Universitȁt von Bonn, 6 Juni 2011 Y-system for the exact spectrum of N = 4 SYM Vladimir Kazakov (ENS,Paris)
Theoretical Challenges in Quantum Gauge Field Theory QFT s in dimension>2, in particular 4D Yang-Mills gauge theories, describe fundamental structures of the Nature but are very difficult to handle Quantum Chromodynamics, or even pure Yang-Mills theories can be treated so far only perturbatively (weak coupling, high energies) or by computer Monte-Carlo simulations on the lattice (bad accuracy, no understanding, problems with super-ym theories). We need a theoretical progress and some exact non-perturbative results Guiding ideas: special gauge theories (N=4 SYM theory!), special limits, AdS/CFT correspondence and integrablility
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 AdSbackground 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
CFT: N=4 SYM as a superconformal 4D QFT 4D superconformal QFT! Global symmetry PSU(2,2 4) Operators in 4D 4D Correlators: scaling dimensions structure constants non-trivial functions of thooft coupling λ! They describe the whole conformal theory via operator product expansion
Anomalous dimensions in various limits Perturbation theory: BFKL approx. for twist-2 operators 1,2,3 -loops: integrable spin chain Minahan,Zarembo Beisert,Kristijanssen,Staudacher Kotikov, Lipatov String (quasi)-classics: Long operators, no wrappings: Strong coupling, short operators: Finite gap method, Bohr-Sommerfeld Frolov,Tseytlin, V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Roiban,Tseytlin, Gromov,Vieira Asymptotic Bethe Ansatz (ABA) Beisert,Staudacher Beisert,Eden,Staudacher Worldsheet perturbation theory Gubser,Klebanov,Polyakov Exact dimensions (all wrappings): Y-system and TBA Gromov,V.K.,Vieira Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov
Weak coupling calculation from SYM Tree level: Δ 0 = L - degeneracy (for scalars) 1-loop 2-loop: nontrivial action on R-indices.
Perturbative integrability Interaction: Vacuum BPS operator Example: SU(2) sector Dilatation operator = Heisenberg Hamiltonian, integrable by Bethe ansatz! Minahan,Zarembo Beisert,Kristijanssen,Staudacher
Exact spectrum at one loop Dilatation operator = Heisenberg Hamiltonian, integrable by Bethe ansatz! - vacuum Minahan, Zarembo Beisert, Kristijansen,Staudacher Rapidity parameterization: Bethe 31 Anomalous dimension:
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... p p 1 2 Light cone gauge breaks the global and world-sheet Lorentz symmetries : psu(2,2 4) Beisert Janik su(2 2) su(2 2) S-matrix of AdS/CFT via bootstrap à-la A.&Al.Zamolodchikov Shastry s R-matrix of Hubbard model
Beisert,Eden,Staudacher Asymptotic Bethe Ansatz (ABA) p j p 1 p M This periodicity condition is diagonalized 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).
Finite size (wrapping) effects Wrapped graphs : beyond S-matrix theory We need to take into account finite size effects - Y-system needed
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
Bound states and TBA in AdS/CFT Strings of Bethe roots labeled by su(2,2 4) Dynkin nodes: Gromov,V.K.,Vieira complex -plane Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov Roots form bound states Takahashi bound states for Hubbard model densities of bound states TBA equations from minimum of free energy at finite temperature T=1/L Inverting the kernels, TBA can be brought to universal Y-system! Bound states organized in T-hook
Dispersion relation Exact one particle dispersion relation at infinite volume Santambrogio,Zanon Beisert,Dippel,Staudacher N.Dorey Bound states (fusion) Parametrization for dispersion relation: via Zhukovsky map: cuts in complex u -plane
Y-system for excited states of AdS/CFT at finite size Gromov,V.K.,Vieira T-hook Extra equation (remnant of classical Z 4 monodromy): Complicated analyticity structure in u dictated by non-relativistic dispersion cuts in complex -plane Energy : (anomalous dimension) obey the exact Bethe eq.:
Konishi operator : numerics from Y-system Beisert, Eden,Staudacher ABA Gubser,Klebanov,Polyakov Y-system numerics Gromov,V.K.,Vieira Gubser Klebanov Polyakov =2! From quasiclassics Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio 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 Y-system passes all known tests
Two loops from string quasiclassics for operators Perfectly reproduces two terms of Y-system numerics for Konishi operator Also works for Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio
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?
Y-systems for other σ-models 3d ABJM model: CP 3 x AdS 4, Gromov,V.K.,Vieira Bombardelli,Fiorvanti,Tateo Gromov,Levkovich-Maslyuk
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!
(Super-)group theoretical origins A curious property of gl(n) representations with rectangular Young tableaux: a = + a-1 a-1 s s s-1 s+1 For characters simplified Hirota eq.: Boundary conditions for Hirota eq.: gl(k M) representations in fat hook : a λ a (a,s) fat hook (K,M) λ 2 λ 1 s Solution: Jacobi-Trudi formula for GL(K M) characters
Super-characters: Fat Hook of U(4 4) and T-hook of U(2,2 4) Generating function for symmetric representations: U(4 4) a U(2,2 4) a s s - dim. unitary highest weight representations of u(2,2 4)! Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi
Gromov,V.K.,Tsuboi 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) Generalization to full T-system with spectral parameter: Hegedus Gromov,Tsuboi,V.K Wronskian determinant solution. Should help to reduce AdS/CFT system to a finite system of equations.
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) Gromov,V.K.,Tsuboi (Quasi)classical solution - psu(2,2 4) character of classical monodromy matrix in Metsaev-Tseytlin superstring sigma-model Zakharov,Mikhailov Bena,Roiban,Polchinski Its eigenvalues (quasi-momenta) encode conservation lows world sheet Finite gap method renders all classical solutions! V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo
Finite gap solution for dual classical superstring 2D ϭ-model on a coset String equations of motion and constraints can be recasted into zero curvature condition V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Monodromy matrix encodes infinitely many conservation lows Zakharov,Mikhailov Bena,Roiban,Polchinski world sheet Eigenvalues conserved quantities Algebraic curve for quasi-momenta: Dimension of YM operator Energy of a string state
From classical to quantum Hirota in U(2,2 4) T-hook Gromov, V.K., Tsuboi Gromov, V.K., Leurent,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)
Asymptotic Bethe ansatz from Y-system From TBA: SU(2 2) SU(2 2) Parameterization in Baxter s Q-functions: SU(2,2 4) where, in terms of Bethe roots: Dressing factor follows from the reality of Y-functions Asymptotic Bethe equation:
For AdS/CFT, as for any sigma model The origins of the Y-system are entirely algebraic: Hirota eq. for PSU(2,2 4) characters in a given hook. Add the spectral parameter dependence and solve Hirota equation in terms of finite number of functions. For AdS/CFT T-hook this solution is known Gromov,Tsuboi,V.K.,Leurent Analyticity in spectral parameter u is the most difficult part of the problem. The problem can be reduce it to a finite system of nonlinear integral equations? (FINLIE analog of Destri-DeVega equations) This program nicely works for SU(N) principal chiral field Gromov, V.K., Vieira V.K.,Leurent Some progress is being made Gromov,V.K.,Leurent,Volin
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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