1. 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

2. Large-N expansion of gauge theory String theory Early examples: 2d QCD Matrix models 4d gauge/string duality: AdS/CFT correspondence

3. Macroscopic strings from planar diagrams Large orders of perturbation theory Large number of constituents or

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

5. λ<<1 Quantum string Classical 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

6. Operator mixing Renormalized operators: Mixing matrix (dilatation operator): Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension

7. N=4 Supersymmetric Yang-Mills Theory Field content: The action:

8. Local operators and spin chains Restrict to SU(2) sector related by SU(2) R-symmetry subgroup a b a b

9. ≈ 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:

10. One loop planar (N→∞) diagrams:

11. 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

12. Spectrum of Heisenberg ferromagnet

13. Excited states: Ground state: flips one spin: (SUSY protected)

14. good approximation if M<<L Exact solution: exact eigenstates are still multi-magnon Fock states (**) stays the same but (*) changes! Non-interacting magnons

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

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

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

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

19. Classical Bethe equations Normalization: Momentum condition: Anomalous dimension:

20. 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 AdS 5 xS 5 But as long as λ>>1 semiclassical approximation is OK Time-periodic classical solutions Quantum states Bohr-Sommerfeld

21. 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

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

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

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

25. 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.

26. Integrability Zero-curvature representation: Equations of motion: equivalent on equations of motion Infinte number of conservation laws

27. Auxiliary linear problem quasimomentum Noether charges are determined by asymptotic behaviour of quasimomentum:

28. 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).

29. Classical string Bethe equation Kazakov, Marshakov, Minahan, K.Z.’04 Normalization: Momentum condition: Anomalous dimension:

30. Normalization: Momentum condition: Anomalous dimension: Take This is classical limit of Bethe equations for spin chain!

31. 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 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;…

32. 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

33. 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

34. x 0 Locally:

35. Anomaly local contribution 1/L correction to classical Bethe equations: Beisert, Tseytlin, Z.’05

36. 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