1. Nikolay Gromov Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939 F. Levkovich-Maslyuk, G. Sizov 1305.1944 Nikolay Gromov Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939 F. Levkovich-Maslyuk, G. Sizov 1305.1944 IGST 2013 Utrecht, Netherlands

2. Historical overview

3. According to Beisert, Kazakov, Sakai and Zarembo, we can map a classical string motion to an 8-sheet Riemann surface Bohr-Sommerfeld quantization condition: Classical spectral curve

4. ABA – Large Length spectrum [Minahan, Zarembo; Beisert, Staudacher; Beisert, Hernandez, Lopez; Beisert, Eden, Staudacher]

5. [N.G., Kazakov, Vieira Bombardelli, Fioravanti, Tateo N.G., Kazakov, Kozak, Vieira Arutynov, Frolov] Thermodynamic Bethe Ansatz

6. Quantum spectral curve

7. The system reduced to 4+5 functions: Analytical continuation to the next sheet: Quadratic branch cuts: Definition of the quantum curve

8. Any T-function can be found Any Y-function can be found Y-functions automatically satisfy TBA equations! In particular: Relation to TBA encodes anomalous dimension Useful: [Cavaglia, Fioravanti, Tateo]

9. Global charges Asymptotic can be read off from the relation to the quasi-momenta General equation see Dima’s talk. For sl2 sector: Having fixed asymptotic for P what are the possible asymptotic for mu: We identify by comparing with TBA:

10. Example: Basso’s Slope function

11. The correct combination has an asymptotic Good for positive integer S, but is obviously symmetric S -> -1-S. So cannot give a singularity at S=-1 [Janik (1,2 loops); NG, Kazakov (1 loop)] Asymptotic There are two independent solutions of sl2 Baxter equation (see Janik’s talk): [Derkachov,Korchemsky,Manashov 2003]

12. Near BPS limit – can solve analytically: 1. 2. 3. Main simplification are smallDerivation Remember the algebraic constraint: [NG. Sizov, Valatka, Levkovich-Maslyuk in prog.] Allare trivial

13. The R-charges of the state are encoded in the asymptotics. For twist L=2: Angle and the energy are in the coefficients of the expansion [Basso]Solution

14. Example2: Slope-to-Slope

15. 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!

16. Tests Weak coupling we get: In agreement with: [Kotikov, Lipatov, Onishenko, Velizhanin] [Moch, Vermaseren, Vogt] [Staudacher] [Kotikov, Lipatov, Rej, Staudacher, Velizhanin] [Bajnok, Janik, Lukowski] [Lukowski, Rej, Velizhanin]

17. Tests Strong coupling we get (so far only numerically): Basso Folded string 0-loop Folded string 1-loop [NG. Valatka]

18. Example3: ABA

19. Our result contains an essential part of the dressing phase: Dressing phase, asymptotic limit These integrals appear in our result. In general we derive the ABA of Beisert-Eden-Staudacher in full generality from System in asymptotic limit when [NG., Kazakov, Leurent, Volin to appear] is an analog of Baxter equation from which ABA follows as an analyticity Condition.

20. Example3: Wilson line with cusp

21. 21 [Drukker, Forrini 2011]

22. Which is in fact log derivative of expectation value of a circular WL [Ericson, Semenoff, Zarembo 2000; Drukker, Gross 2000; Pestun 2010] For L=0 the result is known from localization: [Corea,Maldacena,Sever 2012]

23. Near BPS limit – can solve analytically: 1. 2. 3. Main simplification are small Allare trivial [NG, Sizov, Levkovich-Maslyuk 2013]

24. The R-charges of the state are encoded in the asymptotics Angle and the energy are in the coefficients of the expansion Is a polynomial of degree L Hilbert transform of the r.h.s. For L=0, is a constant and

26. Quantum Classical

27. [Valatka, Sizov 2013]

28. Wave function In separated variables Wave function In separated variables

29. Simple relation to the quasi-momenta: exactly like: Quantum Classical

30. In integrable models it is possible to make a canonical transformation so that the wave function is complitely factorized A natural conjecture that for AdS/CFT the wave function can be build in terms of Ps and fermonic Qs This construction is known explicitly in some cases [Sklyanin 1985; Smirnov 1998; Lukyanov 2000] Measure could be complicated

31. Conclusions/Open questions