Integrability and magnon kinematics in the AdS/CFT correspondence - PowerPoint PPT Presentation

Integrability and magnon kinematics in the AdS/CFT correspondence Integrability and magnon kinematics in the AdS/CFT correspondence † Rafael Hern´ andez, IFT-Madrid † Collaboration with N. Beisert, C. G´ omez and E. L´ opez (See also the talk by E. L´ opez)

Integrability and magnon kinematics in the AdS/CFT correspondence Outline • Introduction • Integrability in the AdS/CFT correspondence • The quantum string Bethe ansatz • Symmetries of the scattering matrix ◦ Crossing symmetry and the dressing phase factor ◦ Quantum-deformed magnon kinematics • Conclusions

Integrability and magnon kinematics in the AdS/CFT correspondence Introduction The AdS/CFT correspondence: The large N limit of N = 4 Yang-Mills is dual to type IIB string theory on AdS 5 × S 5 ⇒ Spectra of both theories should agree → Difficult to test, because the correspondence is a strong/weak coupling duality: we can not use perturbation theory on both sides String energies expanded at large λ E ( λ ) = λ 1 / 4 E 0 + λ − 1 / 4 E 1 + λ − 3 / 4 E 2 + . . . Scaling dimensions of gauge operators at small λ ∆( λ ) = D 0 + λ D 1 + λ 2 D 2 + . . . E ( λ ) ↔ ∆( λ ) → Integrability illuminates both sides of the correspondence → S string should interpolate to S gauge

Integrability and magnon kinematics in the AdS/CFT correspondence Integrability in the AdS/CFT correspondence A complete formulation of the AdS/CFT correspondence ⇒ Precise identification of string states with local gauge invariant operators √ α ′ = ∆ ⇒ E Strong evidence in the supergravity regime , R 2 ≫ α ′ ( R 4 =4 π g 2 YM N α ′ 2 ) ◦ String quantization in AdS 5 × S 5 Difficulties : ◦ Obtaining the whole spectrum of N = 4 is involved An insight: There is a maximally supersymmetric plane-wave background for the IIB string [Blau et al] ⇓ Plane-wave geometry ⇒ Penrose limit

Integrability and magnon kinematics in the AdS/CFT correspondence The Penrose limit shows up on the field theory side [Berenstein, Maldacena, Nastase] ⇓ Operators carrying large charges , tr ( X J 1 . . . ) , J ≫ 1 → Dual description in terms of small closed strings whose center moves with angular momentum J along a circle in S 5 [Gubser, Klebanov, Polyakov] Generalization: Operators of the form tr ( X J 1 1 X J 2 2 X J 3 3 ) are dual to strings with angular momenta J i [Frolov, Tseytlin] ⇒ The energy of these semiclassical strings admits an analytic expansion in λ/ J 2 � λ � J i � � E = J 1 + c 1 J 2 + . . . J ⇒ Comparison with anomalous dimensions of large Yang-Mills operators: • Bare dimension ∆ 0 → J � J i • One-loop anomalous dimension → λ � J c 1 J

Integrability and magnon kinematics in the AdS/CFT correspondence Verifying AdS/CFT in large spin sectors ⇒ Computation of the anomalous dimensions of large operators (Difficult problem due to operator mixing ) Insightful solution: → The one-loop planar dilatation operator of N = 4 Yang-Mills leads to an integrable spin chain ( SO (6) in the scalar sector [Minahan,Zarembo] or PSU (2 , 2 | 4) in the complete theory [Beisert,Staudacher] ) Single trace operators can be mapped to states in a closed spin chain ⇒ BMN impurities : magnon excitations tr( XXXYY X . . . ) ↔ | ↑↑↑ ↓↓ ↑ . . . �

Integrability and magnon kinematics in the AdS/CFT correspondence The Bethe ansatz → The rapidities u j parameterizing the momenta of the magnons satisfy a set of one-loop Bethe equations M M � J � u j + i / 2 u j − u k + i e ip j J ≡ � � = u j − u k − i ≡ S ( u j , u k ) u j − i / 2 k � = j k � = j Thermodynamic limit: integral equations → Assuming integrability an asymptotic long-range Bethe ansatz has been proposed [Beisert,Dippel,Staudacher] � J � x + M M x + j − x − 1 − λ/ 16 π 2 x + j x − u j − u k + i j k k � � = u j − u k − i = x − x − 1 − λ/ 16 π 2 x − j − x + j x + j k k k � = j k � = j where x ± are generalized rapidities j � x ( u ) ≡ u 2 + u 1 − 2 λ 1 x ± ≡ x ( u j ± i / 2) , j 2 8 π 2 u 2

Integrability and magnon kinematics in the AdS/CFT correspondence The quantum string Bethe ansatz String non-linear sigma model on the coset PSU (2 , 2 | 4) SO (4 , 1) × SO (5) Integrable [Mandal,Suryanarayana,Wadia] [Bena,Polchinski,Roiban] Admits a Lax representation : there is a family of flat connections A ( z ), dA ( z ) − A ( z ) ∧ A ( z ) = 0 ⇒ Classical solutions of the sigma model are parameterized by an integral equation [Kazakov,Marshakov,Minahan,Zarembo] dx ′ ρ ( x ′ ) x ∆ � − J + 2 π k = 2 P x ∈ C x 2 − λ x − x ′ C 16 π 2 J 2 Reminds of the thermodynamic Bethe equations for the spin chain ... In fact, it leads to the spin chain equations when λ/ J 2 → 0

