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

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)

Gromov, V.K., Vieira V.K., Leurent

• FiNLIE equations from Y-system for exact planar AdS/CFT spectrum

Gromov, Volin, V.K., Leurent

• Inspiration from Hirota dynamics of gl(K|M) quantum (super)spin chains: mKP hierarchy for T- and Q- operators

V.K., Leurent, Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin

Y-system and T-system

• Y-system • T-system (Hirota eq.)  Related to a property of gl(N|M) irreps with rectangular Young tableaux: a

= +

a-1 a+1 s • Gauge symmetry s s-1 s+1

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:

Master T-operator

  Generating function of characters: Master T-operator:  It is a tau function of mKP hierachy: ( 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

V.K.,Vieira V.K., Leurent,Tsuboi Alexandrov, V.K., Leurent,Tsuboi,Zabrodin

Master Identity and Q-operators

V.K., Leurent,Tsuboi

• Graphically (slightly generalized to any spectral parameters):

The proof in: V.K., Leurent,Tsuboi from the basic identity proved in: V.K, Vieira

Baxter’s Q-operators

V.K., Leurent,Tsuboi

 Generating function for characters of symmetric irreps:

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 Gromov,Vieira Tsuboi,Bazhanov

• Example: gl(2|2) 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 • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s

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: a s • E.g. for gl(2) : • Solution of Hirota equation in a strip: • For gl(N) spin chain (half-strip) we impose:

Inspiring example: principal chiral field

• Y-system

Hirota

dynamics in a in (a,s) strip of width N • Finite volume solution: finite system of NLIE: parametrization fixing the analytic structure:

Gromov, V.K., Vieira V.K., Leurent

polynomials fixing a state • From reality: jumps by • N-1 spectral densities (for L ↔ R symmetric states): Solved numerically by iterations

SU(3) PCF numerics: Energy versus size for vacuum and mass gap

V.K.,Leurent’09

E L/ 2



L

Gromov,V.K.,Vieira

Spectral AdS/CFT Y-system

• 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

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

definitions:

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



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.

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

•

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

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):

Konishi operator : numerics from

Y-system

Beisert, Eden,Staudacher ABA Gubser,Klebanov,Polyakov Y-system numerics Gromov,V.K.,Vieira (confirmed and precised by Frolov) Gubser Klebanov Polyakov From quasiclassics 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

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

Correa, Maldacena, Sever, Drukker Gromov, Sever

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?

END