Hirota integrable dynamics: from quantum spin chains to AdS /CFT integrability

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

2. 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

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

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

5. V.K.,Vieira V.K., Leurent,Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin Master T-operator • Generating function of characters: • Master T-operator: • It is a tau function of mKPhierachy: • (polynomial w.r.t. the mKP charge ) • Satisfies canonical mKP Hirota eq. • Hence - discrete Hirota eq. for T in rectangular irreps: • Commutativity and conservation laws

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

7. s Baxter’s Q-operators V.K., Leurent,Tsuboi • Generating function for characters of symmetric irreps: • 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 Gromov,Vieira Tsuboi,Bazhanov Hasse diagram and QQ-relations (Plücker id.) • Example: gl(2|2) 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 • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s

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

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

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

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

13. 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

14. 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) • Original T-system is in mirror sheet (long cuts) Arutyunov, Frolov • 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 • 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 Alternative approach: Balog, Hegedus

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

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 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

18. Closing FiNLIE: sawing together 3 bands • We have expressedall 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 fromY-system (in TBA form):

19. Konishioperator: numericsfromY-system Beisert, Eden,Staudacher ABA Gubser,Klebanov,Polyakov Gubser Klebanov Polyakov From quasiclassics Y-system numerics Gromov,V.K.,Vieira (confirmed and precised by Frolov) Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio Gromov, Valatka Leurent,Serban,Volin Bajnok,Janik 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 • Uses the TBA form of Y-system • AdS/CFT Y-system passes all known tests Cavaglia, Fioravanti, Tatteo Gromov, V.K., Vieira Arutyunov, Frolov

20. Conclusions • Hirotaintegrable 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 AdS5/CFT4 ? • Why is N=4 SYM integrable? • FiNLIE for another integrableAdS/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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