T-systems andY-systems in integrable systems

Abstract

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 analog of L-operators in KP hierarchy, Stokes phenomena in 1D Schrödinger problem, AdS/CFT correspondence, Toda field equations on discrete spacetime, 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 is a collection of short reviews on these topics which can be read more or less independently.

T-system on T-hook: Grassmannian solution and twisted Quantum Spectral Curve

“…The bi-linear discrete Hirota equation, sometimes also called the Hirota-Miwa equation [25][26][27] T a,s u + i 2 T a,s u − i 2 = T a+1,s (u)T a−1,s (u) + T a,s+1 (u)T a,s−1 (u) (2.1) appears in numerous quantum and classical integrable systems. In these notations, typically used in the context of quantum integrable spin chains and sigma-models, T a,s (u) are complex-valued functions of two integer indices a and s parameterising a Z 2 lattice, and of a parameter u ∈ C usually called spectral parameter.…”

Gluon Scattering Amplitudes From Gauge/String Duality and Integrability

“…Interestingly, with the help of graphical representations as in Figs. 4 and 5, one can show that these cross-ratios are directly expressed by the T-functions [32], which are related to the Y-functions as…”

Yangians,S-matrices and AdS/CFT

“…The converse 63 To give a proper account of the standard literature would be an overwhelming task. We refer to [65,226,227] for reviews. 64 Notice that, in N = 4 SYM, operators that are not protected nevertheless have their anomalous dimensions encoded in short representations of the centrally-extended algebra psl(2|2) c (we are being cavalier on issues related to the infinite length of the operators).…”