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Quantum algebraic curve For N=4 SYM

Nikolay Gromov Based on works with V.Kazakov , S.Leurent , D.Volin 1305.1939 F. Levkovich-Maslyuk , G. Sizov 1305.1944. Quantum algebraic curve For N=4 SYM. IGST 2013 Utrecht, Netherlands. Historical overview. Classical spectral curve.

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Quantum algebraic curve For N=4 SYM

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  1. NikolayGromov Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939 F. Levkovich-Maslyuk, G. Sizov 1305.1944 Quantum algebraic curve For N=4 SYM IGST 2013 Utrecht, Netherlands

  2. Historical overview

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

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

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

  6. Quantum spectral curve

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

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

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

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

  13. Solution 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]

  14. Example2: Slope-to-Slope

  15. Next order Small P’s imply small discontinuity of mu: 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 Basso Folded string 0-loop Folded string 1-loop [NG. Valatka]

  18. Example3: ABA

  19. Dressing phase, asymptotic limit Our result contains an essential part of the dressing phase: 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 is an analog of Baxter equation from which ABA follows as an analyticity Condition. [NG., Kazakov, Leurent, Volin to appear]

  20. Example3: Wilson line with cusp

  21. [Drukker, Forrini 2011]

  22. 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; Pestun 2010]

  23. Near BPS limit – can solve analytically: 1. 2. 3. [NG, Sizov, Levkovich-Maslyuk2013] All are trivial Main simplification are small

  24. 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

  25. Classical Quantum

  26. [Valatka, Sizov 2013]

  27. Wave function In separated variables

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

  29. In integrable models it is possible to make a canonical transformation so that the wavefunction 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 Measure could be complicated

  30. Conclusions/Open questions • Too soon to draw any conclusions • New exact and simple formulation for all operators • There should be a relation to the wave function in the separatedSkylaninvariables • Could a similar set of equations be found for correlation functions? Relation to Thermodynamic Bubble Ansatz? • Solve Pin different regimes – strong coupling systematic expansion, BFKL… • More observables can be studied (available on the market already see ZoltanBajnok talk) • Systematic (diagrammatic?) all loop expansion around BPS

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