“Facets of Integrability: Random Patterns, Stochastic Processes,
Hydrodynamics, Gauge Theories and Condensed Matter Systems”
Simons institute, January 21-27, 2013
Hirota Dynamics of Quantum Integrability
Vladimir Kazakov (ENS, Paris)
Collaborations with
Alexandrov, Gromov, Leurent,
Tsuboi, Vieira, Volin, Zabrodin
New uses of Hirota dynamics in integrability
• Hirota integrable dynamics incorporates the basic properties of all
Miwa,Jimbo
quantum and classical integrable systems.
Sato
• It generates all integrable hierarchies of PDE’s (KdV, KP, Toda etc)
Pierce
• Discrete Hirota eq. (T-system) is an alternative Kluemper,
Kuniba,Nakanishi,Suzuki
Al.Zamolodchikov
approach to quantum integrable systems.
Bazhanov,Lukyanov, A.Zamolodchikov
Krichever,Lipan, Wiegmann, Zabrodin
• Classical KP hierarchy applies to quantum
V.K., Leurent, Tsuboi
Alexandrov, V.K., Leurent,Tsuboi,Zabrodin
T- and Q-operators of (super)spin chains
• Framework for new approach to solution of integrable 2D quantum
sigma-models in finite volume using Y-system, T-system, Baxter’s
Q-functions, Plücker QQ identities, wronskian solutions,…
+
Gromov, V.K., Vieira
V.K., Leurent
Analyticity in spectral parameter!
• First worked out for spectrum of relativistic sigma-models, such as
su(N)×su(N) principal chiral field (PCF), Sine-Gordon, Gross-Neveu
• Provided the complete solution of spectrum of anomalous
dimensions of 4D N=4 SYM theory! AdS/CFT Y-system, recently
reduced to a finite system of non-linear integral eqs (FiNLIE)
Gromov, V.K. Vieira
Gromov, Volin, V.K., Leurent
Discrete Hirota eq.: T-system and Y-system
• Based on a trivial property of Kronecker symbols (and determinants):
• T-system (discrete Hirota eq.)
• Y-system
• Gauge symmetry
(Super-)group theoretical origins of Y- and T-systems
 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. for 𝑈 𝐾1 , 𝐾2 𝑀 T-system (from 𝜒-system):
∞ - dim. unitary highest weight representations of the “T-hook” !
𝑀
a
𝐾2
𝐾1 , 𝐾2 𝑀 -hook
Kwon
Cheng,Lam,Zhang
Gromov, V.K., Tsuboi
𝐾1
s
 Full quantum Hirota equation: extra variable – spectral parameter
 Classical limit: eq. for characters as functions of classical monodromy
Gromov,V.K.,Tsuboi
Quantum (super)spin chains
 Quantum transfer matrices – a natural generalization of group characters
V.K., Vieira
 Co-derivative – left differential w.r.t. group (“twist”) matrix:
Main property:
 Transfer matrix (T-operator) of L spins
R-matrix
 Hamiltonian of Heisenberg quantum spin chain:
Master T-operator and mKP
 Generating function of characters:
 Master T-operator:
 Master T is a tau function of mKP hierachy:
mKP charge is spectral parameter! T is polynomial w.r.t.
 Satisfies canonical mKP Hirota eq.
considered by
Krichever
 Hence - discrete Hirota eq. for T in rectangular irreps:
Baxter’s TQ relations, Backlund transformations etc.
 Commutativity and conservation laws
V.K.,Vieira
V.K., Leurent,Tsuboi
Alexandrov, V.K.,
Leurent,Tsuboi,Zabrodin
Baxter’s Q-operators
V.K., Leurent,Tsuboi
 Generating function for (super)characters of symmetric irreps:
1
•
•
s
Q at level zero of nesting
Definition of Q-operators at 1-st level of nesting:
« removal » of an eigenvalue (example for gl(N)):
Def: complimentary set
• Next levels: multi-pole residues, or « removing » more of eignevalues:
• Nesting (Backlund flow): consequtive « removal » of eigenvalues
Alternative
approaches:
Bazhanov,
Lukowski,
Mineghelli
Rowen
Staudacher
Derkachev,
Manashov
Hasse diagram and QQ-relations (Plücker id.)
•
Tsuboi
V.K.,Sorin,Zabrodin
Tsuboi,Bazhanov
gl(2|2) example: classification of all Q-functions
Hasse diagram: hypercub
• E.g.
- bosonic QQ-rel.
- fermionic QQ rel.
