1. Nikolay Gromov Based on N. G., V. Kazakov, S. Leurent, D. Volin 1305.1939, 1405.4857 N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871 A. Cavaglia, D. Fioravanti, N. G., R. Tateo 1403.1859 N. G., G. Sizov 1403.1894 M. Alfimov,N. G., V. Kazakov appeared recently Nikolay Gromov Based on N. G., V. Kazakov, S. Leurent, D. Volin 1305.1939, 1405.4857 N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871 A. Cavaglia, D. Fioravanti, N. G., R. Tateo 1403.1859 N. G., G. Sizov 1403.1894 M. Alfimov,N. G., V. Kazakov appeared recently Ahrenshoop conference, 2014 29/08/2014

2. Perturbative integrability in N=4 SYM 2002 Minahan,Zarembo, Beisert,Kristijanssen,Staudacher 2003-2008 Bena,Polchinski,Roiban Kazakov,Marshakov, Minahan, Zarembo, Frolov, Tseytlin Schafer-Nameki Beisert,Kazakov,Sakai,Zarembo NG,Vieira Classical integrability of string ϭ -model on AdS5×S5, quasiclassics 2004-2006 Arutyunov,Frolov,Staudacher Staudacher, Beisert Janik Hernandez,Lopez Roiban, Tseytlin Beisert,Eden,Staudacher S-matrix, asymptotic Bethe Ansatz. Cusp dimension 2005-2008 Ambjorn,Janik,Kristijanssen Arutyunov,Frolov Bajnok,Janik, Lukowski Finite size corrections and mirror model 2009-2010 NG, Kazakov, Vieira Bombardelli,Fioravanti, Tateo NG, Kazakov,Vieira Arutyunov, Frolov Covagia,Fioravanti,N.G.,Tateo Analytic Y- system for exact spectrum, TBA 2011-2014 NG, Kazakov, Leurent, Volin Quantum spectral curve for all operators Origins of YM integrability: Lipatov’s BFKL Hamiltonian Lipatov Faddeev,Korchemsky 1993 Origins of YM integrability: Lipatov’s BFKL Hamiltonian Lipatov Faddeev,Korchemsky Alfimov,N.G.,Kazakov to appear 1993 Plan: Review of the QSC construction Examples Applications to ABJM Integrability in gauge theory

3. EOM equivalent to where on EOM current State-dependent cuts Eigenvalues of the monodromy matrix: Analytic properties: Motivation from classics [Bena, Polchinski, Roiban] [Dorey, Vicedo] [Kazakov, Marshakov, Minahan, Zarembo] [Beisert, Kazakov, Sakai, Zarembo]

4. Can be mapped to a spin chain state: The one-loop dilatation operator coincides with Heisenberg spin chain Hamiltonian Sklyanin separation of variables allows to factorize the wave function where In the simplest case Two solutions: polynomialsingular solution From weak coupling [Beisert, Sctaudacher]

5. Beisert, Staudacher; Beisert, Hernandez, Lopez; Beisert,Eden,Staudacher

6. I.e. from the asymptotical spectrum (infinite R) we can compute the Ground state energy for ANY finite volume! …,Matsubara, Zamolodchikov,…

7. Bombardelli, Fioravanti, Tateo N.G., Kazakov, Vieira Arutynov, Frolov

8. N.G., Kazakov, Vieira ‘09 Numerics

9. Is it the final answer to the spectral problem???

10. N.G., Kazakov, Vieira + discontinuity relations Cavaglià, Fioravanti, Tateo

11. 1) We start exploring all DOS of the string 2) Poles open into cuts Heisenberg, SYM Classical string Quantum Spectral Curve 3) Need to know monodromies, when going under the cuts Generalization to finite coupling [N.G., Kazakov, Leuren, Volin]

12. Charges in S 5 are integer Charges in AdS 5 contain anom.dimension “Miraculous” simplification [N.G., Kazakov, Leuren, Volin]

13. The system reduced to 4+6 functions: Analytical continuation to the next sheet: Quadratic branch cuts: -system is a closed system of equtions! - system - system [N.G., Kazakov, Leuren, Volin]

