Symplectic reduction along a submanifold

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.