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Bethe ansatz in String Theory Konstantin Zarembo (Uppsala U.) Integrable Models and Applications, Lyon, 13.09.2006 The European Superstring Theory Network » Members Co-ordinator Chalmers University of Technology (Sweden) Fundamental Physics Department of Physics* (Karlstad Univ.) Other Contractors Uppsala University (Sweden) Theoretical Physics Cosmology Particle Astrophysics and String theory* (Stockholm Univ.) The Chancellor, Masters and Scholars of the University of Cambridge (UK) Theoretical High Energy Particle Physics Group King's College (UK) Theoretical Physics Queen Mary and Westfield College (UK) String Theory Group Theory Group* (Imperial College) Centre National de la Reserche Scientifique (France) LTPENS*(École Normale Supérieure) LPTHE*(Univ. Pierre et Marie Curie) CRNS Universiteit van Amsterdam (The Netherlands) Institute for Theoretical Physics Institute for Theoretical Physics*(Utrecht Univ.) Theoretical Physics* (NIKHEF) Max-Planck-Gesellschaft (Germany) Quantum Gravity & Unified Theories (Max Planck Institute) Centre for Mathematical Physics (Univ. Hamburg & DESY) Universitá degli Studi di Roma "Tor Vergata" (Italy) String Theory Group Gruppo Teorico*(Univ. di Roma La Sapienza) Gruppo Teorico*(Univ. di Pisa) High Energy Group*(ICTP) University of Crete High Energy and Elementary Particle Physics Division *(Univ. of Athens) The Hebrew University of Jerusalem (Israel) Racah Institute of Physics Dep. of Particle Physics*(Weizmann Inst. of Science) Dep. of Particle Physics*(Tel Aviv University) Masarykova univerzita v Brne (Czech Republic) Institute of Theoretical Physics and Astrophysics University of Cyprus (Cyprus) High Energy Physics Group 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 λ 0 SYM perturbation theory 1+ + String perturbation theory +… Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS5 Weakly coupled SYM is reliable if Weakly coupled string is reliable if Can expect an overlap. N=4 Supersymmetric Yang-Mills Theory Gliozzi,Scherk,Olive’77 Field content: Action: Global symmetry: PSU(2,2|4) Spectrum Basis of primary operators: Spectrum = {Δn} Dilatation operator (mixing matrix): 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 Bethe ansatz Bethe’31 Zero momentum (trace cyclicity) condition: Anomalous dimensions: Higher loops Requirments of integrability and BMN scaling uniquely define perturbative scheme to construct dilatation operator through order λL-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 Rej,Serban,Staudacher’05; Z.’05; Feverati,Fiorovanti,Grinza,Rossi’06; Beccaria,DelDebbio’06 Weak coupling: Strong coupling: Q: Is it exact at all λ? 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): Mμν - rotations Pμ - translations Kμ - special conformal transformations D - dilatation Ground state tr ZZZZ… breaks PSU(2,2|4) → P(SU(2|2)xSU(2|2)) Bootstrap: SU(2|2)xSU(2|2) invariant S-matrix Beisert’05 spectrum of an infinite spin chain asymptotic Bethe ansatz Beisert,Staudacher’05 STRINGS String theory in AdS5S5 Metsaev,Tseytlin’98 + constant RR 4-form flux • Finite 2d field theory (¯-function=0) • Sigma-model coupling constant: • Classically integrable Bena,Polchinski,Roiban’03 Classical limit is AdS sigma-models as supercoset S5 = SU(4)/SO(5) AdS5 = SU(2,2)/SO(4,1) AdS superspace: Super(AdS5xS5) = PSU(2,2|4)/SO(5)xSO(4,1) Z4 grading: Coset representative: g(σ) Currents: j = g-1dg = j0 + j1 + j2 + j3 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 AdS5 + 5 in S5) - 2 (reparameterizations) = 8 Fermions: - bifundamentals of su(2,2) x su(4) 4x4x2 = 32 real components : 2 kappa-symmetry : 2 (eqs. of motion are first order) = 8 Quantization • fix light-cone gauge and quantize: Berenstein,Maldacena,Nastase’02 Callan,Lee,McLoughlin,Schwarz, Swanson,Wu’03 Frolov,Plefka,Zamaklar’06 action is VERY complicated perturbation theory for the spectrum, S-matrix,… Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03; Klose,McLoughlin,Roiban,Z.’in progress • study classical equations of motion (gauge unfixed), then guess Kazakov,Marshakov,Minahan,Z.’04; Beisert,Kazakov,Sakai,Z.’05; Arutyunov,Frolov,Staudacher’04; Beisert,Staudacher’05 • quantize near classical string solutions Frolov,Tseytlin’03-04; Schäfer-Nameki,Zamaklar,Z.’05; Beisert,Tseytlin’05; Hernandez,Lopez’06 Consistent truncation String on S3 x R1: Gauge condition: Equations of motion: Zero-curvature representation: equivalent Zakharov,Mikhaikov’78 Classical string Bethe equation Kazakov,Marshakov,Minahan,Z.’04 Normalization: Momentum condition: Anomalous dimension: Quantum string Bethe equations Arutyunov,Frolov,Staudacher’04 extra phase Beisert,Staudacher’05 Arutyunov,Frolov,Staudacher’04 Hernandez,Lopez’06 • Algebraic structure is fixed by symmetries Beisert’05 • The Bethe equations are asymptotic: they describe infinitely long strings / spin chains and do not capture finite-size effects. Schäfer-Nameki,Zamaklar,Z.’06 Open problems • 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 Teschner’s talk • Algebraic structure: Algebraic Bethe ansatz? Yangian symmetries? Baxter equation?