Special polynomials related to the supersymmetric eight-vertex model: A summary

TL;DR

This paper introduces symmetric polynomials connected to elliptic lattice models and affine Lie algebra characters, revealing their relation to integrable systems, Painlevé VI tau functions, and Bäcklund transformations.

Contribution

It presents a new class of symmetric polynomials linked to supersymmetric eight-vertex models and elliptic systems, with connections to Painlevé equations and affine Lie algebra characters.

Findings

Polynomials satisfy a non-stationary Schrödinger equation with elliptic potential.

Specializations correspond to Painlevé VI tau functions.

Results connect elliptic lattice models, integrable systems, and algebraic solutions.

Abstract

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 $\Delta=\pm 1/2$. There is also a close relation to affine Lie algebra characters. After a natural change of variables, our polynomials satisfy a non-stationary Schr\"odinger equation with elliptic potential, which is related to the Knizhnik-Zamolodchikov-Bernard equation and to the canonical quantization of Painlev\'e VI. Moreover, specializations of our polynomials can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund transformations. In the present work, our results on these topics are summarized with a minimum of technical details.