AdS/CFT Correspondence and Integrability Kazuhiro Sakai ( Keio - PowerPoint PPT Presentation

Nagoya U. 2009-01-15 AdS/CFT Correspondence and Integrability Kazuhiro Sakai ( Keio University ) Based on collaborations with N.Beisert, N.Gromov, V .Kazakov, Y .Satoh, P .Vieira, K.Zarembo 1

0 . Introduction AdS/CFT correspondence --- a gauge/string duality ( Maldacena ’97, Gubser - Klebanov - Polyakov ’97, Witten ’97 ) N = 4 U( N ) super Yang-Mills planar • gluon dynamics common to QCDs IIB superstrings on AdS 5 x S 5 free • superstrings in the simplest curved background • limit integrability N → ∞ 2

AdS/CFT correspondence D 3 - brane = black hole like solution l s � R l s � R x 4 ∼ x 9 R R soliton supergravity background 3

D 3 - branes near horizon geometry x 0 ∼ x 3 x 0 ∼ x 4 x 5 ∼ x 9 x x 4 ∼ x 9 S 5 AdS 5            N open string U ( N ) gauge field closed String AdS 5 /CFT 4 correspondence • low energy limit type IIB superstrings = 4- dim N = 4 U ( N ) gauge theory on AdS 5 x S 5 4

gauge theory: string theory: 4 - dim action 2 - dim worldsheet action S = 1 � � √ d 2 σ [ G µν ∂ a X µ ∂ a X ν + · · · ] d 4 x [( F µν ) 2 + · · · ] S σ = λ λ 1 coupling const.: λ coupling const.: √ λ λ � 1 : weakly interacting gauge theory quantum strings λ � 1 : strongly interacting gauge theory ( semi -) classical strings 5

1997-2002 ( BPS states = atypical reps. of supersymmetry algebras ) BPS states: no quantum corrections ( coupling const. independent ) 2003- non - BPS states: quantum corrections ( coupling const. dependent ) N → ∞ limit integrability planar N = 4 gauge theory free strings on AdS g s = 0 ( ) 6

Strong/weak correspondence λ λ � 1 λ → ∞ quantum strings classical strings super Y ang - Mills on AdS 5 x S 5 at one - loop higher loops conventional sigma model spin chain on a coset space ( expected to be ) integrable integrable integrable ( Minahan - Zarembo ’02 ) ( Bena - Roiban - Polchinski ’03 ) at general λ ( Beisert - Staudacher ’03 ) particle model ( Beisert ’03 ) ( Staudacher ’04 ) ( Beisert ’05 ) ( L → ∞ ) 7

Plan of the talk 1. Super Y ang - Mills at one - loop ( spin chain ) 2. Classical string theory ( classical sigma model ) 3. All - order SYM/quantum strings ( particle model ) 8

＝ ＝ ＝ 1 . N = 4 Super Yang-Mills N = 4 gauge multiplet                                N = 2 vector multiplet N = 2 hypermultiplet                       N = 1 N = 1 N = 1 N = 1 vector chiral chiral chiral multiplet multiplet multiplet multiplet                                             A µ Ψ 2 Ψ 3 Ψ 4 Ψ 1 X Z Y Φ 1 + i Φ 2 Φ 3 + i Φ 4 Φ 5 + i Φ 6 9

N = 4 Super Yang-Mills 1 � ( F µν ) 2 + 2( D µ Φ i ) 2 − ([Φ i , Φ j ]) 2 L = − Tr 4 g 2 YM � + 2 i ¯ D Ψ − 2 ¯ Ψ / ΨΓ i [Φ i , Ψ] Global symmetry: SO (4 , 2) × SU (4) ⊂ P SU (2 , 2 | 4) Φ ab  ¯ Q Q +1    ¯ Ψ b  Ψ αb P  α ˙      ¯ F ˙ D ˙ αβ Φ ab F αβ  α ˙  β    V F αβ ¯ Ψ d D ˙ D ˙ αβ Ψ γ d γ ˙     D ¯ F DD Φ DF        DD ¯ Ψ DD Ψ     10

