Hirota Dynamics of Quantum Integrability

1. “Round Table: Frontiers of Mathematical Physics” Dubna, December 16-18, 2012 Hirota Dynamics of Quantum Integrability Vladimir Kazakov (ENS, Paris) Collaborations with Alexandrov, Gromov, Leurent, Tsuboi, Vieira, Volin, Zabrodin

2. New uses of Hirota dynamics in integrability Miwa,Jimbo Sato • Hirota integrable dynamics incorporates the basic properties of all quantum and classical integrable systems. • It generates all integrable hierarchies of PDE’s (KdV, KP, Toda etc) • Discrete Hirota eq. (T-system) is an alternative approach to quantum integrable systems. • Classical KP hierarchy applies to quantum T- and Q-operators of (super)spin chains • Framework for new approach to solution of integrable 2D quantum sigma-models in finite volume using Y-system, T-system, Baxter’s Q-functions, Plücker QQ identities, wronskian solutions,… + Analyticity in spectral parameter! • First worked out for spectrum of relativistic sigma-models, such as su(N)×su(N) principal chiral field (PCF), Sine-Gordon, Gross-Neveu • Provided the complete solution of spectrum of anomalous dimensions of 4D N=4 SYM theory! AdS/CFT Y-system, recently reduced to a finite system of non-linear integral eqs (FiNLIE) Kluemper, Pierce Kuniba,Nakanishi,Suzuki Al.Zamolodchikov Bazhanov,Lukyanov, A.Zamolodchikov V.K., Leurent, Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin Gromov, V.K., Vieira V.K., Leurent Gromov, V.K. Vieira Gromov, Volin, V.K., Leurent

3. Discrete Hirota eq.: T-system and Y-system • Based on a trivial property of Kronecker symbols (and determinants): • T-system (discrete Hirota eq.) • Y-system • Gauge symmetry

4. (Super-)group theoretical origins of Y- and T-systems • A curious property of gl(N|M) representations with rectangular Young tableaux: a-1 = + a a+1 s s s-1 s+1 • For characters – simplified Hirota eq.: • Boundary conditions for Hirota eq. for AdS/CFT T-system: • ∞ - dim. unitary highest weight representations of u(2,2|4)in “T-hook” ! a U(2,2|4) Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi s V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo • Full quantum Hirota equation • Classical (strong coupling) limit: eq. for characters of classical monodromy Gromov,V.K.,Tsuboi

5. Quantum (super)spin chains • Quantum transfer matrices – a natural generalization of group characters V.K., Vieira • Co-derivative – left differential w.r.t. group (“twist”) matrix: Main property: • Transfer matrix (T-operator) of L spins R-matrix • Hamiltonian of Heisenberg quantum spin chain:

6. V.K.,Vieira V.K., Leurent,Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin Master T-operator and mKP • Generating function of characters: • Master T-operator: • Master T is a tau function of mKP hierachy: • mKP charge is spectral parameter! T is polynomial w.r.t. considered by Krichever • Satisfies canonical mKP Hirota eq. • Hence - discrete Hirota eq. for T in rectangular irreps: • Baxter’s TQ relations, Backlund transformations etc. • Commutativity and conservation laws

7. s Baxter’s Q-operators V.K., Leurent,Tsuboi • Generating function for (super)characters of symmetric irreps: • Q at level zero of nesting • Definition of Q-operatorsat 1-st level of nesting: • « removal » of an eigenvalue (example for gl(N)): Def: complimentary set • Next levels: multi-pole residues, or « removing » more of eignevalues: Alternative approaches: Bazhanov, Lukowski, Mineghelli Rowen Staudacher Derkachev, Manashov • Nesting (Backlund flow): consequtive « removal » of eigenvalues

8. Tsuboi V.K.,Sorin,Zabrodin Tsuboi,Bazhanov Hasse diagram and QQ-relations (Plücker id.) • gl(2|2) example: classification of all Q-functions Hasse diagram: hypercub • E.g. - bosonic QQ-rel. - fermionic QQ rel. • Nested Bethe ansatz equations follow from polynomiality of along a nesting path • All Q’s expressed through a few basic ones by determinant formulas • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s

9. Krichever,Lipan, Wiegmann,Zabrodin Tsuboi Wronskian solutions of Hirota equation • We can solve Hirota equations in a band of width N in terms of • differential forms of 2N functions • Solution combines dynamics of gl(N) representations and quantum fusion: Gromov,V.K.,Leurent,Volin • -form encodes all Q-functions with indices: • E.g. for gl(2) : • Solution of Hirota equation in a strip (via arbitrary Q- and P-forms): a • For su(N) spin chain (half-strip) we impose: s

10. Inspiring example: principal chiral field Gromov, V.K., Vieira V.K., Leurent • Y-system Hirota dynamics in a in (a,s) plane. • We know the Wronskian solution in terms of Q-functions • Finite volume solution: finite system of NLIE, • parameterization fixing the analytic structure. • Analyticity strips from large volume asymptotics: -plane a jumps by polynomials fixing a state • From reality: • N-1 TBA equations (for central nodes) on spectral densities s Alternative approach: Balog, Hegedus

