Hirota integrable dynamics: from quantum spin chains to AdS /CFT integrability
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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., VieiraV.K., Leurent
Gromov, Volin, V.K., Leurent
V.K., Leurent, TsuboiAlexandrov, V.K., Leurent,Tsuboi,Zabrodin
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Y-system and T-system
• Y-system
• T-system (Hirota eq.)
• Gauge symmetry
= +a
s s s-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.,VieiraV.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)
TsuboiV.K.,Sorin,Zabrodin
Gromov,VieiraTsuboi,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
polynomialsfixing 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:
Gromov, V.K., VieiraV.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-systemGromov,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.,TsuboiGromov,Tsuboi,V.K.,LeurentTsuboi
Plücker relations express all 256 Q-functionsthrough 8 independent ones
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Solution of AdS/CFT T-system in terms offinite 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, TatteoBalog, 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, TsuboiGromov,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
GubserKlebanovPolyakov
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,ZanonVelizhanin
Bajnok,JanikGromov,V.K.,Vieira
Bajnok,Janik,LukowskiLukowski,Rej,Velizhanin,OrlovaEden,Heslop,Korchemsky,Smirnov,Sokatchev
From quasiclassicsGromov,Shenderovich,Serban, VolinRoiban,TseytlinMasuccato,ValilioGromov, Valatka
Cavaglia, Fioravanti, TatteoGromov, V.K., VieiraArutyunov, Frolov
Leurent,Serban,VolinBajnok,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 AdS5/CFT4 ?• 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, DrukkerGromov, Sever
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