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!