Classification of Rigid Irregular $G_2$-Connections

TL;DR

This paper classifies irreducible rigid irregular connections on the complex projective line with differential Galois group G_2, expanding understanding of such connections beyond known hypergeometric systems.

Contribution

It provides a comprehensive classification of G_2-connections with slopes having numerator 1 using the Katz-Arinkin algorithm, including new families not of hypergeometric type.

Findings

Classification of G_2-connections with specific slopes.

Construction of new G_2-connection families.

Identification of connections beyond hypergeometric systems.

Abstract

Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured $\mathbb{P}^1_{\mathbb{C}}$ having differential Galois group $G_2$, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types.