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INI Cambridge, 31.10.2007

INI Cambridge, 31.10.2007. Quantum Integrability of AdS String Theory: Factorized Scattering in the near-flat limit. Thomas Klose. Princeton Center for Theoretical Physics. based on work with Valentina Giangreco Puletti and Olof Ohlson Sax : hep-th/0707.2082 also thanks to

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INI Cambridge, 31.10.2007

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  1. INI Cambridge, 31.10.2007 Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit Thomas Klose Princeton Center for Theoretical Physics based on work with Valentina Giangreco Puletti and Olof Ohlson Sax: hep-th/0707.2082 also thanks to T. McLoughlin, J. Minahan, R. Roiban, K. Zarembo for further collaborations

  2. Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit String waves in flat space ►Simple Fock spectrum

  3. Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit String waves in curved space ? ? ? ? ►Spectrum unknown

  4. Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit String waves in AdS5 x S5 ►Spectrum from Bethe eq‘s

  5. = = Talk overview AdS/CFT Integrability and Scattering • Conserved charges • No particle production • Factorization  Spectrum Proposed Bethe equations Previous checks of integrability Intro Superstrings on AdS5 x S5 World-sheet scattering and S-matrix Near-flat-space limit ► Factorization of three-particle world-sheet S-matrix in near-flat AdS5 x S5 to one loop in string σ-model (163)2 compoments, but only 4 independent ones !

  6. AdS/CFT spectrum Conformal dimensions String energies Emergence of integrable structures ! IIB Strings on SYM on

  7. Spectrum of string energies ► Dispersion relation dispersionless non-relativistic (propagation of excitation) lattice relativistic AdS/CFT ► Momentum selection (periodicity+level matching) Phase shift

  8. Spectrum of string energies ► If String theory was integrable... ... then the multi-particle phase shifts would be products of ... and the momenta would satisfy Bethe equations like ... and the spectrum would be given by

  9. Proposed Bethe equations for AdS/CFT Nested Bethe equations Bethe roots [Beisert, Staudacher ‘05] Rapidity map Dressing phase [Beisert, Eden, Staudacher ‘07] Dispersion relations [Dorey, Hofman, Maldacena ’07]  planar asymptotic spectrum

  10. Brief history of the dressing phase Dressing phase [BHL ‘06] SYM side String side [HL ‘06] [BES ‘06] [AFS ‘04] [BDS ‘04] “trivial” 0 1 2 3 4 2 1 0 Checks in 4-loop gauge theory Checks in 2-loop string theory [Bern, Czakon, Dixon, Kosower, Smirnov ‘06]  Tristan’s talk tomorrow [Beisert, McLoughlin, Roiban ‘07]

  11. Checks of Integrability in AdS/CFT ► Integrability of planar N=4 SYM theory [Minahan, Zarembo ‘02] Spin chain picture at large N Dilatation operator Hamiltonian of integrable spin-chain [Minahan, Zarembo ‘02]  Algebratic Bethe ansatz at 1-loop [Beisert, Staudacher ‘03] [Beisert, Kristjansen, Staudacher ‘03]  Degeneracies in the spectrum at higher loops  Inozemtsev spin chain up to 3-loops in SU(2) sector [Serban, Staudacher ‘04]  Factorization of 3-impurity S-matrix in SL(2) sector [Eden, Staudacher ‘06]

  12. Checks of Integrability in AdS/CFT [Mandal, Suryanarayana, Wadia ‘02] ► Classical Integrability of planar AdS string theory [Bena, Polchinski, Roiban ‘03] Coset representative conserved Current flat Family of flat currents Monodromy matrix generates conserved charges

  13. Checks of Integrability in AdS/CFT ► Quantum Integrability of planar AdS string theory • Quantum consistency of AdS strings, and existence of higher charges in pure spinor formulation [Berkovits ‘05] • Check energies of multi-excitation states against Bethe equations (at tree-level) [Callan, McLoughlin, Swanson ‘04] [Hentschel, Plefka, Sundin ‘07]  Absence of particle production in bosonic sector in semiclassical limit [TK, McLoughlin, Roiban, Zarembo ‘06]  Quantum consistency of monodromy matrix [Mikhailov, Schäfer-Nameki ‘07]

  14. Integrability in 1+1d QFTs Existence of local higher rank conserved charges [Shankar, Witten ‘78] [Zamolodchikov, Zamolodchikov ‘79] [Parke ‘80] ► No particle production or annihilation ► Conservation of the set of momenta ► -particle S-Matrix factorizes into 2-particle S-Matrices

  15. Conservation laws and Scattering in 1+1 dimensions ► 1 particle ► 2 particles ► 3 particles ► particle

  16. Conservation laws and Scattering in 1+1 dimensions [Parke ‘80] local conserved charges with action conservation implies: “Two mutually commuting local charges of other rank than scalar and tensor are sufficient for S-matrix factorization !”

