Symplectic Reduction of Sheaves of $\mathcal{A}$-modules

TL;DR

This paper develops a sheaf-theoretic framework for symplectic reduction by defining annihilator sheaves of modules over a sheaf of algebras, extending classical module theory to a sheaf context.

Contribution

It introduces the concept of annihilator sheaves for submodules of sheaves of modules and formulates a sheaf-theoretic version of symplectic reduction, extending classical results.

Findings

Defined annihilator sheaves for sub-$\

Established properties analogous to classical annihilators in sheaf context

Formulated a sheaf-theoretic symplectic reduction framework

Abstract

Given an arbitrary sheaf $\mathcal{E}$ of $\mathcal{A}$-modules (or $\mathcal{A}$-module in short) on a topological space $X$, we define \textit{annihilator sheaves} of sub-$\mathcal{A}$-modules of $\mathcal{E}$ in a way similar to the classical case, and obtain thereafter the analog of the \textit{main theorem}, regarding classical annihilators in module theory, see Curtis[\cite{curtis}, pp. 240-242]. The familiar classical properties, satisfied by annihilator sheaves, allow us to set clearly the \textit{sheaf-theoretic version} of \textit{symplectic reduction}, which is the main goal in this paper.