Bethe Ansatz in the AdS/CFT correspondence

Konstantin Zarembo(Uppsala U.)

EuroStrings 2006Cambridge, 4/4/06

Thanks to:Niklas Beisert (Princeton)Johan Engquist (Utrecht)Gabriele Ferretti (Chalmers)Rainer Heise (Potsdam)Vladimir Kazakov (Paris)Thomas Klose (Uppsala)Andrey Marshakov (Moscow)Joe Minahan (Uppsala & Harvard)Kazuhiro Sakai (Paris)Sakura Schäfer-Nameki (Hamburg)Matthias Staudacher (Potsdam)Arkady Tseytlin (Imperial College)Marija Zamaklar (Potsdam)

AdS/CFT correspondence Maldacena’97

Gubser,Klebanov,Polyakov’98Witten’98

Strings in AdS5xS5

World-sheet theory is Green-Schwarz 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, …

RR flux requires manifest space-time supersymmetryGreen,Schwarz’84

Metsaev,Tseytlin’98

Integrability in string theory

But the model is integrable!Bena,Polchinski,Roiban’03

• infinite number of conserved charges• separation of variables in classical

equations of motion• quantum spectrum is determined by

Bethe equations

√

?

√Dorey,Vicedo’06

Integrability in N=4 SYM

• Bethe ansatz for planar anomalous dimensions:rigorous up to three loopsconjectured to all loop orders

• Infinite number of conserved charges associated with local operators (with unclear interpretation)

Minahan,Z.’02Beisert,Kristjansen,Staudacher’03Beisert,Staudacher’03

Beisert,Dippel,Staudacher’04Beisert,Staudacher’05; Rej,Serban,Staudacher’05

N=4 Supersymmetric Yang-Mills Theory

Field content:

The action:

Brink,Schwarz,Scherk’77Gliozzi,Scherk,Olive’77

Operator mixing

Renormalized operators:

Mixing matrix (dilatation operator):

Local operators and spin chains

related by SU(2) R-symmetry subgroup

a b

a b

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; Rej,Serban,Staudacher’05

Beisert,Staudacher’03

Minahan,Z.’02

The spectrum

Ground state:

Excited states (magnons):

Zero momentum (trace cyclicity) condition:

Anomalous dimension:

Bethe’31

Exact spectrum

Rapidity:

Exact periodicity condition:

momentumscattering phase shifts

periodicity of wave function

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 a set of conoturs Ck in the complex plane

Scaling limit:

x

0

Classical Bethe equations

Normalization:

Momentum condition:

Anomalous dimension:

Heisenberg model in Heisenberg representation

Heisenberg operators:

Hiesenberg equations:

Continuum + classical limit

Landau-Lifshitz equation

Consistent truncation

String on S3 x R1:

Conformal/temporal gauge:

Pohlmeyer’76Zakharov,Mikhailov’78Faddeev,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.Kruczenski’03Kruczenski,Ryzhov,Tseytlin’03

Integrability

Zero-curvature representation:

Equations of motion:

equivalent

Conserved charges

time

on equations of motion

Generating function (quasimomentum):

Non-local charges:

Local charges:

Analyticity:

Classical string Bethe equation

Kazakov,Marshakov,Minahan,Z.’04

Normalization:

Momentum condition:

Anomalous dimension:

Quantum Bethe equations

(two particle factorization)

• 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

Successfully used on the gauge-theory sideStaudacher’04; Beisert’05

WZ term:

Landau-Lifshitz model

Perturbation theory

in order to get canonical kinetic term Minahan,Tirziu,Tseytlin’04

= is not renormalized

= 0

…S 2→2 = Σp

p`

p

p`

Summation of bubble diagrams yields

Bethe equations

Heisenberg model: the same equations with

The difference disappears in the low-energy (u→∞) limit.

Klose,Z.’06

AAF model

SU(1|1) sector:

• in SYM:Callan,Heckman,McLoughlin,Swanson’04

• in string theory:

Alday,Arutyunov,Frolov’05

p0

p0

-μ

μ – chemical potential

μ→-∞ All poles are below the real axis.

-m

Empty Fermi sea:

E

+m

-m

E

+m

Physical vacuum:

Berezin,Sushko’65; Bergknoff,Thaker’79; Korepin’79

= 0

…S 2→2 = Σp

p`

p

p`

NO ANTIPARTICLES

Exact S-matrix

θ – rapidity:

Bethe ansatz

Im θj=0: positive-energy states

Im θj=π: negative-energy states

Klose,Z.’06

Ground state

θ

iπ

0

-k+iπ k+iπ

- UV cutoff

Mass renormalization:

This equation also determines the spectrum, the physical S-matrix, ...

• Weak-coupling (-1<g<1):Non-renormalizable (anomalous dimension of mass is complex)

• Strong attraction (g>1) Unstable

(Energy unbounded below)• Strong repulsion (g<-1):

Spectrum consists of fermions and anti-fermionswith non-trivial scattering

Questions

Continuous Discrete(string worldsheet) vs. (spin chain)

in Bethe ansatz:

Possible resolution:• extra hidden d.o.f. (“particles

on the world sheet that create the spin chain”)

• spin chain is thus dynamical

Choice of the reference state, the true vacuum, antiparticles, …

Mann,Polchinski’05Rej,Serban,Staudacher’05Gromov,Kazakov,Sakai,Vieira’06

Berenstein,Maldacena,Nastase’02 Beisert,Dippel,Staudacher’04