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

2. Integrability in AdS/CFT 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 AdS 5 /CFT 4, AdS 4 /CFT 3,...) Conjecture: it calculates exact anomalous dimensions of all local operators of the gauge theory at any coupling Gromov,V.K.,Vieira Further simplification: Y-system as Hirota discrete integrable dynamics

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

4. SYM perturbation and (1+1)D S-matrix  Feynman graphs and asymptotic scattering of “defects” on 1D “spin chain” Light cone gauge breaks the global and world-sheet Lorentz symmetries : S-matrix of AdS/CFT via bootstrap à-la A.&Al.Zamolodchikov Ŝ PSU(2,2|4) (p 1,p 2 ) = S 0  (p 1,p 2 ) × Ŝ SU(2|2) (p 1,p 2 ) ×Ŝ SU(2|2) (p 1,p 2 ) Beisert Janik Shastry’s R-matrix o f Hubbard model psu(2,2|4) su(2|2) On the string side... p1p1 p2p2 Minahan, Zarembo Krisijansen,Beisert,Staudacher Staudacher

5. Asymptotic Bethe Ansatz (ABA) This periodicity condition is diagonalized by nested Bethe ansatz finite size corrections, important for short operators! pjpj p1p1 pMpM Energy of state Results: ABA for dimensions of long YM operators (e.g., cusp dimension). Beisert,Eden,Staudacher

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

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

9. Y-system for excited states of AdS/CFT at finite size T-hook Complicated analyticity structure in u dictated by non-relativistic dispersion Gromov,V.K.,Vieira Extra equation (remnant of classical Z 4 monodromy): cuts in complex -plane obey the exact Bethe eq.: Energy : (anomalous dimension)

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

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 Gromov,V.K.,Vieira Bombardelli,Fiorvanti,Tateo Gromov,Levkovich-Maslyuk CP 3 x AdS 4, … 3d ABJM model: CP 3 x AdS 4, …

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

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

15. Character solution of T-hook for u(2,2|4) Solution in finite 2×2 and 4×4 determinants (analogue of the 1-st Weyl formula) Gromov,V.K.,Tsuboi  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 Gromov,V.K.,Tsuboi  Classical limit: highly excited long strings/operators, strong coupling:  Explicit u-shift in Hirota eq. dropped (only slow parametric dependence)  (Quasi)classical solution - psu(2,2|4) character of classical monodromy matrix in Metsaev-Tseytlin superstring sigma-model  Its eigenvalues (quasimomenta) encode conservation lows world sheet V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo  Finite gap method renders all classical solutions! Zakharov,Mikhailov Bena,Roiban,Polchinski

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

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