[Paper Review] Recent progress in geometric Langlands theory

Dennis Gaitsgory|arXiv (Cornell University)|Jun 30, 2016

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper surveys recent advances in geometric Langlands theory, focusing on the categorical duality between D-modules on the moduli stack of G-bundles and quasi-coherent sheaves on the stack of Langlands dual local systems. It proposes that the stack of local systems on the punctured formal disk is 1-affine, establishing a foundational equivalence that underpins the local geometric Langlands correspondence, with a key result being the full faithfulness of the localization functor in the local setting.

ABSTRACT

The is the English version of the text of the talk at Séminaire Bourbaki on February 16, 2016

Motivation & Objective

• To formulate and motivate the categorical geometric Langlands correspondence as a duality between D-modules on Bun_G(X) and quasi-coherent sheaves on LocSys_{\check{G}}(X).
• To establish the conjecture that the stack of local systems on the punctured formal disk, LocSys_{\check{G}}(\hat{\mathcal{D}}), is 1-affine.
• To clarify the role of 1-affineness in categorically identifying categories over geometric stacks in the context of geometric Langlands.
• To provide a framework for the local geometric Langlands correspondence via the category of sheaves on LocSys_{\check{G}}(\hat{\mathcal{D}}) and its equivalence to QCoh(\cdot) in the local setting.

Proposed method

• Introduces the notion of 1-affineness for prestacks, particularly for stacks like LocSys_{\check{G}}(\hat{\mathcal{D}}), to compare categories of sheaves and quasi-coherent modules.
• Uses the gauge action of G(\mathcal{K}) on \check{\mathfrak{g}} \otimes \omega_{\mathcal{K}} to define the stack LocSys_{\check{G}}(\hat{\mathcal{D}}) as a quotient prestack.
• Applies the theory of ind-schemes and ind-algebraic stacks to handle infinite-dimensional geometric objects arising in the local geometric Langlands setting.
• Leverages the localization functor and generators-and-relations techniques from Beilinson-Drinfeld to construct automorphic D-modules.
• Establishes that the functor \mathbf{Loc} is fully faithful for LocSys_{\check{G}}(\hat{\mathcal{D}}), a key step toward proving 1-affineness.
• Uses the fact that QCoh(LocSys_{\check{G}}(\hat{\mathcal{D}})) is compactly generated, supporting the conjectural equivalence with ShvCat(LocSys_{\check{G}}(\hat{\mathcal{D}})).

Experimental results

Research questions

• RQ1 Is the prestack LocSys_{\check{G}}(\hat{\mathcal{D}}) of \check{G}-local systems on the punctured formal disk 1-affine?
• RQ2 How does the 1-affineness of LocSys_{\check{G}}(\hat{\mathcal{D}}) relate to the categorical geometric Langlands correspondence?
• RQ3 What is the precise relationship between ShvCat(LocSys_{\check{G}}(\hat{\mathcal{D}})) and QCoh(LocSys_{\check{G}}(\hat{\mathcal{D}})) in the local setting?
• RQ4 Why does the infinite-dimensional structure of LocSys_{\check{G}}(\hat{\mathcal{D}}) not obstruct 1-affineness despite the failure of 1-affineness in its components?
• RQ5 To what extent does the local geometric Langlands correspondence generalize the classical and global cases via this 1-affineness conjecture?

Key findings

• The prestack LocSys_{\check{G}}(\hat{\mathcal{D}}) is conjectured to be 1-affine, which would imply that ShvCat(LocSys_{\check{G}}(\hat{\mathcal{D}})) and QCoh(LocSys_{\check{G}}(\hat{\mathcal{D}})) are equivalent as categories.
• The conjecture is known to be true when G is a torus, providing a foundational case for the general conjecture.
• The functor \mathbf{Loc} for LocSys_{\check{G}}(\hat{\mathcal{D}}) is fully faithful, a key step toward proving 1-affineness.
• The category QCoh(LocSys_{\check{G}}(\hat{\mathcal{D}})) is compactly generated, supporting its use as a target for categorical equivalences.
• Despite the ind-scheme nature of \check{\mathfrak{g}} \otimes \omega_{\mathcal{K}} and the infinite-type group G(\mathcal{K}), the quotient LocSys_{\check{G}}(\hat{\mathcal{D}}) avoids typical obstructions to 1-affineness due to the structure of the gauge action.
• The failure of 1-affineness in related objects (e.g., \check{\mathfrak{g}} \otimes \omega_{\mathcal{K}} or quotients under adjoint action) highlights the special role of the gauge action in preserving potential 1-affineness.