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Transcript
Bethe Ansatz in AdS/CFT Correspondence Konstantin Zarembo (Uppsala U.) J. Minahan, K. Z., hep-th/0212208 N. Beisert, J. Minahan, M. Staudacher, K. Z., hep-th/0306139 V. Kazakov, A. Marshakov, J. Minahan, K. Z., hep-th/0402207 N. Beisert, V. Kazakov, K. Sakai, K. Z., hep-th/0503200 N. Beisert, A. Tseytlin, K. Z., hep-th/0502173 S. Schäfer-Nameki, M. Zamaklar, K.Z., hep-th/0507179 DGMTP, Tianjin, 23.08.05 Large-N expansion of gauge theory String theory Early examples: • 2d QCD • Matrix models 4d gauge/string duality: • AdS/CFT correspondence Macroscopic strings from planar diagrams Large orders of perturbation theory or Large number of constituents AdS/CFT correspondence Maldacena’97 Gubser, Klebanov, Polyakov’98 Witten’98 λ<<1 Classical string Quantum string Strong coupling in SYM Way out: consider states with large quantum numbers = operators with large number of constituent fields Price: highly degenerate operator mixing Operator mixing Renormalized operators: Mixing matrix (dilatation operator): Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension N=4 Supersymmetric Yang-Mills Theory Field content: The action: Local operators and spin chains • Restrict to SU(2) sector related by SU(2) R-symmetry subgroup a b a b Operator basis: • ≈ 2L degenerate operators • The space of operators can be identified with the Hilbert space of a spin chain of length L with (L-M) ↑‘s and M ↓‘s One loop planar (N→∞) diagrams: Permutation operator: Minahan, K.Z.’02 Integrable Hamiltonian! Remains such Kristjansen, Staudacher’03 • at higher orders in λ Beisert, Beisert, Dippel, Staudacher’04 • for all operators Beisert, Staudacher’03 Spectrum of Heisenberg ferromagnet Ground state: (SUSY protected) Excited states: flips one spin: Non-interacting magnons • good approximation if M<<L Exact solution: • exact eigenstates are still multi-magnon Fock states • (**) stays the same • but (*) changes! Bethe ansatz Rapidity: Bethe’31 Zero momentum (trace cyclicity) condition: Anomalous dimension: bound states of magnons – Bethe “strings” u 0 mode numbers Macsoscopic spin waves: long strings Sutherland’95; Beisert, Minahan, Staudacher, K.Z.’03 Scaling limit: defined on cuts Ck in the complex plane x 0 Classical Bethe equations Normalization: Momentum condition: Anomalous dimension: Comparison to strings • Need to know the spectrum of string states: - eigenstates of Hamiltonian in light-cone gauge or - (1,1) vertex operators in conformal gauge • Not known how to quantize strings in AdS5xS5 • But as long as λ>>1 semiclassical approximation is OK Time-periodic classical solutions Bohr-Sommerfeld Quantum states String theory in AdS5S5 Metsaev, Tseytlin’98 • Conformal 2d field theory (¯-function=0) • Sigma-model coupling constant: • Classically integrable Bena, Polchinski, Roiban’03 Classical limit is Consistent truncation Keep only String on S3xR1 Conformal/temporal gauge: 2d principal chiral field – well-known intergable model Pohlmeyer’76 Zakharov, Mikhailov’78 Faddeev, Reshetikhin’86 Integrability: Time-periodic solutions of classical equations of motion Spectral data (hyperelliptic curve + meromorphic differential) AdS/CFT correspondence: Noether charges in sigma-model Quantum numbers of SYM operators (L, M, Δ) Noether charges Length of the chain: Total spin: Energy (scaling dimension): Virasoro constraints: BMN scaling BMN coupling Berenstein, Maldacena, Nastase’02 For any classical solution: Frolov-Tseytlin limit: If 1<<λ<<L2: Which can be compared to perturbation theory even though λ is large. Frolov, Tseytlin’03 Integrability Equations of motion: Zero-curvature representation: equivalent on equations of motion Infinte number of conservation laws Auxiliary linear problem quasimomentum Noether charges are determined by asymptotic behaviour of quasimomentum: Analytic structure of quasimomentum p(x) is meromorphic on complex plane with cuts along forbidden zones of auxiliary linear problem and has poles at x=+1,-1 Resolvent: is analytic and therefore admits spectral representation: and asymptotics at ∞ completely determine ρ(x). Classical string Bethe equation Kazakov, Marshakov, Minahan, K.Z.’04 Normalization: Momentum condition: Anomalous dimension: Take Normalization: Momentum condition: Anomalous dimension: This is classical limit of Bethe equations for spin chain! Q: Can we quantize string Bethe equations (undo thermodynamic limit)? A: Yes! Arutyunov, Frolov, Staudacher’04; Staudacher’04;Beisert, Staudacher’05 Quantum strings in AdS: • BMN limit Berenstein, Maldacena, Nastase’02; Metsaev’02;… • Near-BMN limit Callan, Lee,McLoughlin,Schwarz,Swanson,Wu’03;… • Quantum corrections to classical string solutions Frolov, Tseytlin’03 Frolov, Park, Tsetlin’04 Park, Tirziu, Tseytlin’05 Fuji, Satoh’05 Finite-size corrections to Bethe ansatz Beisert, Tseytlin, Z.’05 Hernandez, Lopez, Perianez, Sierra’05 Schäfer-Nameki, Zamaklar, Z.’05 String on AdS3xS1: radial coordinate in AdS angle in AdS angle on S5 Rigid string solution: Arutyunov, Russo, Tseytlin’03 angular momentum on S5 AdS spin One-loop quantum correction: Park, Tirziu, Tseytlin’05 Bethe equations: Even under L→-L First correction is O(1/L2) But singular if simultaneously Local anomaly • cancels at leading order • gives 1/L correction Kazakov’03 Beisert, Kazakov, Sakai, Z.’05 Beisert, Tseytlin, Z.’05 Hernandez, Lopez, Perianez, Sierra’05 x 0 Locally: Anomaly local contribution 1/L correction to classical Bethe equations: Beisert, Tseytlin, Z.’05 Re-expanding the integral: Agrees with the string calculation. Remarks: • anomaly is universal: depends only on singular part of Bethe equations, which is always the same • finite-size correction to the energy can be always Beisert, Freyhult’05 expressed as sum over modes of small fluctuations