• Nested Bethe ansatz equations follow from polynomiality of
along a nesting path
• All Q’s expressed through a few basic ones by determinant formulas
Krichever,Lipan, Wiegmann,Zabrodin
Wronskian solutions of Hirota equation
• We can solve Hirota equations in a band of width N in terms of
Gromov,V.K.,Leurent,Volin
differential forms of 2N functions
Solution combines dynamics of gl(N) representations and quantum fusion:
•
-form encodes all Q-functions with
indices:
a
s
• E.g. for gl(2) :
• Solution of Hirota equation in a strip (via arbitrary
- and
-forms):
definition:
• For su(N) spin chain (half-strip) we impose:
Solution of Hirota in (𝐾|𝑀) fat hook and (𝐾1 , 𝐾2 |𝑀) T-hook
a
𝑀
λa
λ2
λ1
Tsuboi
V.K.,Leurent,Volin
𝑀
a
𝐾
𝐾1 , 𝐾2 𝑀 -hook
𝐾2
𝐾1
s
s
•
Bosonic and fermionic 1-(sub)forms (all
•
Wronskian solution for the (𝐾|𝑀) fat hook:
•
Similar Wronkian solution exists in 𝐾1 , 𝐾2 𝑀 -hook
anticomute):
Inspiring example:
principal chiral field (PCF)
• It is known since long to be integrable:
S-matrix of 𝑁 − 1 types of physical particles
Zamolodchikov&Zamolodchikov
Karowski
Wiegmann
• A limiting case of Thirring model, or WZNW model
Asymptotic 𝐿 ∞ Bethe ansatz constructed.
Interesting explicit large 𝑁 solution at finite density
• Finite 𝐿: TBA → Y-system → Hirota dynamics
in a in (a,s) plane in a band
• Known asymptotics of Y-functions
Polyakov, Wiegmann;
Wiegmann
Fateev, V.K., Wiegmann
Wiegmann, Tsevlik
Al. Zamolodchikov
a
s
-plane
• Analyticity strips of 𝑇𝑎,𝑠 (𝑢) from 𝑢
• 𝑇𝑎,𝑠 (𝑢) is analytic inside the strip
∞ asympotics
Finite volume solution of principal chiral field
Gromov, V.K., Vieira
V.K., Leurent
Alternative approach:
Balog, Hegedus
• Use Wronskian solution in terms of 2𝑁 Q-functions
• It is crucial to know their analyticity properties. The following choice appears
to render the right analyticity strips of Y- and T-functions:
density at analyticity
boundary
-plane
analytic in the upper half-plane
polynomials
fixing a state (for vacuum 𝑆𝑘 𝑢 =
-plane
𝑢𝑘−1
𝑘−1 !
)
• From reality of Y-functions:
analytic in the lower half-plane
• 𝑁 − 1 nonlinear integral equations on spectral densities ρ𝑘 can be obtained
e.g. from the condition of left-right symmetry
true for 𝑆𝑈(𝑁)𝐿 ⟺ 𝑆𝑈(𝑁)𝑅 symmetric states (can be generalized to any state)
• We obtain a finite system of NLIE (somewhat similar to Destri-deVega eqs.)
• Good for analytic study at large or small volume 𝐿 and for numerics at any 𝐿
SU(3) PCF numerics
V.K.,Leurent’09
E / 2
mass gap
ground state
L
Planar N=4 SYM – integrable 4D QFT
Maldacena
Gubser, Polyakov, Klebanov
Witten
• 4D superconformal QFT! Global symmetry PSU(2,2|4)
• AdS/CFT correspondence – duality to Metsaev-Tseytlin superstring
• Integrable for non-BPS states, summing genuine 4D Feynman diagrams!
• Operators via integrable spin chain dual to integrable sigma model
Minahan, Zarembo
Bena,Roiban,Polchinski
Beisert,Kristjanssen,Staudacher
V.K.,Marchakov,Minahan,Zarembo
Beisert, Eden,Staudacher
Janik
• 4D Correlators:
scaling dimensions
non-trivial functions
of ‘tHooft coupling λ!
structure constants
They describe the whole 4D conformal theory via operator product expansion
𝑃𝑆𝑈 2,2 4 T-hook
Spectral AdS/CFT Y-system
Gromov,V.K.,Vieira
cuts in complex
-plane
• Analyticity from large 𝐿 asymptotics via one-particle dispersion relation:
L→∞
Zhukovsky map:
• Extra “corner” equations:
Wronskian solution of u(2,2|4) T-system in T-hook
Gromov,V.K.,Tsuboi
Gromov,Tsuboi,V.K.,Leurent
Tsuboi
definitions:
Plücker relations express all 256 Q-functions
through 8 independent ones
Gromov,V.K.,Leurent,Volin
Solution of AdS/CFT T-system in terms of
finite number of non-linear integral equations (FiNLIE)
• Main tools: integrable Hirota dynamics + analyticity
(inspired by classics and asymptotic Bethe ansatz)
• No single analyticity friendly gauge for T’s of right, left and upper bands.
