T. McLoughlin, J. Minahan, R. Roiban, K. Zarembo 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

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

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

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

Talk overview AdS/CFT Integrability and Scattering  Spectrum  Proposed Bethe equations  Previous checks of integrability Conserved charges No particle production Factorization 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 !

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

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

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

 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

Brief history of the dressing phase [BHL ‘06] SYM side String side [HL ‘06] [BES ‘06] [AFS ‘04] [BDS ‘04] “trivial” 1 2 3 4 2 1 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]

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  Algebratic Bethe ansatz at 1-loop [Minahan, Zarembo ‘02] [Beisert, Staudacher ‘03]  Degeneracies in the spectrum at higher loops [Beisert, Kristjansen, Staudacher ‘03]  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]

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

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]

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

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

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 !”

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

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

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

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

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

fixed up to one function 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

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

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-loop to be shown below !

Only quartic interactions ! 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

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

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

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

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

Second non-triviality Factorization Second non-triviality !? Factorization YBE

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

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

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

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

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 2-loop 2-particle S-Matrix “dog structure” “sun structure”

Summary and open questions ! Proven the factorization of the 3-particle world-sheet S-Matrix to 1-loop in near-flat AdS5xS5 effectively ! 1-loop computation of  the highest-weight amplitudes,  amplitude of mixed processes fixes 3-particle S-matrix 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?