T-systems and Y-systems in integrable systems

Abstract

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schrödinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ansätze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.

156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these corrections

References in corpus (18)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Gluon scattering amplitudes at strong coupling
• The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry
• Thermodynamic Bethe Ansatz for the AdS_5 x S^5 Mirror Model
• The ODE/IM Correspondence
• Four-loop perturbative Konishi from strings and finite size effects for multiparticle states
• Quantum corrections to the string Bethe ansatz
• The Bethe ansatz for superconformal Chern-Simons
• On String S-matrix, Bound States and TBA
• Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal
• Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• From Characters to Quantum (Super)Spin Chains via Fusion
• Towards Baxter equation in supersymmetric Yang-Mills theories
• Perturbed Defects and T-Systems in Conformal Field Theory
• Discrete Hirota dynamics for AdS/CFT
• High temperature expansion of emptiness formation probability for isotropic Heisenberg chain
• Fusion hierarchies for N = 4 superYang-Mills theory

Cited by in corpus (89)

• Fermi gases in one dimension: From Bethe Ansatz to experiments
• Cluster structures on quantum coordinate rings
• Superuniversality of superdiffusion
• Exact steady states for quantum quenches in integrable Heisenberg spin chains
• On form factors in N=4 sym
• From the Quantum Transfer Matrix to the Quench Action: The Loschmidt echo in $XXZ$ Heisenberg spin chains
• Twistor Approach to String Compactifications: a Review
• The Quantum Deformed Mirror TBA I
• Stochastic $R$ matrix for $U_q(A_n^{(1)})$
• The spectral problem for strings on twisted AdS_5 x S^5
• Introduction to the thermodynamic Bethe ansatz
• The Quantum Deformed Mirror TBA II
• Yangians, S-matrices and AdS/CFT
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• Non-commutative rational Yang-Baxter maps
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C
• Geometric aspects of the ODE/IM correspondence
• Thermodynamics of the Topological Kondo Model
• Integrable Kondo problems
• Extended T-systems
• Comments on the Mirror TBA
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Non-commutative lattice modified Gel'fand-Dikii systems
• Discrete KP equation with self-consistent sources
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Thermodynamics, spin-charge separation and correlation functions of spin-1/2 fermions with repulsive interaction
• Categorical relations between Langlands dual quantum affine algebras: Exceptional cases
• The pentagon relation and incidence geometry
• WKB periods for higher order ODE and TBA equations
• Cluster algebras in algebraic Lie theory
• Hyperbolic 3-manifolds and Cluster Algebras
• Web Construction of ABCDEFG and Affine Quiver Gauge Theories
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Universal local pair correlations of Lieb-Liniger bosons at quantum criticality
• Discontinuity relations for the AdS(4)/CFT(3) correspondence
• The Coxeter relations and KP map for non-commuting symbols
• Cartan matrices and integrable lattice Toda field equations
• Randomized box-ball systems, limit shape of rigged configurations and Thermodynamic Bethe ansatz
• The classification of Zamolodchikov periodic quivers
• Integrable boundary conditions for staggered vertex models
• Quantum integrability from non-simply laced quiver gauge theory
• M-systems and Cluster algebras
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Non-commutative Hermite--Padé approximation and integrability
• Analytical results for the low-temperature Drude weight of the XXZ spin chain
• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• A Drinfeld-type presentation of the orthosymplectic Yangians
• Crystal structure on rigged configurations and the filling map
• Generalized Gibbs Ensemble and string-charge relations in nested Bethe Ansatz
• Large deviations and one-sided scaling limit of randomized multicolor box-ball system
• On boundary fusion and functional relations in the Baxterized affine Hecke algebra
• The $A_m^{(1)}$ Q-system
• Desargues maps and their reductions
• On the Expansion Coefficients of Tau-function of the BKP Hierarchy
• Orthosymplectic superoscillator Lax matrices
• MHV amplitudes at strong coupling and linearized TBA equations
• The Staircase Model: Massless Flows and Hydrodynamics
• Tau functions, infinite Grassmannians and lattice recurrences
• $\ell$-weights and factorization of transfer operators
• Integral preserving discretization of 2D Toda lattices
• Quantum groups and functional relations for arbitrary rank
• Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
• Hirota equation and the quantum plane
• Positivity and periodicity of $Q$-systems in the WZW fusion ring
• WKB analysis of the linear problem for modified affine Toda field equations
• Non-linear Transport by Bethe Bound States
• Non-autonomous multidimensional Toda system and multiple interpolation problem
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• On the extended T-system of type $C_3$
• Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
• Generalized hydrodynamics in complete box-ball system for $U_q({\widehat{sl}}_n)$
• Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion
• Fusion for AdS/CFT boundary S-matrices
• Linear recurrence relations in $Q$-systems and difference $L$-operators
• Anisotropic Kondo line defect and ODE/IM correspondence
• Discrete Dynamical Systems From Real Valued Mutation
• Bäcklund transformations as integrable discretization. The geometric approach
• Integrability of planar-algebraic models
• Möbius invariant Y-systems (cluster structures) for Miquel dynamics
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• On the Q operator and the spectrum of the XXZ model at root of unity
• Gluon scattering amplitudes from gauge/string duality and integrability
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
• Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system
• On a determinant formula for some real regular representations
• Addendum: Nonlinear integral equations for the sausage model
• Determinantal approach to multiple orthogonal polynomials, and the corresponding integrable equations