Q-operators and discrete Hirota dynamics for spin chains and sigma models

Vladimir Kazakov (ENS,Paris)

Workshop, “`From sigma models to 4D CFT ” DESY, Hamburg, 1 December 2010

with Nikolay Gromov arXiv:1010.4022 Sebastien Leurent arXiv:1010.2720 ZengoTsuboi arXiv:1002.3981

Outline• Hirota dynamics: attempt of a unified approach to integrability of spin

chains and sigma models

• New approach to quantum gl(K|N) spin chains based on explicit

construction of Baxter’s Q-operators and Backlund flow (nesting)

• Baxter’s TQ and QQ operatorial relations and nested Bethe ansatz equations from new Master identity. Wronskian solutions of Hirota eq.

• Applications of Hirota dynamics in sigma-models :

- spectrum of SU(N) principal chiral field on a finite space circle

- Wronskian solution for AdS/CFT Y-system. Towards a finite system of equations for the full planar spectrum of AdS/CFT

Fused R-matrix in any irrep λ of gl(K|M)

0

=u

v

u

v0

u

0

“”

“f”

fundamental irrep “f” in quantum space

any ““= {a}irrep auxiliary space

generator matrix elementin irrep

Yang-Baxter relations

Co-derivative

• Definition , where

nice representation for R-matrix follows:

V.K., Vieira

• Super-case:

• From action on matrix element

Transfer matrix in terms of left co-derivative• Monodromy matrix of the spin chain:

• Transfer-matrix of N spins

• Transfer-matrix without spins:

• Transfer-matrix of one spin:

V.K., VieiraV.K., Leurent,Tsuboi

(previous particular case )

Grafical representation (slightly generalized to any spectral parameters)

Master Identity and Q-operators

- any class function of

is generating function (super)-characters of symmetric irreps

s

V.K., Leurent,Tsuboi

• level 1 of nesting: T-operators, removed:

Definition of T- and Q-operators

• Level 0 of nesting: transfer-matrix -

• Nesting - Backlund flow: consequtive « removal » of eigenvalues from

Bazhanov,FrassekLukowski,MineghelliStaudacher

• Definition of Q-operators at 1-st level:

For recent alternative approach see

• All T and Q operators commute at any level and act in the same quantum space

Q-operator -

TQ and QQ relations

• Generalizing to any level: « removal » of a subset of eigenvalues

• Operator TQ relation at a level characterized by a subset

• They generalize a relation among characters, e.g.

• Other generalizations: TT relations at any irrep

• From Master identity - the operator Backlund TQ-relation on first level.

notation:

“bosonic”

“fermionic”

QQ-relations (Plücker id., Weyl symmetry…)

bosonic

fermionic

• Example: gl(2|2)

TsuboiV.K.,Sorin,ZabrodinGromov,VieiraTsuboi,Bazhanov

• E.g.

Hasse diagram

Kac-Dynkin dyagram

Wronskians and Bethe equations

• Nested Bethe eqs. from QQ-relations at a nesting step

• All 2K+M Q functions can be expressed through K+M single index Q’sby Wronskian (Casarotian) determinants:

“bosonic” Bethe eq.

“fermionic” Bethe eq.

- polynomial

- polynomial

• All the operatorial TQ and QQ relations are proven from the Master identity!

Determinant formulas and Hirota equation• Jacobi-Trudi formula for general gl(K|M) irrep λ={λ1,λ2,…,λa}

• Generalization to fusion for quantum T-matrix : Bazhanov,ReshetikhinCherednik

V.K.,Vieira• It is proven using Master identity; generalized to super-case, twist

a

s

(K,M)

λ1

λ2

λa (a,s) fat hook

• Boundary conditions for Hirota eq.: gl(K|M) representations in “fat hook”:

• Hirota equation for rectangular Young tableaux follows from BR formula:

• Hirota eq. can be solved in terms of Wronskians of Q

Krichever,Lipan,Wiegmann,Zabrodin Bazhanov,TsuboiTsuboi

• We will see now examples of these wronskians for sigma models…..

