Connections on moduli spaces and infinitesimal Hecke modifications

2021/08/17 by Nick Rozenblyum, Rozenblyum, Nick · 3 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2108.07745

openalex publication_date 2021/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a proper scheme and Z a prestack over X equipped with a flat connection. We give a local-to-global description of D-modules on the prestack S(Z) of flat sections of Z. Examples of S(Z) include the moduli stacks of principal G-bundles and de Rham local systems on X. We show that the category of D-modules is equivalent to the category of ind-coherent sheaves which are equivariant with respect to infinitesimal Hecke groupoids parametrized by finite subsets of X. We describe a number of applications to geometric representation theory and conformal field theory, including a derived enhancement of the Verlinde formula: the derived space of conformal blocks (a.k.a. chiral homology) of the WZW model is isomorphic to the cohomology of the corresponding line bundle on Bun_G, the moduli stack of G-bundles.

Citations

• Outline of the proof of the geometric Langlands conjecture for GL(2)

Cited by

• Moduli of bundles and semiorthogonal decomposition
• Period sheaves via perverse pullbacks
• Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

Related

• Ramifications of the geometric Langlands Program
• Moduli of flat connections in positive characteristic
• Moduli of pre-\cal D-modules, perverse sheaves and the Riemann-Hilbert morphism -I
• A Note On Two Fiber Bundles and The Manifestations Of "Shtuka"
• The moduli stack of G-bundles