Hirota Dynamics of Quantum Integrability Vladimir Kazakov (ENS, Paris) “ Facets of Integrability: Random Patterns, Stochastic Processes, Hydrodynamics, Gauge Theories and Condensed Matter Systems” Simons institute, January 21-27, 2013 Collaborations with Alexandrov, Gromov, Leurent, Tsuboi, Vieira, Volin, Zabrodin

New uses of Hirota dynamics in integrability 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) Gromov, V.K., Vieira V.K., Leurent Gromov, V.K. Vieira Gromov, Volin, V.K., Leurent V.K., Leurent, Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin Miwa,Jimbo Sato Kluemper, Pierce Kuniba,Nakanishi,Suzuki Al.Zamolodchikov Bazhanov,Lukyanov, A.Zamolodchikov Krichever,Lipan, Wiegmann, Zabrodin

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

=+ a sss-1 s+1 a-1 a+1 (Super-)group theoretical origins of Y- and T-systems  A curious property of gl(N|M) representations with rectangular Young tableaux:  For characters – simplified Hirota eq.: Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi  Full quantum Hirota equation: extra variable – spectral parameter  Classical limit: eq. for characters as functions of classical monodromy Gromov,V.K.,Tsuboi s a

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

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

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

Hasse diagram and QQ-relations (Plücker id.) - bosonic QQ-rel. gl(2|2) example: classification of all Q-functions Tsuboi V.K.,Sorin,Zabrodin Tsuboi,Bazhanov Nested Bethe ansatz equations follow from polynomiality of along a nesting path All Q’s expressed through a few basic ones by determinant formulas Hasse diagram: hypercub E.g. - fermionic QQ rel.

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: -form encodes all Q-functions with indices: Solution of Hirota equation in a strip (via arbitrary - and -forms): a s For su(N) spin chain (half-strip) we impose: E.g. for gl(2) : Krichever,Lipan, Wiegmann,Zabrodin Gromov,V.K.,Leurent,Volin definition:

Tsuboi V.K.,Leurent,Volin s a Bosonic and fermionic 1-(sub)forms (all anticomute): a s λ1λ1 λ2λ2 λaλa

Inspiring example: principal chiral field (PCF) a s Wiegmann, Tsevlik Al. Zamolodchikov Polyakov, Wiegmann; Wiegmann Fateev, V.K., Wiegmann -plane Zamolodchikov&Zamolodchikov Karowski Wiegmann

Finite volume solution of principal chiral field From reality of Y-functions: Gromov, V.K., Vieira V.K., Leurent Alternative approach: Balog, Hegedus -plane analytic in the upper half-plane analytic in the lower half-plane density at analyticity boundary

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

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

Spectral AdS/CFT Y-system Gromov,V.K.,Vieira cuts in complex -plane Extra “corner” equations: L→∞ Zhukovsky map:

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

Solution of AdS/CFT T-system in terms of finite number of non-linear integral equations (FiNLIE) 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 Gromov,V.K.,Leurent,Volin Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz) Alternative approach: Balog, Hegedus Inspired by: Bombardelli, Fioravanti, Tatteo Operators/states of AdS/CFT are characterized by certain poles and zeros of Y- and T-functions fixed by exact Bethe equations:

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

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.

Closing FiNLIE: sawing together 3 bands  FiNLIE perfectly reproduces earlier results obtained from Y-system (in TBA form). It is a perfect mean to generate weak and strong coupling expansions of anomalous dimensions in N=4 SYM Dimension can be extracted from the asymptotics: 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).

Konishi dimension to 8-th order Last term is a new structure – multi-index zeta function. Leading transcendentalities can be summed at all orders: Bajnok,Janik Leurent,Serban,Volin Bajnok,Janik,Lukowski Lukowski,Rej, Velizhanin,Orlova Leurent, Volin ’12 (from FiNLIE) Confirmed up to 5 loops by direct graph calculus (6 loops promised) Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Eden,Heslop,Korchemsky,Smirnov,Sokatchev Leurent, Volin ‘12 Integrability allows to sum exactly enormous number of Feynman diagrams of N=4 SYM

Numerics and 3-loops from string quasiclassics for twist-J operators of spin S Gromov,Shenderovich, Serban, Volin Roiban, Tseytlin Vallilo, Mazzucato Gromov, Valatka 3 leading strong coupling terms were calculated: for Konishi operator or even They perfectly reproduce the TBA/Y-system or FiNLIE numerics Gromov, Valatka Gubser, Klebanov, Polyakov Y-system numerics Gromov,V.K.,Vieira Frolov Gromov,Valatka  AdS/CFT Y-system passes all known tests

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 AdS 5 /CFT 4 ? 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.

Happy Birthday Pasha! С ЮБИЛЕЕМ, ПАША!