“Toy” model: SU(N)L x SU(N)R principal chiral field

• Asymptotically free theory with dynamically generated mass• Factorized scattering• S-matrix is a direct product of two SU(N) S-matrices (similar to AdS/CFT).• Result from TBA for finite size: Y-system

a

s

Polyakov, WiegmannFaddeev,ReshetikhinFateev, OnofriFateev,V.K.,WiegmannBalog,Hegedus

• Energy:

Inspiring example: SU(N) principal chiral field at finite volume

• General Wronskian solution in a strip:Krichever,Lipan,Wiegmann,Zabrodin

Gromov,V.K.,VieiraV.K.,Leurent• Y-system Hirota dynamics in a strip of width N in (a,s) plane.

polynomialsfixing a state

jumps by

a

s

• Finite volume solution: define N-1 spectral densities

• well defined in analyticity strip

• For s=-1, the analyticity strip shrinks to zero, giving Im parts of resolvents:

• N-1 middle node Y-eqs. after inversion of difference operator and fixing the zero mode (first term) give N-1 eqs.for spectral densities

Solution of SU(N)L x SU(N)R principal chiral field at finite size

Beccaria , Macorini

Numerics for low-lying states N=3

V.K.,Leurent

• Infinite Y-system reduced to a finite number of non-linear integral equations (a-la Destri-deVega)

• Significantly improved precision for SU(2) PCF

Y-system for AdS CFT and Wronskian solution

Exact one-particle dispersion relation

• Exact one particle dispersion relation: Santambrogio,ZanonBeisert,Dippel,StaudacherN.Dorey

• Bound states (fusion!)

• Parametrization for the dispersion relation (mirror kinematics):

Cassical spectral parameter related to quantum one by Zhukovsky map

cuts in complex -plane

Y-system for excited states of AdS/CFT at finite size

T-hook

• Complicated analyticity structure in u dictated by non-relativistic dispersion

Gromov,V.K.,Vieira

• Extra equation (remnant of classical monodromy):

cuts in complex -plane

• Knowing analyticity one transforms functional Y-system into integral (TBA):Gromov,V.K.,VieiraBombardelli,Fioravanti,TateoGromov,V.K.,Kozak,VieiraArutyunov,FrolovCavaglia, Fioravanti, Tateo

• obey the exact Bethe eq.:

• Energy : (anomalous dimension)

Konishi operator : numerics from Y-system

Gromov,V.K.,Vieira

Frolov

Beisert,Eden,Staudacher

Plot from:Gromov, V.K., Tsuboi

Y-system and Hirota eq.: discrete integrable dynamics

• Relation of Y-system to T-system (Hirota equation) (the Master Equation of Integrability!)

• Discrete classical integrable Hirota dynamics for AdS/CFT!

For spin chains :Klumper,PearceKuniba,Nakanishi,SuzukiFor QFT’s:Al.ZamolodchikovBazhanov,Lukyanov,A.Zamolodchikov

Gromov,V.K.,Vieira

Y-system looks very “simple” and universal! • Similar systems of equations in all known integrable σ-models

• What are its origins? Could we guess it without TBA?

Super-characters: Fat Hook of U(4|4) and T-hook of SU(2,2|4)

∞ - dim. unitary highest weight representations of u(2,2|4) ! KwonCheng,Lam,ZhangGromov, V.K., Tsuboi

SU(2,2|4)

a

s

Generating function for symmetric representations:

Amusing example: u(2) ↔ u(1,1)

SU(4|4)

a

s

a

s s

a

Solving full quantum Hirota in U(2,2|4) T-hook Tsuboi

HegedusGromov, V.K., Tsuboi

• Replace gen. function:

• Parametrization in Baxter’s Q-functions:

by a generating functional

• One can construct the Wronskian determinant solution: all T-functions (and Y-functions) in terms of 7 Q-functions

Gromov, V.K., Leurent, Tsuboi

- expansion in

• Replace eigenvalues by functions of spectral parameter:

Wronskian solution of AdS/CFT Y-system in T-hook

Gromov,Tsuboi,V.K.,Leurent

For AdS/CFT, as for any sigma model…

• (Super)spin chains can be entirely diagonalized by a new method, using the operatorial Backlund procedure, involving (well defined) Q operators

• The underlying Hirota dynamics solved in terms of wronskian determinants of Q functions (operators)

• Application of Hirota dynamics in sigma models. Analyticity in spectral parameter u is the most difficult part of the problem.

• Principal chiral field sets an example of finite size spectrum calculation via Hirota dynamics

• The origins of AdS/CFT Y-system are entirely algebraic: Hirota eq. for characters in T-hook. Analuticity in u is complicated

Some progress is being made…GromovV.K.LeurentVolinTsuboi

END