Integrability and Bethe Ansatz in the AdS/CFT correspondence

Konstantin Zarembo

(Uppsala U.)

Nordic Network MeetingHelsinki, 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)

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 (AdS5)

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 Tμν

• sum rules are automatic

String statesBound 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 theoryLarge number

of constituentsor

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

• ≈ 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:

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:

momentumscattering 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 Ck 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: