Transcript Exactly Solvable Quantum Field Theories: From Two to Four

“Round Table: Frontiers of Mathematical Physics”
Dubna, December 16-18, 2012
Hirota Dynamics of Quantum Integrability
Vladimir Kazakov (ENS, Paris)
Collaborations with
Alexandrov, Gromov, Leurent,
Tsuboi, Vieira, Volin, Zabrodin
New uses of Hirota dynamics in integrability
• Hirota integrable dynamics incorporates the basic properties of all
Miwa,Jimbo
quantum and classical integrable systems.
Sato
• It generates all integrable hierarchies of PDE’s (KdV, KP, Toda etc)
Kluemper, Pierce
• Discrete Hirota eq. (T-system) is an alternative
Kuniba,Nakanishi,Suzuki
Al.Zamolodchikov
approach to quantum integrable systems.
Bazhanov,Lukyanov, A.Zamolodchikov
• Classical KP hierarchy applies to quantum
V.K., Leurent, Tsuboi
Alexandrov, V.K., Leurent,Tsuboi,Zabrodin
T- and Q-operators of (super)spin chains
• Framework for new approach to solution of integrable 2D quantum
sigma-models in finite volume using Y-system, T-system, Baxter’s
Q-functions, Plücker QQ identities, wronskian solutions,…
+
Gromov, V.K., Vieira
V.K., Leurent
Analyticity in spectral parameter!
• First worked out for spectrum of relativistic sigma-models, such as
su(N)×su(N) principal chiral field (PCF), Sine-Gordon, Gross-Neveu
• Provided the complete solution of spectrum of anomalous
dimensions of 4D N=4 SYM theory! AdS/CFT Y-system, recently
reduced to a finite system of non-linear integral eqs (FiNLIE)
Gromov, V.K. Vieira
Gromov, Volin, V.K., Leurent
Discrete Hirota eq.: T-system and Y-system
• Based on a trivial property of Kronecker symbols (and determinants):
• T-system (discrete Hirota eq.)
• Y-system
• Gauge symmetry
(Super-)group theoretical origins of Y- and T-systems
 A curious property of gl(N|M) representations with rectangular Young tableaux:
=
a
s
s
+
s-1
a-1
a+1
s+1
 For characters – simplified Hirota eq.:
 Boundary conditions for Hirota eq. for AdS/CFT T-system:
∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” !
a
Kwon
U(2,2|4)
Cheng,Lam,Zhang
Gromov, V.K., Tsuboi
s
V.K.,Marshakov,Minahan,Zarembo
Beisert,V.K.,Sakai,Zarembo
 Full quantum Hirota equation
 Classical (strong coupling) limit: eq. for characters of classical monodromy
Gromov,V.K.,Tsuboi
Quantum (super)spin chains
 Quantum transfer matrices – a natural generalization of group characters
V.K., Vieira
 Co-derivative – left differential w.r.t. group (“twist”) matrix:
Main property:
 Transfer matrix (T-operator) of L spins
R-matrix
 Hamiltonian of Heisenberg quantum spin chain:
Master T-operator and mKP
 Generating function of characters:
 Master T-operator:
 Master T is a tau function of mKP hierachy:
mKP charge is spectral parameter! T is polynomial w.r.t.
 Satisfies canonical mKP Hirota eq.
considered by
Krichever
 Hence - discrete Hirota eq. for T in rectangular irreps:
Baxter’s TQ relations, Backlund transformations etc.
 Commutativity and conservation laws
V.K.,Vieira
V.K., Leurent,Tsuboi
Alexandrov, V.K.,
Leurent,Tsuboi,Zabrodin
Baxter’s Q-operators
V.K., Leurent,Tsuboi
 Generating function for (super)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
Tsuboi,Bazhanov
gl(2|2) example: classification of all Q-functions
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
Tsuboi
Wronskian solutions of Hirota equation
• We can solve Hirota equations in a band of width N in terms of
Gromov,V.K.,Leurent,Volin
differential forms of 2N functions
Solution combines dynamics of gl(N) representations and quantum fusion:
•
-form encodes all Q-functions with
indices:
a
s
• E.g. for gl(2) :
• Solution of Hirota equation in a strip (via arbitrary Q- and P-forms):
• For su(N) spin chain (half-strip) we impose:
Inspiring example:
principal chiral field
Gromov, V.K., Vieira
V.K., Leurent
• Y-system
Hirota dynamics in a in (a,s) plane.
