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
s
• Gauge symmetry
s
+
s-1
s+1
a-1
a+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:
• Y-system
principal chiral field
Hirota dynamics in a in (a,s) strip of width N
• Finite volume solution: finite system of NLIE:
parametrization fixing the analytic structure:
polynomials
fixing a state
Gromov, V.K., Vieira
V.K., Leurent
jumps
by
• From reality:
• N-1 spectral densities
(for L ↔ R symmetric states):
SU(3) PCF numerics: Energy versus size for vacuum and mass gap
V.K.,Leurent’09
E L/ 2
L
Spectral AdS/CFT Y-system
Gromov,V.K.,Vieira
• Dispersion relation
• Parametrization by Zhukovsky map:
a
• Extra “corner” equations:
•
cuts in complex
s
Type of the operator is fixed by imposing certain analyticity properties in spectral
parameter. Dimension can be extracted from the asymptotics
-plane
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)
• 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
each in its infinite strip.
and
satisfy the Hirota equations,
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
Gubser,Klebanov,Polyakov
ABA
Gubser
Klebanov
Polyakov From
Y-system numerics
Gromov,V.K.,Vieira
(confirmed and precised by Frolov)
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
Future directions
• Better understanding of analyticity of Q-functions.
Correa, Maldacena, Sever,
Drukker
Gromov, Sever
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?
END