Bethe Ansatz in the AdS/CFT correspondence

1. Bethe Ansatz in the AdS/CFT correspondence Thanks to: Niklas Beisert (Princeton) Johan Engquist (Utrecht) Gabriele Ferretti (Chalmers) Rainer Heise (Potsdam) Vladimir Kazakov (Paris) Thomas Klose (Uppsala) Andrey Marshakov (Moscow) Joe Minahan (Uppsala & Harvard) Kazuhiro Sakai (Paris) Sakura Schäfer-Nameki (Hamburg) Matthias Staudacher (Potsdam) Arkady Tseytlin (Imperial College) Marija Zamaklar (Potsdam) Konstantin Zarembo (Uppsala U.) EuroStrings 2006 Cambridge, 4/4/06

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

3. Strings in AdS5xS5 RR flux requiresmanifestspace-time supersymmetry Green,Schwarz’84 World-sheet theory is Green-Schwarzcoset sigma model on SU(2,2|4)/SO(4,1)xSO(5). Metsaev,Tseytlin’98 Conformal gauge is problematic: no kinetic term for fermions, no holomorphic factorization for currents, …

4. Integrability in string theory But the model is integrable! Bena,Polchinski,Roiban’03 √ • infinite number of conserved charges • separation of variables in classical equations of motion • quantum spectrum is determined byBethe equations √ Dorey,Vicedo’06 ?

5. Integrability in N=4 SYM • Bethe ansatz for planar anomalous dimensions: • rigorous up to three loops • conjectured to all loop orders • Infinite number of conserved charges associated with local operators (with unclear interpretation) Minahan,Z.’02 Beisert,Kristjansen,Staudacher’03 Beisert,Staudacher’03 Beisert,Dippel,Staudacher’04 Beisert,Staudacher’05; Rej,Serban,Staudacher’05

6. Field content: N=4 Supersymmetric Yang-Mills Theory Brink,Schwarz,Scherk’77 Gliozzi,Scherk,Olive’77 The action:

7. Operator mixing Renormalized operators: Mixing matrix (dilatation operator):

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

9. One loop planar (N→∞) diagrams:

10. Permutation operator: Minahan,Z.’02 • Integrable Hamiltonian! Remains such • at higher orders in λ • for all operators Beisert,Kristjansen,Staudacher’03; Beisert’03; Beisert,Dippel,Staudacher’04; Rej,Serban,Staudacher’05 Beisert,Staudacher’03

11. The spectrum Ground state: Excited states (magnons):

12. Exact spectrum Rapidity: Bethe’31 Zero momentum (trace cyclicity) condition: Anomalous dimension:

13. scattering phase shifts momentum Exact periodicity condition: periodicity of wave function

14. u 0

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

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

17. x Scaling limit: defined on a set of conoturs Ck in the complex plane 0

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

19. Heisenberg model in Heisenberg representation Heisenberg operators: Hiesenberg equations:

20. Continuum + classical limit Landau-Lifshitz equation

21. Consistent truncation String on S3 x R1:

22. Conformal/temporal gauge: ~energy 2d principal chiral field – well-known intergable model Pohlmeyer’76 Zakharov,Mikhailov’78 Faddeev,Reshetikhin’86

23. Equations of motion Currents: Virasoro constraints:

24. Light-cone currents and spins Classical spins: Virasoro constraints: Equations of motion:

25. High-energy approximation : Approximate solution at The same Landau-Lifshitz equation. Kruczenski’03 Kruczenski,Ryzhov,Tseytlin’03

26. Integrability Equations of motion: Zero-curvature representation: equivalent

27. time Conserved charges Generating function (quasimomentum): on equations of motion

28. Non-local charges: Local charges: Analyticity:

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

30. Quantum Bethe equations (two particle factorization)

31. find the dispersion relation (solve the one-body problem): • find the S-matrix (solve the two-body problem): full spectrum Bethe equations • find the true ground state Successfully used on the gauge-theory side Staudacher’04; Beisert’05

32. Landau-Lifshitz model WZ term:

33. Perturbation theory in order to get canonical kinetic term Minahan,Tirziu,Tseytlin’04

34. … = is not renormalized p p S 2→2 = Σ p` p`

35. Summation of bubble diagrams yields Bethe equations

36. Klose,Z.’06 Heisenberg model: the same equations with The difference disappears in the low-energy (u→∞) limit.

37. AAF model SU(1|1) sector: • in SYM: Callan,Heckman,McLoughlin,Swanson’04 • in string theory: Alday,Arutyunov,Frolov’05

38. p0

39. p0 -μ μ – chemical potential μ→-∞ All poles are below the real axis.

40. Empty Fermi sea: Physical vacuum: E E +m +m -m -m Berezin,Sushko’65; Bergknoff,Thaker’79; Korepin’79

41. … NO ANTIPARTICLES p p S 2→2 = Σ p` p`

42. Exact S-matrix θ – rapidity:

43. Bethe ansatz Klose,Z.’06 Im θj=0: positive-energy states Im θj=π: negative-energy states

44. Ground state θ k+iπ -k+iπ iπ 0 - UV cutoff

45. Mass renormalization: This equation also determines the spectrum, the physical S-matrix, ...

46. Weak-coupling (-1<g<1): • Non-renormalizable • (anomalous dimension of mass is complex) • Strong attraction (g>1) • Unstable • (Energy unbounded below) • Strong repulsion (g<-1): • Spectrum consists of fermions and anti-fermions • with non-trivial scattering

47. Questions Continuous Discrete (string worldsheet) vs. (spin chain) in Bethe ansatz: Beisert,Dippel,Staudacher’04 Berenstein,Maldacena,Nastase’02 • Possible resolution: • extra hidden d.o.f. (“particles • on the world sheet • that create the spin chain”) • spin chain is thus dynamical Mann,Polchinski’05 Rej,Serban,Staudacher’05 Gromov,Kazakov,Sakai,Vieira’06 Choice of the reference state, the true vacuum, antiparticles, …