1. 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 LeurentarXiv:1010.2720 ZengoTsuboi arXiv:1002.3981

2. 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

3. Fused R-matrix in any irrep λ of gl(K|M) 0 = u v u v 0 u 0 “ ” “f” fundamental irrep “f” in quantum space any “ “= {     a } irrep auxiliary space generatormatrix element in irrep Yang-Baxter relations

4. Co-derivative Definition, where nice representation for R-matrix follows: V.K., Vieira Super-case: From action on matrix element

5. 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:

6. V.K., Vieira V.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

7. 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, Frassek Lukowski, Mineghelli Staudacher 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 -

8. 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”

9. QQ-relations (Plücker id., Weyl symmetry…) bosonic fermionic Example: gl(2|2) Tsuboi V.K.,Sorin,Zabrodin Gromov,Vieira Tsuboi,Bazhanov E.g. Hasse diagram Kac-Dynkin dyagram

10. Wronskians and Bethe equations Nested Bethe eqs. from QQ-relations at a nesting step All 2 K+M Q functions can be expressed through K+M single index Q’s by Wronskian (Casarotian) determinants: “bosonic” Bethe eq. “fermionic” Bethe eq. - polynomial All the operatorial TQ and QQ relations are proven from the Master identity!

11. Determinant formulas and Hirota equation Jacobi-Trudi formula for general gl(K|M) irrep λ={λ 1,λ 2,…,λ a } Generalization to fusion for quantum T-matrix : Bazhanov,Reshetikhin Cherednik V.K.,Vieira It is proven using Master identity; generalized to super-case, twist a s (K,M) λ1λ1 λ2λ2 λaλa (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,Tsuboi Tsuboi We will see now examples of these wronskians for sigma models…..

12. “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, Wiegmann Faddeev,Reshetikhin Fateev, Onofri Fateev,V.K.,Wiegmann Balog,Hegedus Energy:

13. Inspiring example: SU(N) principal chiral field at finite volume General Wronskian solution in a strip: Krichever,Lipan, Wiegmann,Zabrodin Gromov,V.K.,Vieira V.K.,Leurent Y-system Hirota dynamics in a strip of width N in (a,s) plane. polynomials fixing 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:

14. 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

15. Y-system for AdS CFT and Wronskian solution

16. Exact one-particle dispersion relation Exact one particle dispersion relation: Santambrogio,Zanon Beisert,Dippel,Staudacher N.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

17. 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.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov Cavaglia, Fioravanti, Tateo obey the exact Bethe eq.: Energy : (anomalous dimension)

18. Konishi operator : numerics from Y-system Gromov,V.K.,Vieira Frolov Beisert,Eden,Staudacher Plot from: Gromov, V.K., Tsuboi

19. 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,Pearce Kuniba,Nakanishi,Suzuki For QFT’s: Al.Zamolodchikov Bazhanov,Lukyanov,A.Zamolodchikov Gromov,V.K.,Vieira

20. 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?

21. 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) ! Kwon Cheng,Lam,Zhang Gromov, 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

22. Solving full quantum Hirota in U(2,2|4) T-hook Tsuboi Hegedus Gromov, 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:

23. Wronskian solution of AdS/CFT Y-system in T-hook Gromov,Tsuboi,V.K.,Leurent

24. 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

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