Bethe Ansatz in the AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) EuroStrings 2006 Cambridge, 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’98 Witten’98

Strings in AdS 5 xS 5 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 supersymmetry Green,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 loops conjectured to all loop orders Infinite number of conserved charges associated with local operators (with unclear interpretation) Minahan,Z.’02 Beisert,Kristjansen,Staudacher’03 Beisert,Staudacher’03 Beisert,Dippel,Staudacher’04 Beisert,Staudacher’05; Rej,Serban,Staudacher’05

N=4 Supersymmetric Yang-Mills Theory Field content: The action: Brink,Schwarz,Scherk’77 Gliozzi,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: momentum scattering 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 C k 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 S 3 x R 1 :

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. Kruczenski’03 Kruczenski,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 side Staudacher’04; Beisert’05

WZ term: Landau-Lifshitz model

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

= is not renormalized … S 2→2 = Σ 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

-μ-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

… S 2→2 = Σ p p` p NO ANTIPARTICLES

Exact S-matrix θ – rapidity:

Bethe ansatz Im θ j =0: positive-energy states Im θ j =π: negative-energy states Klose,Z.’06

Ground state θ iπiπ 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-fermions with 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’05 Rej,Serban,Staudacher’05 Gromov,Kazakov,Sakai,Vieira’06 Berenstein,Maldacena,Nastase’02 Beisert,Dippel,Staudacher’04