Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE

TL;DR

This paper advances the proof of the geometric Langlands conjecture by establishing the Fundamental Local Equivalence at the critical level and exploring the interaction between Kac-Moody localization and the global Langlands functor.

Contribution

It formulates and proves the FLE at the critical level and develops new formalism for ind-coherent sheaves and factorization categories in the context of geometric Langlands.

Findings

Proved the Fundamental Local Equivalence at the critical level.

Analyzed the interaction between Kac-Moody localization and the global Langlands functor.

Developed theory of ind-coherent sheaves in infinite type and formalism of factorization categories.

Abstract

This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction between Kac-Moody localization and the global geometric Langlands functor of ref. [GLC1]. This paper contains an extensive Appendix, whose primary goals are: (a) Development the theory of ind-coherent sheaves in infinite type; (b)Development of the formalism of factorization categories.