Integrability and Bethe Ansatz in the AdS/CFT correspondence

Konstantin Zarembo

(Uppsala U.)

Nordic Network MeetingHelsinki, 28.10.05

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)

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

(+ S5 + fermions)

String theory in AdS5S5Metsaev,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 S3 x R1:

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<<L2:

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 (j0, j1 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 Ck 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 AdS5xS5

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?