Integrability for the Full Spectrum of Planar AdS/CFT

Nikolay GromovPNPI/DESY/HU

V.Kazakov and P.Vieira

Motivation

Spectrum of an interacting field theory is a funny problem by itself

Some quantities are shared with realistic QCD

We can test string/gauge duality

N=4 Supersymmetric Yang-Mills Theory

The action:

Field content:

YM: Local operators and spin chains

- Dilatation operator – integrable Hamiltonian

The spectrumGround state:

Excited states (magnons):

periodicity condition:

momentumscattering phase shifts

periodicity of wave function

©Zarembo

7

2) Get eigenvalues

1) Solve polynomial equation

YM: One-loop

- Integrable Hamiltonian

LepatovFaddeev, KorchemskyMinahan , Zarembo

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Numerical Solution

Till Bargheer, Niklas Beisert, N. G.

String theory

(type IIB super) string theory in AdS5xS5 x

is dual to a 4 dimensional conformal field theory (N=4 SYM)

Local operators String states

Maldacena

AdS/CFT Duality

Anomalous dimensions Spectrum

x

String tension `t Hooft coupling

String coupling Number of colors

AdS/CFT Duality

Summetry:

Beisert, Staudacher;Beisert,Eden,Staudacher

Bethe equations

Vacuum

I.e. from the asymptotical spectrum (R=\infty) we can compute the Ground state energy for ANY finite volume!

SO(4)

Symmetry:

Анзац Бетэ:Zamolodchikov x2Faddeev, Reshetikhin

Ground state from ABAThe typical configuration of roots

isWe define:

Saddle point equation:

Above ground stateDorey, Totteo,Bazhanov

AdS/CFT GeneralizationN.G., Kazakov, Vieira

Large L limit

Use Hirota equation:

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S-matrix

Obeys Yang-Baxter:

Then (see lectures of Faddeev hep-th/9605187 ):

SU(2|2) invariant tensor with 4 fundamental indeses

The eigevalues solves hirota!

Konishi operator

For weak coupling constant:

The simplest operator

In agreement with perturbation theory!!4-loops!

Kotikov, Lipatov, Rej, Staudacher and VelizhaninSieg, TorrielliJanik, Bojnok, N.G., Kazakov, Vieira

CP3xAdS4 / N=6 Chern-Simons

N.G., Kazakov, Vieira

Aharony, Bergman, Jafferis and Maldacena

Conclusions

We can go below BAE nowIntegrability allows to predict

very complicated perturbative calculations

It is possible to compute some quantities for arbitrary coupling

QCD BFKL could be checked

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To apply use www.fc.up.pt/mathematicaMainly for master students