A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves

TL;DR

This paper proves the Dolbeault geometric Langlands correspondence for GL_2 with reduced spectral curves, utilizing limit categories to handle non-quasi-compact moduli stacks, and outlines a strategy for generalization.

Contribution

It establishes the first non-trivial case of the Dolbeault geometric Langlands correspondence for GL_2 with reduced spectral curves using limit categories.

Findings

Proves the correspondence for GL_2 with reduced spectral curves

Introduces a strategy for extending the proof to more general cases

Identifies obstructions to generalization

Abstract

In our previous paper with Tudor P\u{a}durariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\mathrm{GL}_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.