Gauge/gravity duality: an overview Z. Bajnok

Seminar, December 7, 2012, Bratislava 

Gauge/gravity duality: an overview 

Z. Bajnok 

MTA-Lendület Holographic QFT Group 

Wigner Research Centre for Physics, Budapest 

The Illusion of Gravity - Juan Maldacena, Scientific American (2005) 

1

Motivation: Feynman’s legacy 

Richard P. Feynman 

(1918–1988) 

What I cannot create I do not understand. 

Know how to solve every problem that has been solved. 

To learn: Bethe Ansatz Probs., ← Kondo, 2D Hall, accel 

temp, Non linear classical Hydro 

1965 

Quantumelectrodynamics: 

Feynman graphs 

Strong interaction? 

2

Motivation: Organizing matter 

3

Motivation: Organizing matter


Electric interaction (potential Φ(r) = k Zq 

r ) 

Quantum mechanics (Schrödinger eq.) HΨ = (− (∇)2 

2m 

+ Φ(r))Ψ = E Ψ



r ) 


2m 

+ Φ(r))Ψ = E Ψ



r ) 


2m 

+ Φ(r))Ψ = E Ψ

Organizing matter II 

Brookhaven: Relativistic heavy ion collider (gold ion) 

4


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

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 = 1159652180.85(.76) × 10 −12 

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: 


α 

2π = 

2πc e2 = 0.001161 = + +... 

g 

2 

= 1 − 1.3140 α 2π 

momentum-dependent coupling: 

(127) −1 

β(α) = µ ∂α 

∂µ > 0 (137) −1 

Μ Ζ 

α 

5

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: 


0.001 = 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

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 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 2 − 1 2 (DΦ)2 + iΨD/ Ψ + V ] 

V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

no running β = 0 → CFT 

no confinement 

supersymmetric 

heavy ion collision: 

finite T → SUSY is broken 

quark-gluon plasma is not confined 

7

positively curved space 

AdS: string theory on Anti de Sitter ⊃ gravitation 

8



Anti de Sitter: negatively curved space



Anti de Sitter: negatively curved space



Anti de Sitter: negatively curved space 

x 0 

1 

relativistic point particle: ds 2 = −dx 2 0 + dx2 1 + . . . 

S ∝ worldline ∝ ∫ ds = ∫ √ẋ 

· ẋdτ 

x 

2 

x( τ) 

x




x 0 

1 



· ẋdτ 

relativistic string: ds 2 = −dx 2 0 + dx2 1 + . . . 

S ∝ worldsheet ∝ ∫ dA = ∫ √ (ẋ · x ′ ) 2 − ẋ 2 x ′2 dτdσ 

x 

2 

x( τ) 

x 

x 0 

x( τ,σ) 

τ 

x 

2 σ 

x 

1






· ẋdτ 

relativistic string: ds 2 = −dx 2 0 + dx2 1 + . . . 

S ∝ worldsheet ∝ ∫ dA = ∫ √ (ẋ · x ′ ) 2 − ẋ 2 x ′2 dτdσ 

S 5 :Y0 2 + Y 1 2 + Y 2 2 + Y 3 2 + Y 4 2 + Y 2 

x 0 

x( τ) 

x 

2 

x 

1 

S 5 AdS 5 

5 = R2 

x 0 

x( τ,σ) 

τ 

x 

2 σ 

x 

1 

AdS 5 :−X 2 0 + 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)

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 2 − 1 2 (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 

λ2 

+168 λ3 

−(1410+144ζ(3)+ 1 λ4 

(324+864ζ(3)−1440ζ(5))) 

4π 2 (4π 2 ) 2 (4π 2 ) 3 2 (4π 2 ) 4 

9

AdS/CFT correspondence (Maldacena 1998) 

R 2 

α ′ 

II B superstring on AdS 5 × S 5 

4D 

Minkowski 

space 

5D 

anti de Sitter 

space 

5D sphere 

≡ 

space 

time 

extra dimension 

( ) 

∂a X M ∂ a X M + ∂ a Y M ∂ a Y M + . . . 

∑ 6 

1 Y i 2 = R 2 − + + + +− = −R 2 

∫ dτdσ 

4π 

2 

g 2 Y M 

N = 4 D=4 SU(N) SYM 

∫ d 4 xTr [ − 1 4 F 2 − 1 2 (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

R 2 

α ′ 


4D 

Minkowski 

space 

5D 


space 


5D sphere 

≡ 

space 

time 


( ) 

∂a X M ∂ a X M + ∂ a Y M ∂ a Y M + . . . 

∑ 6 

1 Y i 2 = R 2 − + + + +− = −R 2 

∫ 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 

⇓ 


∫ d 4 xTr [ − 1 4 F 2 − 1 2 (DΦ)2 + iΨD/ Ψ + V ] 

V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

P SU(2,2|4) 


SO(5)×SO(1,4) 


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 2 ∆ 2 + . . .

R 2 

α ′ 


4D 

Minkowski 

space 

5D 


space 


5D sphere 

≡ 

space 

time 


( ) 

∂a X M ∂ a X M + ∂ a Y M ∂ a Y M + . . . 

