Y-system for the exact spectrum of N = 4 SYM Vladimir Kazakov - PowerPoint PPT Presentation

Universit ȁ t 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 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 •

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 non-trivial functions of ‘tHooft coupling λ ! structure constants They describe the whole conformal theory via operator product expansion

Anomalous dimensions in various limits 1,2,3…-loops: integrable • Perturbation theory: spin chain Minahan,Zarembo Beisert,Kristijanssen,Staudacher • BFKL approx. for twist-2 operators Kotikov, Lipatov • String (quasi)-classics: Finite gap method, Bohr-Sommerfeld Frolov,Tseytlin, V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Roiban,Tseytlin, Gromov,Vieira • Long operators, no wrappings: Asymptotic Bethe Ansatz (ABA) Beisert,Staudacher Beisert,Eden,Staudacher • Strong coupling, short operators: Worldsheet perturbation theory Gubser,Klebanov,Polyakov • Exact dimensions (all wrappings): Y-system and TBA Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Gromov,V.K.,Vieira Arutyunov,Frolov

Weak coupling calculation from SYM Δ 0 = L - degeneracy (for scalars) • Tree level: • 1-loop nontrivial action • 2-loop: 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 2 p 1 • Light cone gauge breaks the global and world-sheet Lorentz symmetries : psu(2,2|4) Beisert Janik su(2|2) su(2|2) Shastry’s R-matrix • S-matrix of AdS/CFT via bootstrap à-la A.&Al.Zamolodchikov o f Hubbard model

Asymptotic Bethe Ansatz (ABA) Beisert,Eden,Staudacher 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 Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov complex -plane 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 Santambrogio,Zanon • Exact one particle dispersion relation at infinite volume 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 • Complicated analyticity structure in u dictated by non-relativistic dispersion cuts in complex -plane • Extra equation (remnant of classical Z 4 monodromy): • Energy : (anomalous dimension) • obey the exact Bethe eq. :

Konishi operator : numerics from Y-system Beisert, Eden,Staudacher Gubser,Klebanov,Polyakov ABA Gubser =2! From Klebanov Polyakov quasiclassics Y-system numerics Gromov,Shenderovich, Gromov,V.K.,Vieira 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

Two loops from string quasiclassics for operators Gromov,Shenderovich, Serban, Volin • Perfectly reproduces two terms of Y-system numerics Roiban,Tseytlin for Konishi operator Masuccato,Valilio • Also works for

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-1 = + a 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(2,2|4) U(4|4) a a s s ∞ - dim. unitary highest weight representations of u(2,2|4) ! Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi

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: 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 world sheet  Its eigenvalues (quasi-momenta) encode conservation lows  Finite gap method renders all classical solutions! V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo

Finite gap solution for dual classical superstring V.K.,Marshakov,Minahan,Zarembo  2D ϭ -model on a Beisert,V.K.,Sakai,Zarembo coset  String equations of motion and constraints can be recasted into zero curvature condition Zakharov,Mikhailov world sheet Bena,Roiban,Polchinski  Monodromy matrix encodes infinitely many conservation lows  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 ov, V , V.K., , Tsuboi oi Gromov ov, V , V.K., , Leuren ent,Tsuboi boi • 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 ov, , V.K.,Le Leur uren ent,Volin (in p progr ogres ess)