Symplectic reduction along a submanifold

TL;DR

This paper introduces a unified symplectic reduction method along submanifolds applicable across various geometric categories, generalizing many existing reduction techniques and constructing new Hamiltonian spaces with universal properties.

Contribution

It develops a systematic approach to symplectic reduction along submanifolds, unifying multiple reduction methods and introducing new Hamiltonian G-spaces with universal properties.

Findings

Unified framework for symplectic reduction along submanifolds.

Construction of Hamiltonian G-spaces with universal reduction properties.

Generalization of existing reduction techniques like Marsden--Weinstein and symplectic cutting.

Abstract

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in TQFT. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak{M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq\mathrm{Lie}(G)^*$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak{M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. While these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.