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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
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
 Spectrum
 Proposed Bethe equations
 Previous checks of integrability
Integrability and Scattering
 Conserved charges
 No particle production
 Factorization
 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
Emergence of
integrable structures !
IIB Strings
SYM on
Spectrum of string energies
► Dispersion relation
(propagation of excitation)
► 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
Dressing phase
[BHL ‘06]
SYM side
String side
[HL ‘06]
[BES ‘06]
[AFS ‘04]
[BDS ‘04]
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
 Degeneracies in the spectrum at higher loops
 Inozemtsev spin chain up to 3-loops in SU(2) sector
 Factorization of 3-impurity S-matrix in SL(2) sector
[Minahan, Zarembo ‘02]
[Beisert, Staudacher ‘03]
[Beisert, Kristjansen,
Staudacher ‘03]
[Serban, Staudacher ‘04]
[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
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
[Berkovits ‘05]
in pure spinor formulation
 Check energies of multi-excitation states against Bethe equations
[Callan, McLoughlin, Swanson ‘04]
(at tree-level)
[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
► No particle production or annihilation
[Shankar, Witten ‘78]
[Zamolodchikov, Zamolodchikov ‘79]
[Parke ‘80]
► 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
Conservation laws and Scattering in 1+1 dimensions
local conserved charges
[Parke ‘80]
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]
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
such that
world-sheet size
, no
loop counting parameter
to define asymptotic states
Superstrings on AdS5xS5
[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
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
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 !
Near-flat-space limit
► Boost in the world-sheet theory:
Highly interacting
giant magnons
[Hofman, Maldacena ‘06]
Only quartic interactions !
Free massive theory
[Maldacena, Swanson ‘06]
[Berenstein, Maldacena, Nastase ‘02]
Near-flat-space limit
Coupling constant:
, fermions
Non-Lorentz invariant interactions
Decoupling of right-movers
quantum mechanically consistent reduction
at least to two-loops
 Tristan’s talk tomorrow
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
S-Matrix from Feynman diagrams
► 3-particle S-matrix
► 6-point amplitude
Emergence of factorization
light-cone momenta
► Tree-level amplitudes
compare to sinh-Gordon:
sets the internal propagator on-shell
Emergence of factorization
► Example
from Feynman diagrams
... agrees with the predicted factorized 3-particle S-Matrix
Emergence of factorization
► 1-loop amplitudes
phase space
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...
► General 3-particle S-Matrix
 works for symmetric processes like
 fails for mixed processes like
 cannot hold at higher loops, e.g.
► Contributions at order
2-loop 2-particle
“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
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?