Integrability and Bethe Ansatz in the AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) Nordic Network Meeting Helsinki, 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)

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

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

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

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

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

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

Anti-de-Sitter space (AdS 5 ) 5D bulk 4D boundary z 0

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

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

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

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

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.

“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

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

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

Price: highly degenerate operator mixing

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

Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension

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

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

≈ 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:

One loop planar (N→∞) diagrams:

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

Spectrum of Heisenberg ferromagnet

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

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

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

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

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

u 0

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

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

defined on cuts C k in the complex plane Scaling limit: x 0

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

Classical Bethe equations Normalization: Momentum condition: Anomalous dimension: