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

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 AdS5/CFT4 , AdS4/CFT3 ,...)

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

N=4 SYM as a superconformal 4D QFT

• 4D Correlators (superconformal!):

• Operators in 4D

non-trivial functionsof ‘tHooft coupling λ!

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

ŜPSU(2,2|4)(p1,p2) = S02(p1,p2) × ŜSU(2|2) (p1,p2) ×ŜSU(2|2) (p1,p2) Beisert

Janik

Shastry’s R-matrix of Hubbard model

psu(2,2|4)

su(2|2)su(2|2)

On the string side...

p1p2

Minahan, ZaremboKrisijansen,Beisert,StaudacherStaudacher

Asymptotic Bethe Ansatz (ABA)

• This periodicity condition is diagonalized by nested Bethe ansatz

finite size corrections,important for short operators!

pj

p1

pM

• Energy of state

• Results: ABA for dimensions of long YM operators (e.g., cusp dimension).

Beisert,Eden,Staudacher

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)

ϭ-model in physical channelon 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.,VieiraBombardelli,Fioravanti,TateoGromov,V.K.,Kozak,VieiraArutyunov,Frolov

Dispersion relation• Exact one particle dispersion relation at infinite volume

• Bound states (fusion)

• Parametrization for dispersion relation:

cuts in complex u -plane

Santambrogio,ZanonBeisert,Dippel,Staudacher

N.Dorey

via Zhukovsky map:

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 Z4 monodromy):

cuts in complex -plane

• obey the exact Bethe eq.:

• Energy : (anomalous dimension)

Konishi operator : numerics from Y-system

GubserKlebanovPolyakov

Beisert, Eden,Staudacher ABA

Y-system numericsGromov,V.K.,Vieira

Gubser,Klebanov,Polyakov

Y-system passes all known tests

millions of 4D Feynman graphs!

5 loops and BFKL from stringFiamberti,Santambrogio,Sieg,Zanon

VelizhaninBajnok,Janik

Gromov,V.K.,VieiraBajnok,Janik,LukowskiLukowski,Rej,Velizhanin,Orlova

=2! From quasiclassicsGromov,Shenderovich,Serban, VolinRoiban,TseytlinMasuccato,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

Gromov,V.K.,VieiraBombardelli,Fiorvanti,TateoGromov,Levkovich-Maslyuk

3d ABJM model: CP3 x AdS4, …

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

× = × + ×a

s s s-1 s+1

a-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

KwonCheng,Lam,Zhang

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)

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.

HegedusGromov,Tsuboi,V.K.

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 lowsworld sheet

V.K.,Marshakov,Minahan,ZaremboBeisert,V.K.,Sakai,Zarembo

Finite gap method renders all classical solutions!

Zakharov,MikhailovBena,Roiban,Polchinski

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,ReshetikhinCherednikV.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)

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?

END