Toda hierarchies and their applications

Abstract

The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as reductions. These integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s. A recent example is a series of studies on models of statistical mechanics called the melting crystal model. This research has revealed that the aforementioned two reductions of the 2D Toda hierarchy underlie two different melting crystal models. Technical clues are a fermionic realization of the quantum torus algebra, special algebraic relations therein called shift symmetries, and a matrix factorization problem. The two melting crystal models thus exhibit remarkable similarity with the Hermitian and unitary matrix models for which the two reductions of the 2D Toda hierarchy play the role of fundamental integrable structures.

47 pages, no figure, contribution to JPhysA Special Issue "Fifty years of the Toda lattice"; (v2) Final version for publication. The last part of sects. 5.2 and 5.3 are slightly modified, and the bibliographic data are updated

References in corpus (6)

• Matrix Models for Random Partitions
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Phase transitions, double-scaling limit, and topological strings
• Hilbert schemes, integrable hierarchies, and Gromov-Witten theory
• Orbifold melting crystal models and reductions of Toda hierarchy
• The Equivariant Gromov--Witten Theory of CP^1 and Integrable hierarchies

Cited by in corpus (15)

• Direct linearisation of the discrete-time two-dimensional Toda lattices
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Extended lattice Gelfand-Dickey hierarchy
• Cubic Hodge integrals and integrable hierarchies of Volterra type
• Rank shift conditions and reductions of 2d-Toda theory
• Tropical limit of matrix solitons and entwining Yang-Baxter maps
• Dressing operators in equivariant Gromov-Witten theory of ${\mathbb{CP}}^1$
• $Q$-Boson model and relations with integrable hierarchies
• Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Integrable structures of specialized hypergeometric tau functions
• Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy and its constraint
• Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
• Intertwining operator and integrable hierarchies from topological strings
• Solitons and Normal Random Matrices