We parameterize T’s of 3 bands in different, analyticity friendly gauges,
also respecting their reality and certain symmetries.
• Quantum analogue of classical
symmetry:
continued on special magic sheet in labels
can be analytically
• Operators/states of AdS/CFT are characterized by certain poles and zeros
of Y- and T-functions fixed by exact Bethe equations:
Inspired by:
Bombardelli, Fioravanti, Tatteo
Alternative approach:
Balog, Hegedus
Magic sheet and solution for the right band
• The property
suggests that certain T-functions are much simpler
on the “magic” sheet, with only short cuts:
•
Wronskian solution for the right band in terms of two Q-functions with one
magic cut on ℝ
parameterized by a polynomial 𝑃𝑀−1 and two spectral densities 𝜌1 , 𝜌2
Parameterization of the upper band: continuation
• Remarkably, choosing the upper band Q-functions analytic in a half-plane
we get all T-functions with the right analyticity strips!
 All Q’s in the upper band of T-hook can be parametrized by 2 densities.
Closing FiNLIE: sawing together 3 bands
• Finally, we can close the FiNLIE system by using reality of T-functions
and certain symmetries. For example, for left-right symmetric states
• The states/operators are fixed by introducing certain zeros and poles
to Y-functions, and hence to T- and Q-functions (exact Bethe roots).
• Dimension can be extracted from the asymptotics:
 FiNLIE perfectly reproduces earlier results obtained
from Y-system (in TBA form). It is a perfect mean to generate
weak and strong coupling expansions of anomalous dimensions
in N=4 SYM
Konishi
dimension to 8-th order
• Integrability allows to sum exactly enormous number
of Feynman diagrams of N=4 SYM
Bajnok,Janik
Leurent,Serban,Volin
Bajnok,Janik,Lukowski
Lukowski,Rej,
Velizhanin,Orlova
Leurent, Volin ’12
(from FiNLIE)
• Last term is a new structure – multi-index zeta function.
• Leading transcendentalities can be summed at all orders:
Leurent, Volin ‘12
• Confirmed up to 5 loops by direct graph calculus (6 loops promised)
Fiamberti,Santambrogio,Sieg,Zanon
Velizhanin
Eden,Heslop,Korchemsky,Smirnov,Sokatchev
Numerics and 3-loops from string quasiclassics
for twist-J operators of spin S
• 3 leading strong coupling terms were calculated:
for Konishi operator
or even
They perfectly reproduce the TBA/Y-system or FiNLIE numerics
Y-system numerics
Gromov,V.K.,Vieira
Frolov
Gromov,Valatka
Gubser, Klebanov, Polyakov

Gromov,Shenderovich,
Serban, Volin
Roiban, Tseytlin
Vallilo, Mazzucato
Gromov, Valatka
AdS/CFT Y-system passes all known tests
Gromov, Valatka
Conclusions
•
Hirota integrable dynamics, supplied by analyticity in spectral parameter, is a powerful method
of solving integrable 2D quantum sigma models.
•
For spin chains (mKP structure): a curious alternative to the algebraic Bethe ansatz of
Leningrad school
•
Y-system for sigma-models can be reduced to a finite system of non-linear integral eqs
(FiNLIE) in terms of Wronskians of Q-functions.
•
For the spectral problem in AdS/CFT, FiNLIE represents the most efficient way for numerics
and weak/strong coupling expansions.
•
Recently Y-system and FiNLIE used to find quark-antiquark potential in N=4 SYM
Future directions
Correa, Maldacena, Sever,
Drukker
Gromov, Sever
•
Better understanding of analyticity of Q-functions. Quantum algebraic curve for AdS5/CFT4 ?
•
•
BFKL limit from Y-system and FiNLIE
Hirota dynamics for structure constants of OPE and correlators?
•
Why is N=4 SYM integrable? Can integrability be used to prove AdS/CFT correspondence?
Recent advances:
Gromov, Sever, Vieira,
Kostov, Serban, Janik etc.
Happy Birthday Pasha!
С ЮБИЛЕЕМ, ПАША!