Nikolay Gromov. Based on

Transcription

1 Nikolay Gromov Based on N. G., V. Kazakov, S. Leurent, D. Volin , N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka A. Cavaglia, D. Fioravanti, N. G., R. Tateo N. G., G. Sizov M. Alfimov,N. G., V. Kazakov LPT ENS, /01/2015 Paris

2 Motivation Solve exactly at least the simplest gauge theory in 4D (i.e. N=4 SYM) Test AdS/CFT for no supersymmetric observables Approach real physics (QCD) from this side

3 Use methods of Integrability for D>2 field theories 2005: Summer Institute, ENS Paris 2006: int06 Workshop, AEI Potsdam 2007: 12th Claude Itzykson Meeting, Saclay/ENS Paris 2008: IGST08 Workshop, Universiteit Utrecht 2009: int09 Conference, AEI Potsdam 2010: IGST10 Conference, NORDITA Stockholm 2011: IGST11 Conference, Perimeter Institute for Theoretical Physics 2012: IGST12 Conference, ETH Zürich 2013: IGST13 Conference, Universiteit Utrecht 2014: IGST14 Conference, DESY Hamburg 2015: IGST15 Conference, KCL London (

4 Integrability in gauge theory Origins of YM integrability: Lipatov s BFKL Hamiltonian Perturbative integrability in N=4 SYM Classical integrability of string ϭ-model on AdS5 S5, quasiclassics Lipatov Faddeev,Korchemsky Brown, Derkachov, Manashov Minahan,Zarembo, Beisert,Kristijanssen,Staudacher Bena,Polchinski,Roiban Kazakov,Marshakov, Minahan, Zarembo, Frolov, Tseytlin Schafer-Nameki Beisert,Kazakov,Sakai,Zarembo NG,Vieira

5 N=4 SYM The simplest generalization of QCD: Parameters: and Plus extra scalar fields and fermions Symmetries: Lorentz: Conformal: - rotation of the scalars Super (graded) Lie algebra:

6 Planar limit Expansion in large N organized in 1/N 2 series. For real QCD 1/N 2 =1/9. Teper 2008 N=8 N=3

7 Dual description String in AdS 5 xs 5 x Symmetries: Paramiter: String tension:

8 AdS/CFT correspondence x String tension `t Hooft coupling String coupling Number of colors

9 Dictionary Weak coupling Strong coupling Effectively store the information in the nodes of the wave function Simplifies to the classical motion

10 Dictionary Weak coupling Strong coupling Perturbative YM Classical string x Schrödinger equation: String has infinitely many modes and several polarizations

11 Strong coupling current [Bena, Polchinski, Roiban] EOM equivalent to where on EOM Eigenvalues of the monodromy matrix: [Kazakov, Marshakov, Minahan, Zarembo] Analytic properties: [Beisert, Kazakov, Sakai, Zarembo] State-dependent cuts

12 From weak coupling [Brown, Derkachov, Manashov] [Beisert, Sctaudacher] In the simplest case Two solutions: polynomial singular solution Equivalent description: Easy to generalize SU(3): - polynomial - meromorphic - polynomial - polynomial - meromorphic

13 From weak coupling Two main ingredients: QQ-relations Analyticity - polynomial - meromorphic What are the analtic properties of the Q s at finite coupling?

14 Generalization to finite coupling [N.G., Kazakov, Leuren, Volin] Poles open into cuts Heisenberg, SYM Quantum Spectral Curve Classical string Need to know monodromies, when going under the cuts

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

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

17 Examples: near-bps expansion

18 Twist-2 operators Important class of single trace operators: Spectrum for different spins:

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

20 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!

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

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

23 BFKL regime

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

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

26 ABJM Theory

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

28 Spectral curve for ABJM Algebraically and interchanged their roles, but not analytically SYM: i-periodic ABJM: i-(anti)periodic Another important difference is the position of the branch points: SYM: ABJM: 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 [N.G., Sizov] Integrability: Elliptic type integral ABJM Matric model integral in its planar limit: [Pestun][Kapustin, Willett, Yaakov] [Marino, Putrov] Localization: Comparing cross-ratios of the branch points:

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

31 Other recent applications

32 Recent applications 10 loops (!) Konishi operator at weak coupling: [C. Marboe, D. Volin Nov 2014] [A. Cavaglià, M. Cornagliotto, M. Mattelliano, R. Tateo 2 days ago] Quantum Spectral Curve for Hubbard model

33 Future directions

34 Wave function Wave Structure function? Sklyanin tells us that

35 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? [N.G., Kazakov, Leuren, Volin] 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?

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