We know the Wronskian solution in terms of Q-functions
a
• Finite volume solution: finite system of NLIE,
parameterization fixing the analytic structure.
• Analyticity strips from large volume asymptotics:
s
-plane
• From reality:
polynomials
fixing a state
jumps
by
• N-1 TBA equations (for central nodes) on spectral densities
Alternative approach:
Balog, Hegedus
SU(3) PCF numerics
V.K.,Leurent’09
E / 2
mass gap
ground state
L
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
Planar N=4 SYM – integrable 4D QFT
• 4D superconformal QFT! Global symmetry PSU(2,2|4)
• AdS/CFT correspondence – duality to Metsaev-Tseytlin superstring
• Integrable for non-BPS states, summing genuine 4D Feynman diagrams!
• Operators in 4D
• 4D Correlators:
scaling dimensions
non-trivial functions
of ‘tHooft coupling λ!
structure constants
They describe the whole 4D conformal theory via operator product expansion
Spectral AdS/CFT Y-system
Gromov,V.K.,Vieira
cuts in complex
-plane
T-hook
• Analyticity from large L symptotics: from one-particle dispersion relation:
L→∞
Zhukovsky map:
• Extra “corner” equations:
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)
• 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.
• Quantum analogue of classical
symmetry:
continued on special magic sheet in labels
can be analytically
• Operators/states of AdS/CFT are characterized by certain poles and zeros
of Y- and T-functions fixed by exact Bethe equations:
Inspired by:
Bombardelli, Fioravanti, Tatteo
Balog, Hegedus
Alternative approach:
Balog, Hegedus
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 of T-hook in terms of a spectral densities.
 The rest of Q’s restored from Plucker QQ relations
Closing FiNLIE: sawing together 3 bands
• Finally, we can close the FiNLIE system by using reality of T-functions
and certain symmetries. For example, for left-right symmetric states
• Dimension can be extracted from the asymptotics:
 FiNLIE perfectly reproduces earlier results obtained
from Y-system (in TBA form). It is a perfect mean to generate
weak and strong coupling expansions of anomalous dimensions
in N=4 SYM
Konishi
dimension to 8-th order
• Integrability allows to sum exactly enormous number
of Feynman diagrams of N=4 SYM
Bajnok,Janik
Leurent,Serban,Volin
Bajnok,Janik,Lukowski
Lukowski,Rej,
Velizhanin,Orlova
Leurent, Volin ’12
(from FiNLIE)
• Last term is a new structure – multi-index zeta function.
• Leading transcendentalities can be summed at all orders:
Leurent, Volin ‘12
• Confirmed up to 5 loops by direct graph calculus
Fiamberti,Santambrogio,Sieg,Zanon
Velizhanin
Eden,Heslop,Korchemsky,Smirnov,Sokatchev
Numerics and 3-loops from string quasiclassics
for twist-J operators of spin S
• Perfectly reproduces 3 terms of Y-system numerics
for Konishi operator
or even
Y-system numerics
Gromov,V.K.,Vieira
Frolov
Gromov,Valatka
Gubser, Klebanov, Polyakov
Gromov,Shenderovich,
Serban, Volin
Roiban, Tseytlin
Vallilo, Mazzucato
Gromov, Valatka
Gromov, Valatka
 Numerics uses the TBA or FiNLIE forms of Y-system

AdS/CFT Y-system passes all known tests
Gromov, V.K., Vieira
Cavaglia, Fioravanti, Tatteo
Arutyunov, Frolov
Gromov, V.K., Leurent, Volin
Conclusions
•
Hirota integrable dynamics, supplied by analyticity in spectral parameter, is a powerful method
of solving integrable 2D quantum sigma models.
•
For spin chains (mKP structure): a curious alternative to the algebraic Bethe ansatz of
Leningrad school
•
Y-system for sigma-models 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 AdS5/CFT4 ?
•
•
BFKL limit from Y-system and FiNLIE
Why is N=4 SYM integrable? Can integrability be used to prove AdS/CFT correspondence?
END