  17. Strings on AdS5xS5 (bosonic) [Metsaev, Tseytlin ‘98] AdS5 x S5

  18. Strings on AdS5xS5 (bosonic) ►Fixing reparametization invariance in uniform lightcone gauge [Arutyunov, Frolov, Zamaklar ‘06] worldsheet Hamiltonian density eliminated by Virasoro constraints ►Back to Lagrangian formulation

  19. Strings on AdS5xS5 (bosonic) ► Decompactification limit rescale such that world-sheet size no , no loop counting parameter send to define asymptotic states

  20. Superstrings on AdS5xS5 [Metsaev, Tseytlin ‘98] ► Sigma model on Gauge fixing (L.C. + -symmetry) [Frolov, Plefka, Zamaklar ‘06] Manifest symmetries

  21. Worldsheet S-Matrix ► 4 types of particles, 48=65536 Matrix elements ► Group factorization ► Each factor has manifest invariance

  22. Symmetry constraints on the S-Matrix In infinite volume, the symmetry algebra gets centrally extended to [Beisert ‘06] ► 2-particle S-Matrix: for one S-Matrix factor: irrep of of centrally extended algreba relate the two irreps of the total S-Matrix is: fixed up to one function

  23. Symmetry constraints on the S-Matrix ► 3-particle S-matrix: fixed up to four functions

  24. 3-particle S-matrix ► Eigenstates Extract coefficient functions from: ► Disconnected piece: factorizes trivially, 2-particle S-matrix checked to 2-loops [TK, McLoughlin, Minahan, Zarembo ‘07] ► Connected piece: factorization at 1-loopto be shown below !

  25. Near-flat-space limit ► Boost in the world-sheet theory: Highly interacting giant magnons [Hofman, Maldacena ‘06] Only quartic interactions ! near-flat-space [Maldacena, Swanson ‘06] plane-wave [Berenstein, Maldacena, Nastase ‘02] Free massive theory

  26. Near-flat-space limit Coupling constant: Propagators: bosons , fermions Non-Lorentz invariant interactions ! Decoupling of right-movers ! ! UV-finiteness  Tristan’s talk tomorrow ! quantum mechanically consistent reduction at least to two-loops

  27. 2-particle S-Matrix in Near-Flat-Space limit Overall phase: Exact coefficients for one PSU(2|2) factor:

  28. S-Matrix from Feynman diagrams ► 2-particle S-matrix ► 4-point amplitude compare for

  29. S-Matrix from Feynman diagrams ► 3-particle S-matrix ► 6-point amplitude First non-triviality !?

  30. Factorization Second non-triviality !? Factorization YBE

  31. Emergence of factorization light-cone momenta ► Tree-level amplitudes finite divergencies sets the internal propagator on-shell compare to sinh-Gordon:

  32. Emergence of factorization ► Example from Feynman diagrams ... agrees with the predicted factorized 3-particle S-Matrix disconnected

  33. Emergence of factorization phase space ► 1-loop amplitudes “dog”: “sun”:

  34. Emergence of factorization Cutting rule in 2d for arbitrary 1-loop diagrams [Källén, Toll ‘64] Applied to “sun-diagram”:

  35. Emergence of factorization The below identification...  works for symmetric processes like ►General 3-particle S-Matrix  fails for mixed processes like  cannot hold at higher loops, e.g. ►Contributions at order “dog structure” “sun structure” 2-loop 2-particle S-Matrix

  36. Summary and open questions ! Proven the factorization of the 3-particle world-sheet S-Matrix to 1-loop in near-flat AdS5xS5 effectively fixes 3-particle S-matrix 1-loop computation of the highest-weight amplitudes, amplitude of mixed processes ! checks supersymmetries Direct check of quantum integrability of AdS string theory (albeit in the NFS limit) ! Extenstions of the above: higher loops, more particles, full theory ? ? Finite size corrections Asymptotic states?

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