Integrability and Bethe Ansatz in the AdS/CFT correspondence

1. Integrability and Bethe Ansatz in the AdS/CFT correspondence Thanks to: Niklas Beisert (Princeton) Johan Engquist (Utrecht) Gabriele Ferretti (Chalmers) Rainer Heise (AEI, Potsdam) Vladimir Kazakov (ENS) Andrey Marshakov (ITEP, Moscow) Joe Minahan (Uppsala & Harvard) Kazuhiro Sakai (ENS) Sakura Schäfer-Nameki (Hamburg) Matthias Staudacher (AEI, Potsdam) Arkady Tseytlin (Imperial College & Ohio State) Marija Zamaklar (AEI, Potsdam) Konstantin Zarembo (Uppsala U.) Nordic Network Meeting Helsinki, 27.10.05

2. Large-N expansion of gauge theory String theory Early examples: • 2d QCD • Matrix models ‘t Hooft’74 Brezin,Itzykson,Parisi,Zuber’78 4d gauge/string duality: • AdS/CFT correspondence Maldacena’97

3. Plan I. GAUGE THEORY • Large-N limit and planar diagrams • Instead of an introduction: local operators=closed string states • Operator mixing and intergable spin chains • Basics of Bethe ansatz • Thermodynamic limit II. STRING THEORY • Classical integrability • Classical Bethe ansatz • (time permitting) Quantum corrections

4. Yang-Mills theory anti-Hermitean traceless NxN matrices But we keep N as a parameter Interesting case: N=3

5. Large-N limit ‘t Hooft’74 “Index conservation law”:

6. Planar diagrams and strings time (kept finite) ‘t Hooft coupling: String coupling constant = (goes to zero)

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

8. Anti-de-Sitter space (AdS5) z 5D bulk strings 0 gauge fields 4D boundary

9. Two-point correlation functions z string propagator in the bulk 0

10. Scale invariance leaves metric invariant dual gauge theory is scale invariant (conformal)

11. Breaking scale invariance “IR wall” asymptotically AdS metric UV boundary approximate scale invariance at short distances

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

13. Perturbation theory: Spectral representation: 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.

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

15. Quantum strings λ<<1 Strong coupling in SYM Classical strings Way out: consider states with large quantum numbers = operators with large number of constituent fields

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

17. Price: highly degenerate operator mixing

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

19. Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension

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

21. Local operators and spin chains related by SU(2) R-symmetry subgroup j i j i

22. 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

23. One loop planar (N→∞) diagrams:

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

25. Spectrum of Heisenberg ferromagnet

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

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

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

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

30. How to solve Bethe equations? Non-interactions magnons: mode number Thermodynamic limit (L→∞):

31. u 0

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

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

34. x Scaling limit: defined on cuts Ck in the complex plane 0

35. In the scaling limit, Taking the logarithm and expanding in 1/L: determines the branch of log

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