Hirota integrable dynamics: from quantum spin chains to AdS/CFT integrability

Vladimir Kazakov (ENS, Paris)

International Symposium Ahrenshoop “Recent Developments in String and Field Theory”

Schmöckwitz, August 27-31, 2012

Collaborations with Alexandrov, Gromov, Leurent, Tsuboi, Vieira, Volin, Zabrodin

Hirota equations in quantum integrability

• New approach to solution of integrable 2D quantum sigma-models in finite volume

• Based on discrete classical Hirota dynamics (Y-system, T-system , Baxter’s Q-functions, Plücker QQ identities, wronskian solutions,…)

+ Analyticity in spectral parameter!

• Important examples already worked out, such as su(N)×su(N) principal chiral field (PCF)

• FiNLIE equations from Y-system for exact planar AdS/CFT spectrum • Inspiration from Hirota dynamics of gl(K|M) quantum (super)spin

chains: mKP hierarchy for T- and Q- operators

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

Gromov, Volin, V.K., Leurent

V.K., Leurent, TsuboiAlexandrov, V.K., Leurent,Tsuboi,Zabrodin

Y-system and T-system

• Y-system

• T-system (Hirota eq.)

• Gauge symmetry

= +a

s s s-1 s+1

a-1

a+1

Related to a property of gl(N|M) irreps with rectangular Young tableaux:

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

It is a tau function of mKP hierachy: (polynomial w.r.t. the mKP charge )

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:

V.K., Leurent,Tsuboi

• Graphically (slightly generalized to any spectral parameters):

Master Identity and Q-operators

The proof in:V.K., Leurent,Tsuboi

from the basic identity proved in:V.K, Vieira

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 characters of symmetric irreps:

s

Hasse diagram and QQ-relations (Plücker id.)

- bosonic QQ-rel.

-- fermionic QQ rel.

• Example: gl(2|2)

TsuboiV.K.,Sorin,Zabrodin

Gromov,VieiraTsuboi,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.

Wronskian solutions of Hirota equation• We can solve Hirota equations in a strip of width N in terms of differential forms of N functions . Solution combines dynamics of gl(N) representations and the quantum fusion:

• -form encodes all Q-functions with indices:

• Solution of Hirota equation in a strip:

a

s

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

• E.g. for gl(2) :

Krichever,Lipan,Wiegmann,Zabrodin

Gromov,V.K.,Leurent,Volin

Inspiring example: principal chiral field

• Y-system Hirota dynamics in a in (a,s) strip of width N

polynomialsfixing a state

jumps by

• Finite volume solution: finite system of NLIE: parametrization fixing the analytic structure:

• N-1 spectral densities (for L ↔ R symmetric states):

• From reality:

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

SU(3) PCF numerics: Energy versus size for vacuum and mass gap

E L/ 2

L

V.K.,Leurent’09

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

• Type of the operator is fixed by imposing certain analyticity properties in spectral parameter. Dimension can be extracted from the asymptotics

cuts in complex -plane

• Extra “corner” equations:

s

a

• Parametrization by Zhukovsky map:

• Dispersion relation

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

Gromov,V.K.,Leurent,Volin

• Original T-system is in mirror sheet (long cuts)

• Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz)

Alternative approach:Balog, Hegedus

We found and checked from TBA the following relation between the upper and right/left bands Inspired by:

Bombardelli, Fioravanti, TatteoBalog, Hegedus

• Irreps (n,2) and (2,n) are in fact the same typical irrep, so it is natural to impose for our physical gauge

• From unimodularity of the quantum monodromy matrix

Arutyunov, Frolov

Quantum symmetry

can be analytically continued on special magic sheet in labels

Analytically continued and satisfy the Hirota equations, each in its infinite strip.

Gromov,V.K. Leurent, TsuboiGromov,V.K.Leurent,Volin

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 in terms of a spectral density , the “wing exchange” function and gauge function and two polynomials P(u) and (u) encoding Bethe roots

The rest of q’s restored from Plucker QQ relations

Closing FiNLIE: sawing together 3 bands

We have expressed all T (or Y) functions through 6 functions

From analyticity of and we get, via spectral Cauchy representation, extra equations fixing all unknown functions

　 Numerics for FiNLIE perfectly reproduces earlier results obtained from Y-system (in TBA form):

Konishi operator : numerics from Y-system

GubserKlebanovPolyakov

Beisert, Eden,Staudacher ABA

Y-system numerics Gromov,V.K.,Vieira(confirmed and precised by Frolov)

Gubser,Klebanov,Polyakov

Uses the TBA form of Y-system AdS/CFT Y-system passes all known tests

zillions of 4D Feynman graphs! Fiamberti,Santambrogio,Sieg,ZanonVelizhanin

Bajnok,JanikGromov,V.K.,Vieira

Bajnok,Janik,LukowskiLukowski,Rej,Velizhanin,OrlovaEden,Heslop,Korchemsky,Smirnov,Sokatchev

From quasiclassicsGromov,Shenderovich,Serban, VolinRoiban,TseytlinMasuccato,ValilioGromov, Valatka

Cavaglia, Fioravanti, TatteoGromov, V.K., VieiraArutyunov, Frolov

Leurent,Serban,VolinBajnok,Janik

Conclusions • Hirota integrable dynamics, supplied by analyticity in spectral parameter, is a powerful method

of solving integrable 2D quantum sigma models.

• Y-system 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 ?• Why is N=4 SYM integrable?• FiNLIE for another integrable AdS/CFT duality: 3D ABJM gauge theory• BFKL limit from Y-system?• 1/N – expansion integrable?• Gluon amlitudes, correlators …integrable?

Correa, Maldacena, Sever, DrukkerGromov, Sever

END