T-systems and Y-systems in integrable systems

TL;DR

This paper reviews the widespread role of T- and Y-systems across various areas of integrable systems, highlighting their mathematical structures and applications in physics and geometry.

Contribution

It provides a comprehensive collection of short reviews on T- and Y-systems, connecting diverse topics in integrable models and related mathematical frameworks.

Findings

T- and Y-systems appear in numerous integrable models and mathematical structures.

They relate to quantum groups, cluster algebras, and conformal field theory.

The review summarizes key developments and applications in the field.

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 Schr\"odinger 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 ans\"atze, 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.