Scheme-theoretic coisotropic reduction
Peter Crooks, Maxence Mayrand
TL;DR
The paper develops an affine scheme-theoretic variant of Hamiltonian reduction driven by an affine symplectic groupoid, establishing that a Poisson bracket descends to the invariants $\Bbbk[\mu^{-1}(S)]^{\mathcal{H}}$ and that $\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ serves as the Hamiltonian reduction of $M$ along a coisotropic $S$ with stabilizer $\mathcal{H}$. It then shows that, under suitable hypotheses, this reduced space inherits a residual Hamiltonian action of another affine symplectic groupoid, with a descended action map and moment map that preserve the Hamiltonian Poisson structure. The results provide scheme-theoretic analogues of Marsden–Ratiu, Śniatycki–Weinstein, Mikami–Weinstein, and general symplectic reduction along coisotropic submanifolds, thus unifying several classical reductions in an affine setting. The work is motivated by and contributes to generalizations of the Moore– Tachikawa conjecture, by enlarging the target category to include affine Poisson varieties with Hamiltonian groupoid actions, and by providing tools to construct new TQFTs valued in this expanded category. The paper also develops the foundational theory of affine algebraic Lie groupoids and their actions, proves the two main reduction theorems, and offers concrete examples such as level-zero reduction and affine closures of cotangent bundles, illustrating the framework’s scope and potential applications in geometric representation theory and Poisson geometry.
Abstract
We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{C}$, and is formulated for an affine symplectic groupoid $\mathcal{G}\rightrightarrows X$, an affine Hamiltonian $\mathcal{G}$-scheme $μ:M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal{H}\rightrightarrows S$. Our first main result is that the Poisson bracket on $\Bbbk[M]$ induces a Poisson bracket on the subquotient $\Bbbk[μ^{-1}(S)]^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to an earlier paper, where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction, Śniatycki-Weinstein reduction, and symplectic reduction along general coisotropic submanifolds. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore-Tachikawa conjecture.