Integrability and Bethe Ansatz in the AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) Nordic Network Meeting Helsinki, 28.10.05 Thanks to: Niklas.

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Presentation transcript:

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)

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

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

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

Multiplicatively renormalizable operators with definite scaling dimension: anomalous dimension

Mixing matrix Heisenberg Hamiltonian

Heisenberg model in Heisenberg representation Heisenberg operators: Hiesenberg equations:

Continuum + classical limit Landau-Lifshitz equation

COMPARISON TO STRINGS

5D bulk 4D boundary z 0 (+ S 5 + fermions)

String theory in AdS 5  S 5 Metsaev,Tseytlin’98 Bena,Polchinski,Roiban’03 Conformal 2d field theory ( ¯ -function=0) Sigma-model coupling constant: Classically integrable Classical limit is

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 Quantum states Bohr-Sommerfeld

Consistent truncation String on S 3 x R 1 :

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

Equations of motion Currents: Virasoro constraints:

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

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

Integrability: AdS/CFT correspondence: Time-periodic solutions of classical equations of motion Spectral data (hyperelliptic curve + meromorphic differential) Noether charges in sigma-model Quantum numbers of SYM operators (L, M, Δ)

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

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

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

BMN scaling Frolov,Tseytlin’03 For any classical solution: Frolov-Tseytlin limit: If 1<<λ<<L 2 : Which can be compared to perturbation theory even though λ is large.

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

Integrability Zero-curvature representation: Equations of motion: equivalent

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

Non-local charges: Local charges:

Auxiliary linear problem

quasimomentum Dirac equation in 1d (j 0, j 1 are 2x2 matrices) with spectral parameter x Quasi-periodic boundary conditions:

Noether charges:

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).

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

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

defined on cuts C k in the complex plane x 0

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

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

Quantizing strings in AdS 5 xS 5 Solving N=4, D=4 SYM at large N!

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