Infinitesimal Star Products Compatible with Coisotropic Reduction
Marvin Dippell
TL;DR
The paper addresses when a star product on a Poisson manifold $M$ can descend to the reduced space $\mathcal{M}_{\mathrm{red}}=C/D$ for a coisotropic submanifold $C$ with simple distribution $D$. It develops a constraint symbol calculus and uses a prolongation construction to compute the second constraint Hochschild cohomology $\operatorname{H}_{\mathrm{diff}}^2(\mathcal{M})_{\mathrm{N}}$, proving an explicit isomorphism $\operatorname{H}_{\mathrm{diff}}^2(\mathcal{M})_{\mathrm{N}} \cong \mathfrak{X}^2(\mathcal{M})_{\mathrm{N}} \oplus \cc{\Sym} \Secinfty(D) \vee \Secinfty(TC^{\perp})$ via $\mathcal{U}(X,\psi) = \operatorname{hkr}(X) + [\Op^\nabla(\mathcal{D}\psi)]$. This decomposition shows that beyond bivector terms, symmetric constraint contributions classify infinitesimal star products that are not compatible with reduction, although all such deformations become equivalent once reduction is performed. The results provide a complete differential-constraint deformation theory up to second order and clarify how quantization interacts with coisotropic reduction in Poisson geometry. Overall, the work yields explicit tools to determine when quantization commutes with reduction and how constraint structures shape the space of admissible infinitesimal deformations.
Abstract
We determine infinitesimal star products on Poisson manifolds compatible with coisotropic reduction. This is achieved by computing the second constraint Hochschild cohomology of the constraint algebra of functions associated to any submanifold equipped with a simple distribution.