Conformal Field Theory ● Correlation function of local operators �O 1 ( x 1 ) O 2 ( x 2 ) � B 12 = δ D 1 D 2 | x 12 | D 1 + D 2 �O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) � C 123 = | x 12 | D 1 + D 2 − D 3 | x 23 | D 2 + D 3 − D 1 | x 31 | D 3 + D 1 − D 2 O i D i : scaling dimension of the local operator 11

● Single trace operators O = Tr[ W A 1 W A 2 · · · W A J ] W A ∈ { D k Φ , D k Ψ , D k ¯ Ψ , D k F } ( Beisert ’03 ) - dominant in the large N limit ● Scaling dimension: ˆ eigenvalue of the Dilatation operator D ● At tree level: ˆ D 0 O = dim( O ) O [Ψ] = 3 [Φ] = 1 , [ F ] = 2 , [ D ] = 1 2 , 12

● Quantum correction: operator mixing O � = ˆ DO Φ Ψ Φ Φ Ψ F O Φ Ψ Φ Ψ Φ F ˆ : non - diagonal matrix spectral problem D ∞ λ n ˆ � ˆ λ = g 2 D = D n YM N ( ’t Hooft coupling ) n =0 su (2 , 2 | 4) ˆ ⇔ Hamiltonian of spin chain D 1 ( Minahan - Zarembo ’02 ) ( Beisert - Staudacher ’03 ) 13

⇔ ⇔ ● SU(2) subsector ↓ ↓ ↓ ↓ ↓ ↓ ↓ Tr Z L ↓ ↓ ↓ ↓ ↓ ferromagnetic vacuum X = Φ 1 + i Φ 2 Z = Φ 5 + i Φ 6 ↑ ↓ ↓ ↑ ↓ ↓ ↓ Tr( ZZZXZXZZ · · · ) ↓ ↓ ↓ ↓ ↓ XXX Heisenberg Spin chain L � ) H = ( − l +1 l +1 l =1 l l 14

Bethe ansatz equation ( coordinate Bethe ansatz ) One - magnon states L l � | Ψ( p ) � = ψ ( l ) | ↑ · · · ↑ ↓ ↑ · · · ↑� l =1 ψ ( l ) = e ipl Schrödinger Eq. H | Ψ � = E | Ψ � L � ) H = ( − l +1 l +1 l =1 l l 2 − e ip − e − ip E = 4 sin 2 p = : Dispersion relation 2 15

Two - magnon states l 1 l 2 � | Ψ( p 1 , p 2 ) � = ψ ( l 1 , l 2 ) | ↑ · · · ↑ ↓ ↑ · · · ↑ ↓ ↑ · · · ↑� 1 ≤ l 1 <l 2 ≤ L H | Ψ � = E | Ψ � Schrödinger Eq. 2 4 sin 2 p k � ( dispersion relation ) E = 2 k =1 ψ ( l 1 , l 2 ) = e ip 1 l 1 + ip 2 l 2 + S ( p 2 , p 1 ) e ip 1 l 2 + ip 2 l 1 ( Bethe’s ansatz ) S ( p 1 , p 2 ) = − e ip 1 + ip 2 − e 2 ip 1 + 1 S - matrix e ip 1 + ip 2 − e 2 ip 2 + 1 16

Integrability of 2 D particle models ‣ Factorization of multi - particle scattering amplitudes = = 3 1 3 1 3 1 2 2 2 E ( p ) � dispersion relation for 1 particle ˆ scattering matrix for 2 particles S ( p 1 , p 2 ) � � all scattering amplitudes are determined spectra of conserved charges 17

Factorized scattering ψ ( p 2 , p 1 , p 3 , . . . , p J ) = S ( p 1 , p 2 ) ψ ( p 1 , p 2 , p 3 , . . . , p J ) Periodic boundary condition ψ ( p 2 , . . . , p J , p 1 ) = e − ip 1 L ψ ( p 1 , . . . , p J ) Y ang equations J e ip k L = � S ( p k , p l ) l � = k � ∂ p u = 1 � 2 cot p rapidity variable ∂ u = E 2 Bethe ansatz equations � L � J u k + i u k − u l + i � 2 ( k = 1 , . . . , J ) = u k − i u k − u l − i 2 l � = k 18

Local Charges Momentum i ln u k + i 1 � 2 P = Q 1 = u k − i k 2 Energy � � i i � E = Q 2 = − u k + i u k − i k 2 2 Higher charges � � 1 i i � Q r = 2 ) r − 1 − ( u k + i ( u k − i r − 1 2 ) r − 1 k 19

