1. Nikolay Gromov Based on works with V.Kazakov, P.Vieira & A.Kozak Nikolay Gromov Based on works with V.Kazakov, P.Vieira & A.Kozak Symposium on Theoretical and Mathematical Physics July 3-8, 2009 Symposium on Theoretical and Mathematical Physics July 3-8, 2009

2. AdS/CFT duality: Anomalous dimensions = spectrum of 2D integrable field theories String tension`t Hooft coupling Symmetry:

3. - Mixing matrix – integrable Hamiltonian The problem of diagonalization of the huge matrix can be solved by Bethe ansatz [Minahan, Zarembo 2002&2008] N=4 Super Yang-Mills:

4. Infinitely many Integrals of motion: Motion of the strin in AdS:

5. Operator corresponding to an integral of motion Between in and out states The outgoing momenta are constrained: The only solution:

6. S p1 p2 p3 p1 p2 p3 S S S p1 p2 p3 S S S Yang-Baxter equation

7. For spectral density we need finite volume From periodicity of the wave function -Bethe equations

8. Beisert, Staudacher; Beisert, Hernandez, Lopez; Beisert,Eden,Staudacher Arutyunov, Frolov

9. I.e. from the asymptotical spectrum (infinite R) we can compute the Ground state energy for ANY finite volume! …,Matsubara, Zamolodchikov,… In our case the wick rotated theory is different – ”mirror” theory Ambjorn, Janik, Kristjansen; Arutynov, Frolov

10. N.Dorey Takahashi Arutyunov, Frolov Adopted for mirror theory Till Bargheer, Niklas Beisert, N. G.

11. N.G., Kazakov, Vieira …Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo… …Destri de Vega, Bytsko,Teschner….

12. Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo Inspiration #1: Analytical continuation Inspiration #2: Integrable lattice models Destri de Vega, Teschner

13. Use Hirota equation: N.G., Kazakov, Vieira

14. The eigevalues solve Hirota! for rep = S SSS

15. Instead of computing the transfer matrices explicitly one can use: Beisert Arutyunov, de Leeuw, Suzuki, Torrielli Checked by explicit computation for some T’s By a short code we can generate any Tas Bazhanov, Reshitikhin

16. The simplest operator In agreement with perturbation theory!! 4-loops! Kotikov, Lipatov, Rej, Staudacher and Velizhanin Sieg, Torrielli; Janik, Bojnok,; N.G., Kazakov, Vieira Bianchi, Kovacs,Rossi, Stanev; Eden, Jarczak, Sokatchev,Stanev…

17. Bombardelli, Fioravanti, Tateo N.G., Kazakov, Vieira Arutynov, Frolov Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo Destri de Vega, Teschner

18. From mirror to physical Precision is excellent!!!

19. Fit: From string side: Roiban,Tseytlin

20. 20 computers 4 nights Emergence of integers – a miracle Exact BAE – modified by phase Porto university Wolfram

22. Y-system is proven to be a computational tool We reproduced several leading terms at strong coupling for a simple operator Any sl2 operator could be computed readily for any g Generalization to other sectors to be done Analytical results – coming soon? Other AdS/CFT’s? ABJM N.G., Kazakov, Vieira