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

Y-system for the exact spectrum of N = 4 SYM

Vladimir Kazakov (ENS,Paris) Universitȁt von Bonn, 6 Juni 2011

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 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

CFT: N=4 SYM as a superconformal 4D QFT

• 4D Correlators:
• Operators in 4D

scaling dimensions non-trivial functions

• f ‘tHooft coupling λ!

structure constants

They describe the whole conformal theory via operator product expansion

• 4D superconformal QFT! Global symmetry PSU(2,2|4)

Anomalous dimensions in various limits

• Perturbation theory:

Minahan,Zarembo Beisert,Kristijanssen,Staudacher

• String (quasi)-classics:
• Long operators, no wrappings:
• Exact dimensions (all wrappings): Y-system and TBA

1,2,3…-loops: integrable spin chain Finite gap method, Bohr-Sommerfeld

Frolov,Tseytlin, V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Roiban,Tseytlin, Gromov,Vieira Beisert,Staudacher Beisert,Eden,Staudacher

Gromov,V.K.,Vieira

Asymptotic Bethe Ansatz (ABA)

• BFKL approx. for twist-2 operators

Kotikov, Lipatov Gubser,Klebanov,Polyakov

Worldsheet perturbation theory

• Strong coupling, short operators:

Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov

• Tree level:

Δ0 = L - degeneracy (for scalars)

Weak coupling calculation from SYM

nontrivial action

• n R-indices.
• 1-loop
• 2-loop:

Perturbative integrability

• “Vacuum” – BPS operator
• Dilatation operator = Heisenberg Hamiltonian, integrable by Bethe ansatz!
• Interaction:

Minahan,Zarembo Beisert,Kristijanssen,Staudacher

• Example: SU(2) sector

Anomalous dimension:

Bethe’31

Exact spectrum at one loop

Rapidity parameterization:

• Dilatation operator = Heisenberg Hamiltonian, integrable by Bethe ansatz!

Minahan, Zarembo

• vacuum

Beisert, Kristijansen,Staudacher

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

Beisert Janik

Shastry’s R-matrix

• f Hubbard model

psu(2,2|4) su(2|2) su(2|2) On the string side... p1 p2

Minahan, Zarembo Krisijansen,Beisert,Staudacher Staudacher

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 channel

• n small space circle L

world sheet

• Large R : cross channel momenta localize
• n poles of S-matrix →

bound states

ϭ-model in cross channel

• n large circle R

Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov

Bound states and TBA in AdS/CFT

Takahashi bound states for Hubbard model

Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov

• “Strings” of Bethe roots labeled by su(2,2|4) Dynkin nodes:
• 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

Roots form bound states

complex -plane

densities of bound states

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:

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

• bey the exact Bethe eq.:
• Energy :

(anomalous dimension)

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

Two loops from string quasiclassics for operators

Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio

• Perfectly reproduces two terms of Y-system numerics

for Konishi operator

• 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: 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) representations with rectangular Young tableaux:

a s

(K,M)

λ1 λ2 λa (a,s)

fat hook

• For characters – simplified Hirota eq.:
• Boundary conditions for Hirota eq.: gl(K|M) representations in “fat hook”:
• 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)

∞ - dim. unitary highest weight representations of u(2,2|4) !

Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi

U(2,2|4) a s

• Generating function for symmetric representations:

U(4|4) a s

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

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 (quasi-momenta) 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

Finite gap solution for dual classical superstring

• Algebraic curve for quasi-momenta:

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

• Eigenvalues – conserved quantities
• Monodromy matrix

encodes infinitely many conservation lows Energy of a string state

• Dimension of YM operator
• String equations of motion and constraints

can be recasted into zero curvature condition

• 2D ϭ-model on a

coset

world sheet Zakharov,Mikhailov Bena,Roiban,Polchinski

From classical to quantum Hirota in U(2,2|4) T-hook

Gromov

• v, V

, V.K., , Tsuboi

• i

Gromov

• v, V

, V.K., , Leuren ent,Tsuboi boi

• 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,

, V.K.,Le Leur uren ent,Volin (in p progr

• gres

ess)

Asymptotic Bethe ansatz from Y-system

SU(2,2|4)

SU(2|2) SU(2|2)

• From TBA:

where, in terms of Bethe roots:

• Dressing factor follows from the reality of Y-functions
• Asymptotic Bethe equation:
• Parameterization in Baxter’s Q-functions:

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

• 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

Some progress is being made…

Gromov,Tsuboi,V.K.,Leurent Gromov, V.K., Vieira V.K.,Leurent 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?

END