Tate's thesis in the de Rham Setting

TL;DR

This paper establishes a derived equivalence between D-modules on the loop space of the affine line and ind-coherent sheaves on a moduli space of de Rham local systems, connecting geometric Langlands and mirror symmetry.

Contribution

It proves a conjecture linking D-modules and de Rham local systems, advancing the understanding of the geometric Langlands program through quantum field theory insights.

Findings

Derived equivalence between D-modules and ind-coherent sheaves.

Connection of the result to 3d mirror symmetry and quantum field theory.

Supports conjectures in the geometric Langlands program.

Abstract

We calculate the category of D-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.