1. Bethe Ansatz and Integrability in AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) “Constituents, Fundamental Forces and Symmetries of the Universe”, Napoli, 9.10.2006 Thanks to: Niklas Beisert Johan Engquist Gabriele Ferretti Rainer Heise Vladimir Kazakov Thomas Klose Andrey Marshakov Tristan McLoughlin Joe Minahan Radu Roiban Kazuhiro Sakai Sakura Schäfer-Nameki Matthias Staudacher Arkady Tseytlin Marija Zamaklar

2. AdS/CFT correspondence Yang-Mills theory with N=4 supersymmetry String theory on AdS 5 xS 5 background Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98 Exact equivalence

3. Planar diagrams and strings time Large-N limit:

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

5. λ<<1 Quantum strings Classical strings Strong coupling in SYM Spectrum of SYM = String spectrum but

6. Strong-weak coupling interpolation Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 0 λ SYM perturbation theory 1 + + … + String perturbation theory Minimal area law in AdS 5 Gubser,Klebanov,Tseytlin’98; …

7. SYM is weakly coupled if String theory is weakly coupled if There is an overlap!

9. Q:HOW TO COMAPARE SYM AND STRINGS? A(?): SOLVE EACH WITH THE HELP OF BETHE ANSATZ

10. Plan 1.Integrability in SYM 2.Integrability in AdS string theory 3.Integrability and Bethe ansatz 4.Bethe ansatz in AdS/CFT 5.Testing Bethe ansatz against string quantum corrections

11. N=4 Supersymmetric Yang-Mills Theory Field content: Action: Gliozzi,Scherk,Olive’77 Global symmetry: PSU(2,2|4)

12. Spectrum Basis of primary operators: Dilatation operator (mixing matrix): Spectrum = {Δ n }

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

14. One loop: Tree level: Δ=L (huge degeneracy)

15. One loop planar dilatation generator: Minahan,Z.’02 Heisenberg Hamiltonian

16. Integrability Lax operator: Monodromy matrix: Faddeev et al.’70-80s Transfer “matrix”:

17. Infinite tower of conserved charges: U – lattice translation generator: U=e iP

18. Algebraic Bethe Ansatz Spectrum: are eigenstates of the Hamiltonian with eigenvalues (anomalous dimension) (total momentum) Provided Bethe equations

19. Strings in AdS 5 xS 5 Green-Schwarz-type coset sigma model on SU(2,2|4)/SO(4,1)xSO(5). Conformal gauge is problematic: no kinetic term for fermions, no holomorphic factorization for currents, … Light-cone gauge is OK. Metsaev,Tseytlin’98 The action is complicated, but the model is integrable! Bena,Polchinski,Roiban’03

20. Consistent truncation String on S 3 x R 1 :

21. Zero-curvature representation: Equations of motion: equivalent Zakharov,Mikhaikov’78 Gauge condition :

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

23. Non-local charges: Local charges:

24. Bethe ansatz Algebraic Bethe ansatz: quantum Lax operator + Yang-Baxter equations → spectrum Coordinate Bethe ansatz: direct construction of the wave functions in the Schrödinger representation Asymptotic Bethe ansatz: S-matrix ↔ spectrum (infinite L) ? (finite L)

25. Spectrum and scattering phase shifts periodic short-range potential

26. exact only for V(x) = g δ(x)

27. Continuity of periodized wave function

28. where is (eigenvalue of) the S-matrix correct up to O(e -L/R ) works even for bound states via analytic continuation to complex momenta

29. Multy-particle states

30. Bethe equations Assumptions: R<<L particles can only exchange momenta no inelastic processes

31. 2→2 scattering in 2d p1p1 p2p2 k1k1 k2k2 Energy and momentum conservation:

32. I II Momentum conservation Energy conservation k1k1 k2k2 I: k 1 =p 1, k 2 =p 2 (transition) II: k 1 =p 2, k 2 =p 1 (reflection)

33. n→n scattering 2 equations for n unknowns (n-2)-dimensional phase space pipi kiki

34. Unless there are extra conservation laws! Integrability: No phase space: No particle production (all 2→many processes are kinematically forbidden)

35. Factorization: Consistency condition (Yang-Baxter equation): 1 2 3 1 2 3 =

36. Strategy: find the dispersion relation (solve the one-body problem): find the S-matrix (solve the two-body problem): Bethe equations full spectrum find the true ground state Integrability + Locality Bethe ansatz

37. What are the scattering states? SYM: magnons String theory: “giant magnons” Staudacher’04 Hofman,Maldacena’06 Common dispersion relation: S-matrix is highly constrained by symmetries Beisert’05

38. Zero momentum (trace cyclicity) condition: Anomalous dimension: Algebraic BA: one-loop su(2) sector Rapidity: Minahan,Z.’02

39. Algebraic BA: one loop, complete spectrum Beisert,Staudacher’03 Nested BAE: - Cartan matrix of PSU(2,2|4) - highest weight of the field representation

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

41. Sutherland’95; Beisert,Minahan,Staudacher,Z.’03 Semiclassical states

42. defined on a set of conoturs C k in the complex plane Scaling limit: x 0

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

44. Algebraic BA: classical string Bethe equation Kazakov,Marshakov,Minahan,Z.’04 Normalization: Momentum condition: String energy: su(2) sector: General classical BAE are known and have the nested structure consistent with the PSU(2,2|4) symmetry of AdS 5 xS 5 superstring Beisert,Kazakov,Sakai,Z.’05

45. Asymptotic BA: SYM Beisert,Staudacher’05

46. Asymptotic BA: string extra phase

47. Arutyunov,Frolov,Staudacher’04 Hernandez,Lopez’06 Algebraic structure is fixed by symmetries The Bethe equations are asymptotic: they describe infinitely long strings / spin chains. Beisert’05 Schäfer-Nameki,Zamaklar,Z.’06

48. Testing BA: semiclassical string in AdS 3 xS 1 - radial coordinate in AdS - angle in AdS - angle on S 5 - global time

49. Rigid string solution Arutyunov,Russo,Tseytlin’03 AdS 5 S5S5 winds k times and rotates winds m times and rotates

50. Internal length of the string is Perturbative SYM regime: (string is very long) For simplicity, I will consider (large-winding limit) Schäfer-Nameki,Zamaklar,Z.’05

51. string fluctuation frequencies Explicitly, Park,Tirziu,Tseytlin’05 classical energy one loop correction

52. Quantum-corrected Bethe equations classical BE Kazakov,Z.’04 Anomaly Kazakov’04;Beisert,Kazakov,Sakai,Z.’05 Beisert,Tseytlin,Z.’05; Schäfer-Nameki,Zamaklar,Z.’05 Quantum correction to the scattering phase Hernandez,Lopez’06

53. Large (long strings): Comparison String BA BA misses exponential terms Schäfer-Nameki,Zamaklar,Z.’05

54. Conclusions Large-N SYM / string sigma-model on AdS 5 xS 5 are probably solvable by Bethe ansatz Open problems:  Interpolation from weak to strong coupling  Finite-size effects  Appropriate reference state / ground state  Algebraic formulation: –Transfer matrix –Yang-Baxter equation –Pseudo-vacuum