Corpus ID: 119636740

Special polynomials related to the supersymmetric eight-vertex model. III. Painlevé VI equation

  title={Special polynomials related to the supersymmetric eight-vertex model. III. Painlev{\'e} VI equation},
  author={H. Rosengren},
  journal={arXiv: Mathematical Physics},
  • H. Rosengren
  • Published 2014
  • Mathematics, Physics
  • arXiv: Mathematical Physics
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painleve VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of Backlund transformations. For particular lines in the lattice, this proves conjectures of Bazhanov and Mangazeev. As applications, we describe the behaviour of the corresponding solutions near the singular points of Painleve VI, and obtain several new properties of our polynomials… Expand
6 Citations
Special Polynomials Related to the Supersymmetric Eight-Vertex Model: A Summary
We introduce and study symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $${\Delta=\pmExpand
Sum rules for the supersymmetric eight-vertex model
The eight-vertex model on the square lattice with vertex weights a, b, c, d obeying the relation (a 2 + ab)(b 2 + ab) = (c 2 + ab)(d 2 + ab) is considered. Its transfer matrix with L = 2n + 1, n ⩾ 0,Expand
On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions
The square-lattice eight-vertex model with vertex weights $a,b,c,d$ obeying the relation $(a^2+ab)(b^2+ab) = (c^2+ab)(d^2+ab)$ and periodic boundary conditions is considered. It is shown that theExpand
Series Solutions of the Non-Stationary Heun Equation
We consider the non-stationary Heun equation, also known as quantum PainleveVI, which has appeared in dierent works on quantum integrable models and conformaleld theory. We use a generalized kernelExpand
On the elliptic $\mathfrak{gl}_2$ solid-on-solid model: functional relations and determinants
In this work we study an elliptic solid-on-solid model with domain-wall boundaries having the elliptic quantum group $\mathcal{E}_{p, \gamma}[\widehat{\mathfrak{gl}_2}]$ as its underlying symmetryExpand
A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
We study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. ByExpand


The eight-vertex model and Painlevé VI
In this paper we establish a connection of Picard-type elliptic solutions of the Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the studyExpand
An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices
We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin-Korepin formula for theExpand
Eight-vertex Model and Non-stationary Lame Equation
We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the six-vertex model. We show that theseExpand
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III
Abstract A unified treatment of monodromy and spectrum preserving deformations is presented. In particular a general procedure is described to reduce the latter into the former consistently. TheExpand
Ground-state properties of a supersymmetric fermion chain
We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. ByExpand
On a Class of Algebraic Solutions to the Painlevé VI Equation, Its Determinant Formula and Coalescence Cascade
A determinant formula for a class of algebraic solutions to Painlev\'e VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entriesExpand
Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations
Rational solutions of the second, third and fourth Painlevé equations can be expressed in terms of special polynomials defined through second order bilinear differential-difference equations whichExpand
A staggered fermion chain with supersymmetry on open intervals
A strongly interacting fermion chain with supersymmetry on the lattice and open boundary conditions is analysed. The local coupling constants of the model are staggered, and the properties of theExpand
The three-colour model with domain wall boundary conditions
The method generalizes Kuperberg’s proof of the alternating sign matrix theorem, replacing the six-vertex model used byKuperberg with the eight- Vertex-solid-on-solid model, and conjecture an explicit formula for the free energy of the model. Expand
A possible combinatorial point for the XYZ spin chain
We formulate and discuss several conjectures related to the ground state vectors of odd-length XYZ spin chains with periodic boundary conditions and a special choice of the Hamiltonian parameters. InExpand