Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps

Abstract

We study certain symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $\Delta =\pm 1/2$. In this paper, which is the first part of a series, we study the behaviour of the polynomials at special parameter values, which can be identified with cusps of the modular group ${\Gamma }_0(12)$. In subsequent papers, we will show that the polynomials satisfy a non-stationary Schrödinger equation related to the Knizhnik--Zamolodchikov--Bernard equation and that they give a four-dimensional lattice of tau functions of Painlevé VI.

53 pages. Minor changes from previous version

References in corpus (5)

• On Rationally Parametrized Modular Equations
• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• A possible combinatorial point for XYZ-spin chain
• Ground-state properties of a supersymmetric fermion chain
• Special polynomials related to the supersymmetric eight-vertex model. II. Schrödinger equation

Cited by in corpus (8)

• Special polynomials related to the supersymmetric eight-vertex model: A summary
• Special polynomials related to the supersymmetric eight-vertex model. II. Schrödinger equation
• On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions
• Elliptic pfaffians and solvable lattice models
• A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
• Sum rules for the supersymmetric eight-vertex model
• On the elliptic ${\mathfrak{gl}}_2$ solid-on-solid model: functional relations and determinants
• A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials $p_n$