Bethe ansatz in String Theory Konstantin Zarembo (Uppsala U.) Integrable Models and Applications, Lyon,

AdS/CFT correspondence Maldacena’97 Gubser,Klebanov,Polyakov’98 Witten’98

Planar diagrams and strings time ‘t Hooft coupling: String coupling constant = (kept finite) (goes to zero)

Strong-weak coupling interpolation Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 0 λ SYM perturbation theory … + String perturbation theory Minimal area law in AdS 5

Weakly coupled SYM is reliable if Weakly coupled string is reliable if Can expect an overlap.

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 = {Δ n }

Local operators and spin chains related by SU(2) R-symmetry subgroup a b a b

Tree level: Δ=L (huge degeneracy) One loop: Minahan,Z.’02

Zero momentum (trace cyclicity) condition: Anomalous dimensions: Bethe’31 Bethe ansatz

Higher loops Requirments of integrability and BMN scaling uniquely define perturbative scheme to construct dilatation operator through order λ L-1 : Beisert,Kristjansen,Staudacher’03

The perturbative Hamiltonian turns out to coincide with strong-coupling expansion of Hubbard model at half-filling: Rej,Serban,Staudacher’05

Asymptotic Bethe ansatz Beisert,Dippel,Staudacher’04 In Hubbard model, these equations are approximate with O(e -f(λ)L ) corrections at L→∞

Anti-ferromagnetic state Weak coupling: Strong coupling: Q: Is it exact at all λ? Rej,Serban,Staudacher’05; Z.’05; Feverati,Fiorovanti,Grinza,Rossi’06; Beccaria,DelDebbio’06

Arbitrary operators Bookkeeping: “letters”: “words”: “sentences”: Spin chain: infinite-dimensional representation of PSU(2,2|4)

Length fluctuations: operators (states of the spin chain) of different length mix Hamiltonian is a part of non-abelian symmetry group: conformal group SO(4,2)~SU(2,2) is part of PSU(2,2|4) so(4,2): M μν - rotations P μ - translations K μ - special conformal transformations D - dilatation Bootstrap: SU(2|2)xSU(2|2) invariant S-matrix asymptotic Bethe ansatz spectrum of an infinite spin chain Ground state tr ZZZZ… breaks PSU(2,2|4) → P(SU(2|2)xSU(2|2)) Beisert’05

Beisert,Staudacher’05

STRINGS

String theory in AdS 5  S 5 Metsaev,Tseytlin’98 + constant RR 4-form flux Bena,Polchinski,Roiban’03 Finite 2d field theory ( ¯ -function=0) Sigma-model coupling constant: Classically integrable Classical limit is

AdS sigma-models as supercoset S 5 = SU(4)/SO(5) AdS 5 = SU(2,2)/SO(4,1) Super(AdS 5 xS 5 ) = PSU(2,2|4)/SO(5)xSO(4,1) AdS superspace: Z 4 grading:

Coset representative: g(σ) Currents: j = g -1 dg = j 0 + j 1 + j 2 + j 3 Action: Metsaev,Tseytlin’98 In flat space: Green,Schwarz’84 no kinetic term for fermions!

Degrees of freedom Bosons: 15 (dim. of SU(2,2)) + 15 (dim. of SU(4)) - 10 (dim. of SO(4,1)) - 10 (dim. of SO(5)) = 10 (5 in AdS in S 5 ) - 2 (reparameterizations) = 8 Fermions: - bifundamentals of su(2,2) x su(4) 4 x 4 x 2 = 32 real components : 2 kappa-symmetry : 2 (eqs. of motion are first order) = 8

Quantization fix light-cone gauge and quantize: action is VERY complicated perturbation theory for the spectrum, S-matrix,… study classical equations of motion (gauge unfixed), then guess quantize near classical string solutions Berenstein,Maldacena,Nastase’02 Callan,Lee,McLoughlin,Schwarz, Swanson,Wu’03 Frolov,Plefka,Zamaklar’06 Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03; Klose,McLoughlin,Roiban,Z.’in progress Kazakov,Marshakov,Minahan,Z.’04; Beisert,Kazakov,Sakai,Z.’05; Arutyunov,Frolov,Staudacher’04; Beisert,Staudacher’05 Frolov,Tseytlin’03-04; Schäfer-Nameki,Zamaklar,Z.’05; Beisert,Tseytlin’05; Hernandez,Lopez’06

Consistent truncation String on S 3 x R 1 :

Zero-curvature representation: Equations of motion: equivalent Zakharov,Mikhaikov’78 Gauge condition :

Classical string Bethe equation Kazakov,Marshakov,Minahan,Z.’04 Normalization: Momentum condition: Anomalous dimension:

Quantum string Bethe equations extra phase Beisert,Staudacher’05 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 and do not capture finite-size effects. Beisert’05 Schäfer-Nameki,Zamaklar,Z.’06

Interpolation from weak to strong coupling in the dressing phase How accurate is the asymptotic BA? (Probably up to e -f(λ)L ) Eventually want to know closed string/periodic chain spectrum need to understand finite-size effects Algebraic structure: Algebraic Bethe ansatz? Yangian symmetries? Baxter equation? Open problems Teschner’s talk