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

1. 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 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. “f” “l” u 0 u u = v v 0 0 Fused R-matrix in any irrepλ of gl(K|M) fundamental irrep “f” in quantum space any “l“= {l1,l2,..., la} irrep auxiliary space generator matrix element in irrep l Yang-Baxter relations

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

5. 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

6. V.K., Vieira V.K., Leurent,Tsuboi Master Identity and Q-operators is generating function (super)-characters of symmetric irreps (previous particular case ) - any class function of s Grafical representation (slightly generalized to any spectral parameters)

7. For recent alternative approach see Definition of T- and Q-operators Bazhanov, Frassek Lukowski, Mineghelli Staudacher V.K., Leurent,Tsuboi • 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

8. 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

9. 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

10. 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!

11. Determinant formulas and Hirota equation • Jacobi-Trudi formula for generalgl(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: a • Boundary conditions for Hirota eq.: gl(K|M) representations in “fat hook”: λa Krichever,Lipan, Wiegmann,Zabrodin Bazhanov,Tsuboi Tsuboi • Hirota eq. can be solved • in terms of Wronskians of Q (a,s) fat hook (K,M) • We will see now examples of these • wronskians for sigma models….. λ2 λ1 s

12. “Toy” model: SU(N)L x SU(N)Rprincipal 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:

13. Inspiring example: SU(N) principal chiral field at finite volume Gromov,V.K.,Vieira V.K.,Leurent • Y-system Hirota dynamics in a strip of width N in (a,s) plane. Krichever,Lipan, Wiegmann,Zabrodin • General Wronskian solution in a strip: • Finite volume solution: define N-1 spectral densities jumps by polynomials fixing a state • well defined in analyticity strip a • For s=-1, the analyticity strip shrinks to zero, giving Im parts of resolvents: s

14. Solution of SU(N)L x SU(N)Rprincipal 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 Numerics for low-lying states N=3 • Infinite Y-system reduced to a finite • number of non-linear integral equations • (a-la Destri-deVega) V.K.,Leurent • Significantly improved precision for SU(2) PCF Beccaria , Macorini

15. Y-system for AdS CFT and Wronskian solution

16. 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 -plane • Parametrization for the dispersion relation (mirror kinematics):

17. Y-system for excited states of AdS/CFT atfinite size Gromov,V.K.,Vieira T-hook • Complicated analyticity structure in u • dictated by non-relativistic dispersion • Extra equation (remnant of classical monodromy): • Energy : • (anomalous dimension) • obey the exact Bethe eq.: 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

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!) 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

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) • Generating function for symmetric representations: SU(4|4) SU(2,2|4) a a s ∞ - dim. unitary highest weight representations of u(2,2|4) ! Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi s a • Amusing example: u(2) ↔ u(1,1) a s s

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

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 operatorialBacklund 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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