1. Integrability for the Full Spectrum of Planar AdS/CFT Nikolay Gromov DESY/HU/PNPI V.Kazakov and P.Vieira

2. 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

3. N=4 Supersymmetric Yang-Mills Theory The action: Field content:

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

5. The spectrum Ground state: Excited states (magnons):

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

7. 7 2) Get eigenvalues 1) Solve polynomial equation YM: One-loop - Integrable Hamiltonian Lepatov Faddeev, Korchemsky Minahan, Zarembo

8. 8 Numerical Solution Till Bargheer, Niklas Beisert, N. G.

9. String theory

10. (type IIB super ) string theory in AdS 5 xS 5 x is dual to a 4 dimensional conformal field theory (N=4 SYM) Local operators String states Maldacena AdS/CFT Duality Anomalous dimensions Spectrum

11. x String tension`t Hooft coupling String couplingNumber of colors AdS/CFT Duality Summetry:

12. Beisert, Staudacher; Beisert,Eden,Staudacher Bethe equations

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

14. SO(4) Symmetry: Анзац Бетэ: Zamolodchikov x2 Faddeev, Reshetikhin

15. Ground state from ABA The typical configuration of roots is We define: Saddle point equation:

16. Above ground state Dorey, Totteo, Bazhanov

17. Spectrum of SO(4)

18. AdS/CFT Generalization N.G., Kazakov, Vieira

19. Large L limit Use Hirota equation:

20. 20 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!

21. Konishi operator For weak coupling constant: The simplest operator In agreement with perturbation theory!! 4-loops! Kotikov, Lipatov, Rej, Staudacher and Velizhanin Sieg, Torrielli Janik, Bojnok, N.G., Kazakov, Vieira

22. CP3xAdS4 / N=6 Chern- Simons N.G., Kazakov, Vieira Aharony, Bergman, Jafferis and Maldacena

23. 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