1. T. McLoughlin, J. Minahan, R. Roiban, K. Zarembo
INI Cambridge,
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/
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. ►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

5. 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 !

6. AdS/CFT spectrum Conformal String dimensions 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
[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]

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
 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]

12. 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

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 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 ► Sigma model on
[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. 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

23. fixed up to four functions
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-loop to be shown below !

25. 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

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. Second non-triviality
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
2-loop 2-particle
S-Matrix
“dog structure”
“sun structure”

36. 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?