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
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Q-operators and discrete Hirota dynamics for spin chains and sigma models Vladimir Kazakov (ENS,Paris) Workshop, “`From sigma models to 4D CFT ” DESY,
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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.
• 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
• 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)