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Hirota integrable dynamics: from quantum spin chains to AdS/CFT integrability Vladimir Kazakov (ENS, Paris) International Symposium Ahrenshoop “Recent Developments in String and Field Theory” Schmöckwitz, August 27-31, 2012 Collaborations with Alexandrov, Gromov, Leurent, Tsuboi, Vieira, Volin, Zabrodin

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Hirota equations in quantum integrability New approach to solution of integrable 2D quantum sigma-models in finite volume Based on discrete classical Hirota dynamics (Y-system, T-system, Baxter’s Q-functions, Plücker QQ identities, wronskian solutions,…) + Analyticity in spectral parameter! Important examples already worked out, such as su(N)×su(N) principal chiral field (PCF) FiNLIE equations from Y-system for exact planar AdS/CFT spectrum Inspiration from Hirota dynamics of gl(K|M) quantum (super)spin chains: mKP hierarchy for T- and Q- operators Gromov, V.K., Vieira V.K., Leurent Gromov, Volin, V.K., Leurent V.K., Leurent, Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin

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Y-system and T-system Y-system T-system (Hirota eq.) Gauge symmetry =+ a sss-1 s+1 a-1 a+1 Related to a property of gl(N|M) irreps with rectangular Young tableaux:

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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

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Master T-operator It is a tau function of mKP hierachy: ( polynomial w.r.t. the mKP charge ) 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:

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V.K., Leurent,Tsuboi Graphically (slightly generalized to any spectral parameters): Master Identity and Q-operators The proof in: V.K., Leurent,Tsuboi from the basic identity proved in: V.K, Vieira

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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 characters of symmetric irreps: s

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Hasse diagram and QQ-relations (Plücker id.) - bosonic QQ-rel. -- fermionic QQ rel. Example: gl(2|2) Tsuboi V.K.,Sorin,Zabrodin Gromov,Vieira 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 T-operators obey Hirota equation: solved by Wronskian determinants of Q’s Hasse diagram: hypercub E.g.

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

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Inspiring example: principal chiral field Y-system Hirota dynamics in a in (a,s) strip of width N polynomials fixing a state jumps by Finite volume solution: finite system of NLIE: parametrization fixing the analytic structure: N-1 spectral densities (for L ↔ R symmetric states): From reality: Solved numerically by iterations Gromov, V.K., Vieira V.K., Leurent

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SU(3) PCF numerics: Energy versus size for vacuum and mass gap E L/ 2 L V.K.,Leurent’09

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Spectral AdS/CFT Y-system Gromov,V.K.,Vieira Type of the operator is fixed by imposing certain analyticity properties in spectral parameter. Dimension can be extracted from the asymptotics cuts in complex -plane Extra “corner” equations: s a Parametrization by Zhukovsky map: Dispersion relation

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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

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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 Gromov,V.K.,Leurent,Volin Original T-system is in mirror sheet (long cuts) Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz) Alternative approach: Balog, Hegedus We found and checked from TBA the following relation between the upper and right/left bands Inspired by: Bombardelli, Fioravanti, Tatteo Balog, Hegedus Irreps (n,2) and (2,n) are in fact the same typical irrep, so it is natural to impose for our physical gauge From unimodularity of the quantum monodromy matrix Arutyunov, Frolov

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Quantum symmetry can be analytically continued on special magic sheet in labels Analytically continued and satisfy the Hirota equations, each in its infinite strip. Gromov,V.K. Leurent, Tsuboi Gromov,V.K.Leurent,Volin

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Magic sheet and solution for the right band 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 The property suggests that certain T-functions are much simpler on the “magic” sheet, with only short cuts:

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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 in terms of a spectral density, the “wing exchange” function and gauge function and two polynomials P(u) and (u) encoding Bethe roots The rest of q’s restored from Plucker QQ relations

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Closing FiNLIE: sawing together 3 bands We have expressed all T (or Y) functions through 6 functions From analyticity of and we get, via spectral Cauchy representation, extra equations fixing all unknown functions Numerics for FiNLIE perfectly reproduces earlier results obtained from Y-system (in TBA form):

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Konishi operator : numerics from Y-system Gubser Klebanov Polyakov Beisert, Eden,Staudacher ABA Y-system numerics Gromov,V.K.,Vieira (confirmed and precised by Frolov) Gubser,Klebanov,Polyakov Uses the TBA form of Y-system AdS/CFT Y-system passes all known tests zillions of 4D Feynman graphs! Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Bajnok,Janik Gromov,V.K.,Vieira Bajnok,Janik,Lukowski Lukowski,Rej,Velizhanin,Orlova Eden,Heslop,Korchemsky,Smirnov,Sokatchev From quasiclassics Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio Gromov, Valatka Cavaglia, Fioravanti, Tatteo Gromov, V.K., Vieira Arutyunov, Frolov Leurent,Serban,Volin Bajnok,Janik

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Conclusions Hirota integrable dynamics, supplied by analyticity in spectral parameter, is a powerful method of solving integrable 2D quantum sigma models. Y-system 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 ? Why is N=4 SYM integrable? FiNLIE for another integrable AdS/CFT duality: 3D ABJM gauge theory BFKL limit from Y-system? 1/N – expansion integrable? Gluon amlitudes, correlators …integrable? Correa, Maldacena, Sever, Drukker Gromov, Sever

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