Towards automorphic to differential correspondence for vertex algebras

Abstract

In these notes we propose a version of geometric correspondence between parameter spaces for projectively flat connections in vector bundles and automorphic representations of modular groups over Riemann surfaces. Principal role of vertex algebras is discussed. We then formulate a conjecture concerning an extended correspondence between categories of twisted D-modules and Hecke eigensheaves both defined on the moduli stack of modular group bundles, and obtained as sheaves of conformal blocks for vertex operator algebras with formal parameters on complex algebraic curves.