Bethe Ansatz and Integrability in AdS/CFT correspondence

Konstantin Zarembo

(Uppsala U.)

“Constituents, Fundamental Forces and Symmetries of the Universe”,Napoli, 9.10.2006

Thanks to:Niklas BeisertJohan EngquistGabriele FerrettiRainer HeiseVladimir KazakovThomas KloseAndrey MarshakovTristan McLoughlinJoe MinahanRadu RoibanKazuhiro SakaiSakura Schäfer-NamekiMatthias StaudacherArkady TseytlinMarija Zamaklar

AdS/CFT correspondence

Yang-Mills theory with

N=4 supersymmetry

String theory on

AdS5xS5 background

Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

Exact equivalence

Planar diagrams and strings

time

Large-N limit:

AdS/CFT correspondence Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

λ<<1 Quantum strings

Classical strings Strong coupling in SYM

Spectrum of SYM = String spectrum

but

Strong-weak coupling interpolation

Circular Wilson loop (exact):Erickson,Semenoff,Zarembo’00

Drukker,Gross’00

0 λSYM perturbation

theory

1 + + …+

String perturbation

theory

Minimal area law in AdS5

Gubser,Klebanov,Tseytlin’98; …

SYM is weakly coupled if

String theory is weakly coupled if

There is an overlap!

Q:HOW TO COMAPARE

SYM AND STRINGS?

A(?): SOLVE EACH

WITH THE HELP OF BETHE ANSATZ

Plan

1. Integrability in SYM

2. Integrability in AdS string theory

3. Integrability and Bethe ansatz

4. Bethe ansatz in AdS/CFT

5. Testing Bethe ansatz against string quantum corrections

N=4 Supersymmetric Yang-Mills Theory

Field content:

Action:

Gliozzi,Scherk,Olive’77

Global symmetry: PSU(2,2|4)

Spectrum

Basis of primary operators:

Dilatation operator (mixing matrix):

Spectrum = {Δn}

Local operators and spin chains

related by SU(2) R-symmetry subgroup

i j

i j

One loop:

Tree level: Δ=L (huge degeneracy)

One loop planar dilatation generator:

Minahan,Z.’02

Heisenberg Hamiltonian

Integrability

Lax operator:

Monodromy matrix:

Faddeev et al.’70-80s

Transfer “matrix”:

Infinite tower of conserved charges:

U – lattice translation generator: U=eiP

Algebraic Bethe Ansatz

Spectrum:

are eigenstates of the Hamiltonian

with eigenvalues

(anomalous dimension)

(total momentum)

ProvidedBethe equations

Strings in AdS5xS5

Green-Schwarz-type coset sigma model

on SU(2,2|4)/SO(4,1)xSO(5).

Conformal gauge is problematic:

no kinetic term for fermions, no holomorphic

factorization for currents, …

Light-cone gauge is OK.

Metsaev,Tseytlin’98

The action is complicated, but the model is integrable!Bena,Polchinski,Roiban’03

Consistent truncation

String on S3 x R1:

Zero-curvature representation:

Equations of motion:

equivalent

Zakharov,Mikhaikov’78

Gauge condition:

Conserved charges

time

on equations of motion

Generating function (quasimomentum):

Non-local charges:

Local charges:

Bethe ansatz

• Algebraic Bethe ansatz: quantum Lax operator + Yang-Baxter equations → spectrum

• Coordinate Bethe ansatz: direct construction of the wave functions in the Schrödinger representation

• Asymptotic Bethe ansatz: S-matrix ↔ spectrum (infinite L) ? (finite L)

Spectrum and scattering phase shifts

periodic short-range potential

• exact only for V(x) = g δ(x)

Continuity of periodized wave function

where

is (eigenvalue of) the S-matrix

• correct up to O(e-L/R)• works even for bound states via analytic

continuation to complex momenta

Multy-particle states

Bethe equations

Assumptions:• R<<L• particles can only exchange momenta• no inelastic processes

2→2 scattering in 2d

p1

p2

k1

k2

Energy and momentum conservation:

I

II

Momentum conservation

Energy conservation

k1

k2

I: k1=p1, k2=p2 (transition)

II: k1=p2, k2=p1 (reflection)

n→n scattering

2 equations for n unknowns

(n-2)-dimensional phase space

pi ki

Unless there are extra conservation laws!

Integrability:

• No phase space:

• No particle production (all 2→many processes are kinematically forbidden)

Factorization:

Consistency condition (Yang-Baxter equation):

1

2

3

1

2

3=

Strategy:

• find the dispersion relation (solve the one-body problem):

• find the S-matrix (solve the two-body problem):

Bethe equations full spectrum

• find the true ground state

Integrability + Locality Bethe ansatz

What are the scattering states?

SYM: magnons

String theory: “giant magnons”

Staudacher’04

Hofman,Maldacena’06

Common dispersion relation:

S-matrix is highly constrained by symmetriesBeisert’05

Zero momentum (trace cyclicity) condition:

Anomalous dimension:

Algebraic BA: one-loop su(2) sector

Rapidity:Minahan,Z.’02

Algebraic BA: one loop, complete spectrumBeisert,Staudacher’03

Nested BAE:

- Cartan matrix of PSU(2,2|4)

- highest weight of the field representation

bound states of magnons – Bethe “strings”

mode numbers

u

0

Sutherland’95;

Beisert,Minahan,Staudacher,Z.’03

Semiclassical states

defined on a set of conoturs Ck in the complex plane

Scaling limit:

x

0

Classical Bethe equations

Normalization:

Momentum condition:

Anomalous dimension:

Algebraic BA: classical string Bethe equation

Kazakov,Marshakov,Minahan,Z.’04

Normalization:

Momentum condition:

String energy:

su(2) sector:

General classical BAE are known and have the nested structure

consistent with the PSU(2,2|4) symmetry of AdS5xS5 superstringBeisert,Kazakov,Sakai,Z.’05

Asymptotic BA: SYM

Beisert,Staudacher’05

Asymptotic BA: string

extra phase

Arutyunov,Frolov,Staudacher’04

Hernandez,Lopez’06

• Algebraic structure is fixed by symmetries

• The Bethe equations are asymptotic: they describe infinitely long strings / spin chains.

Beisert’05

Schäfer-Nameki,Zamaklar,Z.’06

Testing BA: semiclassical string in AdS3xS1

- radial coordinate in AdS

- angle in AdS

- angle on S5

- global time

Rigid string solution

Arutyunov,Russo,Tseytlin’03

AdS5S5

winds k times

and rotates

winds m times

and rotates

Internal length of the string is

Perturbative SYM regime:

(string is very long)

For simplicity, I will consider

(large-winding limit) Schäfer-Nameki,Zamaklar,Z.’05

string fluctuation frequencies

Explicitly,

Park,Tirziu,Tseytlin’05

classical energy one loop correction

Quantum-corrected Bethe equations

classical BEKazakov,Z.’04

AnomalyKazakov’04;Beisert,Kazakov,Sakai,Z.’05

Beisert,Tseytlin,Z.’05; Schäfer-Nameki,Zamaklar,Z.’05

Quantum correction to the scattering phaseHernandez,Lopez’06

Large (long strings):

Comparison

• String

• BA

BA misses exponential termsSchäfer-Nameki,Zamaklar,Z.’05

Conclusions

• Large-N SYM / string sigma-model on AdS5xS5 are probably solvable by Bethe ansatz

• Open problems: Interpolation from weak to strong coupling Finite-size effects Appropriate reference state / ground state Algebraic formulation:

– Transfer matrix

– Yang-Baxter equation

– Pseudo-vacuum