Rank three representations of Painlev\'e systems: I. Wild character varieties

TL;DR

This paper derives explicit cubic equations for wild character varieties associated with rank 3 Painlevé systems and compares them to classical rank 2 cases, advancing understanding of their geometric structures.

Contribution

It provides the first explicit cubic equations for rank 3 wild character varieties and compares these with classical rank 2 representations, revealing new structural insights.

Findings

Explicit cubic equations for rank 3 wild character varieties

Comparison with classical rank 2 representations

Enhanced understanding of geometric structures of Painlevé systems

Abstract

We give explicit cubic equations for the wild character varieties corresponding to the rank 3 representations of Painlev\'e equations, and compare them to the ones of their classical rank 2 representations.