Connections on moduli spaces and infinitesimal Hecke modifications

TL;DR

This paper provides a local-to-global description of D-modules on prestacks of flat sections, linking geometric representation theory and conformal field theory through a derived Verlinde formula.

Contribution

It introduces a novel equivalence between D-modules and equivariant ind-coherent sheaves on moduli stacks, with applications to the Verlinde formula and conformal blocks.

Findings

Equivalence between D-modules and equivariant ind-coherent sheaves.

Derived enhancement of the Verlinde formula.

Isomorphism of conformal blocks with cohomology of line bundles on Bun_G.

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.