Derived Symplectic Reduction and L-Equivariant Geometry

TL;DR

This paper explores derived symplectic reduction and L-equivariant geometry motivated by physics, generalizing classical notions and providing new geometric insights into the BV complex through shifted symplectic reduction.

Contribution

It introduces a generalized framework for derived symplectic reduction and moment maps, extending classical concepts to Lie algebroids and shifted symplectic geometry.

Findings

Development of derived symplectic reduction techniques

Generalization of moment maps for Lie algebroids

New shifted examples via derived Lagrangian intersection

Abstract

Motivated by the cohomological construction for the BV formalism from physics, this thesis asks how to perform the intersections and quotients appearing in the BV construction. This leads to the study of the derived symplectic reduction and the extension of these classical notions for group action and infinitesimal actions of Lie algebroids. In this context, we generalize the classical notion of moment maps for global and infinitesimal actions. We then study examples of such moment maps and their derived reductions and produce new shifted examples through a procedure of derived Lagrangian intersection. We then motivate why these new examples can bring a new geometric understanding of the BV complex as a generalized shifted symplectic reduction.