hirota dynamics of quantum integrability
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Hirota Dynamics of Quantum Integrability
Vladimir Kazakov (ENS, Paris)
“Facets of Integrability: Random Patterns, Stochastic Processes, Hydrodynamics, Gauge Theories and Condensed Matter Systems”
Simons institute, January 21-27, 2013
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
quantum and classical integrable systems.• It generates all integrable hierarchies of PDE’s (KdV, KP, Toda etc)• Discrete Hirota eq. (T-system) is an alternative approach to quantum integrable systems. • Classical KP hierarchy applies to quantum 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,…
+ 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., VieiraV.K., Leurent
Gromov, V.K. VieiraGromov, Volin, V.K., Leurent
V.K., Leurent, TsuboiAlexandrov, V.K., Leurent,Tsuboi,Zabrodin
Miwa,JimboSato
Kluemper, PierceKuniba,Nakanishi,SuzukiAl.ZamolodchikovBazhanov,Lukyanov, A.ZamolodchikovKrichever,Lipan, Wiegmann, Zabrodin
Discrete Hirota eq.: T-system and Y-system
• Y-system
• T-system (discrete Hirota eq.)
• Based on a trivial property of Kronecker symbols (and determinants):
• Gauge symmetry
= +a
s s s-1 s+1
a-1
a+1
(Super-)group theoretical origins of Y- and T-systems A curious property of gl(N|M) representations with rectangular Young tableaux:
For characters – simplified Hirota eq.:
KwonCheng,Lam,Zhang
Gromov, V.K., Tsuboi
Full quantum Hirota equation: extra variable – spectral parameter Classical limit: eq. for characters as functions of classical monodromy
Gromov,V.K.,Tsuboi
Boundary conditions for Hirota eq. for T-system (from -system): ∞ - dim. unitary highest weight representations of the “T-hook” !
s
-hooka
𝑀
𝐾 1𝐾 2
Quantum (super)spin chains
Co-derivative – left differential w.r.t. group (“twist”) matrix:
Transfer matrix (T-operator) of L spins
Hamiltonian of Heisenberg quantum spin chain:
V.K., Vieira
Quantum transfer matrices – a natural generalization of group characters
Main property:
R-matrix
Master T-operator and mKP
Master T is a tau function of mKP hierachy: mKP charge is spectral parameter! T is polynomial w.r.t.
Commutativity and conservation laws
Generating function of characters: Master T-operator:
V.K.,VieiraV.K., Leurent,Tsuboi
Alexandrov, V.K., Leurent,Tsuboi,Zabrodin
Satisfies canonical mKP Hirota eq.
Hence - discrete Hirota eq. for T in rectangular irreps: Baxter’s TQ relations, Backlund transformations etc.
considered byKrichever
V.K., Leurent,Tsuboi
• Definition of Q-operators at 1-st level of nesting: « removal » of an eigenvalue (example for gl(N)):
Baxter’s Q-operators
• Nesting (Backlund flow): consequtive « removal » of eigenvalues
Alternative approaches:Bazhanov,Lukowski,Mineghelli
Rowen Staudacher
Derkachev,Manashov
Def: complimentary set
• Q at level zero of nesting
• Next levels: multi-pole residues, or « removing » more of eignevalues:
Generating function for (super)characters of symmetric irreps:
s 1
Hasse diagram and QQ-relations (Plücker id.)
- bosonic QQ-rel.
• gl(2|2) example: classification of all Q-functions
TsuboiV.K.,Sorin,ZabrodinTsuboi,Bazhanov
• Nested Bethe ansatz equations follow from polynomiality of along a nesting path
• All Q’s expressed through a few basic ones by determinant formulas
Hasse diagram: hypercub
• E.g.
- fermionic QQ rel.
