Hirota Dynamics of Quantum Integrability

1. “Facets of Integrability: Random Patterns, Stochastic Processes, Hydrodynamics, Gauge Theories and Condensed Matter Systems” Simons institute, January 21-27, 2013 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 Krichever,Lipan, Wiegmann, Zabrodin 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 T-system (from -system): • ∞ - dim. unitary highest weight representations of the “T-hook” ! a -hook Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi s • Full quantum Hirota equation: extra variable – spectral parameter • Classical limit: eq. for characters as functions 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 mKPhierachy: • 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: 1 • Q atlevelzero of nesting • Definition of Q-operatorsat 1-st level of nesting: • « removal » of an eigenvalue (example for gl(N)): Def: complimentary set • Nextlevels: 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 Hassediagram: 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

9. Krichever,Lipan, Wiegmann,Zabrodin 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 - and -forms): a • For su(N) spin chain (half-strip) we impose: definition: s

10. Solution of Hirota in fat hook and T-hook Tsuboi V.K.,Leurent,Volin • Bosonic and fermionic 1-(sub)forms (all anticomute): a -hook • Wronskian solution for the fat hook: a s λa • Similar Wronkian solution exists in -hook λ2 λ1 s

11. Inspiring example: principal chiral field (PCF) Zamolodchikov&Zamolodchikov Karowski Wiegmann • It is known since long to be integrable: • S-matrix of types of physical particles • A limiting case of Thirring model, or WZNW model • Asymptotic Bethe ansatz constructed. • Interesting explicit large solution at finite density Polyakov, Wiegmann; Wiegmann Fateev, V.K., Wiegmann • Finite : TBA → Y-system →Hirota dynamics • in a in (a,s) planein a band • Known asymptotics of Y-functions Wiegmann, Tsevlik Al. Zamolodchikov -plane a • Analyticity strips of from asympotics • is analytic inside the strip s

12. Gromov, V.K., Vieira V.K., Leurent Finite volume solution of principal chiral field Alternative approach: Balog, Hegedus • UseWronskian solution in terms of 2Q-functions • It is crucial to know their analyticity properties. The following choice appears • to render the right analyticity strips of Y- and T-functions: density at analyticity boundary analytic in the upper half-plane polynomials fixing a state (for vacuum ) • From reality of Y-functions: analytic in the lower half-plane -plane -plane • nonlinear integral equations on spectral densities can be obtained • e.g. from the condition of left-right symmetry • true for symmetric states (can be generalized to any state) • We obtain a finite system of NLIE (somewhat similar to Destri-deVegaeqs.) • Good for analytic study at large or small volume and for numerics at any

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

14. Planar N=4 SYM – integrable 4D QFT Maldacena Gubser, Polyakov, Klebanov Witten • 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 via integrable spin chain dual to integrable sigma model Minahan, Zarembo Bena,Roiban,Polchinski Beisert,Kristjanssen,Staudacher V.K.,Marchakov,Minahan,Zarembo Beisert, Eden,Staudacher Janik • 4D Correlators: scaling dimensions non-trivial functions of ‘tHooft coupling λ! structure constants They describe the whole 4D conformal theory via operator product expansion

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

16. 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

17. 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 Alternative approach: Balog, Hegedus

18. 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: • Wronskian solution for the right band in terms of two Q-functions with one magic cut on ℝ parameterized by a polynomial and two spectral densities

19. Parameterization of the upper band: continuation • Remarkably, choosing the upper band Q-functions analytic in a half-plane • we get all T-functions with the right analyticity strips! • All Q’s in the upper band of T-hook can be parametrized by 2 densities.

20. 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 • The states/operators are fixed by introducing certain zeros and poles • to Y-functions, and hence to T- and Q-functions (exact Bethe roots). • 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

21. 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 transcendentalitiescan be summed at all orders: Leurent, Volin ‘12 • Confirmed up to 5 loops by direct graph calculus (6 loops promised) Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Eden,Heslop,Korchemsky,Smirnov,Sokatchev

22. Numerics and 3-loops from string quasiclassics for twist-J operators of spin S • 3 leading strong coupling terms were calculated: • for Konishi operator • or even • They perfectly reproduce the TBA/Y-system or FiNLIEnumerics 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 • AdS/CFT Y-system passes all known tests

23. Conclusions • Hirota integrable 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 • Hirota dynamics for structure constants of OPE and correlators? • Why is N=4 SYM integrable? Can integrability be used to prove AdS/CFT correspondence? Correa, Maldacena, Sever, Drukker Gromov, Sever Recent advances: Gromov, Sever, Vieira, Kostov, Serban, Janik etc.

24. Happy Birthday Pasha!СЮБИЛЕЕМ, ПАША!