Hirota Dynamics of Quantum Integrability

Vladimir Kazakov (ENS, Paris)

“Round Table: Frontiers of Mathematical Physics” Dubna, December 16-18, 2012

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.Zamolodchikov

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.:

Boundary conditions for Hirota eq. for AdS/CFT T-system: ∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” !

U(2,2|4)a

s

KwonCheng,Lam,Zhang

Gromov, V.K., Tsuboi

Full quantum Hirota equation Classical (strong coupling) limit: eq. for characters of classical monodromy

Gromov,V.K.,Tsuboi

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

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

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 • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s

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 Q- and P-forms):

a

s

• For su(N) spin chain (half-strip) we impose:

• E.g. for gl(2) :

Krichever,Lipan,Wiegmann,Zabrodin Tsuboi

Gromov,V.K.,Leurent,Volin

Inspiring example: principal chiral field

• Y-system Hirota dynamics in a in (a,s) plane. We know the Wronskian solution in terms of Q-functions

• Finite volume solution: finite system of NLIE, parameterization fixing the analytic structure. •Analyticity strips from large volume asymptotics:

a

s

polynomialsfixing a state

jumps by

• N-1 TBA equations (for central nodes) on spectral densities

• From reality:

Gromov, V.K., VieiraV.K., Leurent

Alternative approach:Balog, Hegedus

-plane

SU(3) PCF numerics

E / 2

L

V.K.,Leurent’09

ground state

mass gap

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

Planar N=4 SYM – integrable 4D QFT

• 4D Correlators:

• Operators in 4D

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!

Spectral AdS/CFT Y-systemGromov,V.K.,Vieira

cuts in complex -plane

• Extra “corner” equations:

L→∞

• Analyticity from large L symptotics: from one-particle dispersion relation:

Zhukovsky map:

T-hook

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, TatteoBalog, Hegedus

• 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

• Only two cuts left on the magic sheet for ! • Right band parameterized: by a polynomial S(u), a gauge

function with one magic cut on ℝ and a density

• The property suggests that certain T-functions are much simpler

on the “magic” sheet, with only short cuts:

Parameterization of the upper band: continuation

• Remarkably, choosing the q-functions analytic in a half-plane we get all T-functions with the right analyticity strips!

We parameterize the upper band of T-hook in terms of a spectral densities.

The rest of Q’s restored from Plucker QQ relations

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

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 calculusFiamberti,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

• Perfectly reproduces 3 terms of Y-system numerics for Konishi operator or even

Gromov, ValatkaGubser, Klebanov, Polyakov

Y-system numerics Gromov,V.K.,VieiraFrolovGromov,Valatka

Numerics uses the TBA or FiNLIE forms of Y-system

AdS/CFT Y-system passes all known tests

Gromov, V.K., VieiraCavaglia, Fioravanti, TatteoArutyunov, FrolovGromov, V.K., Leurent, Volin

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• Why is N=4 SYM integrable? Can integrability be used to prove AdS/CFT correspondence?

Correa, Maldacena, Sever, DrukkerGromov, Sever

END