1 / 38

Integrability and Bethe Ansatz in the AdS/CFT correspondence

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)

morwen
Download Presentation

Integrability and Bethe Ansatz in the AdS/CFT correspondence

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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, 28.10.05

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

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

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

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

  6. Mixing matrix Heisenberg Hamiltonian

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

  8. Continuum + classical limit Landau-Lifshitz equation

  9. COMPARISON TO STRINGS

  10. (+ S5 + fermions) z 5D bulk strings 0 gauge fields 4D boundary

  11. 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

  12. Need to know the spectrum of string states: • - eigenstates of Hamiltonian in light-cone gauge • or • - (1,1) vertex operators in conformal gauge • Nothing of that is known • But as long as λ>>1 semiclassical approximation is OK Time-periodic classical solutions Bohr-Sommerfeld Quantum states

  13. Consistent truncation String on S3 x R1:

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

  15. Equations of motion Currents: Virasoro constraints:

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

  17. High-energy approximation : Approximate solution at The same (Landau-Lifshitz) equation describes the spin chain in the classical limit! Kruczenski’03

  18. 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)

  19. Global symmetries of the sigma-model Left shifts: Right shifts: Time translations: World-sheet reparameterization invariance

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

  21. “Dimensional analysis” Q – any charge:energy Δ; spins L, M; … Dimensionless variables: • BMN coupling: • filling fraction: Berenstein,Maldacena,Nastase’02

  22. BMN scaling 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

  23. String energy (strong-coupling calculation): Anomalous dimension (weak-coupling calculation): • three-loop discrepancy • structural difference of finite-size/quantum corrections Callan et al’03; Beisert,Kristjansen,Staudacher’03; Beisert,Dippel,Staudacher’04 Beisert,Tseytlin’05; Schäfer-Nameki,Zamaklar’05

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

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

  26. Non-local charges: Local charges:

  27. Auxiliary linear problem

  28. Dirac equation in 1d (j0, j1 are 2x2 matrices) with spectral parameter x Quasi-periodic boundary conditions: quasimomentum

  29. Noether charges:

  30. 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).

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

  32. Take Normalization: Momentum condition: Anomalous dimension: This is the classical limit of Bethe equations for spin chain!

  33. x defined on cuts Ck in the complex plane 0

  34. Exact quantum Bethe equations: In the scaling limit, Taking the logarithm and expanding in 1/L:

  35. Bethe equations for quantum strings? Arutyunov,Frolov,Staudacher’04 Staudacher’04; Beisert,Staudacher’05 Mann,Polchinski’05 Ambjørn,Janik,Kristjansen’05

  36. Quantizing strings in AdS5xS5 Solving N=4, D=4 SYM at large N!

  37. IS N=4 SYM SOLVABLE? STRINGS SPIN CHAINS PLANAR DIAGRAMS Universal relationship for large-N gauge theories?

More Related