“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 New uses of Hirota dynamics in integrability • Hirota integrable dynamics incorporates the basic properties of all Miwa,Jimbo quantum and classical integrable systems. Sato • It generates all integrable hierarchies of PDE’s (KdV, KP, Toda etc) Pierce • Discrete Hirota eq. (T-system) is an alternative Kluemper, Kuniba,Nakanishi,Suzuki Al.Zamolodchikov approach to quantum integrable systems. Bazhanov,Lukyanov, A.Zamolodchikov Krichever,Lipan, Wiegmann, Zabrodin • Classical KP hierarchy applies to quantum V.K., Leurent, Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin 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,… + Gromov, V.K., Vieira V.K., Leurent 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 Gromov, Volin, V.K., Leurent 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 (Super-)group theoretical origins of Y- and T-systems A curious property of gl(N|M) representations with rectangular Young tableaux: = a s s + s-1 a-1 a+1 s+1 For characters – simplified Hirota eq.: Boundary conditions for Hirota eq. for 𝑈 𝐾1 , 𝐾2 𝑀 T-system (from 𝜒-system): ∞ - dim. unitary highest weight representations of the “T-hook” ! 𝑀 a 𝐾2 𝐾1 , 𝐾2 𝑀 -hook Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi 𝐾1 s Full quantum Hirota equation: extra variable – spectral parameter Classical limit: eq. for characters as functions of classical monodromy Gromov,V.K.,Tsuboi 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: 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. Satisfies canonical mKP Hirota eq. considered by Krichever Hence - discrete Hirota eq. for T in rectangular irreps: Baxter’s TQ relations, Backlund transformations etc. Commutativity and conservation laws V.K.,Vieira V.K., Leurent,Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin Baxter’s Q-operators V.K., Leurent,Tsuboi Generating function for (super)characters of symmetric irreps: 1 • • s Q at level zero of nesting Definition of Q-operators at 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: • Nesting (Backlund flow): consequtive « removal » of eigenvalues Alternative approaches: Bazhanov, Lukowski, Mineghelli Rowen Staudacher Derkachev, Manashov Hasse diagram and QQ-relations (Plücker id.) • Tsuboi V.K.,Sorin,Zabrodin Tsuboi,Bazhanov 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 Krichever,Lipan, Wiegmann,Zabrodin Wronskian solutions of Hirota equation • We can solve Hirota equations in a band of width N in terms of Gromov,V.K.,Leurent,Volin differential forms of 2N functions Solution combines dynamics of gl(N) representations and quantum fusion: • -form encodes all Q-functions with indices: a s • E.g. for gl(2) : • Solution of Hirota equation in a strip (via arbitrary - and -forms): definition: • For su(N) spin chain (half-strip) we impose: Solution of Hirota in (𝐾|𝑀) fat hook and (𝐾1 , 𝐾2 |𝑀) T-hook a 𝑀 λa λ2 λ1 Tsuboi V.K.,Leurent,Volin 𝑀 a 𝐾 𝐾1 , 𝐾2 𝑀 -hook 𝐾2 𝐾1 s s • Bosonic and fermionic 1-(sub)forms (all • Wronskian solution for the (𝐾|𝑀) fat hook: • Similar Wronkian solution exists in 𝐾1 , 𝐾2 𝑀 -hook anticomute): Inspiring example: principal chiral field (PCF) • It is known since long to be integrable: S-matrix of 𝑁 − 1 types of physical particles Zamolodchikov&Zamolodchikov Karowski Wiegmann • A limiting case of Thirring model, or WZNW model Asymptotic 𝐿 ∞ Bethe ansatz constructed. Interesting explicit large 𝑁 solution at finite density • Finite 𝐿: TBA → Y-system → Hirota dynamics in a in (a,s) plane in a band • Known asymptotics of Y-functions Polyakov, Wiegmann; Wiegmann Fateev, V.K., Wiegmann Wiegmann, Tsevlik Al. Zamolodchikov a s -plane • Analyticity strips of 𝑇𝑎,𝑠 (𝑢) from 𝑢 • 𝑇𝑎,𝑠 (𝑢) is analytic inside the strip ∞ asympotics Finite volume solution of principal chiral field Gromov, V.K., Vieira V.K., Leurent Alternative approach: Balog, Hegedus • Use Wronskian solution in terms of 2𝑁 Q-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 -plane analytic in the upper half-plane polynomials fixing a state (for vacuum 𝑆𝑘 𝑢 = -plane 𝑢𝑘−1 𝑘−1 ! ) • From reality of Y-functions: analytic in the lower half-plane • 𝑁 − 1 nonlinear integral equations on spectral densities ρ𝑘 can be obtained e.g. from the condition of left-right symmetry true for 𝑆𝑈(𝑁)𝐿 ⟺ 𝑆𝑈(𝑁)𝑅 symmetric states (can be generalized to any state) • We obtain a finite system of NLIE (somewhat similar to Destri-deVega eqs.) • Good for analytic study at large or small volume 𝐿 and for numerics at any 𝐿 SU(3) PCF numerics V.K.,Leurent’09 E / 2 mass gap ground state L 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 𝑃𝑆𝑈 2,2 4 T-hook Spectral AdS/CFT Y-system Gromov,V.K.,Vieira cuts in complex -plane • Analyticity from large 𝐿 asymptotics via one-particle dispersion relation: L→∞ Zhukovsky map: • Extra “corner” equations: 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 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: continued on special magic sheet in labels can be analytically • 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 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 𝑃𝑀−1 and two spectral densities 𝜌1 , 𝜌2 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 • 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 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 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 (6 loops promised) Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Eden,Heslop,Korchemsky,Smirnov,Sokatchev 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 FiNLIE numerics Y-system numerics Gromov,V.K.,Vieira Frolov Gromov,Valatka Gubser, Klebanov, Polyakov Gromov,Shenderovich, Serban, Volin Roiban, Tseytlin Vallilo, Mazzucato Gromov, Valatka AdS/CFT Y-system passes all known tests Gromov, Valatka 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 Correa, Maldacena, Sever, Drukker Gromov, Sever • 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? Recent advances: Gromov, Sever, Vieira, Kostov, Serban, Janik etc. Happy Birthday Pasha! С ЮБИЛЕЕМ, ПАША!