Integrability and magnon kinematics in the AdS/CFT correspondence The previous string integral equations are classical/thermodynamic equations ⇓ Assuming integrability survives at the quantum level, a discretization would provide a quantum string Bethe ansatz [Arutyunov,Frolov,Staudacher]

Integrability and magnon kinematics in the AdS/CFT correspondence The quantum string Bethe ansatz is [Arutyunov,Frolov,Staudacher] � J � x + M x + j − x − 1 − λ/ 16 π 2 x + j x − j k k � e 2 i θ ( x j , x k ) = x − x − j − x + 1 − λ/ 16 π 2 x − j x + j k k k � = j The string and gauge theory ans¨ atze differ by a dressing phase factor!! The phase factor is given by ∞ � � � θ 12 = 2 c r ( λ ) q r ( x 1 ) q r +1 ( x 2 ) − q r +1 ( x 1 ) q r ( x 2 ) r =2 � q r ( p i ) are the conserved magnon charges � 1 1 � � i q r ( x ± ) = ( x + ) r − 1 − ( x − ) r − 1 r − 1

Integrability and magnon kinematics in the AdS/CFT correspondence → The dressing phase coefficients c r ( λ ) should interpolate from the string to the gauge theory (strong/weak-interpolation) → To recover the integrable structure of the classical string the coefficients must satisfy c r ( λ ) → 1 as λ → ∞ ⇓ Explicit form of c r ( λ ) ... To constrain the string Bethe ansatz and find the structure of the dressing phase we can compare with one-loop corrections to semiclassical strings ⇓ The classical limit c r ( ∞ ) = 1 needs to be modified in order to include quantum corrections to the string

Integrability and magnon kinematics in the AdS/CFT correspondence The S-matrix of AdS/CFT The S -matrices of the (discrete) quantum string and the long-range gauge Bethe ans¨ atze differ simply by a phase [Arutyunov,Frolov,Staudacher] S string ( p 1 , p 2 ) = e i θ ( p 1 , p 2 ) S gauge ( p 1 , p 2 ) The S-matrix can be determined explicitly ⇓ The spin chain vacuum breaks the PSU (2 , 2 | 4) symmetry algebra down to ( PSU (2 | 2) × PSU (2 | 2) ′ ) ⋉ R , with R a shared central charge The S-matrix is determined up to a scalar (dressing phase) factor [Beisert] [Klose,McLoughlin,Roiban,Zarembo] S SU (2 | 2) ′ 12 S SU (2 | 2) S 12 = S 0 12 12 12 = x + 1 − x − 1 − 1 / x − 1 x + S 0 2 2 e 2 i θ 12 x − 1 x − 1 − x + 1 − 1 / x + 2 2

Integrability and magnon kinematics in the AdS/CFT correspondence Symmetries of the scattering matrix One-loop corrections to semiclassical strings → One-loop corrections are obtained from the spectrum of quadratic fluctuations around a classical solution [Frolov,Tseytlin] [Frolov,Park,Tseytlin] → They amount to empirical constraints on the string Bethe ansatz → Careful analysis of the one-loop sums over bosonic and fermionic frequencies [Sch¨ afer-Nameki,Zamaklar,Zarembo] [Beisert,Tseytlin] [RH,L´ opez] [Freyhult,Kristjansen] provides a compact form of the first quantum correction [RH,L´ opez] [Gromov,Vieira] 1 c r , s = δ r +1 , s + √ a r , s λ ( r − 1)( s − 1) − 8 a r , s = ( r + s − 2)( s − r )

Integrability and magnon kinematics in the AdS/CFT correspondence Crossing symmetry and the dressing phase factor Crossing symmetry The structure of the complete S-matrix is [Beisert] S SU (2 | 2) ′ S SU (2 | 2) S 12 = S 0 � � 12 12 12 ◦ Term in the bracket: determined by the symmetries (Yang-Baxter) ◦ The scalar coefficient is the dressing factor: constrained by unitarity and crossing ( → dynamics ) [Janik] , which implies θ ( x ± 1 , x ± 2 ) + θ (1 / x ± 1 , x ± 2 ) = − 2 i log h ( x ± 1 , x ± 2 ) with h ( x 1 , x 2 ) = x − x − 1 − x + 1 − 1 / x − 1 x − 2 2 2 x + x + 1 − x + 1 − 1 / x + 1 x − 2 2 2 An expansion of both sides has been shown to agree, using the explicit form of the one-loop correction in θ ( x 1 , x 2 ) [Arutyunov,Frolov] ⇓ The θ one-loop ( λ ) phase is a solution of the crossing equations

Integrability and magnon kinematics in the AdS/CFT correspondence Higher corrections Idea: Search for coefficients to fit the expansion of the crossing function h ( x 1 , x 2 ) This provides a strong-coupling expansion [Beisert,RH,L´ opez] ∞ � c ( n ) r , s g 1 − n c r , s = n =0 √ � g ≡ � for the coefficients in the dressing phase λ/ 4 π The all-order proposal is c ( n ) r , s = ( r − 1)( s − 1) B n A ( r , s , n ) with ( − 1) r + s − 1 � � A ( r , s , n ) = 2 π n ) Γ[ n + 1] Γ[ n − 1] × 4 cos( 1 Γ[ 1 Γ[ 1 2 ( s + r + n − 3)] 2 ( s − r + n − 1)] Γ[ 1 Γ[ 1 2 ( s + r − n + 1)] 2 ( s − r − n + 3)]