The category of singularities as a crystal and global Springer fibers

TL;DR

This paper proves the Gluing Conjecture in categorical geometric Langlands by leveraging the crystal structure on singularities, reducing the problem to homological triviality of certain homotopy types derived from Springer fibers.

Contribution

It introduces a novel approach using crystal structures on singularities to prove a major conjecture in geometric Langlands theory.

Findings

Proof of the Gluing Conjecture on the spectral side

Reduction of the conjecture to homological triviality of homotopy types

Application of Springer fibers in the proof

Abstract

We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of homological triviality of certain homotopy types. These homotopy types are obtained by gluing from a global version of Springer fibers.