Diagrams and irregular connections on the Riemann sphere

TL;DR

This paper introduces a new diagrammatic framework for algebraic connections on vector bundles over the Riemann sphere, generalizing previous work to include multiple irregular singularities and ramification, and links these diagrams to Lax representations of Painlevé-type equations.

Contribution

It extends the diagram concept to more general irregular connections, proves its invariance under symplectic automorphisms, and connects diagrams to different Lax forms of Painlevé equations.

Findings

Diagram invariance under symplectic automorphisms

New cases linking diagrams to Painlevé Lax representations

Generalization to irregular and ramified singularities

Abstract

We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities, possibly ramified. We prove that the diagram is invariant under the symplectic automorphisms of the Weyl algebra, encompassing the Fourier-Laplace transform. As an application, we establish several new cases of the observation that different Lax representations of a given Painlev\'e-type equation may be read off directly from the diagram, corresponding to connections with different formal data, usually on different rank bundles.