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Konishi : from 4 loops to infinity

Nikolay Gromov Based on works with V.Kazakov , P.Vieira & A.Kozak. Konishi : from 4 loops to infinity. Symposium on Theoretical and Mathematical Physics July 3-8, 2009. AdS /CFT correspondence. AdS/CFT duality:. String tension. `t Hooft coupling.

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Konishi : from 4 loops to infinity

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  1. NikolayGromov Based on works with V.Kazakov, P.Vieira & A.Kozak Konishi: from 4 loops to infinity Symposium on Theoretical and Mathematical Physics July 3-8, 2009

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

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

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

  5. Integrability in 2D Operator corresponding to an integral of motion Between in and out states The outgoing momenta are constrained: The only solution:

  6. Factorization of S-matrix S S S S S Yang-Baxter equation S S p3 p3 p3 p2 p2 p2 p1 p1 p1

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

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

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

  10. AdS/CFT bound states Till Bargheer, NiklasBeisert, N. G. N.Dorey Takahashi Adopted for mirror theory Arutyunov, Frolov

  11. AdS/CFT Y-system N.G., Kazakov, Vieira …Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo… …Destri de Vega, Bytsko ,Teschner….

  12. Large L limit or weak coupling N.G., Kazakov, Vieira Use Hirota equation:

  13. Hirota – large L The eigevalues solve Hirota! for rep = S S S S

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

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

  16. Integral form of Y-system for exitations Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo Destri de Vega, Teschner Bombardelli, Fioravanti, Tateo N.G., Kazakov, Vieira Arutynov, Frolov

  17. Exact BAE • From mirror to physical Precision is excellent!!!

  18. Konishi – from weak to strong Fit: From string side: Roiban,Tseytlin

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

  20. Conclutions • 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

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