Algebraic theory of formal regular-singular connections with parameters

TL;DR

This paper provides an algebraic and categorical framework for understanding regular-singular differential systems over complex fields and parameterized rings, establishing their equivalence to group representations.

Contribution

It extends classical classification of regular-singular systems to parameterized settings using algebraic and categorical methods, linking them to group representations over rings.

Findings

Classical regular-singular systems are equivalent to representations of a4 over complex fields.

Extended the classification to systems over rings with parameters, maintaining the group representation equivalence.

Unified the theory using categorical perspectives, bridging classical and parameterized cases.

Abstract

This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is algebraically closed and of characteristic zero. It aims at reading the existing classification results as an equivalence between regular-singular systems and representations of the group $\mathbb Z$. In the second part, we deal with regular-singular connections over $R((x))$ and $\mathbb P_R^1\smallsetminus\{0,\infty\}$, where $R=C[[t_1,\ldots,t_r]]/I$. The picture we offer shows that regular-singular connections are equivalent to representations of $\mathbb Z$, now over $R$.