q-operators and discrete hirota dynamics for spin chains and sigma models
DESCRIPTION
Workshop, “ `From sigma models to 4D CFT ” DESY, Hamburg, 1 December 2010. Q-operators and discrete Hirota dynamics for spin chains and sigma models. Vladimir Kazakov (ENS,Paris). with Nikolay Gromov arXiv:1010.4022 Sebastien Leurent arXiv:1010.2720 - PowerPoint PPT PresentationTRANSCRIPT
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