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World-sheet Scattering in AdS 5 xS 5

World-sheet Scattering in AdS 5 xS 5. Konstantin Zarembo (Uppsala U.). T.Klose, T.McLoughlin, R.Roiban, K.Z., hep-th/0611169 T.Klose, K.Z., hep-th/0701240 T.Klose, T.McLoughlin, J.Minahan, K.Z., 0704.3891. Random Matrix Theory: Recent Applications, Copenhagen, 14.05.07.

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World-sheet Scattering in AdS 5 xS 5

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  1. World-sheet Scattering in AdS5xS5 Konstantin Zarembo (Uppsala U.) T.Klose, T.McLoughlin, R.Roiban, K.Z., hep-th/0611169 T.Klose, K.Z., hep-th/0701240 T.Klose, T.McLoughlin, J.Minahan, K.Z., 0704.3891 Random Matrix Theory: Recent Applications, Copenhagen, 14.05.07

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

  3. Anti-de-Sitter space (AdS5) z 5D bulk strings 0 gauge fields 4D boundary

  4. Spectral problem in large-N SYM: • can be reformulated in terms of an integrable spin system Minahan,Z.’02 Beisert,Kristjansen,Staudacher’03 Beisert,Staudacher’03 • Sigma-model in AdS5xS5: • well-defined theory with the known (but complicated) Lagrangian • also integrable! • Exactly solvable? • How does the (discrete) spin chain arise from the (continuous) world-sheet? Bena,Polchinski,Roiban’03

  5. The non-perturbative S-matrix for the spin chain is known* Beisert’05 Beisert,Eden,Staudacher’06 Goal:check if the same S-matrix describes scattering on the world sheet *) But the Hamiltonian is not!

  6. Sigma-model in AdS5xS5 Metsaev,Tseytlin’98 Green-Schwarz-type coset PSU(2,2|4)/SO(4,1)xSO(5) Sigma-model coupling constant: Classical limit is

  7. φ zi yi’ t S5 AdS5

  8. The metric S5 AdS5 Transverse coordinates: + RR flux

  9. RR flux requiresmanifestspace-time supersymmetry Fermion part of the sigma-model is of Green-Schwarztype. • Conformal gauge is problematic: • no kinetic term for fermions • no holomorphic factorization for currents, … Standard CFT does not work…

  10. Kinetic term for fermions: IfX=const,ρa=0and the kinetic term vanishes. Need to fix the light-cone gauge: X+=τ (thenρ0=Γ+ and the kin. term is not degenerate)

  11. Gauge fixing Light-like geodesics: angular momentum Berenstein,Maldacen,Nastase’02 Parnachev,Ryzhov’02 Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03 Arutyunov,Frolov,Plefka,Zamaklar’06 Gubser,Klebanov,Polyakov’02

  12. Unbroken bosonic subgroup: Taking into account supersymmetry: This also gets central extension with Beisert’05 Arutyunov,Frolov,Plefka,Zamaklar’06 p – world-sheet momentum

  13. World-sheet fields World-sheet fields (embedding coordinates in AdS5xS5): global time in AdS5 and angle on S5 – eliminated by gauge condition Fields form (2|2)x(2|2) multiplet of PSU(2|2)2:

  14. Gauge-fixed action Loop counting parameter: Internal string length: Decompatification (infinite string) limit:

  15. Gauge-fixed Lagrangian (the bosonic part): is the gauge parameter

  16. Bosonic Lagrangian up to quartic order in fields: • Massive, integrable 2d field theory • Lorentz invariant kin. terms • Lorentz invariance is broken by interactions • Gauge-dependent: a=1/2 (pure l.c. gauge) is the simplest case

  17. World-sheet scattering S-matrix: Integrability (consistency with Yang-Baxter equations) requires: SU(2)4 – invariace:

  18. Tree level Klose,McLoughlin,Roiban,Z.’06 SU(2)4 form of T:

  19. (gauge dependence affects only the overall phase) Coincides with the expansion of the spin-chain S-matrix: Beisert’05

  20. Dressing phase Beisert,Hernandez,Lopez’06 Beisert,Eden,Staudacher’06 To leading order in : agrees with tree-level scattering amplitudes in the sigma-model

  21. Near-flat limit Maldacena,Swanson’06 perturbative string modes giant magnons Hofman,Maldacena’06 talk by Semenoff “Left-moving” sector:

  22. Reduced Sigma Model Consistent truncation of the full sigma-model: Maldacena,Swanson’06 - chiral SO(8) spinor

  23. Dispersion relation Exact dispersion relation: Beisert,Dippel,Staudacher’04 At λ→∞: Mass receives radiative corrections starting from two loops In the near-flat limit: (exact)

  24. Tadpole cancellation = 0 + Yi’ (S5) Zi (AdS5) = 0 (fermions) The same mechanism renders the model finite to any order in perturbation theory

  25. Mass renormalization = Mass-shell condition:

  26. S-matrix One loop: Two loops: + mass renormalization + two-loop corrections to Z-factors

  27. Tree-level amplitude: One-loop correction: Klose,Z.’07 Two loops: Klose,McLoughlin,Minahan,Z.’07 Agrees!

  28. “Lorentz-invariant” form of the S-matrix All Lorentz non-invariance of amplitudes in the reduced model can be hidden in the momentum-dependent redefinition of the coupling: In the full σ-model/spin chain, LI is possibly q-deformed rather than violated Gomez,Hernandez’06; Young’07

  29. At strong coupling, the SU(2|2)xSU(2|2) S-matrix describes the scattering of the string modes in the light-cone gauge Perturbative calculations of scattering amplitudes agree with the conjectured BHL/BES phase up to two loops at strong coupling (and to four loops at week coupling Bern,Czakon,Dixon,Kosower,Smirnov’06) Ultimately we want to quantize the σ-model with periodic bounary conditions, when S-matrix is not well defined. S-matrix → asymptotic Bethe ansatz ???????? → exact Bethe ansatz Conclusions talk by Staudacher

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