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Integrability and Bethe Ansatz in the AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) Nordic Network Meeting Helsinki, 27.10.05 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)
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Large-N expansion of gauge theory String theory Early examples: 2d QCD Matrix models 4d gauge/string duality: AdS/CFT correspondence ‘t Hooft’74 Brezin,Itzykson,Parisi,Zuber’78 Maldacena’97
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Plan 1.Large-N limit and planar diagrams 2.Instead of an introduction: local operators=closed string states 3.Operator mixing and intergable spin chains 4.Basics of Bethe ansatz 5.Thermodynamic limit I. GAUGE THEORY II. STRING THEORY 1.Classical integrability 2.Classical Bethe ansatz 3.(time permitting) Quantum corrections
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Yang-Mills theory anti-Hermitean traceless NxN matrices Interesting case: N=3 But we keep N as a parameter
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Large-N limit ‘t Hooft’74 “Index conservation law”:
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Planar diagrams and strings time ‘t Hooft coupling: String coupling constant = (kept finite) (goes to zero)
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AdS/CFT correspondence Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98
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Anti-de-Sitter space (AdS 5 ) 5D bulk 4D boundary z 0
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z 0 string propagator in the bulk Two-point correlation functions
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Scale invariance leaves metric invariant dual gauge theory is scale invariant (conformal)
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Breaking scale invariance “IR wall” UV boundary asymptotically AdS metric approximate scale invariance at short distances
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If there is a string dual of QCD, this resolves many puzzles: graviton is not a massless glueball, but the dual of T μν sum rules are automatic String states Bound states in QFT (mesons, glueballs) String states Local operators
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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.
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“IR wall” UV boundary asymptotically AdS The simplest phenomenological model describes all data in the vector meson channel to 4% accuracy (Spectral representation of bulk-to-boundary propagator) Erlich,Katz,Son,Stephanov’05
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λ<<1 Quantum strings Classical strings Strong coupling in SYM Way out: consider states with large quantum numbers = operators with large number of constituent fields
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Macroscopic strings from planar diagrams Large orders of perturbation theory Large number of constituents or
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Price: highly degenerate operator mixing
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Operator mixing Renormalized operators: Mixing matrix (dilatation operator):
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Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension
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N=4 Supersymmetric Yang-Mills Theory Field content: The action: Brink,Schwarz,Scherk’77 Gliozzi,Scherk,Olive’77
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Local operators and spin chains related by SU(2) R-symmetry subgroup i j i j
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≈ 2 L 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 Operator basis:
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One loop planar (N→∞) diagrams:
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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
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Spectrum of Heisenberg ferromagnet
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Excited states: Ground state: flips one spin: (SUSY protected)
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good approximation if M<<L Exact solution: exact eigenstates are still multi-magnon Fock states (**) stays the same only (*) changes! Non-interacting magnons
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Exact periodicity condition: momentum scattering phase shifts periodicity of wave function
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Zero momentum (trace cyclicity) condition: Anomalous dimension: Bethe’31 Bethe ansatz Rapidity:
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How to solve Bethe equations? Non-interactions magnons: mode number Thermodynamic limit (L→∞):
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u 0
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bound states of magnons – Bethe “strings” mode numbers u 0
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Sutherland’95; Beisert,Minahan,Staudacher,Z.’03 Macroscopic spin waves: long strings
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defined on cuts C k in the complex plane Scaling limit: x 0
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In the scaling limit, determines the branch of log Taking the logarithm and expanding in 1/L:
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Classical Bethe equations Normalization: Momentum condition: Anomalous dimension:
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