INI Cambridge, 31.10.2007

Thomas Klose

Princeton Center for Theoretical Physics

based on work withValentina Giangreco Puletti and Olof Ohlson Sax: hep-th/0707.2082

also thanks toT. McLoughlin, J. Minahan, R. Roiban, K. Zarembo

for further collaborations

Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit

String wavesin flat space

Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit

►Simple Fock spectrum

?

? ?

?

String wavesin curved space

Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit

►Spectrum unknown

String wavesin AdS5 x S5

Quantum Integrability of AdS String Theory:Factorized Scattering in the near-flat limit

►Spectrum from Bethe eq‘s

Talk overview

► 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 !

Intro

Spectrum Proposed Bethe equations Previous checks of integrability

AdS/CFT Integrability and Scattering Conserved charges No particle production Factorization

Superstrings on AdS5 x S5

World-sheet scattering and S-matrix Near-flat-space limit

AdS/CFT spectrum

SYM on

IIB Strings

Emergence ofintegrable structures !

on

Conformaldimensions

Stringenergies

Spectrum of string energies

► Dispersion relation

► Momentum selection

dispersionless

non-relativistic

lattice

relativistic

AdS/CFT

(propagation of excitation)

(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

Rapidity map

Dressing phase

Dispersion relations

planar asymptotic spectrum

[Beisert, Staudacher ‘05]

[Beisert, Eden, Staudacher ‘07]

[Dorey, Hofman, Maldacena ’07]

[BDS ‘04]

String side

Dressing phase

SYM side

0 1 2 3 4 012

“trivial”

[AFS ‘04]

[HL ‘06]

[BHL ‘06]

[BES ‘06]

Checks in 4-loop gauge theory[Bern, Czakon, Dixon, Kosower, Smirnov ‘06] Tristan’s talk tomorrow

Checks in 2-loop string theory

[Beisert, McLoughlin, Roiban ‘07]

Brief history of the dressing phase

Dilatation operator Hamiltonian of integrable spin-chain

Checks of Integrability in AdS/CFT

► Integrability of planar N=4 SYM theory[Minahan, Zarembo ‘02]

Spin chain picture at large N

[Beisert, Staudacher ‘03]

[Serban, Staudacher ‘04]

Algebratic Bethe ansatz at 1-loop

Inozemtsev spin chain up to 3-loops in SU(2) sector

Factorization of 3-impurity S-matrix in SL(2) sector [Eden, Staudacher ‘06]

[Minahan, Zarembo ‘02]

[Beisert, Kristjansen,Staudacher ‘03] Degeneracies in the spectrum at higher loops

Checks of Integrability in AdS/CFT

[Mandal, Suryanarayana, Wadia ‘02][Bena, Polchinski, Roiban ‘03]

Coset representative

generates conserved charges

► Classical Integrability of planar AdS string theory

Monodromy matrix

Current conserved

flat

Family of flat currents

Checks of Integrability in AdS/CFT

► Quantum Integrability of planar AdS string theory

Quantum consistency of monodromy matrix

Absence of particle production in bosonic sector in semiclassical limit

Quantum consistency of AdS strings, and existence of higher charges in pure spinor formulation [Berkovits ‘05]

[Callan, McLoughlin, Swanson ‘04]

[TK, McLoughlin, Roiban, Zarembo ‘06]

[Hentschel, Plefka, Sundin ‘07]

[Mikhailov, Schäfer-Nameki ‘07]

Check energies of multi-excitation states against Bethe equations (at tree-level)

Integrability in 1+1d QFTs

► No particle production or annihilation

► Conservation of the set of momenta

► -particle S-Matrix factorizes into 2-particle S-Matrices

Existence of local higher rank conserved charges

[Zamolodchikov, Zamolodchikov ‘79]

[Shankar, Witten ‘78]

[Parke ‘80]

Conservation laws and Scattering in 1+1 dimensions

► 2 particles

► 1 particle

► 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 tensorare sufficient for S-matrix factorization !”

Strings on AdS5xS5 (bosonic)

AdS5 x S5

[Metsaev, Tseytlin ‘98]

Strings on AdS5xS5 (bosonic)

► Fixing reparametization invariance in uniform lightcone gauge[Arutyunov, Frolov, Zamaklar ‘06]

worldsheet Hamiltonian density

eliminated byVirasoro constraints

► Back to Lagrangian formulation

Strings on AdS5xS5 (bosonic)

► Decompactification limit

rescale such that

send

no , no

loop counting parameter

world-sheet size

to define asymptotic states

Superstrings on AdS5xS5

Gauge fixing (L.C. + -symmetry)

Manifest symmetries

► Sigma model on[Metsaev, Tseytlin ‘98]

[Frolov, Plefka,Zamaklar ‘06]

Worldsheet S-Matrix

► 4 types of particles, 48=65536 Matrix elements

► Group factorization

► Each factor has manifest invariance

Symmetry constraints on the S-Matrix

of centrally extended algrebarelate the two irreps of

fixed up to one function

► 2-particle S-Matrix:

irrep of

[Beisert ‘06]

In infinite volume, the symmetry algebra gets centrally extended to

for one S-Matrix factor:

the total S-Matrix is:

Symmetry constraints on the S-Matrix

fixed up to four functions

► 3-particle S-matrix:

3-particle S-matrix

► Eigenstates

Extract coefficient functions from:

► Disconnected piece:

► Connected piece:

factorizes trivially,2-particle S-matrix checked to 2-loops

[TK, McLoughlin, Minahan, Zarembo ‘07]

factorization at 1-loop to be shown below !

near-flat-space

Near-flat-space limit

[Maldacena, Swanson ‘06]

giant magnons

plane-wave

[Hofman, Maldacena ‘06]

[Berenstein, Maldacena, Nastase ‘02]

Highly interacting

► Boost in the world-sheet theory:

Only quartic interactions !

Free massive theory

Near-flat-space limit

Non-Lorentz invariant interactions

Decoupling of right-movers

!

!

UV-finiteness

quantum mechanically consistent reductionat least to two-loops

!

! Tristan’s talk tomorrow

Propagators: bosons , fermions

Coupling constant:

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

for

compare

S-Matrix from Feynman diagrams

► 3-particle S-matrix

► 6-point amplitude

!?First non-triviality

Factorization

Factorization

!?Second non-triviality

YBE

Emergence of factorization

► Tree-level amplitudes

light-cone momenta

finite divergencies

sets the internal propagator on-shellcompare to sinh-Gordon:

Emergence of factorization

► Example

from Feynman diagrams

... agrees with the predicted factorized 3-particle S-Matrix

disconnected

Emergence of factorization

► 1-loop amplitudes

“dog”:

“sun”:

phase space

[Källén, Toll ‘64]

Cutting rule in 2d for arbitrary 1-loop diagrams

Applied to “sun-diagram”:

Emergence of factorization

2-loop 2-particleS-Matrix

Emergence of factorization

“dog structure” “sun structure”

► General 3-particle S-Matrix

► Contributions at order

works for symmetric processes like

fails for mixed processes like

cannot hold at higher loops, e.g.

The below identification...

Summary and open questions

1-loop computation of the highest-weight amplitudes, amplitude of mixed processes

Finite size corrections

Proven the factorization of the 3-particle world-sheet S-Matrixto 1-loop in near-flat AdS5xS5

!

?

Asymptotic states?

!

?Extenstions of the above: higher loops, more particles, full theory

fixes 3-particle S-matrix

checks supersymmetries

Direct check of quantum integrability of AdS string theory(albeit in the NFS limit)

!

effectively