Quivers and difference Painleve equations

TL;DR

This work constructs Lax-pair descriptions for difference Painlevé equations with affine Weyl symmetry types $E_6$, $E_7$, and $E_8$ by embedding them in Kronheimer’s ALE quiver varieties and Sakai’s geometric framework of rational surfaces. It links moduli spaces of Fuchsian systems (via star-shaped quivers) to two-dimensional Painlevé phase spaces and shows how translations in the affine Weyl group arise from Schlesinger transformations, producing birational symmetries that extend to the Lax pair setting. The main contributions include explicit connections between Kronheimer ALE spaces, quiver varieties, and moduli spaces of logarithmic connections, yielding a unified, geometric account of the Weyl-group actions underlying discrete Painlevé dynamics. The results clarify how discrete Painlevé equations emerge as symmetries of linear systems and illuminate the E-type symmetry cases within a coherent algebraic-geometric framework.

Abstract

We will describe natural `Lax pairs' for the difference Painleve equations with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations.

Figures (14)

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Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1: Nakaj-quiver.dukeRumpCB-HNakaj-refl
• Lemma 5
• Remark 6
• Remark 7
• Proposition 8
• Remark 9