2. Classical strings • AdS/CFT Correspondence IIB Superstrings on N = 4 U( N ) x Super Y ang - Mills S 5 AdS 5 SO (4 , 2) × SO (6) ⊂ P SU (2 , 2 | 4) R 4 = 4 πg s α � 2 N λ = g 2 Y M N g 2 Y M = g s N → ∞ 4 πλ = R 4 α � 2 20

E = L L � �� � x O = Tr( ZZZ · · · ZZZ ) L O = Tr( Z · · · X · · · ¯ Y · · · Z ) x + · · · O = Tr( Z · · · ∇ s Z · · · ∇ s � Z · · · Z ) x + · · · 21

R t × S n Sigma model on √ λ � dσdτ [ − ∂ a X 0 ∂ a X 0 + ∂ a X i ∂ a X i + Λ ( X i X i − 1)] S = 4 π ( i = 1 , . . . , n ) Equations of motion ∂ + ∂ − X i + ( ∂ + X j ∂ − X j ) X i = 0 , ∂ + ∂ − X 0 = 0 Gauge: X 0 = κτ ∆ : energy of the string κ = ∆ � 2 π � � √ λ √ ∆ = dσ∂ τ X 0 = λκ √ λ 2 π 0 Virasoro constraints ( ∂ ± X i ) 2 = ( ∂ ± X 0 ) 2 = κ 2 22

Examples of classical string solutions in S 2 × R t point - like string pulsating string folded string circular string 23

giant magnon ∆ ϕ ( Hofman - Maldacena ’06 ) p 24

circular string   sn( κσ | k ) cos ωτ k = ω � sn( κσ | k ) sin ωτ X =   κ cn( κσ | k ) folded string   k sn( ωσ | k ) cos ωτ k = κ � k sn( ωσ | k ) sin ωτ X =   ω dn( ωσ | k ) 25

R t × S 3 Sigma model on SU ( 2 ) Principal Chiral Field Model ∼ = � X ∈ S 3 g ∈ SU(2) ↔ � � X 1 + iX 2 X 3 + iX 4 g = − X 3 + iX 4 X 1 − iX 2 Right current j = − g − 1 dg d j − j ∧ j = 0 , d ∗ j = 0 Virasoro constraints 1 2Tr j 2 ± = − κ 2 26

Lax Connection 1 x a ( x ) = 1 − x 2 j + 1 − x 2 ∗ j x : spectral parameter d j − j ∧ j = 0 d a ( x ) − a ( x ) ∧ a ( x ) = 0 d ∗ j = 0 [ L ( x ) , M ( x )] = 0 Lax pair L ( x ) = ∂ σ − a σ ( x ) = ∂ σ − 1 � j + j − � 1 − x − 2 1 + x M ( x ) = ∂ τ − a τ ( x ) = ∂ τ − 1 � j + j − � 1 − x + 2 1 + x 27

Auxiliary Linear Problem � ∂ σ Ψ = a σ Ψ � L ( x )Ψ( x ; τ , σ ) = 0 M ( x )Ψ( x ; τ , σ ) = 0 ∂ τ Ψ = a τ Ψ � σ Ψ( x ; τ, σ ) = P exp a σ dσ 0 Monodromy matrix Ψ( x ; τ, σ + 2 π ) = Ω( x ; τ, σ )Ψ( x ; τ, σ ) � 2 π Ω ( x ; τ, σ ) = P exp a σ dσ 0 28

τ Monodromy matrix σ Ω ( x ; ˜ τ , ˜ σ ) (˜ τ , ˜ σ ) (˜ τ , ˜ σ + 2 π ) σ ) = U − 1 Ω ( x ; τ, σ ) U Ω ( x ; ˜ τ, ˜ Ω ( x ; τ, σ ) ( τ , σ + 2 π ) ( τ , σ ) e ip 1 ( x ) � � 0 Ω( x ) ∼ e ip 2 ( x ) 0 p 1 ( x ) = − p 2 ( x ) =: p ( x ) quasi - momentum 29

Spectral curve ( Kazakov - Marshakov - Minahan - Zarembo ’04 ) p 1 ( x ) x p 2 ( x ) x = − 1 x = +1 30