Remarks on $\tau$-functions for the difference Painlev\'e equations of type $E_8$

TL;DR

This paper explores the structure of $ au$-functions for the elliptic difference Painlevé equation of type $E_8$, introducing new notions and constructing solutions using elliptic hypergeometric integrals, linking to geometric configurations.

Contribution

It introduces ORG $ au$-functions for the $E_8$ lattice and constructs particular solutions expressed via elliptic hypergeometric integrals, connecting algebraic and geometric frameworks.

Findings

Construction of specific solutions using elliptic hypergeometric integrals

Introduction of ORG $ au$-functions for $E_8$ lattice

Relation to lattice $ au$-functions and geometric configurations

Abstract

We investigate the structure of $\tau$-functions for the elliptic difference Painlev\'e equation of type $E_8$. Introducing the notion of ORG $\tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $\tau$-functions associated with the configuration of generic nine points in the projective plane.