Realization of symmetry of $A^{(1)*}_2$-surfaces as transformations of logarithmic connections

TL;DR

This paper demonstrates how the symmetries of $A^{(1)*}_2$-surfaces, related to difference Painlevé equations, can be understood as transformations of parabolic logarithmic connections, linking geometric and analytical frameworks.

Contribution

It explicitly realizes the symmetry of $A^{(1)*}_2$-surfaces as transformations of parabolic logarithmic connections, connecting moduli spaces with difference Painlevé equations.

Findings

Symmetry of $A^{(1)*}_2$-surfaces is realized as transformations of connections.

Moduli spaces of parabolic logarithmic connections are linked to initial conditions.

Provides a geometric interpretation of difference Painlevé symmetries.

Abstract

An $A^{(1)*}_2$-surface is a space of initial conditions of certain difference Painlev\'e equations. $A^{(1)*}_2$-surfaces are realized as the moduli spaces of parabolic logarithmic connections. In this paper, we realize the symmetry of $A^{(1)*}_2$-surfaces as transformations of parabolic logarithmic connections.