Bethe Ansatz in AdS/CFT Correspondence

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. Quantum string λ<<1 Strong coupling in SYM Classical string 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. Field content: N=4 Supersymmetric Yang-Mills Theory The action:

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

9. 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

10. One loop planar (N→∞) diagrams:

11. Minahan, K.Z.’02 Permutation operator: • 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. Ground state: (SUSY protected) Excited states: flips one spin:

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

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

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

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

18. x Scaling limit: defined on cuts Ck in the complex plane 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 AdS5xS5 • But as long as λ>>1 semiclassical approximation is OK Time-periodic classical solutions Bohr-Sommerfeld Quantum states

21. String theory in AdS5S5 Metsaev, Tseytlin’98 • Conformal 2d field theory (¯-function=0) • Sigma-model coupling constant: • Classically integrable Classical limit is Bena, Polchinski, Roiban’03

22. 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

23. 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)

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

25. 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

26. Integrability Equations of motion: Zero-curvature representation: 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. Take Normalization: Momentum condition: Anomalous dimension: 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 Berenstein, Maldacena, Nastase’02; Metsaev’02;… Callan, Lee,McLoughlin,Schwarz,Swanson,Wu’03;… 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

32. String on AdS3xS1: angle in AdS angle on S5 radial coordinate in AdS Rigid string solution: Arutyunov, Russo, Tseytlin’03 AdS spin angular momentum on S5 One-loop quantum correction: Park, Tirziu, Tseytlin’05

33. Bethe equations: Even under L→-L First correction is O(1/L2) 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