Wronskian solutions of Hirota equation• We can solve Hirota equations in a band of width N in terms of differential forms of 2N functions Solution combines dynamics of gl(N) representations and quantum fusion:
• -form encodes all Q-functions with indices:
• Solution of Hirota equation in a strip (via arbitrary - and -forms):
a
s
• For su(N) spin chain (half-strip) we impose:
• E.g. for gl(2) :
Krichever,Lipan, Wiegmann,Zabrodin
Gromov,V.K.,Leurent,Volin
definition:
Solution of Hirota in fat hook and T-hookTsuboi
V.K.,Leurent,Volin
s
-hooka
𝑀
𝐾 1𝐾 2
• Bosonic and fermionic 1-(sub)forms (all anticomute):
a
sλ1
λ2
λa
• Wronskian solution for the fat hook:
• Similar Wronkian solution exists in -hook
𝑀
𝐾
Inspiring example: principal chiral field (PCF)
• Finite : TBA → Y-system → Hirota dynamics in a in (a,s) plane in a band• Known asymptotics of Y-functions
a
s
• It is known since long to be integrable: S-matrix of types of physical particles
Wiegmann, TsevlikAl. Zamolodchikov
• A limiting case of Thirring model, or WZNW model Asymptotic Bethe ansatz constructed. Interesting explicit large solution at finite density
Polyakov, Wiegmann; WiegmannFateev, V.K., Wiegmann
• Analyticity strips of from asympotics• is analytic inside the strip
-plane
Zamolodchikov&ZamolodchikovKarowskiWiegmann
Finite volume solution of principal chiral field
true for 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 and for numerics at any
polynomialsfixing a state (for vacuum )
• nonlinear integral equations on spectral densities can be obtained e.g. from the condition of left-right symmetry
• From reality of Y-functions:
Gromov, V.K., VieiraV.K., LeurentAlternative approach:Balog, Hegedus
-plane
• Use Wronskian solution in terms of 2 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:
-plane
analytic in the upper half-plane
analytic in the lower half-plane
density at analyticityboundary
SU(3) PCF numerics
E / 2
L
V.K.,Leurent’09
ground state
mass gap
Planar N=4 SYM – integrable 4D QFT
• 4D Correlators:
• Operators via integrable spin chain dual to integrable sigma model
scaling dimensions non-trivial functions
of ‘tHooft coupling λ!structure constants
They describe the whole 4D conformal theory via operator product expansion
• 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!
MaldacenaGubser, Polyakov, KlebanovWitten
Minahan, ZaremboBena,Roiban,PolchinskiBeisert,Kristjanssen,StaudacherV.K.,Marchakov,Minahan,ZaremboBeisert, Eden,StaudacherJanik
Spectral AdS/CFT Y-systemGromov,V.K.,Vieira
cuts in complex -plane
• Extra “corner” equations:
L→∞
• Analyticity from large asymptotics via one-particle dispersion relation:
Zhukovsky map:
T-hook
definitions:
Wronskian solution of u(2,2|4) T-system in T-hook Gromov,V.K.,TsuboiGromov,Tsuboi,V.K.,LeurentTsuboi
Plücker relations express all 256 Q-functionsthrough 8 independent ones
Solution of AdS/CFT T-system in terms offinite number of non-linear integral equations (FiNLIE)
• 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: can be analytically continued on special magic sheet in labels
Gromov,V.K.,Leurent,Volin
• Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz)
Alternative approach:Balog, Hegedus
Inspired by:Bombardelli, Fioravanti, Tatteo
• Operators/states of AdS/CFT are characterized by certain poles and zeros
of Y- and T-functions fixed by exact Bethe equations:
Magic sheet and solution for the right band
parameterized by a polynomial and two spectral densities
• 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
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
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
• Dimension can be extracted from the asymptotics:
• 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).
Konishi dimension to 8-th order
• Last term is a new structure – multi-index zeta function.
• Leading transcendentalities can be summed at all orders:
Bajnok,JanikLeurent,Serban,VolinBajnok,Janik,LukowskiLukowski,Rej,Velizhanin,Orlova
Leurent, Volin ’12(from FiNLIE)
• Confirmed up to 5 loops by direct graph calculus (6 loops promised)Fiamberti,Santambrogio,Sieg,Zanon
VelizhaninEden,Heslop,Korchemsky,Smirnov,Sokatchev
Leurent, Volin ‘12
• Integrability allows to sum exactly enormous number of Feynman diagrams of N=4 SYM
Numerics and 3-loops from string quasiclassics for twist-J operators of spin S
Gromov,Shenderovich,Serban, VolinRoiban, TseytlinVallilo, MazzucatoGromov, Valatka
• 3 leading strong coupling terms were calculated: for Konishi operator or even They perfectly reproduce the TBA/Y-system or FiNLIE numerics
Gromov, ValatkaGubser, Klebanov, Polyakov
Y-system numerics Gromov,V.K.,VieiraFrolovGromov,Valatka
AdS/CFT Y-system passes all known tests
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 • 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?
Correa, Maldacena, Sever, DrukkerGromov, Sever
Recent advances:Gromov, Sever, Vieira, Kostov, Serban, Janik etc.
Happy Birthday Pasha!
С ЮБИЛЕЕМ, ПАША!
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