Gauge/gravity duality: an overview. Z. Bajnok

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1 Conference of the Middle European Cooperation in Statistical Physics, MECO37, March 18-22, 2012, Tatranské Matliare, Slovak Republik Gauge/gravity duality: an overview Z. Bajnok Theoretical Physics Research Group of the Hungarian Academy of Sciences, Eötvös University, Budapest 1

2 Conference of the Middle European Cooperation in Statistical Physics, MECO37, March 18-22, 2012, Tatranské Matliare, Slovak Republik Gauge/gravity duality: an overview Z. Bajnok Theoretical Physics Research Group of the Hungarian Academy of Sciences, Eötvös University, Budapest 4D Minkowski space 5D sphere 5D anti de Sitter space time space extra dimension Alias: AdS/CFT correspondencce gauge/gravity duality

3 Motivation: AdS=CFT J. Polchinski: TASI lectures, arxiv: : Physics World, reader poll: What is the GREATEST EQUATION EVER? 2

4 Motivation: AdS=CFT J. Polchinski: TASI lectures, arxiv: : Physics World, reader poll: What is the GREATEST EQUATION EVER? Leonhard Euler ( ) James Clerk Maxwell ( ) e i π + 1 = 0 d F = j ; df = 0 i, π, e, 1, 0 and +,, ˆ unifies: electric+magnetic int.

5 Motivation: AdS=CFT J. Polchinski: TASI lectures, arxiv: : Physics World, reader poll: What is the GREATEST EQUATION EVER? Leonhard Euler ( ) James Clerk Maxwell ( ) e i π + 1 = 0 d F = j ; df = 0 i, π, e, 1, 0 and +,, ˆ unifies: electric+magnetic int. Juan Martín Maldacena (1968-) AdS = CF T unifies: all forces+quantum theory

6 Motivation: AdS=CFT J. Polchinski: TASI lectures, arxiv: : Physics World, reader poll: What is the GREATEST EQUATION EVER? Leonhard Euler ( ) James Clerk Maxwell ( ) e i π + 1 = 0 d F = j ; df = 0 i, π, e, 1, 0 and +,, ˆ unifies: electric+magnetic int. Juan Martín Maldacena (1968-) AdS = CF T unifies: all forces+quantum theory J. Maldacena, Adv.Theor.Math.Phys. 2 (1998) : more than 8000 citations so far

7 Motivation: Organizing matter 3

8 Motivation: Organizing matter

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10 Motivation: Organizing matter Electric interaction (potential Φ(r) = k Zq r ) Quantum mechanics (Schrödinger eq.) HΨ = ( ( )2 2m + Φ(r))Ψ = E Ψ

11 Motivation: Organizing matter Electric interaction (potential Φ(r) = k Zq r ) Quantum mechanics (Schrödinger eq.) HΨ = ( ( )2 2m + Φ(r))Ψ = E Ψ

12 Motivation: Organizing matter Electric interaction (potential Φ(r) = k Zq r ) Quantum mechanics (Schrödinger eq.) HΨ = ( ( )2 2m + Φ(r))Ψ = E Ψ

13 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion) 4

14 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion)

15 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion).

16 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion) Number of elementary particles > number of atoms classification.

17 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion) Number of elementary particles > number of atoms classification.

18 Organizing matter II Brookhaven: Relativistic heavy ion collider (gold ion) Number of elementary particles > number of atoms classification. Decay strong, weak interaction Standard Model Interaction γ photon electromagnetic I. W ±, Z weak bosons weak g gluon strong II. gr graviton gravitational

19 Quantum electrodynamics Relativity theory: A µ = (Φ, A) electric + magnetic int: F µν +Quantum theory QED U(1) gauge theory: A µ (x) A µ (x) + µ Λ(x) electron electron photon L = 4 1 F 2 + Ψ(i / m)ψ e ΨA/Ψ experiment: µ = g2mc e s where g = 2(1 + a) Gabrielse et.al.: a = (.76) Feynman: If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. Quantum gauge theory perturbation theory: Feynman graphs α 2π = 2π c e2 = = g 2 = α 2π momentum-dependent coupling: (127) 1 β(α) = µ α µ > 0 (137) 1 Μ Ζ α 5

20 Quantum Chromodynamics photon A µ G 1..8 µ gluon Fµν 1..8 electron Ψ e Ψ kvark quark SU(3) gauge theory: G µ g 1 G µ g + g 1 µ g quark quark gluon gluon L = 1 4 F 2 + Ψ(i / m)ψ g ΨG/Ψ gluon experiments: hadron spectrum Quantum gauge theory asymptotic freedom perturbation theory: Feynman graphs = α 2π α s 4π = O(1) = momentum-dependent coupling: β(α s ) = µ α s µ < 0 α asymptotic freedom confinement (127) 1 (137) 1 Μ Ζ 6

