Bethe Ansatz and Integrability in AdS/CFT correspondence

1. Bethe Ansatz and Integrability in AdS/CFT correspondence 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 Konstantin Zarembo (Uppsala U.) “Constituents, Fundamental Forces and Symmetries of the Universe”, Napoli, 9.10.2006

2. AdS/CFT correspondence Yang-Mills theory with N=4 supersymmetry Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98 Exact equivalence String theory on AdS5xS5 background

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

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

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

6. Strong-weak coupling interpolation Gubser,Klebanov,Tseytlin’98; … λ 0 SYM perturbation theory String perturbation theory 1 + + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS5

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. Integrability in SYM Integrability in AdS string theory Integrability and Bethe ansatz Bethe ansatz in AdS/CFT Testing Bethe ansatz against string quantum corrections Plan

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

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

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

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

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

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

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

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

19. Strings in AdS5xS5 Green-Schwarz-type coset 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, … Light-cone gauge is OK. The action is complicated, but the model is integrable! Bena,Polchinski,Roiban’03

20. Consistent truncation String on S3 x R1:

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

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

23. Non-local charges: Local charges:

24. 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) Bethe ansatz

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 p1 k1 k2 p2 Energy and momentum conservation:

32. k2 Energy conservation I Momentum conservation k1 II I: k1=p1, k2=p2 (transition) II: k1=p2, k2=p1 (reflection)

33. n→n scattering pi ki 2 equations for n unknowns (n-2)-dimensional phase space

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 1 2 2 = 3 3

36. Bethe ansatz Integrability + Locality Strategy: • 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

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

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

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

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

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

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

44. Algebraic BA: classical string Bethe equation su(2) sector: Kazakov,Marshakov,Minahan,Z.’04 Normalization: Momentum condition: String energy: General classical BAE are known and have the nested structure consistent with the PSU(2,2|4) symmetry of AdS5xS5 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 AdS3xS1 - global time - radial coordinate in AdS - angle in AdS - angle on S5

49. Rigid string solution Arutyunov,Russo,Tseytlin’03 S5 AdS5 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