap spin J

Numerical Bootstrap (Conformal Oracle) NO Tentative CFT data MAYBE [Talk by Slava Rychkov ’14] Input: a) Z 2 symmetry b) 1 even relevant scalar c) 1 odd relevant scalar [Kos, Poland, Simmons-Duffin, Vichi ’16]

Analytic Bootstrap

Analytic Methods Protected Observables [Dolan, Osborn; Beem, Lemos, Liendo, Peelaers, Rastelli, van Rees; Chesler, Lee, Pufu, Yacoby, …] Integrability [Escobedo, Gromov, Sever, Vieira; Basso, Coronado, Komatsu, Tat Lam, Vieira, Zhong; Bargheer, Caetano, Fleury, Komatsu, …] Large Central Charge [Heemskerk, Penedones, Polchinski, Sully; Fitzpatrick, Kaplan; Alday, Bissi, Lukowski; Rastelli, Zhou, …] Bootstrap in 2d [Cardy; Hellerman; Hartman, Keller, Stoica; Fitzpatrick, Kaplan, Walters; Lin, Shao, Simmons-Duffin, Wang, Yin, …] Crossing in Mellin space [Mack; Penedones; Gopakumar, Kaviraj, Sen, Sinha; Dey; Rastelli, Zhou; Alday, Bissi, Lukowski, …] Large Global Charge [Hellerman, Orlando, Reffert, Watanabe; Alvarez-Gaume, Loukas; Monin, Pirtskhalava, Rattazzi, Seibold; Jafferis, Mukhametzhanov, AZ …] S-matrix bootstrap [Caron-Huot, Komargodski, Sever, AZ; Paulos, Penedones, Toledo, van Rees, Vieira]

Analytic Bootstrap Study of the crossing equations in the Lorentzian regime. Analytic/Light-Cone Bootstrap Analyticity in Spin Regge limit, ANEC, chaos, gravity, etc

Analytic Bootstrap 101: Minimal Solution to Crossing

Analytic Bootstrap Consider the crossing equation in the light-cone limit v ⌧ u ⌧ 1 z Lorentzian regime 1 z = x 2 12 x 2 34 u = z ¯ x 2 13 x 2 • 24 • • z ) = x 2 14 x 2 0 23 v = (1 − z )(1 − ¯ x 2 13 x 2 24 1 ¯ z

Analytic Bootstrap We can use the OPE in one channel ∆ − J 2 ) G ( v, u ) = 1 + O ( v v ⌧ u ⌧ 1 v ∆ G ( u, v ) = u ∆ G ( v, u ) Crossing equation becomes ∆ , J g ∆ , J ( u, v ) = u ∆ X λ 2 G ( u, v ) = 1 + v ∆ (1 + ... ) ∆ , J diverges! Puzzle 1: v ⌧ 1 g ∆ , J ( u, v ) ∼ log v lim

Generalized Free Field (GFF) Example: Generalized Free Field h OOOO i = h OO ih OO i + permutations ⌘ ∆ G (0) ( u, v ) = 1 + u ∆ + ⇣ u v Spectrum contains operators O ⇤ n ∂ µ 1 ... ∂ µ J O (double-twist operators) ∆ n,J = 2 ∆ O + 2 n + J u ∆ Sum over spins produces the divergence v ∆

Analytic Bootstrap Resolution: Every solution to crossing equations has an infinite number of operators of every spin.

Analytic Bootstrap 201: Large Spin Universality

Analytic Bootstrap Impact parameter b is dual to spin J . Flat Space: b ∼ J b [Cornalba, Costa, Penedones, Schiappa] AdS (CFT): b ∼ log J [Alday, Maldacena] scattering phase shift CFT energy levels 1 δ ( s, b ) ∼ e − mb δ ∆ ( J ) ∼ J m

Large Spin Universality This mechanism of reproducing operators on one side by summing large spin operators on the other side is completely universal. (inner workings of crossing equations) [Fitzpatrick, Kaplan, Poland, Simmons-Duffin 12’] [Komargodski, AZ 12’] Every CFT is GFF at large spin Every CFT admits an infinite family of operators with the properties ∆ n,J = ∆ O 1 + ∆ O 2 + 2 n + J + O ( 1 J ) O ⇤ n ∂ µ 1 ... ∂ µ J O ✓ ◆ 1 + O ( 1 λ n,J = λ GF F J ) n,J

Analytic Bootstrap Let us add a first nontrivial correction to the previous exercise ∆ ,J g ∆ ,J ( u, v ) = u ∆ d 2 ∆ 2 ✓ ◆ d − 2 X 2 log u + ... λ 2 1 + v v ∆ ( d − 1) 2 c T ∆ ,J leading ‘correction GFF result due to stress tensor u ∆ (1 + γ J X X 2 log u ) λ GF F λ 2 ∆ ,J g ∆ ,J ( u, v ) ' f J ( v ) + ... J ∆ ,J J anomalous known collinear dimension conformal block By matching the two we get d 2 ∆ 2 Γ ( ∆ ) 2 Γ ( d + 2) 1 γ J = − � 2 J d − 2 2( d − 1) 2 c T Γ ( d +2 2 ) Γ ( ∆ − d − 2 � 2 )

Analytic Bootstrap The method works not only for singular terms, but also for Casimir-singular terms (act on the Casimir equation on the crossing equations). [Alday, Bissi, Lukowski] These correspond to terms that become singular upon acting on them with the Casimir operator v a Casimir-regular terms are v n , v n log v Equivalently, these are terms with non-zero double discontinuity dDisc[ f ( v )] = f ( v ) − 1 f ( ve 2 π i ) − f ( ve − 2 π i ) � � 2