T-systems and Y-systems in integrable systems

By A. Kuniba, Tomoki Nakanishi, J. Suzuki. 2010

271 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/35f1a6f8679db0a3bada4c6e59bc9299a52faef4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. X. Guan
2. Bernard Leclerc
3. A. Goncharov
4. Chaohong Lee
5. J. Schröer
6. M. Batchelor
7. C. Geiss
8. Msri Postdoctoral Fellow

In Geometry and Topology

1. Bernard Leclerc
2. C. Geiss
3. J. Schröer
4. B. Keller
5. Osamu Iyama
6. I. Reiten
7. Philipp Lampe
8. Sergey Fomin

In Statistical and Nonlinear Physics

1. C. Geiss
2. J. Schröer
3. Bernard Leclerc
4. A. Goncharov
5. M. Batchelor
6. Max Glick
7. Richard Kenyon
8. B. Keller

Citations in the corpus

• Dimers and cluster integrable systems (2011, background): “on of the octaherdon reccurence with the Hirota bilinear difference 61 equation [Hir81] is discussed there. A solution of the octaherdon reccurence on a simplex was found in [FG03a]. See also [DFK10], [Naka10] for other examples which are special cases. The cluster transformation Atogether with the q”
• Fermi gases in one dimension: From Bethe ansatz to experiments (2013, background): “T - systems and integrability in general also play a fundamental role in the gauge/string theories of high energy physics (Beisert et al., 2012; Kuniba et al., 2011).”
• The pentagram map and Y-patterns (2010, background): “+ yj and 1 + 1/yj are also given by cross ratios. The equations (2.3)–(2.4) then become multiplicative cross ratio identities which can be proven geometrically. Remark 2.5. Section 12.2 of the survey [6] provides formulas analogous to (2.3)– (2.4) in the setting of quadrilateral lattices in 3-space.”
• Cluster algebras and derived categories (2012, unknown): “• discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] .”
• Cluster structures on quantum coordinate rings (2011, methodology): “.5) allowing to calculate certain quantum minors depending on w in a recursive way. This system is a q-analogue of aT-system arising in various problems of mathematical physics and combinatorics (see [KNS]), and we believe it could be of independent interest. It will turn out that all quantum minors”