Bethe Ansatz in AdS/CFT Correspondence

Konstantin Zarembo

(Uppsala U.)

J. Minahan, K. Z., hep-th/0212208N. Beisert, J. Minahan, M. Staudacher, K. Z., hep-th/0306139V. Kazakov, A. Marshakov, J. Minahan, K. Z., hep-th/0402207N. Beisert, V. Kazakov, K. Sakai, K. Z., hep-th/0503200N. Beisert, A. Tseytlin, K. Z., hep-th/0502173S. Schäfer-Nameki, M. Zamaklar, K.Z., hep-th/0507179

DGMTP, Tianjin, 23.08.05

Large-N expansion of gauge theory

String theory

Early examples:

• 2d QCD

• Matrix models

4d gauge/string duality:

• AdS/CFT correspondence

Macroscopic strings from planar diagrams

Large orders

of perturbation theoryLarge number

of constituentsor

AdS/CFT correspondence Maldacena’97

Gubser, Klebanov, Polyakov’98

Witten’98

λ<<1 Quantum string

Classical string Strong coupling in SYM

Way out: consider states with large quantum numbers

= operators with large number of constituent fields

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:

Local operators and spin chains

• Restrict to SU(2) sector

related by SU(2) R-symmetry subgroup

a b

a b

• ≈ 2L 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:

Minahan, K.Z.’02

Integrable Hamiltonian! Remains such

• at higher orders in λ

• for all operators

Beisert, Kristjansen, Staudacher’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

• but (*) changes!

Non-interacting magnons

Zero momentum (trace cyclicity) condition:

Anomalous dimension:

Bethe’31

Bethe ansatz

Rapidity:

bound states of magnons – Bethe “strings”

mode numbers

u

0

Sutherland’95;

Beisert, Minahan, Staudacher, K.Z.’03

Macsoscopic spin waves: long strings

defined on cuts Ck in the complex plane

Scaling limit:

x

0

Classical Bethe equations

Normalization:

Momentum condition:

Anomalous dimension:

Comparison to strings

• Need to know the spectrum of string states:

- eigenstates of Hamiltonian in light-cone gauge

or

- (1,1) vertex operators in conformal gauge

• Not known how to quantize strings in AdS5xS5

• But as long as λ>>1 semiclassical approximation is OK

Time-periodic classical solutions

Quantum states

Bohr-Sommerfeld

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

Consistent truncation

Conformal/temporal gauge:

Pohlmeyer’76

Zakharov, Mikhailov’78

Faddeev, Reshetikhin’86

Keep only

String on S3xR1

2d principal chiral field – well-known intergable model

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)

Noether charges

Length of the chain:

Total spin:

Energy (scaling dimension):

Virasoro constraints:

BMN scalingBerenstein, Maldacena, Nastase’02

Frolov, Tseytlin’03

For any classical solution:

Frolov-Tseytlin limit:

If 1<<λ<<L2:

BMN coupling

Which can be compared to perturbation theory even

though λ is large.

Integrability

Zero-curvature representation:

Equations of motion:

equivalent

on equations of motion

Infinte number of conservation laws

Auxiliary linear problem

quasimomentumNoether charges are determined by asymptotic

behaviour of quasimomentum:

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, K.Z.’04

Normalization:

Momentum condition:

Anomalous dimension:

Normalization:

Momentum condition:

Anomalous dimension:

Take

This is classical limit of Bethe equations for spin chain!

Q: Can we quantize string Bethe equations

(undo thermodynamic limit)?A: Yes! Arutyunov, Frolov, Staudacher’04; Staudacher’04;Beisert, Staudacher’05

Quantum strings in AdS:

• BMN limit

• Near-BMN limit

• Quantum corrections to classical string solutions

Finite-size corrections to Bethe ansatz

Frolov, Tseytlin’03

Frolov, Park, Tsetlin’04

Park, Tirziu, Tseytlin’05

Fuji, Satoh’05

Beisert, Tseytlin, Z.’05

Hernandez, Lopez, Perianez, Sierra’05

Schäfer-Nameki, Zamaklar, Z.’05

Berenstein, Maldacena, Nastase’02; Metsaev’02;…

Callan, Lee,McLoughlin,Schwarz,Swanson,Wu’03;…

String on AdS3xS1:

radial coordinate in AdS

angle in AdS

angle on S5

Rigid string solution:

Arutyunov, Russo, Tseytlin’03

One-loop quantum correction:

Park, Tirziu, Tseytlin’05

AdS spinangular momentum on S5

Bethe equations:

Even under L→-L First correction is O(1/L2)

But singular if simultaneously

Local anomaly Kazakov’03

• cancels at leading order

• gives 1/L correction

Beisert, Kazakov, Sakai, Z.’05

Beisert, Tseytlin, Z.’05

Hernandez, Lopez, Perianez, Sierra’05

x

0

Locally:

Anomaly

local contribution

1/L correction to classical Bethe equations:

Beisert, Tseytlin, Z.’05

Re-expanding the integral:

Agrees with the string calculation.

Remarks:

• anomaly is universal: depends only on singular part

of Bethe equations, which is always the same

• finite-size correction to the energy can be always

expressed as sum over modes of small fluctuationsBeisert, Freyhult’05