Topological sheaves and spaces of distributions in the global case

TL;DR

This paper generalizes the theory of fields and distributions to a global scheme setting, introducing topological sheaves and reconstructing vertex algebra theory with global distributions.

Contribution

It develops a new framework for topological sheaves on schemes and extends vertex algebra theory to the global distribution context.

Findings

Constructed analogues of spaces of fields on schemes

Reformulated multiplication of fields in the global setting

Rebuilt the theory of vertex algebras using global distributions

Abstract

We extend the theory of fields/distributions developed the paper "A Feigin-Frenkel theorem with n singularities" to a general base scheme. In order to do so we introduce suitable notions of topological sheaves on schemes and study their basic properties. We then construct appropriate analogues of the spaces of fields, consider multiplication of fields between them and rebuild the basic theory of vertex algebras in the setting of global distributions in place of formal power series, which takes the form of chiral algebras introduced Beilinson and Drinfeld in "Chiral Algebras".