Bethe Ansatz in AdS/CFT: from local operators to classical strings Konstantin Zarembo (Uppsala U.) J. Minahan, K. Z., hep-th/ N. Beisert, J. Minahan, M. Staudacher, K. Z., hep-th/ V. Kazakov, A. Marshakov, J. Minahan, K. Z., hep-th/ N. Beisert, V. Kazakov, K. Sakai, K. Z., hep-th/ N. Beisert, A. Tseytlin, K. Z., hep-th/ S. Schäfer-Nameki, M. Zamaklar, K.Z., hep-th/ ENS,

AdS/CFT correspondence Maldacena’97 Gubser, Klebanov, Polyakov’98 Witten’98

If there is a string dual of QCD, this resolves many puzzles: graviton is not a massless glueball, but the dual of T μν sum rules are authomatic String states Bound states in QFT (mesons, glueballs) String states Local operators

Perturbation theory: Unitarity: Hence the sum rule: If {n} are all string states with right quantum numbers, the sum is likely to diverge because of the Hagedorn spectrum.

“IR wall” UV boundary asymptotically AdS The simplest phenomenological model describes all data in the vector meson channel to 4% accuracy (Spectral representation of bulk-to-boundary propagator) Erlich, Katz, Son, Stephanov’05

Operator/string state correspondence is worth studying. But it is also hard. λ<<1 Quantum string Classical stringStrong coupling in SYM or need to consider operators with many constituent fields (long operators)

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

≈ 2 L 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 Operator basis:

One loop planar (N→∞) diagrams:

Permutation operator: Minahan, K.Z.’02 Integrable Hamiltonian! Remains such at higher orders in λ for all operators Beisert,Kristjansen, Staudacher’03; Beisert,Dippel,Staudacher’04 Beisert, Staudacher’03

Spectrum of Heisenberg ferromagnet

Excited states: Non-intereacting magnons (good approximation if M<<L): Ground state: flips one spin: Exact solution: only (**) changes; (*), (***) stay the same

Zero momentum (trace cyclicity) condition: Anomalous dimension: Bethe’31 Bethe ansatz Rapidity:

bound states of magnons – Bethe “strings” mode numbers u 0

Sutherland’95; Beisert, Minahan, Staudacher, K.Z.’03 Macsoscopic spin waves: long strings

defined on cuts C k in the complex plane Scaling limit: x 0

Classical Bethe equations Normalization: Momentum condition: Anomalous dimension:

String theory in AdS 5  S 5 Metsaev, Tseytlin’98 Bena, Polchinski, Roiban’03 Conformal 2d field theory ( ¯ -function=0) Sigma-model coupling constant: Classically integrable Classical limit is

Consistent truncation Conformal/temporal gauge: Pohlmeyer’76; Zakharov, Mikhailov’78; Reshetikhin, Faddeev’86 Keep only String on S 3 xR 1 2d principal chiral field – well-known intergable model

Integrability: AdS/CFT correspondence: Time-periodic solutions of classical equations of motion Spectral data (hyperelliptic curve + meromorphic differential) Noether charges in the sigma-model Quantum numbers of the SYM operators (L, M, Δ)

Noether charges Length of the chain: Total spin: Energy (scaling dimension): Virasoro constraints:

BMN scaling Berenstein, Maldacena, Nastase’02 Frolov, Tseytlin’03 For any classical solution: Frolov-Tseytlin limit: If 1<<λ<<L 2 : BMN coupling Which can be compared to perturbation theory even though λ is large.

Integrability Zero-curvature representation: Equations of motion: equivalent on equations of motion Infinte number of conservatin 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:

Normalization: Momentum condition: Anomalous dimension: Take 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 Near-BMN limit Quantum corrections to classical string solutions Finite-size corrections to Bethe ansatz Frolov, Tseytlin’03 Frolov, Park, Tsetlin’04 Park, Tirziu, Tseytlin’05 Fuji, Satoh’05 Lübcke, Z.’04 Freyhult, Kristjansen’05 Beisert, Tseytlin, Z.’05 Hernandez, Lopez, Perianez, Sierra’05 Schäfer-Nameki, Zamaklar, Z.’05 Berenstein, Maldacena, Nastase’02; Metsaev’02;… Callan, Lee,McLoughlin,Schwarz,Swanson,Wu’03;…

String on AdS 3 xS 1 : radial coordinate in AdS angle in AdS angle on S 5 Rigid string solution: Arutyunov, Russo, Tseytlin’03 One-loop quantum correction: Park, Tirziu, Tseytlin’05 AdS spin angular momentum on S 5

Bethe equations: Even under L→-L First correction is O(1/L 2 ) But singular if simultaneously Local anomaly Kazakov’03 cancels at leading order gives 1/L correction 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 expressed as sum over modes of small fluctuations Beisert, Freyhult’05