21 Fundamental interactions CFT: maximally supersymmetric gauge theory interaction particles gauge theory electromagnetic photon+electron U(1) electroweak W ±, Z µ,ν+higgs SU(2) U(1) strong gluon+quarks SU(3) only analytical tool: perturbation theory maximally supersymmetric (N=4) gauge theory interaction particles gauge theory N = 4 supersymm. gluon+quarks+scalars SU(N) Coupling SU(3) SU(2) U(1) non perturbative no analytical tool perturbative strong force electroweak force Energy L = 2 g 2 Y M all fields N 2 1 component matrix Ψ 1,2,3,4 A µ Φ 1,2,3,4,5,6 Ψ 1,2,3,4 d 4 xtr [ 1 4 F (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] no running β = 0 CFT 7

22 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 8

23 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 Anti de Sitter: negatively curved space

24 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 Anti de Sitter: negatively curved space

25 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 Anti de Sitter: negatively curved space relativistic point particle: ds 2 = dx dx S worldline ds = ẋ ẋdτ x 0 x 2 x( τ) x 1

26 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 Anti de Sitter: negatively curved space relativistic point particle: ds 2 = dx dx S worldline ds = ẋ ẋdτ relativistic string: ds 2 = dx dx x 0 x 2 x( τ) x 1 x 0 S worldsheet da = (ẋ x ) 2 ẋ 2 x 2 dτdσ x( τ,σ) x 2 τ σ x 1

27 AdS: string theory on Anti de Sitter gravitation positively curved space S 1 Anti de Sitter: negatively curved space relativistic point particle: ds 2 = dx dx S worldline ds = ẋ ẋdτ relativistic string: ds 2 = dx dx x 0 x 2 x( τ) x 1 x 0 S worldsheet da = (ẋ x ) 2 ẋ 2 x 2 dτdσ S 5 S 5 :Y Y Y Y Y Y 2 5 = R2 5 AdS x( τ,σ) x 2 τ σ x 1 AdS 5 : X X2 1 + X2 2 + X2 3 + X2 4 X2 5 = R2 S = R2 dτdσ α 4π ( a X M a X M + a Y M a Y M ) + fermionok supercoset P SU(2,2 4) SO(5) SO(1,4)

28 CFT: Observables maximally supersymmetric gauge theory Ψ 1,2,3,4 A Φ 1,2,3,4,5,6 S = 2 g 2 Y M Ψ 1,2,3,4 fields SU(N) matrices d 4 xtr [ 1 4 F (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] observables parameters: g Y M, N observables: partition function gauge-invariant operators O(x) = Tr(A L 1Ψ L 2Φ L 3..) correlators: O 1 (x)o 2 (0) correlators: O 1 (x)o 2 (0) = [da...]e is O 1 (x)o 2 (0) = O 1 (x)o 2 (0)e iv 0 perturbation: g 2 Y M g 2 Y M g 2 Y M N genus exp. g 2 Y M N 3 = N 2 λ λ = g 2 Y M N partition func. Z(λ, 1 N ) = N 2 g( 1 N )2g n α(g, n)λ n string interactions? (t Hooft) conformal field theory: O i (x)o j (0) = δ ij x 2 i scale dim.: i Konishi op. O K = Tr(Φ 2 i ) K (λ) = 2+6 λ 24 λ λ3 ( ζ(3)+ 1 λ4 ( ζ(3) 1440ζ(5))) 4π 2 (4π 2 ) 2 (4π 2 ) 3 2 (4π 2 ) 4 9

29 AdS/CFT correspondence (Maldacena 1998) R 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space 5D sphere 6 1 Y i 2 = R = R ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π 2 g 2 Y M N = 4 D=4 SU(N) SYM Ψ 1,2,3,4 A µ Φ 1,2,3,4,5,6 Ψ 1,2,3,4 d 4 xtr [ 1 4 F (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = 0 superconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ 2 ), det( ) 10

30 R 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space AdS/CFT correspondence (Maldacena 1998) 5D sphere 6 1 Y i 2 = R = R ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π Couplings: λ = R2 α, g s = λ N 0 2D QFT String energy levels: E(λ) E(λ) = E( ) + E 1 + E 2 λ λ +... Dictionary 2 g 2 Y M strong weak N = 4 D=4 SU(N) SYM Ψ 1,2,3,4 A µ Φ 1,2,3,4,5,6 Ψ 1,2,3,4 d 4 xtr [ 1 4 F (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = 0 superconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ 2 ), det( ) λ = g 2 Y M N, N planar limit O n (x)o m (0) = δ nm x 2 n(λ) Anomalous dim (λ) (λ) = (0) + λ 1 + λ

31 R 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space AdS/CFT correspondence (Maldacena 1998) 5D sphere 6 1 Y i 2 = R = R ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π Couplings: λ = R2 α, g s = λ N 0 2D QFT String energy levels: E(λ) E(λ) = E( ) + E 1 + E 2 λ λ +... Dictionary 2 g 2 Y M strong weak N = 4 D=4 SU(N) SYM Ψ 1,2,3,4 A µ Φ 1,2,3,4,5,6 Ψ 1,2,3,4 d 4 xtr [ 1 4 F (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = 0 superconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ 2 ), det( ) λ = gy 2 MN, N planar limit O n (x)o m (0) = δ nm x 2 n(λ) Anomalous dim (λ) (λ) = (0) + λ 1 + λ D integrable QFT