∑ 6 

1 Y i 2 = R 2 − + + + +− = −R 2 

∫ 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 

⇓ 


∫ d 4 xTr [ − 1 4 F 2 − 1 2 (DΦ)2 + iΨD/ Ψ + V ] 

V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

P SU(2,2|4) 


SO(5)×SO(1,4) 


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 2 ∆ 2 + . . . 

2D integrable QFT 

√ 

spectrum: Q = 1, 2, . . . , ∞ dispersion: ɛ Q (p) = Q 2 + λ π 2 sin2 p 

2 

Exact scattering matrix: S Q1 Q 2 

(p 1 , p 2 , λ)

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| 2 + . . . ] 

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) = 4 + 6 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

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 2 + . . . = J I + 

λ 

8π 2H XXX + λ 2 H 2 + . . . 

H 2 : next-to-nearest neighbour integrable! → use Bethe ansatz 

1. choose a groundstate: Z = Z 1 → Tr [ Z J] = Tr [ZZZZZ . . . ZZZZ] ↔| ↑ . . . ↑〉 

2. 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

AdS/CFT correspondence: confirmation 

supersymmetric BPS operators 

V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

Z = Φ 1 + iΦ 2 , X = Φ 3 + iΦ 4 

O BP S = Tr(Z J ) ↔ | ↑↑ . . . ↑〉 

∆ BP S = J 

weak↔strong 

BPS string configuration 

S 5 

J 

E BP S (λ) = J 

13



V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

Z = Φ 1 + iΦ 2 , X = Φ 3 + iΦ 4 

O BP S = Tr(Z J ) ↔ | ↑↑ . . . ↑〉 

∆ BP S = J 

weak↔strong 


S 5 

J 



supersymmetric groundstate E 0 (J) = ∆(λ) − J = 0



V (Φ, Ψ) = 1 4 [Φ, Φ]2 + Ψ[Φ, Ψ] 

Z = Φ 1 + iΦ 2 , X = Φ 3 + iΦ 4 

O BP S = Tr(Z J ) ↔ | ↑↑ . . . ↑〉 

∆ BP S = J 

weak↔strong 


S 5 

J 



supersymmetric groundstate E 0 (J) = ∆(λ) − J = 0 

Nontrivial anomalous dimension 

supersymmetric theory: Excited state 

O K = Tr(ZXZX + ...) ↔ | ↑↓↑↓〉 + .

Confirmation: excited state - Konishi operator 

nonsupersymmetric operator: Konishi 

O K = Tr(ZXZX + ...) ↔ | ↑↓↑↓〉 + . 

operator mixing 

string configuration 

S 5 

−p 

∆(λ) = ∆(0) + λ∆ 1 + . . . + λ 4 ∆ 4 + 

≡ 

J 

p 

[Fiamberti ..’08] 

∆ 4 = −2496 + 576ζ 3 − 1440ζ 5 

moving bumps (sine-Gordon) [Hofman .. ’07] 

string action=saddle point+loop corr. 

E(λ) = E(∞) + √ E 1 

+ E 2 

λ λ + . . . 

14

Confirmation: excited state - Konishi operator 

nonsupersymmetric operator: Konishi 

O K = Tr(ZXZX + ...) ↔ | ↑↓↑↓〉 + . 

operator mixing 

string configuration 

S 5 

−p 

∆(λ) = ∆(0) + λ∆ 1 + . . . + λ 4 ∆ 4 + 

≡ 

J 

p 

[Fiamberti ..’08] 

∆ 4 = −2496 + 576ζ 3 − 1440ζ 5 

moving bumps (sine-Gordon) [Hofman .. ’07] 

string action=saddle point+loop corr. 

E(λ) = E(∞) + √ E 1 

+ E 2 

λ λ + . . . 

two particle state 

E = E BA + E F SC 

Bethe Ansatz: e ipJ S(p, −p) = 1 

E BA = 2E(p, λ) = 2 1 + λ π 2(sin p 2 )2 

E F SC = ∑ ∫ dq 

Q 2π S Q1(q, p)S Q1 (q, −p)e −ɛ QL 

E 4 = ∆ 4 = −2496 + 576ζ 3 − 1440ζ 5 [Z.B., R. Janik ’09] 

√

AdS/CFT spectral problem 

15

Konishi dimension: Tr(ZXZX − ZZXX) 

AdS/CFT spectral problem

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

AdS/CFT correspondence: applications 

Minimal surface 

quark-antiquark potential 

anti−quark 

L 

quark 

space 

time 


exact for strong coupling λ → ∞ 

≡ 

Wilson loop: 〈 ∮ C A µdx µ 〉 

non-perturbative 

V (r) = − 4π2√ 2λ 

Γ( 1 4 )4 1 

r 

16

AdS/CFT correspondence: applications 

Minimal surface 

quark-antiquark potential 

anti−quark 

L 

quark 

space 

time 


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 


metric δg(x, 0) ∝ 〈T µν 〉 

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 +...

Gauge/gravity duality: an overview Z. Bajnok

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?