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A window into 4D integrability: the exact spectrum of N = 4 SYM from Y-system

“Great Lakes Strings” Conference 2011 Chicago University, April 29. A window into 4D integrability: the exact spectrum of N = 4 SYM from Y-system. Vladimir Kazakov (ENS,Paris). Integrability in AdS /CFT.

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A window into 4D integrability: the exact spectrum of N = 4 SYM from Y-system

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  1. “Great Lakes Strings” Conference 2011 Chicago University, April 29 A window into 4D integrability: the exact spectrum of N = 4 SYM from Y-system Vladimir Kazakov (ENS,Paris)

  2. Integrability in AdS/CFT Conjecture: itcalculatesexact anomalousdimensions of all local operatorsof the gauge theoryatanycoupling Gromov,V.K.,Vieira • Further simplification: Y-system asHirota discrete integrable dynamics • Integrable planar superconformal 4D N=4 SYM and 3D N=8 Chern-Simons... (non-BPS, summing genuine 4D Feynman diagrams!) • Based on AdS/CFT duality to very special 2D superstring ϭ-models on AdS-background • Most of 2D integrability tools applicable: S-matrix, TBA for finite volume spectrum, etc. • .... Y-system (for planar AdS5/CFT4 , AdS4/CFT3 ,...)

  3. N=4 SYM as a superconformal 4D QFT • Operators in 4D • 4D Correlators (superconformal!): non-trivial functions of ‘tHooft coupling λ!

  4. SYM perturbation and (1+1)D S-matrix Minahan, Zarembo Krisijansen,Beisert,Staudacher Staudacher • Feynman graphs and asymptotic scattering of “defects” on 1D “spin chain” On the string side... psu(2,2|4) p2 p1 • Light cone gauge breaks the global and world-sheet Lorentz symmetries : su(2|2) su(2|2) Shastry’s R-matrix of Hubbard model • S-matrix of AdS/CFT via bootstrap à-la A.&Al.Zamolodchikov ŜPSU(2,2|4)(p1,p2) = S02(p1,p2) × ŜSU(2|2) (p1,p2)×ŜSU(2|2) (p1,p2) Beisert Janik

  5. Asymptotic Bethe Ansatz (ABA) Beisert,Eden,Staudacher pj p1 pM • This periodicity condition isdiagonalized by nested Bethe ansatz • Energy of state finite size corrections, important for short operators! • Results: ABA for dimensions of long YM operators (e.g., cusp dimension).

  6. Finite size (wrapping) effects • Wrapped graphs : beyond S-matrix theory • We need to take into account finite size effects - Y-system needed

  7. TBA for finite size (Al.Zamolodchikov trick) Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov ϭ-model in cross channel on large circle R world sheet ϭ-model in physical channel on small space circle L • Large R : cross channel momenta localize • on poles of S-matrix → bound states

  8. Dispersion relation Santambrogio,Zanon Beisert,Dippel,Staudacher N.Dorey • Exact one particle dispersion relation at infinite volume • Bound states (fusion) • Parametrizationfor dispersion relation: via Zhukovskymap: cuts in complex u -plane

  9. Y-system for excited states of AdS/CFT atfinite size Gromov,V.K.,Vieira T-hook • Complicated analyticity structure in u • dictated by non-relativistic dispersion • Extra equation (remnant of classical Z4monodromy): • Energy : • (anomalous dimension) • obey the exact Bethe eq.: cuts in complex -plane

  10. Konishioperator: numericsfromY-system Beisert, Eden,Staudacher ABA Gubser,Klebanov,Polyakov Gubser Klebanov Polyakov =2! From quasiclassics Y-system numerics Gromov,V.K.,Vieira Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio 5 loops and BFKL from string Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Bajnok,Janik Gromov,V.K.,Vieira Bajnok,Janik,Lukowski Lukowski,Rej,Velizhanin,Orlova millions of 4D Feynman graphs! • Y-system passes all known tests

  11. Y-system looks very “simple” and universal! • Similar systems of equations in all known integrable σ-models • What are its origins? Could we guess it without TBA?

  12. Y-systems for other σ-models 3d ABJM model: CP3 x AdS4, … Gromov,V.K.,Vieira Bombardelli,Fiorvanti,Tateo Gromov,Levkovich-Maslyuk

  13. Y-system and Hirota eq.: discrete integrable dynamics • Relation of Y-system to T-system (Hirota equation) • (the Master Equation of Integrability!) Hirota eq. in T-hook for AdS/CFT Gromov, V.K., Vieira Discrete classical integrable dynamics!

  14. (Super-)group theoretical origins • A curious property of gl(N|M)representations with rectangular Young tableaux: a-1 = + × × a × a-1 s s s-1 s+1 • For characters – simplified Hirota eq.: • Boundary conditions for Hirota eq.: • ∞ - dim. unitary highest weight representations of u(2,2|4)in “T-hook” ! a U(2,2|4) Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi s • Solution of Hirota for any irrep: Jacobi-Trudi formula for GL(K|M)characters:

  15. Character solution of T-hook for u(2,2|4) Gromov,V.K.,Tsuboi • Solution in finite 2×2 and 4×4 determinants • (analogue of the 1-st Weyl formula) • Generalization to full T-system with spectral parameter: • Wronskian determinant solution. • Should help to reduce AdS/CFT system to a finite system of equations. Hegedus Gromov,Tsuboi,V.K.

  16. Quasiclassical solution of AdS/CFT Y-system • Classical limit: highly excited long strings/operators, strong coupling: • Explicit u-shift in Hirota eq. dropped (only slow parametric dependence) world sheet Gromov,V.K.,Tsuboi • (Quasi)classical solution - psu(2,2|4) character of classical monodromymatrix • in Metsaev-Tseytlin superstring sigma-model Zakharov,Mikhailov Bena,Roiban,Polchinski • Its eigenvalues (quasimomenta) encode conservation lows • Finite gap method renders all classical solutions! V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo

  17. From classical to quantum Hirota in U(2,2|4) T-hook Gromov, V.K., Tsuboi • Quantization: replace classical spectral function by a spectral functional • More explicitly: - expansion in • The solution for any T-function is then given in terms of 7 independent functions by For spin chains : Bazhanov,Reshetikhin Cherednik V.K.,Vieira (for the proof) • Using analyticity in u one can transform Y-system to a Cauchi-Riemann problem • for 7 functions! Gromov, V.K.,Leurent,Volin (in progress)

  18. Conclusions • Non-trivial D=2,3,4,… dimensional solvable QFT’s! • Y-system for exact spectrum of a few AdS/CFT dualities has passed many important checks. • Y-system obeys integrable Hirota dynamics – can be reduced to a finite system of non-linear integral eqs (FiNLIE). General method of solving quantum ϭ-models Future directions • Why is N=4 SYM integrable? • What lessons for less supersymmetric SYM and QCD? • 1/N – expansion integrable? • Gluon amlitudes, correlators …integrable? • BFKL from Y-system?

  19. END

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