Related papers: Toda lattice representation for random matrix mode…
We show how to construct semi-invariants and integrals of the full symmetric sl(n) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates…
This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to…
In this paper, we use Flaschka's change of variables of the open Toda lattice and its interpretation in term of the group structure of the LU factorisation as a coadjoint motion on a certain dual of Lie algebra to implement a structure…
We present the way the Lorentz invariant canonical partition function for Matrix Theory as a light-cone formulation of M-theory can be computed. We explicitly show how when the eleventh dimension is decompactified, the N = 1 eleven…
We present the extension of the Lagrangian $loop$ representation in such a way to introduce matter fields. The partition function of lattice compact U(1) Gauge-Higgs model is expressed as a sum over closed as much as open surfaces. These…
We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $\tau$-function and discuss various implications of non-vanishing "negative"- and "zero"-time variables: the appear to modify the…
We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral…
By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as…
We present a method for calculating transfer matrices for the $q$-state Potts model partition functions $Z(G,q,v)$, for arbitrary $q$ and temperature variable $v$, on strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc)…
The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr\"odinger equations. The well-known…
We introduce a so-called `coprimeness-preserving non-integrable' extension (another terminology is `quasi-integrable' extension) to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such…
We consider the reproducing kernel function of the theta Bargmann-Fock Hilbert space associated to given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the…
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be…
We consider the Potts model in a magnetic field on an arbitrary graph $G$. Using a formula of F. Y. Wu for the partition function $Z$ of this model as a sum over spanning subgraphs of $G$, we prove some properties of $Z$ concerning…
The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that…
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N=3 the $\tau$-function Hirota form is presented that allows to construct an exast solutions of the equations of…
We consider a matrix space based on the spin degree of freedom, describing both a Hilbert state space, and its corresponding symmetry operators. Under the requirement that the Lorentz symmetry be kept, at given dimension, scalar symmetries,…
The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan…
In this paper, we introduce a novel and general method for computing partition functions of solvable lattice models with free fermionic Boltzmann weights. The method is based on the ``permutation graph'' and the ``$F$-matrix'': the…
The aim of this work is focused on linearizing and found the Lax Pairs of the algebraic complete integrability (a.c.i) Toda lattice associated with the twisted affine Lie algebra \(a_4^{\left(2\right)}\). Firstly, we recall that our case of…