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Bethe ansatz in String Theory

Bethe ansatz in String Theory. Konstantin Zarembo (Uppsala U.). Integrable Models and Applications, Lyon, 13.09.2006. AdS/CFT correspondence. Maldacena’97. Gubser,Klebanov,Polyakov’98 Witten’98. Planar diagrams and strings. time. (kept finite). ‘t Hooft coupling:

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Bethe ansatz in String Theory

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  1. Bethe ansatz in String Theory Konstantin Zarembo (Uppsala U.) Integrable Models and Applications, Lyon, 13.09.2006

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

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

  4. Strong-weak coupling interpolation λ 0 SYM perturbation theory String perturbation theory 1 + + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS5

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

  6. Field content: N=4 Supersymmetric Yang-Mills Theory Gliozzi,Scherk,Olive’77 Action: Global symmetry: PSU(2,2|4)

  7. Spectrum Basis of primary operators: Spectrum = {Δn} Dilatation operator (mixing matrix):

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

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

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

  11. Higher loops Requirments ofintegrability and BMN scaling uniquely define perturbative scheme to construct dilatation operator through order λL-1: Beisert,Kristjansen,Staudacher’03

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

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

  14. 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?

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

  16. 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 asymptotic Bethe ansatz spectrum of an infinite spin chain Beisert’05

  17. Beisert,Staudacher’05

  18. STRINGS

  19. String theory in AdS5S5 Metsaev,Tseytlin’98 + constant RR 4-form flux • Finite 2d field theory (¯-function=0) • Sigma-model coupling constant: • Classically integrable Classical limit is Bena,Polchinski,Roiban’03

  20. 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:

  21. 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!

  22. 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) 4 x 4 x 2 = 32 real components : 2 kappa-symmetry : 2 (eqs. of motion are first order) = 8

  23. 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 Quantization 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

  24. Consistent truncation String on S3 x R1:

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

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

  27. Quantum string Bethe equations Arutyunov,Frolov,Staudacher’04 extra phase Beisert,Staudacher’05

  28. 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

  29. 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

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