Nikolay Gromov. Based on works with V.Kazakov, S.Leurent, D.Volin F. Levkovich-Maslyuk, G. Sizov

Transcription

1 Nikolay Gromov Based on works with V.Kazakov, S.Leurent, D.Volin F. Levkovich-Maslyuk, G. Sizov Gauge/Gravity duality July 29- August 2, 2013 Munich, Germany

2 Motion of the string: The scalar fields are constrained

3 on equations of motion Eigenvalues = integrals of motion Bena, Polchinski, Roiban; Kazakov, Marshakov, Minahan, Zarembo;

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

5 For the simplest 1D system we can define quasi-momentum: Schrodinger equation becomes for oscillator we have: In the quasi-classical limit goes to zero: Quantum Classical

6 Thermodynamic Bethe Ansatz give exact solution for the spectral problem: Bombardelli, Fioravanti, Tateo N.G., Kazakov, Vieira Arutynov, Frolov

7 N.G., Kazakov, Vieira 09 Numerics Gubser, Klebanov, Polyakov `98 Gromov, Serban, Shenderovich, Volin`11; Roiban, Tseytlin`11; Vallilo, Mazzucato`11 String theory N.G., Valatka`11 Agrees with weak coupling gauge theory up to 5 loop! Fieamberti, Fantambrogio, Sieg, Zanon `08 Eden, Heslop, Korchemsky, Smirnov, Sokatchev `12 Bajnok, Janik, Lukowski `08 Bajnok, Hegedus, Janik, Lukowski `09 Arutyunov, Frolov, Suzuki `10

8 Quantum spectral curve

9 The system reduced to 4+5 functions: Analytical continuation to the next sheet: Quadratic branch cuts:

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

11 Example: Wilson line with cusp

12 12

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

14 Near BPS limit can solve analytically: Main simplification are small All are trivial

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

16

17 [NG, Levkovich-Maslyuk, Sizov 2013]

18 Quantum Classical

19

20 [Valatka, Sizov 2013]

21 Wave function In separated variables

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

23 9-loops from system [Volin, last night] [Gromov, Kazakov, Vieira`09] 4 loops (first test of Y-system) [Arutyunov, Frolov,Suzuki 09] 5 loops numerically [Balog, Hegedus` 10] 5 loops analytically [Fiamberti, Santambrogio, Sieg, Zanon,, 08] - 4 loops [Eden, Heslop, Korchemsky, Smirnov, Sokatchev `12] - 5/6 loops [Bajnok,Janik 08] 4 loops [Bajnok,Janik,Lukowski 08] 4 loops, arbitrary S [Bajnok, Hegedus, Janik, Lukowski 09] 5 loops [Lukowski, Rej, Velizhanin 09] 5 loops, arbitrary S [Bajnok, Janik 12] 6 and 7 loops [Leurent, Serban, D.V 12] 6 loops [Leurent, D.V. 13] 8 loops

24 Conclusions Can iterate and compute more orders in the near BPS expansion Solve P! in different regimes strong coupling systematic expansion, BFKL More observables can be studied (available on the market already) Full string theory as a matrix integral (multimatrix integral)? Skylanin variables? Correlation functions? Relation to Bubble ansatz

25 BFKL

26 At weak coupling (one-loop) the anomalous dimension for any integer S is given by For J=2 one can solve it explicitly: Easy to guess the general form: Has a simple pole at S=-1 as predicted by BFKL

27 For us important question is what is the analytical continuation of Q Good to positive integer S, but is obviously symmetric S -> -1-S. So cannot give a singularity at S=-1 [Janik 2013; NG, Kazakov 2013] The correct combination has an asymptotic