An example of the Langlands correspondence for irregular rank two connections on P^1

TL;DR

This paper constructs a categorical equivalence linking rank 2 irregular connections on P^1 with modules over a TDO ring, exemplifying the Langlands correspondence in a specific geometric setting.

Contribution

It establishes a new instance of the categorical Langlands correspondence for irregular rank 2 connections on P^1, connecting moduli stacks and TDO modules.

Findings

Derived category equivalence between sheaves on moduli stack and TDO modules.

Identification of the curve with the coarse moduli space of parabolic bundles.

An explicit example of the categorical Langlands correspondence.

Abstract

Special kinds of rank 2 vector bundles with (possibly irregular) connections on P^1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on certain non-separated curve. We identify this curve with the coarse moduli space of some parabolic bundles on P^1. Then our equivalence becomes an example of the categorical Langlands correspondence.