11. SU(3) PCF numerics V.K.,Leurent’09 E / 2 mass gap ground state L

12. Wronskian solution of u(2,2|4) T-system in T-hook Gromov,V.K.,Tsuboi Gromov,Tsuboi,V.K.,Leurent Tsuboi definitions: Plücker relations express all 256 Q-functions through 8 independent ones

13. Planar N=4 SYM – integrable 4D QFT • 4D superconformal QFT! Global symmetry PSU(2,2|4) • AdS/CFT correspondence – duality to Metsaev-Tseytlin superstring • Integrable for non-BPS states, summing genuine 4D Feynman diagrams! • Operators in 4D • 4D Correlators: scaling dimensions non-trivial functions of ‘tHooft coupling λ! structure constants They describe the whole 4D conformal theory via operator product expansion

14. Spectral AdS/CFT Y-system Gromov,V.K.,Vieira cuts in complex -plane T-hook • Analyticity from large L symptotics: from one-particle dispersion relation: L→∞ Zhukovsky map: • Extra “corner” equations:

15. Gromov,V.K.,Leurent,Volin Solution of AdS/CFT T-system in terms of finite number of non-linear integral equations (FiNLIE) • Main tools: integrable Hirota dynamics + analyticity • (inspired by classics and asymptotic Bethe ansatz) • No single analyticity friendly gauge for T’s of right, left and upper bands. • We parameterize T’s of 3 bands in different, analyticity friendly gauges, • also respecting their reality and certain symmetries. • Quantum analogue of classical symmetry: can be analytically continued on special magic sheet in labels • Operators/states of AdS/CFT are characterized by certain poles and zeros • of Y- and T-functions fixed by exact Bethe equations: Inspired by: Bombardelli, Fioravanti, Tatteo Balog, Hegedus Alternative approach: Balog, Hegedus

16. Magic sheet and solution for the right band • The property • suggests that certain T-functions are much simpler • on the “magic” sheet, with only short cuts: • Only two cuts left on the magic sheet for ! • Right band parameterized: by a polynomial S(u), a gauge function • with one magic cut on ℝ and a density

17. Parameterization of the upper band: continuation • Remarkably, choosing the q-functions analytic in a half-plane • we get all T-functions with the right analyticity strips! • We parameterize the upper band of T-hook in terms of a spectral densities. • The rest of Q’s restored from Plucker QQ relations

18. Closing FiNLIE: sawing together 3 bands • Finally, we can close the FiNLIE system by using reality of T-functions • and certain symmetries. For example, for left-right symmetric states • Dimension can be extracted from the asymptotics: • FiNLIE perfectly reproduces earlier results obtained • fromY-system (in TBA form). It is a perfect mean to generate • weak and strong coupling expansions of anomalous dimensions • in N=4 SYM

19. Konishi dimension to 8-th order • Integrability allows to sum exactly enormous number of Feynman diagrams of N=4 SYM Bajnok,Janik Leurent,Serban,Volin Bajnok,Janik,Lukowski Lukowski,Rej, Velizhanin,Orlova Leurent, Volin ’12 (from FiNLIE) • Last term is a new structure – multi-index zeta function. • Leading transcendentalities can be summed at all orders: Leurent, Volin ‘12 • Confirmed up to 5 loops by direct graph calculus Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Eden,Heslop,Korchemsky,Smirnov,Sokatchev

20. Numerics and 3-loops from string quasiclassics for twist-J operators of spin S • Perfectly reproduces 3 terms of Y-system numerics • for Konishi operator • or even Y-system numerics Gromov,V.K.,Vieira Frolov Gromov,Valatka Gubser, Klebanov, Polyakov Gromov, Valatka Gromov,Shenderovich, Serban, Volin Roiban, Tseytlin Vallilo, Mazzucato Gromov, Valatka • Numerics uses the TBA or FiNLIE forms of Y-system • AdS/CFT Y-system passes all known tests Gromov, V.K., Vieira Cavaglia, Fioravanti, Tatteo Arutyunov, Frolov Gromov, V.K., Leurent, Volin

21. Conclusions • Hirotaintegrable dynamics, supplied by analyticity in spectral parameter, is a powerful method of solving integrable 2D quantum sigma models. • For spin chains (mKP structure): a curious alternative to the algebraic Bethe ansatz of Leningrad school • Y-system for sigma-models can be reduced to a finite system of non-linear integral eqs (FiNLIE) in terms of Wronskians of Q-functions. • For the spectral problem in AdS/CFT, FiNLIE represents the most efficient way for numerics and weak/strong coupling expansions. • Recently Y-system and FiNLIE used to find quark-antiquark potential in N=4 SYM Future directions • Better understanding of analyticity of Q-functions. Quantum algebraic curve for AdS5/CFT4 ? • BFKL limit from Y-system and FiNLIE • Why is N=4 SYM integrable? Can integrability be used to prove AdS/CFT correspondence? Correa, Maldacena, Sever, Drukker Gromov, Sever

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