Integrability for the Full Spectrum of Planar AdS/CFT

Nikolay Gromov

PNPI/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 Field content: The action:

YM: Local operators and spin chains - Dilatation operator – integrable Hamiltonian

The spectrum Ground state: Excited states (magnons):

©Zarembo scattering phase shifts momentum periodicity condition: periodicity of wave function

YM: One-loop - Integrable Hamiltonian 1) Solve polynomial equation

Lepatov Faddeev, Korchemsky Minahan , Zarembo

2) Get eigenvalues 7

Numerical Solution

Till Bargheer, Niklas Beisert, N. G.

8

String theory

AdS/CFT Duality (type IIB super ) string theory in AdS 5 xS 5

Maldacena

is dual to a 4 dimensional conformal field theory (N=4 SYM) x Local operators String states Anomalous dimensions Spectrum

AdS/CFT Duality String tension String coupling x `t Hooft coupling Number of colors Summetry:

Bethe equations

Beisert, Staudacher; Beisert,Eden,Staudacher

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 x2 Faddeev, Reshetikhin

Ground state from ABA  Saddle point equation:

Above ground state

Dorey, Totteo, Bazhanov

AdS/CFT Generalization

N.G., Kazakov, Vieira

Large L limit Use Hirota equation:

S-matrix SU(2|2) invariant tensor with 4 fundamental indeses Obeys Yang-Baxter : Then (see lectures of Faddeev

hep-th/9605187

) : The eigevalues solves hirota!

19

Konishi operator The simplest operator For weak coupling constant: In agreement with perturbation theory!!

4-loops!

Kotikov, Lipatov, Rej, Staudacher and Velizhanin Sieg, Torrielli

Janik, Bojnok, N.G., Kazakov, Vieira

CP3xAdS4 / N=6 Chern Simons Aharony, Bergman, Jafferis and Maldacena

N.G., Kazakov, Vieira

Conclusions     We can go below BAE now Integrability allows to predict very complicated perturbative calculations It is possible to compute some quantities for arbitrary coupling QCD BFKL could be checked

Advertisement To apply use www.fc.up.pt/mathematica Mainly for master students