QCD Pomeron from ADS/CFT Quantum spectral curve NikolayGromov Based onN. G., V. Kazakov, S. Leurent, D. Volin1305.1939 , 1405.4857 N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871 A. Cavaglia, D. Fioravanti, N. G., R. Tateo1403.1859 N. G., G. Sizov 1403.1894 M. Alfimov,N. G., V. Kazakovappeared last week Strings, Matrices, Integrability, 2014 18/08/2014 Paris
Integrability in gauge theory Plan: Review of the QSC construction Origins of YM integrability: Lipatov’s BFKL Hamiltonian Perturbativeintegrability in N=4 SYM Classical integrability of string ϭ-model on AdS5×S5, quasiclassics S-matrix, asymptotic Bethe Ansatz. Cusp dimension Finite size corrections and mirror model Analytic Y-system for exact spectrum, TBA Quantum spectral curve for all operators Origins of YM integrability: Lipatov’s BFKL Hamiltonian Examples: 1) near BPS 2) BFKL limit 1993 2002 2003-2008 2004-2006 2005-2008 2009-2010 2011-2014 1993 Applications to ABJM Bena,Polchinski,Roiban Kazakov,Marshakov, Minahan, Zarembo, Frolov, Tseytlin Schafer-Nameki Beisert,Kazakov,Sakai,Zarembo NG,Vieira Arutyunov,Frolov,Staudacher Staudacher, Beisert Janik Hernandez,Lopez Roiban, Tseytlin Beisert,Eden,Staudacher Ambjorn,Janik,Kristijanssen Arutyunov,Frolov Bajnok,Janik, Lukowski NG , Kazakov, Vieira Bombardelli,Fioravanti, Tateo NG , Kazakov,Vieira Arutyunov, Frolov Covagia,Fioravanti,N.G.,Tateo Minahan,Zarembo, Beisert,Kristijanssen,Staudacher Lipatov Faddeev,Korchemsky Lipatov Faddeev,Korchemsky Alfimov,N.G.,Kazakov to appear NG, Kazakov, Leurent, Volin
Motivation from classics [Bena, Polchinski, Roiban] current where EOM equivalent to on EOM Eigenvalues of the monodromy matrix: [Kazakov, Marshakov, Minahan, Zarembo] [Beisert, Kazakov, Sakai, Zarembo] Analytic properties: State-dependent cuts [Dorey, Vicedo]
From weak coupling [Beisert, Sctaudacher] 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: polynomial singular solution
Beisert, Staudacher; Beisert, Hernandez, Lopez; Beisert,Eden,Staudacher
…,Matsubara, Zamolodchikov,… I.e. from the asymptotical spectrum (infinite R) we can compute the Ground state energy for ANY finite volume!
Bombardelli, Fioravanti, Tateo N.G., Kazakov, Vieira Arutynov, Frolov
N.G., Kazakov, Vieira ‘09 Numerics
N.G., Kazakov, Vieira + discontinuity relations Cavaglià, Fioravanti, Tateo
Generalization to finite coupling [N.G., Kazakov, Leuren, Volin] 1) We start exploring all DOS of the string 2) Poles open into cuts Classical string Heisenberg, SYM Quantum Spectral Curve 3) Need to know monodromies, when going under the cuts
“Miraculous” simplification [N.G., Kazakov, Leuren, Volin] Charges in S5 are integer Charges in AdS5 contain anom.dimension
- 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:
-system ↔ -system [N.G., Kazakov, Leuren, Volin] Before: 1) Find ‘‘rotation” matrix (from a 4th order FDE): 2) Find and Short-cut: Solve 4th order FDE: [Alfimov, N.G., Kazakov] where:
Twist-2 operators Important class of single trace operators: Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06]
Near BPS limit: small S [NG. Sizov, Valatka, Levkovich-Maslyuk] In the BPS limit: entire periodic function For in the small S limit: - simple Riemann-Hilbert problem Solution: [Basso] [Zarembo; Pestun] Result: Similar to the localization results!
Next order Result: [NG. Sizov, Valatka, Levkovich-Maslyuk] Similar to the dressing phase
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 Gubser, Klebanov, Polyakov `98 Kotikov, Lipatov`13 Gromov, Serban, Shenderovich, Volin`11; Roiban, Tseytlin`11; Vallilo, Mazzucato`11 Plefka, Frolov`13 Costa, Goncalves, Penedones` 12 We also extract pomeron intercept: Extrapolating results to finite spin
BFKL regime Important class of single trace operators: Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06] [Kotikov, Lipatov, Rej, Staudacher, Velizhanin] BFKL regime: So that: Resumming to all loops terms In this regime SYM is undistinguishable from the real QCD
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!
NLO Q-function It is not hard to get more orders [Alfimov, N.G., Kazakov to appear] where This equation can still be solved explicitly
Generalization to more general states For twist L states we expect [N.G., Sizov in preparation] Another All-loop Bethe ansatz All-loop BES Bethe ansatz All-loop BES ansatzresums L-loops in the perturbation theory. Features: Zhukovsky variable, BES dressing phase Similarly in the BFKL regime We can resum L/2-loops and get BFKL-ABA with its own dressing phase.
Spectral curve for ABJM [A. Cavaglia , D. Fioravanti, N. G., R. Tateo] SYM ABJM PSU(2,2|4) OSP(2,2|6) Constrains define Discontinuities
Spectral curve for ABJM Algebraically and interchanged their roles, but not analytically i-periodic SYM: ABJM: i-(anti)periodic Another important difference is the position of the branch points: ABJM: SYM: enters into many important quantities: cusp dimension, magnondispertion
Finding Interpolation function h In the near BPS limit we should be able to match with localization [N.G., Sizov] Integrability: Elliptic type integral [Pestun][Kapustin, Willett, Yaakov] [Marino, Putrov] ABJM Matric model integral in its planar limit: Localization: Comparing cross-ratios of the branch points:
Interpolation function h Minahan, Ohlsson Sax, Sieg & Leoni, Mauri, Minahan, Ohlsson Sax, Santambrogio, Sieg, Tartaglino- Mazzucchelli, Minahan, Zarembo Bergman, Hirano L.Bianchia, M. Bianchia, Bresa, Forinia,Vescovia McLoughlin, RoibanTseytlin Abbott, Aniceto, Bombardelli Lopez-Arcos, Nastase ? Reproduces ~4 nontrivial coefficients!
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 AdS3? • 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]