14. Asymptotics – global charges R-charges Lorentz, Conformal charges Similarly to the classical curve

15. 1) Find ‘‘rotation” matrix (from a 4 th order FDE): [Alfimov, N.G., Kazakov] -system ↔ -system -system ↔ -system [N.G., Kazakov, Leuren, Volin] Before: 2) Find and Short-cut: Solve 4 th order FDE: where:

16. Examples: near-BPS expansion

17. Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06] Important class of single trace operators: Twist-2 operators

18. In the BPS limit:entire periodic function For in the small S limit: Solution: - simple Riemann-Hilbert problem Near BPS limit: small S Result: Similar to the localization results! [NG. Sizov, Valatka, Levkovich-Maslyuk] [Basso] [Zarembo; Pestun]

19. Small P’s imply small discontinuity of mu: Next order Small P’s imply small discontinuity of mu: Result: [NG. Sizov, Valatka, Levkovich-Maslyuk in prog.] Dressing phase!

20. Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06] Twist-2 operators

21. Extrapolating results to finite spin Gromov, Serban, Shenderovich, Volin`11; Roiban, Tseytlin`11; Vallilo, Mazzucato`11 Plefka, Frolov`13 Gubser, Klebanov, Polyakov `98 Not hard to iterate the procedure and go further away from BPS. Kotikov, Lipatov`13 Costa, Goncalves, Penedones` 12 We also extract pomeron intercept: More orders in small S Gubser, Klebanov, Polyakov `98 [Basso][NG. Sizov, Valatka, Levkovich-Maslyuk]

22. BFKL regime

23. Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06] Important class of single trace operators: BFKL regime BFKL regime: So that: Resumming to all loops terms In this regime SYM is undistinguishable from the real QCD [Kotikov, Lipatov, Rej, Staudacher, Velizhanin]

24. Small coupling no branch cuts The problem is essentially about gluons, i.e. it is more natural to pass to AdS S= -1 is when for the first time this ansatz is consistent for non-integer ∆ Can be solved explicitly Enters into the Q-function of Lipatov, de Vega; Korchemsky, Faddeev! BFKL limit of -system [Alfimov, N.G., Kazakov to appear] [Kotikov, Lipatov] Plugging it into - system we get:

25. It is not hard to get more orders NLO Q-function [Alfimov, N.G., Kazakov to appear] where This equation can still be solved explicitly

26. ABJM Theory

27. Spectral curve for ABJM SYM ABJM PSU(2,2|4) OSP(2,2|6) define Discontinuities Constrains [A. Cavaglia, D. Fioravanti, N. G., R. Tateo]

28. Spectral curve for ABJM Algebraically and interchanged their roles, but not analytically Another important difference is the position of the branch points: SYM: ABJM: i-(anti)periodicABJM: i-periodic SYM: enters into many important quantities: cusp dimension, magnon dispertion

29. Finding Interpolation function h In the near BPS limit we should be able to match with localization Integrability: Elliptic type integral ABJM Matric model integral in its planar limit: Localization: Comparing cross-ratios of the branch points: [N.G., Sizov] [Pestun][Kapustin, Willett, Yaakov] [Marino, Putrov]

30. Interpolation function h Minahan, Ohlsson Sax, Sieg & Leoni, Mauri, Minahan, Ohlsson Sax, Santambrogio, Sieg, Tartaglino- Mazzucchelli, Minahan, Zarembo ? McLoughlin, Roiban Tseytlin Abbott, Aniceto, Bombardelli Lopez-Arcos, Nastase Bergman, Hirano L.Bianchia, M. Bianchia, Bresa, Forinia,Vescovia Reproduces ~4 nontrivial coefficients!

31. Conclusions QSC unifies all integrable structures: BFKL/ local operators, classical strings/spin chains. Beisert-Eden- Staudascher equations can be derived from it in complete generality. Mysterious relation between ABJM and N=4 SYM integrable structures. Sign for an unifying theory? What is QSC for AdS 3 ? Q-functions should give a way to the exact wave function in separated variables. Can we use it to compute general 3-point correlation functions to all loops? Established links between exact results in integrability and localization. Does there exist a unified structure which works for both non-BPS and non-planar? Discretization of Zhukovsky cut? [N.G., Kazakov, Leuren, Volin]