32 CFT: Integrablity Perturbative correlator: O 1 (x)o 2 (0) = O 1 (x)o 2 (0)e i(1 4 [Φ,Φ]2 +Ψ[Φ,Ψ]) 0 Conformal (scale invariant) field theory: = δ ij x 2 (λ) = 1 x 2 (0) [ 1 + λ 1 log 1 x ] Scalar sector: Z 1 = Φ 1 +iφ 2, Z 2 = Φ 3 +iφ 4 SUSY st: O = Tr [ Z J i ] O (λ) = J Operator mixing: O 1 = Tr [Z 1 Z 1 Z 2 Z 2 ] O 2 = Tr [Z 1 Z 2 Z 1 Z 2 ] diagonalize the 1-loop mixing matrix: O ± = O 1 ± O 2 O+ (λ) = 4 O (λ) = λ 4π 2 generic state at size J: O i1...i J = Tr [ Z i1... Z ij ] i1... i J j 1 j j j j 2 k k+1 J j 1 j j j j 2 k k+1 J j 1 j j j j 2 k k+1 J j 1 j j j j 2 k k+1 J = J I + λ Jk=1 8π 2 (I P k,k+1 ) Heisenberg spin chain i i i i i 1 2 k k+1 J i i i i i 1 2 k k+1 J i i i i i 1 2 k k+1 J i i i i i 1 2 k k+1 J 11

33 CFT: Integrablity + Bethe Ansatz Mixing matrix on the subspace Tr [ Z i1... Z ij ] of dim 2 J : Minahan-Zarembo 2002 = H 0 + λh 1 + λ 2 H = J I + λ 8π 2H XXX + λ 2 H H 2 : next-to-nearest neighbour integrable! use Bethe ansatz 1. choose a groundstate: Z = Z 1 Tr [ Z J] = Tr [ZZZZZ... ZZZZ] excitations Z...ZXZ...X with SUSY multiplet X = Z 2, Z 3, Ψ α a, Ψa, α D µ n {}}{ 3. plane wave: n e ipn Tr( Z...Z XZ... ZZ) 4. scattering states: n 1 n 2 a 1 a 2 e ip 1n 1 +ip 2 n 2 Tr( Z...Z }{{} X a1 Z...Z X a2 Z... Z)+S(12) b 1b 2 n 1 n 2 {}}{ symmetry completely fixes the S-matrix for any λ (satisfies unitarity, crossing, Yang-Baxter) Bethe ansatz follows from S-matrix: Shastry s Hubbard S-matrix a 1 a 2 12

34 AdS/CFT correspondence: confirmation two particle states E BP S (λ) = E 0 S 5 J p E K (λ) = 2E(p, λ) dispersion relation E(p, λ) = 1 + λ π 2(sin p 2 )2 elastic scattering S(p, p) Bethe Ansatz: e ipj S(p, p) = 1 finite size corrections p Konishi anomalous dimension Z = Φ 1 + iφ 2, X = Φ 3 + iφ 4 supersymmetric BPS operators O BP S = Tr(Z J )... nonsupersymmetric operator: Konishi O K = Tr(ZXZX +...) +. operator mixing integrable spinchain O i (x)o j (0) = δ ij x 2 i (λ) Bethe Ansatz + wrapping (λ) = (0) + λ λ E 4 = 4 = ζ ζ 5 perturbatively 10 5 diagrams E 4 = 4 = ζ ζ 5 13

35 AdS/CFT spectral problem 14

36 Konishi dimension: Tr(ZXZX ZZXX) AdS/CFT spectral problem

37 AdS/CFT spectral problem gauge coupling Classical string/gravity theory Semiclassical string/gravity theory quantum corrections Quantum string/gravity theory TBA Strongly coupled gauge theory loop corrections finite size corrections Lüscher correction Asymptotic Bethe Ansatz S matrix 1 loop perturbative gauge theory 0 size

38 AdS/CFT correspondence: applications Minimal surface quark-antiquark potential anti quark L quark space time extra dimension exact for strong coupling λ Wilson loop: C A µdx µ non-perturbative V (r) = 4π2 2λ Γ( 1 4 )4 1 r 15

39 AdS/CFT correspondence: applications Minimal surface quark-antiquark potential anti quark L quark space time extra dimension exact for strong coupling λ Wilson loop: C A µdx µ non-perturbative V (r) = 4π2 2λ Γ( 1 4 )4 1 r growing black hole Heavy ion collision: expansion quark gluon plasma expands, cools grows black hole heavy ion collision thermalization isotropization Expansion Cooling Hadronization time heavy ions space extra dimension metric ds 2 = 1 z 2 (g(x, z) µν dx µ dx ν + dz 2 ) Einstein equation R ab 1 2 g abr 6g ab = 0 growing black hole g tt = (1 z4 /z 4 0 )2 (1+z 4 /z 4 0 )2 ; g xx = 1 + z4 z 4 0 T µν matter distribution relativistic hydrodynamics µ T µν = 0 and T µ µ = 0 viscous quark-gluon plasma expansion in time: perfect fluid + η s = 1 4π +...