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“Great Lakes Strings” Conference 2011
Chicago University, April 29
A window into 4D integrability:
the exact spectrum of N = 4 SYM from Y-system
Vladimir Kazakov (ENS,Paris)
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 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
• Operators in 4D
• 4D Correlators (superconformal!):
non-trivial functions
of ‘tHooft coupling λ!
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...
p2
p1
• Light cone gauge breaks the global and world-sheet Lorentz symmetries :
psu(2,2|4)
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
ŜPSU(2,2|4)(p1,p2) = S02(p1,p2) × ŜSU(2|2) (p1,p2) ×ŜSU(2|2) (p1,p2)
Beisert
Janik
Asymptotic Bethe Ansatz (ABA)
Beisert,Eden,Staudacher
pj
p1
pM
• 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
Dispersion relation
• Exact one particle dispersion relation at infinite volume
• Bound states (fusion)
• Parametrization for dispersion relation:
via Zhukovsky map:
cuts in complex
u -plane
Santambrogio,Zanon
Beisert,Dippel,Staudacher
N.Dorey
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
• Extra equation (remnant of
classical Z4 monodromy):
• Energy :
(anomalous dimension)
•
obey the exact Bethe eq.:
-plane
Konishi operator
: numerics from Y-system
Beisert, Eden,Staudacher
Gubser,Klebanov,Polyakov
ABA
Gubser
Klebanov
Polyakov
Y-system numerics
Gromov,V.K.,Vieira
millions of 4D Feynman graphs!
 Y-system passes all known tests
=2! From
quasiclassics
Gromov,Shenderovich,
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
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, …
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|M) representations with rectangular Young tableaux:
×
a
s
=
s
+
×
s-1
a-1
×
a-1
s+1
 For characters – simplified Hirota eq.:
 Boundary conditions for Hirota eq.:
∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” !
U(2,2|4)
a
Kwon
Cheng,Lam,Zhang
Gromov, V.K., Tsuboi
s
• Solution of Hirota for any irrep: Jacobi-Trudi formula for GL(K|M) characters:
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
 Its eigenvalues (quasimomenta) encode conservation lows
 Finite gap method renders all classical solutions!
world sheet
V.K.,Marshakov,Minahan,Zarembo
Beisert,V.K.,Sakai,Zarembo
From classical to quantum Hirota in U(2,2|4) T-hook
Gromov, V.K., 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
Gromov, V.K.,Leurent,Volin
for 7 functions!
(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