Transcript Kazakov - From Sigma Models to Four-dimensional QFT

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
Sebastien Leurent
ZengoTsuboi
arXiv:1010.4022
arXiv:1010.2720
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)
fundamental irrep “f” in quantum space
“f”
any “l“= {l1,l2,..., la}
irrep auxiliary space
“l”
u
0
generator
matrix element
in irrep l
u
u
=
v
v
0
0
Yang-Baxter relations
Co-derivative
V.K., Vieira
• Definition
• Super-case:
• From action on matrix element
nice representation for R-matrix follows:
,
where
Transfer matrix in terms of left co-derivative
• Monodromy matrix of the spin chain:
• Transfer-matrix without spins:
• Transfer-matrix of one spin:
• Transfer-matrix of N spins
Master Identity and Q-operators
V.K., Vieira
V.K., Leurent,Tsuboi
is generating function (super)-characters
of symmetric irreps
s
- any class function of
Grafical representation (slightly generalized
to any spectral parameters)
(previous particular case
)
Definition of T- and Q-operators
V.K., Leurent,Tsuboi
For recent alternative
approach see
Bazhanov,
Frassek
Lukowski,
Mineghelli
Staudacher
• Nesting - Backlund flow: consequtive « removal » of eigenvalues from
•
Level 0 of nesting: transfer-matrix Q-operator
•
level 1 of nesting:
-
T-operators,
removed:
• Definition of Q-operators at 1-st level:
•
All T and Q operators commute at any level and act in the same quantum space
TQ and QQ relations
• From Master identity - the operator Backlund TQ-relation on first level.
notation:
• Generalizing to any level: « removal » of a subset of eigenvalues
• Operator TQ relation at a level characterized by a subset
“bosonic”
“fermionic”
• They generalize a relation among characters, e.g.
•
Other generalizations: TT relations at any irrep
QQ-relations (Plücker id., Weyl symmetry…)
• Example: gl(2|2)
Hasse diagram
Kac-Dynkin dyagram
• E.g.
bosonic
fermionic
Tsuboi
V.K.,Sorin,Zabrodin
Gromov,Vieira
Tsuboi,Bazhanov
Wronskians and Bethe equations
• Nested Bethe eqs. from QQ-relations at a nesting step
“bosonic” Bethe eq.
- polynomial
“fermionic” Bethe eq.
- polynomial
• All 2K+M Q functions can be expressed through K+M single index Q’s
by Wronskian (Casarotian) determinants:
• 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,Reshetikhin
Cherednik
• It is proven using Master identity; generalized to super-case, twist
V.K.,Vieira
• Hirota equation for rectangular Young tableaux follows from BR formula:
• Boundary conditions for Hirota eq.: gl(K|M) representations in “fat hook”:
Krichever,Lipan,
a
• Hirota eq. can be solved
Wiegmann,Zabrodin
λa
(a,s)
fat hook
in terms of Wronskians of Q
(K,M)
λ2
λ1
Bazhanov,Tsuboi
Tsuboi
• We will see now examples of these
wronskians for sigma models…..
s
“Toy” model: SU(N)L x SU(N)R principal chiral field
Polyakov, Wiegmann
Faddeev,Reshetikhin
Fateev, Onofri
Fateev,V.K.,Wiegmann
Balog,Hegedus
• 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
• Energy:
Inspiring example: SU(N) principal chiral field at finite volume
• Y-system
Hirota dynamics in a strip of width N in (a,s) plane.
• General Wronskian solution in a strip:
Krichever,Lipan,
Wiegmann,Zabrodin
Gromov,V.K.,Vieira
V.K.,Leurent
a
s
• Finite volume solution: define N-1 spectral densities
polynomials
fixing a state
•
jumps
by
well defined in analyticity strip
• For s=-1, the analyticity strip shrinks to zero, giving Im parts of resolvents:
Solution of SU(N)L x SU(N)R principal chiral field at finite size
• 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
• Infinite Y-system reduced to a finite
number of non-linear integral equations
(a-la Destri-deVega)
Numerics for
low-lying states N=3
V.K.,Leurent
• Significantly improved precision for SU(2) PCF
Beccaria , Macorini
Y-system for AdS CFT and Wronskian solution
Exact one-particle dispersion relation
Santambrogio,Zanon
Beisert,Dippel,Staudacher
N.Dorey
• Exact one particle dispersion relation:
• Bound states
(fusion!)
 Cassical spectral parameter
related to quantum one by Zhukovsky map
cuts in complex
• Parametrization for the dispersion relation (mirror kinematics):
-plane
Y-system for excited states of AdS/CFT at finite size
Gromov,V.K.,Vieira
T-hook
• Complicated analyticity structure in u
dictated by non-relativistic dispersion
cuts in complex
-plane
• Extra equation (remnant of
classical
monodromy):
• Energy :
(anomalous dimension)
•
obey the exact Bethe eq.:
• Knowing analyticity one transforms functional Y-system into integral (TBA):
Gromov,V.K.,Vieira
Bombardelli,Fioravanti,Tateo
Gromov,V.K.,Kozak,Vieira
Arutyunov,Frolov
Cavaglia, Fioravanti, Tateo
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!)
Gromov,V.K.,Vieira
• Discrete classical integrable Hirota dynamics for AdS/CFT!
For spin chains :
Klumper,Pearce
Kuniba,Nakanishi,Suzuki
For QFT’s:
Al.Zamolodchikov
Bazhanov,Lukyanov,A.Zamolodchikov
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)
 Generating function for symmetric representations:
SU(2,2|4)
SU(4|4)
a
a
s
s
∞ - dim. unitary highest weight representations of u(2,2|4) !
 Amusing example:
a
u(2)
↔ u(1,1)
s
a
Kwon
Cheng,Lam,Zhang
Gromov, V.K., Tsuboi
s
Solving full quantum Hirota in U(2,2|4) T-hook
• Replace eigenvalues by functions of spectral parameter:
• Replace gen. function:
by a generating functional
- expansion in
• Parametrization in Baxter’s Q-functions:
Gromov, V.K., Leurent, Tsuboi
• One can construct the Wronskian determinant solution:
all T-functions (and Y-functions) in terms of 7 Q-functions
Tsuboi
Hegedus
Gromov, V.K., Tsuboi
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…
Gromov
V.K.
Leurent
Volin
Tsuboi
END